@article{12183,
abstract = {We consider a gas of n bosonic particles confined in a box [−ℓ/2,ℓ/2]3 with Neumann boundary conditions. We prove Bose–Einstein condensation in the Gross–Pitaevskii regime, with an optimal bound on the condensate depletion. Moreover, our lower bound for the ground state energy in a small box [−ℓ/2,ℓ/2]3 implies (via Neumann bracketing) a lower bound for the ground state energy of N bosons in a large box [−L/2,L/2]3 with density ρ=N/L3 in the thermodynamic limit.},
author = {Boccato, Chiara and Seiringer, Robert},
issn = {1424-0637},
journal = {Annales Henri Poincare},
publisher = {Springer Nature},
title = {{The Bose Gas in a box with Neumann boundary conditions}},
doi = {10.1007/s00023-022-01252-3},
year = {2023},
}
@phdthesis{11473,
abstract = {The polaron model is a basic model of quantum field theory describing a single particle
interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned
with models applicable in the context of an impurity atom in a Bose-Einstein condensate as
well as the problem of electrons moving in polar crystals.
The model has a simple structure in which the interaction of the particle with the field is given
by a term linear in the field’s creation and annihilation operators. In this work, we investigate
the properties of this model by providing rigorous estimates on various energies relevant to the
problem. The estimates are obtained, for the most part, by suitable operator techniques which
constitute the principal mathematical substance of the thesis.
The first application of these techniques is to derive the polaron model rigorously from first
principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an
impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas
in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak
interactions as a low-energy effective theory for this problem.
In the second part, we investigate rigorously the ground state of the model at fixed momentum
and for large values of the coupling constant. Qualitatively, the system is expected to display
a transition from the quasi-particle behavior at small momenta, where the dispersion relation
is parabolic and the particle moves through the medium dragging along a cloud of phonons, to
the radiative behavior at larger momenta where the polaron decelerates and emits free phonons.
At the same time, in the strong coupling regime, the bosonic field is expected to behave purely
classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to
be asymptotically equal to the one obtained from the semiclassical counterpart of the problem,
first studied by Landau and Pekar in the 1940s. For polaron models with regularized form
factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear
function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove
that for a large window of momenta below the radiation threshold, the energy-momentum
relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the
Landau–Pekar effective mass, as expected.
For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is
of the optical type and the form factor is formally UV–singular due to the nature of the point
charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to
the regular case, this requires the inclusion of the quantum fluctuations of the phonon field,
which makes the problem considerably more difficult.
The results are supplemented by studies on the absolute ground-state energy at strong coupling,
a proof of the divergence of the effective mass with the coupling constant for a wide class of
polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model
and the application of the techniques used for its removal for the energy estimates.
},
author = {Mysliwy, Krzysztof},
issn = {2663-337X},
pages = {138},
publisher = {Institute of Science and Technology Austria (ISTA)},
title = {{Polarons in Bose gases and polar crystals: Some rigorous energy estimates}},
doi = {10.15479/at:ista:11473},
year = {2022},
}
@article{10564,
abstract = {We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.},
author = {Mysliwy, Krzysztof and Seiringer, Robert},
issn = {1572-9613},
journal = {Journal of Statistical Physics},
number = {1},
publisher = {Springer Nature},
title = {{Polaron models with regular interactions at strong coupling}},
doi = {10.1007/s10955-021-02851-w},
volume = {186},
year = {2022},
}
@article{10850,
abstract = {We study two interacting quantum particles forming a bound state in d-dimensional free
space, and constrain the particles in k directions to (0, ∞)k ×Rd−k, with Neumann boundary
conditions. First, we prove that the ground state energy strictly decreases upon going from k
to k+1. This shows that the particles stick to the corner where all boundary planes intersect.
Second, we show that for all k the resulting Hamiltonian, after removing the free part of the
kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper
generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413–1444,
2020) to dimensions d > 1.},
author = {Roos, Barbara and Seiringer, Robert},
issn = {0022-1236},
journal = {Journal of Functional Analysis},
keywords = {Analysis},
number = {12},
publisher = {Elsevier},
title = {{Two-particle bound states at interfaces and corners}},
doi = {10.1016/j.jfa.2022.109455},
volume = {282},
year = {2022},
}
@article{10755,
abstract = {We provide a definition of the effective mass for the classical polaron described by the Landau–Pekar (LP) equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by LP (1948 J. Exp. Theor. Phys. 18 419–423).},
author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {1751-8121},
journal = {Journal of Physics A: Mathematical and Theoretical},
number = {1},
publisher = {IOP Publishing},
title = {{The effective mass problem for the Landau-Pekar equations}},
doi = {10.1088/1751-8121/ac3947},
volume = {55},
year = {2022},
}
@article{11783,
abstract = {We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/ N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix.},
author = {Bossmann, Lea},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {6},
publisher = {AIP Publishing},
title = {{Low-energy spectrum and dynamics of the weakly interacting Bose gas}},
doi = {10.1063/5.0089983},
volume = {63},
year = {2022},
}
@article{11732,
abstract = {We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.},
author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{The BCS energy gap at high density}},
doi = {10.1007/s10955-022-02965-9},
volume = {189},
year = {2022},
}
@article{11917,
abstract = {We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.},
author = {Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {1572-9613},
journal = {Journal of Statistical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{Large deviation estimates for weakly interacting bosons}},
doi = {10.1007/s10955-022-02940-4},
volume = {188},
year = {2022},
}
@article{12083,
abstract = {We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both a law of large numbers and a central limit theorem.},
author = {Rademacher, Simone Anna Elvira},
issn = {0022-2488},
journal = {Journal of Mathematical Physics},
number = {8},
publisher = {AIP Publishing},
title = {{Dependent random variables in quantum dynamics}},
doi = {10.1063/5.0086712},
volume = {63},
year = {2022},
}
@phdthesis{12390,
abstract = {The scope of this thesis is to study quantum systems exhibiting a continuous symmetry that
is broken on the level of the corresponding effective theory. In particular we are going to
investigate translation-invariant Bose gases in the mean field limit, effectively described by
the Hartree functional, and the Fröhlich Polaron in the regime of strong coupling, effectively
described by the Pekar functional. The latter is a model describing the interaction between a
charged particle and the optical modes of a polar crystal. Regarding the former, we assume in
addition that the particles in the gas are unconfined, and typically we will consider particles
that are subject to an attractive interaction. In both cases the ground state energy of the
Hamiltonian is not a proper eigenvalue due to the underlying translation-invariance, while on
the contrary there exists a whole invariant orbit of minimizers for the corresponding effective
functionals. Both, the absence of proper eigenstates and the broken symmetry of the effective
theory, make the study significantly more involved and it is the content of this thesis to
develop a frameworks which allows for a systematic way to circumvent these issues.
It is a well-established result that the ground state energy of Bose gases in the mean field limit,
as well as the ground state energy of the Fröhlich Polaron in the regime of strong coupling, is
to leading order given by the minimal energy of the corresponding effective theory. As part
of this thesis we identify the sub-leading term in the expansion of the ground state energy,
which can be interpreted as the quantum correction to the classical energy, since the effective
theories under consideration can be seen as classical counterparts.
We are further going to establish an asymptotic expression for the energy-momentum relation
of the Fröhlich Polaron in the strong coupling limit. In the regime of suitably small momenta,
this asymptotic expression agrees with the energy-momentum relation of a free particle having
an effectively increased mass, and we find that this effectively increased mass agrees with the
conjectured value in the physics literature.
In addition we will discuss two unrelated papers written by the author during his stay at ISTA
in the appendix. The first one concerns the realization of anyons, which are quasi-particles
acquiring a non-trivial phase under the exchange of two particles, as molecular impurities.
The second one provides a classification of those vector fields defined on a given manifold
that can be written as the gradient of a given functional with respect to a suitable metric,
provided that some mild smoothness assumptions hold. This classification is subsequently
used to identify those quantum Markov semigroups that can be written as a gradient flow of
the relative entropy.
},
author = {Brooks, Morris},
issn = {2663-337X},
pages = {196},
publisher = {Institute of Science and Technology Austria},
title = {{Translation-invariant quantum systems with effectively broken symmetry}},
doi = {10.15479/at:ista:12390},
year = {2022},
}
@article{12246,
abstract = {The Lieb–Oxford inequality provides a lower bound on the Coulomb energy of a classical system of N identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree–Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.},
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
issn = {0377-9017},
journal = {Letters in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {5},
publisher = {Springer Nature},
title = {{Improved Lieb–Oxford bound on the indirect and exchange energies}},
doi = {10.1007/s11005-022-01584-5},
volume = {112},
year = {2022},
}
@article{12276,
abstract = {Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of nontrivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states (MPSs). We compare counterdiabatic and leakage minimization approaches to the so-called local steering problem that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counterdiabatic framework and use it to construct a Floquet model with quantum scars. We perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model. Finally, we apply our leakage minimization approach to construct quantum scars in the periodically driven nonintegrable Ising model.},
author = {Ljubotina, Marko and Roos, Barbara and Abanin, Dmitry A. and Serbyn, Maksym},
issn = {2691-3399},
journal = {PRX Quantum},
keywords = {General Medicine},
number = {3},
publisher = {American Physical Society},
title = {{Optimal steering of matrix product states and quantum many-body scars}},
doi = {10.1103/prxquantum.3.030343},
volume = {3},
year = {2022},
}
@article{7900,
abstract = {Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic approximation, deriving an effective Hamiltonian. We then show that for systems with Coulomb interaction this effective theory predicts collective excitations (plasmons) in accordance with the random phase approximation of Bohm and Pines, and with experimental observation.},
author = {Benedikter, Niels P},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
number = {1},
publisher = {World Scientific},
title = {{Bosonic collective excitations in Fermi gases}},
doi = {10.1142/s0129055x20600090},
volume = {33},
year = {2021},
}
@article{8603,
abstract = {We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem.},
author = {Frank, Rupert and Seiringer, Robert},
issn = {10970312},
journal = {Communications on Pure and Applied Mathematics},
number = {3},
pages = {544--588},
publisher = {Wiley},
title = {{Quantum corrections to the Pekar asymptotics of a strongly coupled polaron}},
doi = {10.1002/cpa.21944},
volume = {74},
year = {2021},
}
@article{7685,
abstract = {We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.},
author = {Boccato, Chiara},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
number = {1},
publisher = {World Scientific},
title = {{The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime}},
doi = {10.1142/S0129055X20600065},
volume = {33},
year = {2021},
}
@article{10585,
abstract = {Recently it was shown that anyons on the two-sphere naturally arise from a system of molecular impurities exchanging angular momentum with a many-particle bath (Phys. Rev. Lett. 126, 015301 (2021)). Here we further advance this approach and rigorously demonstrate that in the experimentally realized regime the lowest spectrum of two linear molecules immersed in superfluid helium corresponds to the spectrum of two anyons on the sphere. We develop the formalism within the framework of the recently experimentally observed angulon quasiparticle},
author = {Brooks, Morris and Lemeshko, Mikhail and Lundholm, Douglas and Yakaboylu, Enderalp},
issn = {2218-2004},
journal = {Atoms},
keywords = {anyons, quasiparticles, Quantum Hall Effect, topological states of matter},
number = {4},
publisher = {MDPI},
title = {{Emergence of anyons on the two-sphere in molecular impurities}},
doi = {10.3390/atoms9040106},
volume = {9},
year = {2021},
}
@article{10738,
abstract = {We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant α. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to α2.},
author = {Leopold, Nikolai K and Rademacher, Simone Anna Elvira and Schlein, Benjamin and Seiringer, Robert},
issn = {1948-206X},
journal = {Analysis and PDE},
number = {7},
pages = {2079--2100},
publisher = {Mathematical Sciences},
title = {{ The Landau–Pekar equations: Adiabatic theorem and accuracy}},
doi = {10.2140/APDE.2021.14.2079},
volume = {14},
year = {2021},
}
@article{10852,
abstract = { We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.},
author = {Seiringer, Robert},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {01},
publisher = {World Scientific Publishing},
title = {{The polaron at strong coupling}},
doi = {10.1142/s0129055x20600120},
volume = {33},
year = {2021},
}
@article{9246,
abstract = {We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.},
author = {Leopold, Nikolai K and Mitrouskas, David Johannes and Seiringer, Robert},
issn = {14320673},
journal = {Archive for Rational Mechanics and Analysis},
pages = {383--417},
publisher = {Springer Nature},
title = {{Derivation of the Landau–Pekar equations in a many-body mean-field limit}},
doi = {10.1007/s00205-021-01616-9},
volume = {240},
year = {2021},
}
@article{10224,
abstract = {We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging.},
author = {Feliciangeli, Dario and Seiringer, Robert},
issn = {1432-0673},
journal = {Archive for Rational Mechanics and Analysis},
number = {3},
pages = {1835–1906},
publisher = {Springer Nature},
title = {{The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics}},
doi = {10.1007/s00205-021-01715-7},
volume = {242},
year = {2021},
}
@article{10537,
abstract = {We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle–hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasi-free bosonic evolution of the collective particle–hole excitations.},
author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
issn = {1424-0637},
journal = {Annales Henri Poincaré},
publisher = {Springer Nature},
title = {{Bosonization of fermionic many-body dynamics}},
doi = {10.1007/s00023-021-01136-y},
year = {2021},
}
@article{9318,
abstract = {We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N.},
author = {Bossmann, Lea and Petrat, Sören P and Seiringer, Robert},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Asymptotic expansion of low-energy excitations for weakly interacting bosons}},
doi = {10.1017/fms.2021.22},
volume = {9},
year = {2021},
}
@article{9333,
abstract = {We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state φ0⊗ξα, where φ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.},
author = {Mitrouskas, David Johannes},
issn = {15730530},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{A note on the Fröhlich dynamics in the strong coupling limit}},
doi = {10.1007/s11005-021-01380-7},
volume = {111},
year = {2021},
}
@article{9348,
abstract = {We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the L2 spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.},
author = {Brooks, Morris and Di Gesù, Giacomo},
issn = {1096-0783},
journal = {Journal of Functional Analysis},
number = {3},
publisher = {Elsevier},
title = {{Sharp tunneling estimates for a double-well model in infinite dimension}},
doi = {10.1016/j.jfa.2021.109029},
volume = {281},
year = {2021},
}
@article{9351,
abstract = {We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years. },
author = {Kirkpatrick, Kay and Rademacher, Simone Anna Elvira and Schlein, Benjamin},
issn = {1424-0637},
journal = {Annales Henri Poincare},
pages = {2595--2618},
publisher = {Springer Nature},
title = {{A large deviation principle in many-body quantum dynamics}},
doi = {10.1007/s00023-021-01044-1},
volume = {22},
year = {2021},
}
@article{9462,
abstract = {We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional – a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose–Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions.},
author = {Deuchert, Andreas and Seiringer, Robert},
issn = {1096-0783},
journal = {Journal of Functional Analysis},
number = {6},
publisher = {Elsevier},
title = {{Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons}},
doi = {10.1016/j.jfa.2021.109096},
volume = {281},
year = {2021},
}
@unpublished{9787,
abstract = {We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging.},
author = {Feliciangeli, Dario and Seiringer, Robert},
booktitle = {arXiv},
title = {{The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics}},
year = {2021},
}
@unpublished{9792,
abstract = {This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on careful a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.},
author = {Feliciangeli, Dario and Gerolin, Augusto and Portinale, Lorenzo},
booktitle = {arXiv},
pages = {34},
title = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}},
year = {2021},
}
@article{9891,
abstract = {Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations.},
author = {Lauritsen, Asbjørn Bækgaard},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {8},
publisher = {AIP},
title = {{Floating Wigner crystal and periodic jellium configurations}},
doi = {10.1063/5.0053494},
volume = {62},
year = {2021},
}
@article{7901,
abstract = {We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.},
author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
issn = {1432-1297},
journal = {Inventiones Mathematicae},
pages = {885--979},
publisher = {Springer},
title = {{Correlation energy of a weakly interacting Fermi gas}},
doi = {10.1007/s00222-021-01041-5},
volume = {225},
year = {2021},
}
@article{9225,
abstract = {The Landau–Pekar equations describe the dynamics of a strongly coupled polaron.
Here, we provide a class of initial data for which the associated effective Hamiltonian
has a uniform spectral gap for all times. For such initial data, this allows us to extend the
results on the adiabatic theorem for the Landau–Pekar equations and their derivation
from the Fröhlich model obtained in previous works to larger times.},
author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {15730530},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{Persistence of the spectral gap for the Landau–Pekar equations}},
doi = {10.1007/s11005-020-01350-5},
volume = {111},
year = {2021},
}
@article{9256,
abstract = {We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2. We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.},
author = {Napiórkowski, Marcin M and Seiringer, Robert},
issn = {15730530},
journal = {Letters in Mathematical Physics},
number = {2},
publisher = {Springer Nature},
title = {{Free energy asymptotics of the quantum Heisenberg spin chain}},
doi = {10.1007/s11005-021-01375-4},
volume = {111},
year = {2021},
}
@phdthesis{9733,
abstract = {This thesis is the result of the research carried out by the author during his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich polaron model, specifically to its regime of strong coupling. This model, which is rigorously introduced and discussed in the introduction, has been of great interest in condensed matter physics and field theory for more than eighty years. It is used to describe an electron interacting with the atoms of a solid material (the strength of this interaction is modeled by the presence of a coupling constant α in the Hamiltonian of the system). The particular regime examined here, which is mathematically described by considering the limit α →∞, displays many interesting features related to the emergence of classical behavior, which allows for a simplified effective description of the system under analysis. The properties, the range of validity and a quantitative analysis of the precision of such classical approximations are the main object of the present work. We specify our investigation to the study of the ground state energy of the system, its dynamics and its effective mass. For each of these problems, we provide in the introduction an overview of the previously known results and a detailed account of the original contributions by the author.},
author = {Feliciangeli, Dario},
issn = {2663-337X},
pages = {180},
publisher = {IST Austria},
title = {{The polaron at strong coupling}},
doi = {10.15479/at:ista:9733},
year = {2021},
}
@unpublished{9791,
abstract = {We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar.},
author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert},
booktitle = {arXiv},
title = {{The effective mass problem for the Landau-Pekar equations}},
year = {2021},
}
@article{9005,
abstract = {Studies on the experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. Here, we have undertaken the first step toward realizing the emerging fractional statistics for particles restricted to move on the sphere instead of on the plane. We show that such a model arises naturally in the context of quantum impurity problems. In particular, we demonstrate a setup in which the lowest-energy spectrum of two linear bosonic or fermionic molecules immersed in a quantum many-particle environment can coincide with the anyonic spectrum on the sphere. This paves the way toward the experimental realization of anyons on the sphere using molecular impurities. Furthermore, since a change in the alignment of the molecules corresponds to the exchange of the particles on the sphere, such a realization reveals a novel type of exclusion principle for molecular impurities, which could also be of use as a powerful technique to measure the statistics parameter. Finally, our approach opens up a simple numerical route to investigate the spectra of many anyons on the sphere. Accordingly, we present the spectrum of two anyons on the sphere in the presence of a Dirac monopole field.},
author = {Brooks, Morris and Lemeshko, Mikhail and Lundholm, D. and Yakaboylu, Enderalp},
issn = {10797114},
journal = {Physical Review Letters},
number = {1},
publisher = {American Physical Society},
title = {{Molecular impurities as a realization of anyons on the two-sphere}},
doi = {10.1103/PhysRevLett.126.015301},
volume = {126},
year = {2021},
}
@article{7790,
abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 .},
author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
doi = {10.1017/fms.2020.17},
volume = {8},
year = {2020},
}
@article{8042,
abstract = {We consider systems of N bosons in a box of volume one, interacting through a repulsive two-body potential of the form κN3β−1V(Nβx). For all 0<β<1, and for sufficiently small coupling constant κ>0, we establish the validity of Bogolyubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N.},
author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
issn = {14359855},
journal = {Journal of the European Mathematical Society},
number = {7},
pages = {2331--2403},
publisher = {European Mathematical Society},
title = {{The excitation spectrum of Bose gases interacting through singular potentials}},
doi = {10.4171/JEMS/966},
volume = {22},
year = {2020},
}
@article{8091,
abstract = {In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [6], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level.},
author = {Seiringer, Robert and Yngvason, Jakob},
issn = {15729613},
journal = {Journal of Statistical Physics},
pages = {448--464},
publisher = {Springer},
title = {{Emergence of Haldane pseudo-potentials in systems with short-range interactions}},
doi = {10.1007/s10955-020-02586-0},
volume = {181},
year = {2020},
}
@article{8130,
abstract = {We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N, while its range is proportional to (ε/N)β with scaling parameter β∈(0,1]. We consider the simultaneous limit (N,ε)→(∞,0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈(0,1), we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.},
author = {Bossmann, Lea},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {11},
pages = {541--606},
publisher = {Springer Nature},
title = {{Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons}},
doi = {10.1007/s00205-020-01548-w},
volume = {238},
year = {2020},
}
@article{8134,
abstract = {We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]2+) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion.},
author = {Mayer, Simon and Seiringer, Robert},
issn = {00222488},
journal = {Journal of Mathematical Physics},
number = {6},
publisher = {AIP},
title = {{The free energy of the two-dimensional dilute Bose gas. II. Upper bound}},
doi = {10.1063/5.0005950},
volume = {61},
year = {2020},
}
@article{8587,
abstract = {Inspired by the possibility to experimentally manipulate and enhance chemical reactivity in helium nanodroplets, we investigate the effective interaction and the resulting correlations between two diatomic molecules immersed in a bath of bosons. By analogy with the bipolaron, we introduce the biangulon quasiparticle describing two rotating molecules that align with respect to each other due to the effective attractive interaction mediated by the excitations of the bath. We study this system in different parameter regimes and apply several theoretical approaches to describe its properties. Using a Born–Oppenheimer approximation, we investigate the dependence of the effective intermolecular interaction on the rotational state of the two molecules. In the strong-coupling regime, a product-state ansatz shows that the molecules tend to have a strong alignment in the ground state. To investigate the system in the weak-coupling regime, we apply a one-phonon excitation variational ansatz, which allows us to access the energy spectrum. In comparison to the angulon quasiparticle, the biangulon shows shifted angulon instabilities and an additional spectral instability, where resonant angular momentum transfer between the molecules and the bath takes place. These features are proposed as an experimentally observable signature for the formation of the biangulon quasiparticle. Finally, by using products of single angulon and bare impurity wave functions as basis states, we introduce a diagonalization scheme that allows us to describe the transition from two separated angulons to a biangulon as a function of the distance between the two molecules.},
author = {Li, Xiang and Yakaboylu, Enderalp and Bighin, Giacomo and Schmidt, Richard and Lemeshko, Mikhail and Deuchert, Andreas},
issn = {0021-9606},
journal = {The Journal of Chemical Physics},
keywords = {Physical and Theoretical Chemistry, General Physics and Astronomy},
number = {16},
publisher = {AIP Publishing},
title = {{Intermolecular forces and correlations mediated by a phonon bath}},
doi = {10.1063/1.5144759},
volume = {152},
year = {2020},
}
@article{7235,
abstract = {We consider the Fröhlich model of a polaron, and show that its effective mass diverges in thestrong coupling limit.},
author = {Lieb, Elliott H. and Seiringer, Robert},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
pages = {23--33},
publisher = {Springer Nature},
title = {{Divergence of the effective mass of a polaron in the strong coupling limit}},
doi = {10.1007/s10955-019-02322-3},
volume = {180},
year = {2020},
}
@article{7508,
abstract = {In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N d-dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form (N−1)−1Ndβv(Nβ·)forβ∈[0,14d). We derive a sequence of N-body functions which approximate the true many-body dynamics in L2(RdN)-norm to arbitrary precision in powers of N−1. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.},
author = {Bossmann, Lea and Pavlović, Nataša and Pickl, Peter and Soffer, Avy},
issn = {1572-9613},
journal = {Journal of Statistical Physics},
pages = {1362--1396},
publisher = {Springer Nature},
title = {{Higher order corrections to the mean-field description of the dynamics of interacting bosons}},
doi = {10.1007/s10955-020-02500-8},
volume = {178},
year = {2020},
}
@phdthesis{7514,
abstract = {We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case.
We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks.},
author = {Mayer, Simon},
issn = {2663-337X},
pages = {148},
publisher = {IST Austria},
title = {{The free energy of a dilute two-dimensional Bose gas}},
doi = {10.15479/AT:ISTA:7514},
year = {2020},
}
@article{7611,
abstract = {We consider a system of N bosons in the limit N→∞, interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.},
author = {Rademacher, Simone Anna Elvira},
issn = {0377-9017},
journal = {Letters in Mathematical Physics},
pages = {2143--2174},
publisher = {Springer Nature},
title = {{Central limit theorem for Bose gases interacting through singular potentials}},
doi = {10.1007/s11005-020-01286-w},
volume = {110},
year = {2020},
}
@article{7650,
abstract = {We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ20). Here ϱ denotes the density of the system and ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.},
author = {Deuchert, Andreas and Seiringer, Robert},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {6},
pages = {1217--1271},
publisher = {Springer Nature},
title = {{Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature}},
doi = {10.1007/s00205-020-01489-4},
volume = {236},
year = {2020},
}
@article{6649,
abstract = {While Hartree–Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree–Fock state given by plane waves and introduce collective particle–hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.
},
author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
pages = {2097–2150},
publisher = {Springer Nature},
title = {{Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime}},
doi = {10.1007/s00220-019-03505-5},
volume = {374},
year = {2020},
}
@article{8769,
abstract = {One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter, is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hall systems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantum impurity model, where bosonic or fermionic impurities turn into anyons as a consequence of their interaction with the surrounding many-particle bath. A cloud of phonons dresses each impurity in such a way that it effectively attaches fluxes or vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impurity model, first, provides a different approach to the numerical solution of the many-anyon problem, along with a concrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. More importantly, the model paves the way toward realizing anyons using impurities in crystal lattices as well as ultracold gases. In particular, we consider two heavy electrons interacting with a two-dimensional lattice crystal in a magnetic field, and show that when the impurity-bath system is rotated at the cyclotron frequency, impurities behave as anyons as a consequence of the angular momentum exchange between the impurities and the bath. A possible experimental realization is proposed by identifying the statistics parameter in terms of the mean-square distance of the impurities and the magnetization of the impurity-bath system, both of which are accessible to experiment. Another proposed application is impurities immersed in a two-dimensional weakly interacting Bose gas.},
author = {Yakaboylu, Enderalp and Ghazaryan, Areg and Lundholm, D. and Rougerie, N. and Lemeshko, Mikhail and Seiringer, Robert},
issn = {2469-9950},
journal = {Physical Review B},
number = {14},
publisher = {American Physical Society},
title = {{Quantum impurity model for anyons}},
doi = {10.1103/physrevb.102.144109},
volume = {102},
year = {2020},
}
@article{9781,
abstract = {We consider the Pekar functional on a ball in ℝ3. We prove uniqueness of minimizers, and a quadratic lower bound in terms of the distance to the minimizer. The latter follows from nondegeneracy of the Hessian at the minimum.},
author = {Feliciangeli, Dario and Seiringer, Robert},
issn = {1095-7154},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {Applied Mathematics, Computational Mathematics, Analysis},
number = {1},
pages = {605--622},
publisher = {Society for Industrial & Applied Mathematics },
title = {{Uniqueness and nondegeneracy of minimizers of the Pekar functional on a ball}},
doi = {10.1137/19m126284x},
volume = {52},
year = {2020},
}
@article{8705,
abstract = {We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model.},
author = {Mysliwy, Krzysztof and Seiringer, Robert},
issn = {1424-0637},
journal = {Annales Henri Poincare},
number = {12},
pages = {4003--4025},
publisher = {Springer Nature},
title = {{Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit}},
doi = {10.1007/s00023-020-00969-3},
volume = {21},
year = {2020},
}
@article{80,
abstract = {We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.},
author = {Deuchert, Andreas and Seiringer, Robert and Yngvason, Jakob},
journal = {Communications in Mathematical Physics},
number = {2},
pages = {723--776},
publisher = {Springer},
title = {{Bose–Einstein condensation in a dilute, trapped gas at positive temperature}},
doi = {10.1007/s00220-018-3239-0},
volume = {368},
year = {2019},
}
@article{5856,
abstract = {We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗ ∗≈ 0.36 , which is the same regime for which we recently showed stability of the system.},
author = {Moser, Thomas and Seiringer, Robert},
issn = {14240637},
journal = {Annales Henri Poincare},
number = {4},
pages = {1325–1365},
publisher = {Springer},
title = {{Energy contribution of a point-interacting impurity in a Fermi gas}},
doi = {10.1007/s00023-018-00757-0},
volume = {20},
year = {2019},
}
@article{7015,
abstract = {We modify the "floating crystal" trial state for the classical homogeneous electron gas (also known as jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground-state energy of jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in density functional theory. Finally, in the third system each electron interacts with a periodic image of itself; that is, periodic boundary conditions are imposed on the interaction potential.},
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
issn = {2469-9950},
journal = {Physical Review B},
number = {3},
publisher = {APS},
title = {{Floating Wigner crystal with no boundary charge fluctuations}},
doi = {10.1103/physrevb.100.035127},
volume = {100},
year = {2019},
}
@article{7100,
abstract = {We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.},
author = {Jeblick, Maximilian and Leopold, Nikolai K and Pickl, Peter},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {1--69},
publisher = {Springer Nature},
title = {{Derivation of the time dependent Gross–Pitaevskii equation in two dimensions}},
doi = {10.1007/s00220-019-03599-x},
volume = {372},
year = {2019},
}
@article{7226,
author = {Jaksic, Vojkan and Seiringer, Robert},
issn = {00222488},
journal = {Journal of Mathematical Physics},
number = {12},
publisher = {AIP},
title = {{Introduction to the Special Collection: International Congress on Mathematical Physics (ICMP) 2018}},
doi = {10.1063/1.5138135},
volume = {60},
year = {2019},
}
@article{7413,
abstract = {We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions.},
author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
issn = {0001-5962},
journal = {Acta Mathematica},
number = {2},
pages = {219--335},
publisher = {International Press of Boston},
title = {{Bogoliubov theory in the Gross–Pitaevskii limit}},
doi = {10.4310/acta.2019.v222.n2.a1},
volume = {222},
year = {2019},
}
@unpublished{7524,
abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit
$a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.},
author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
booktitle = {arXiv:1910.03372},
pages = {61},
publisher = {ArXiv},
title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
year = {2019},
}
@article{6788,
abstract = {We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations.},
author = {Leopold, Nikolai K and Petrat, Sören P},
issn = {1424-0661},
journal = {Annales Henri Poincare},
number = {10},
pages = {3471–3508},
publisher = {Springer Nature},
title = {{Mean-field dynamics for the Nelson model with fermions}},
doi = {10.1007/s00023-019-00828-w},
volume = {20},
year = {2019},
}
@article{6840,
abstract = {We discuss thermodynamic properties of harmonically trapped
imperfect quantum gases. The spatial inhomogeneity of these systems imposes
a redefinition of the mean-field interparticle potential energy as compared
to the homogeneous case. In our approach, it takes the form a
2N2 ωd, where
N is the number of particles, ω—the harmonic trap frequency, d—system’s
dimensionality, and a is a parameter characterizing the interparticle interaction.
We provide arguments that this model corresponds to the limiting case of
a long-ranged interparticle potential of vanishingly small amplitude. This
conclusion is drawn from a computation similar to the well-known Kac scaling
procedure, which is presented here in a form adapted to the case of an isotropic
harmonic trap. We show that within the model, the imperfect gas of trapped
repulsive bosons undergoes the Bose–Einstein condensation provided d > 1.
The main result of our analysis is that in d = 1 the gas of attractive imperfect
fermions with a = −aF < 0 is thermodynamically equivalent to the gas of
repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill
the relation aB + aF = . This result supplements similar recent conclusion
about thermodynamic equivalence of two-dimensional (2D) uniform imperfect
repulsive Bose and attractive Fermi gases.},
author = {Mysliwy, Krzysztof and Napiórkowski, Marek},
issn = {1742-5468},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
number = {6},
publisher = {IOP Publishing},
title = {{Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps}},
doi = {10.1088/1742-5468/ab190d},
volume = {2019},
year = {2019},
}
@article{6906,
abstract = {We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.},
author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
pages = {1311--1395},
publisher = {Springer},
title = {{Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime}},
doi = {10.1007/s00220-019-03555-9},
volume = {376},
year = {2019},
}
@article{295,
abstract = {We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter (Formula presented.). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.},
author = {Lundholm, Douglas and Seiringer, Robert},
journal = {Letters in Mathematical Physics},
number = {11},
pages = {2523--2541},
publisher = {Springer},
title = {{Fermionic behavior of ideal anyons}},
doi = {10.1007/s11005-018-1091-y},
volume = {108},
year = {2018},
}
@article{446,
abstract = {We prove that in Thomas–Fermi–Dirac–von Weizsäcker theory, a nucleus of charge Z > 0 can bind at most Z + C electrons, where C is a universal constant. This result is obtained through a comparison with Thomas-Fermi theory which, as a by-product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.},
author = {Frank, Rupert and Phan Thanh, Nam and Van Den Bosch, Hanne},
journal = {Communications on Pure and Applied Mathematics},
number = {3},
pages = {577 -- 614},
publisher = {Wiley-Blackwell},
title = {{The ionization conjecture in Thomas–Fermi–Dirac–von Weizsäcker theory}},
doi = {10.1002/cpa.21717},
volume = {71},
year = {2018},
}
@article{455,
abstract = {The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities},
author = {Benedikter, Niels P and Sok, Jérémy and Solovej, Jan},
journal = {Annales Henri Poincare},
number = {4},
pages = {1167 -- 1214},
publisher = {Birkhäuser},
title = {{The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations}},
doi = {10.1007/s00023-018-0644-z},
volume = {19},
year = {2018},
}
@phdthesis{52,
abstract = {In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.},
author = {Moser, Thomas},
pages = {115},
publisher = {IST Austria},
title = {{Point interactions in systems of fermions}},
doi = {10.15479/AT:ISTA:th_1043},
year = {2018},
}
@article{554,
abstract = {We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit (Formula presented.), we determine the critical temperature to be (Formula presented.) to leading order. Here, (Formula presented.) is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit (Formula presented.).},
author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan},
issn = {00103616},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {347--403},
publisher = {Springer},
title = {{The Bogoliubov free energy functional II: The dilute Limit}},
doi = {10.1007/s00220-017-3064-x},
volume = {360},
year = {2018},
}
@article{5983,
abstract = {We study a quantum impurity possessing both translational and internal rotational degrees of freedom interacting with a bosonic bath. Such a system corresponds to a “rotating polaron,” which can be used to model, e.g., a rotating molecule immersed in an ultracold Bose gas or superfluid helium. We derive the Hamiltonian of the rotating polaron and study its spectrum in the weak- and strong-coupling regimes using a combination of variational, diagrammatic, and mean-field approaches. We reveal how the coupling between linear and angular momenta affects stable quasiparticle states, and demonstrate that internal rotation leads to an enhanced self-localization in the translational degrees of freedom.},
author = {Yakaboylu, Enderalp and Midya, Bikashkali and Deuchert, Andreas and Leopold, Nikolai K and Lemeshko, Mikhail},
issn = {2469-9950},
journal = {Physical Review B},
number = {22},
publisher = {American Physical Society},
title = {{Theory of the rotating polaron: Spectrum and self-localization}},
doi = {10.1103/physrevb.98.224506},
volume = {98},
year = {2018},
}
@article{6002,
abstract = {The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.},
author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan Philip},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {3},
pages = {1037--1090},
publisher = {Springer Nature},
title = {{The Bogoliubov free energy functional I: Existence of minimizers and phase diagram}},
doi = {10.1007/s00205-018-1232-6},
volume = {229},
year = {2018},
}
@inproceedings{11,
abstract = {We report on a novel strategy to derive mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. The technique combines the method of counting and the coherent state approach to study the growth of the correlations among the particles and in the radiation field. As an instructional example, we derive the Schrödinger–Klein–Gordon system of equations from the Nelson model with ultraviolet cutoff and possibly massless scalar field. In particular, we prove the convergence of the reduced density matrices (of the nonrelativistic particles and the field bosons) associated with the exact time evolution to the projectors onto the solutions of the Schrödinger–Klein–Gordon equations in trace norm. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the nonrelativistic particles in Sobolev norm.},
author = {Leopold, Nikolai K and Pickl, Peter},
location = {Munich, Germany},
pages = {185 -- 214},
publisher = {Springer},
title = {{Mean-field limits of particles in interaction with quantised radiation fields}},
doi = {10.1007/978-3-030-01602-9_9},
volume = {270},
year = {2018},
}
@article{154,
abstract = {We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m2 ≈ 0.58 such that the system is stable, i.e., the energy is bounded from below, for m∈[m2,m2−1]. So far it was not known whether this 2 + 2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N + M system.},
author = {Moser, Thomas and Seiringer, Robert},
issn = {15729656},
journal = {Mathematical Physics Analysis and Geometry},
number = {3},
publisher = {Springer},
title = {{Stability of the 2+2 fermionic system with point interactions}},
doi = {10.1007/s11040-018-9275-3},
volume = {21},
year = {2018},
}
@article{180,
abstract = {In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.},
author = {Lewi, Mathieu and Lieb, Élliott and Seiringer, Robert},
journal = {Journal de l'Ecole Polytechnique - Mathematiques},
pages = {79 -- 116},
publisher = {Ecole Polytechnique},
title = {{Statistical mechanics of the uniform electron gas}},
doi = {10.5802/jep.64},
volume = {5},
year = {2018},
}
@article{399,
abstract = {Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.},
author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan},
journal = {EPL},
number = {1},
publisher = {IOP Publishing Ltd.},
title = {{Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation}},
doi = {10.1209/0295-5075/121/10007},
volume = {121},
year = {2018},
}
@article{400,
abstract = {We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum, our results carry over to the three-dimensional case.},
author = {Deuchert, Andreas and Geisinge, Alissa and Hainzl, Christian and Loss, Michael},
journal = {Annales Henri Poincare},
number = {5},
pages = {1507 -- 1527},
publisher = {Springer},
title = {{Persistence of translational symmetry in the BCS model with radial pair interaction}},
doi = {10.1007/s00023-018-0665-7},
volume = {19},
year = {2018},
}
@article{484,
abstract = {We consider the dynamics of a large quantum system of N identical bosons in 3D interacting via a two-body potential of the form N3β-1w(Nβ(x - y)). For fixed 0 = β < 1/3 and large N, we obtain a norm approximation to the many-body evolution in the Nparticle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.},
author = {Nam, Phan and Napiórkowski, Marcin M},
issn = {10950761},
journal = {Advances in Theoretical and Mathematical Physics},
number = {3},
pages = {683 -- 738},
publisher = {International Press},
title = {{Bogoliubov correction to the mean-field dynamics of interacting bosons}},
doi = {10.4310/ATMP.2017.v21.n3.a4},
volume = {21},
year = {2017},
}
@article{739,
abstract = {We study the norm approximation to the Schrödinger dynamics of N bosons in with an interaction potential of the form . Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large N limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all . The range of β is expected to be optimal for this large class of initial states.},
author = {Nam, Phan and Napiórkowski, Marcin M},
issn = {00217824},
journal = {Journal de Mathématiques Pures et Appliquées},
number = {5},
pages = {662 -- 688},
publisher = {Elsevier},
title = {{A note on the validity of Bogoliubov correction to mean field dynamics}},
doi = {10.1016/j.matpur.2017.05.013},
volume = {108},
year = {2017},
}
@article{741,
abstract = {We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m*. The value of m* is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.},
author = {Moser, Thomas and Seiringer, Robert},
issn = {00103616},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {329 -- 355},
publisher = {Springer},
title = {{Stability of a fermionic N+1 particle system with point interactions}},
doi = {10.1007/s00220-017-2980-0},
volume = {356},
year = {2017},
}
@article{1079,
abstract = {We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.},
author = {Nam, Phan and Van Den Bosch, Hanne},
issn = {13850172},
journal = {Mathematical Physics, Analysis and Geometry},
number = {2},
publisher = {Springer},
title = {{Nonexistence in Thomas Fermi-Dirac-von Weizsäcker theory with small nuclear charges}},
doi = {10.1007/s11040-017-9238-0},
volume = {20},
year = {2017},
}
@article{1120,
abstract = {The existence of a self-localization transition in the polaron problem has been under an active debate ever since Landau suggested it 83 years ago. Here we reveal the self-localization transition for the rotational analogue of the polaron -- the angulon quasiparticle. We show that, unlike for the polarons, self-localization of angulons occurs at finite impurity-bath coupling already at the mean-field level. The transition is accompanied by the spherical-symmetry breaking of the angulon ground state and a discontinuity in the first derivative of the ground-state energy. Moreover, the type of the symmetry breaking is dictated by the symmetry of the microscopic impurity-bath interaction, which leads to a number of distinct self-localized states. The predicted effects can potentially be addressed in experiments on cold molecules trapped in superfluid helium droplets and ultracold quantum gases, as well as on electronic excitations in solids and Bose-Einstein condensates. },
author = {Li, Xiang and Seiringer, Robert and Lemeshko, Mikhail},
issn = {24699926},
journal = {Physical Review A},
number = {3},
publisher = {American Physical Society},
title = {{Angular self-localization of impurities rotating in a bosonic bath}},
doi = {10.1103/PhysRevA.95.033608},
volume = {95},
year = {2017},
}
@article{632,
abstract = {We consider a 2D quantum system of N bosons in a trapping potential |x|s, interacting via a pair potential of the form N2β−1 w(Nβ x). We show that for all 0 < β < (s + 1)/(s + 2), the leading order behavior of ground states of the many-body system is described in the large N limit by the corresponding cubic nonlinear Schrödinger energy functional. Our result covers the focusing case (w < 0) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X. Chen and J. Holmer for harmonic traps (s = 2). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all 0 < β < 3/4. },
author = {Lewin, Mathieu and Nam, Phan and Rougerie, Nicolas},
journal = {Proceedings of the American Mathematical Society},
number = {6},
pages = {2441 -- 2454},
publisher = {American Mathematical Society},
title = {{A note on 2D focusing many boson systems}},
doi = {10.1090/proc/13468},
volume = {145},
year = {2017},
}
@article{912,
abstract = {We consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We measure the free energy of the whole sample with respect to the free energy of a reference state which allows us to define a BCS functional with boundary conditions at infinity. Our main result is a lower bound for this energy functional in terms of expressions that typically appear in Ginzburg-Landau functionals.
},
author = {Deuchert, Andreas},
issn = {00222488},
journal = { Journal of Mathematical Physics},
number = {8},
publisher = {AIP},
title = {{A lower bound for the BCS functional with boundary conditions at infinity}},
doi = {10.1063/1.4996580},
volume = {58},
year = {2017},
}
@article{1198,
abstract = {We consider a model of fermions interacting via point interactions, defined via a certain weighted Dirichlet form. While for two particles the interaction corresponds to infinite scattering length, the presence of further particles effectively decreases the interaction strength. We show that the model becomes trivial in the thermodynamic limit, in the sense that the free energy density at any given particle density and temperature agrees with the corresponding expression for non-interacting particles.},
author = {Moser, Thomas and Seiringer, Robert},
issn = {03779017},
journal = {Letters in Mathematical Physics},
number = {3},
pages = { 533 -- 552},
publisher = {Springer},
title = {{Triviality of a model of particles with point interactions in the thermodynamic limit}},
doi = {10.1007/s11005-016-0915-x},
volume = {107},
year = {2017},
}
@article{997,
abstract = {Recently it was shown that molecules rotating in superfluid helium can be described in terms of the angulon quasiparticles (Phys. Rev. Lett. 118, 095301 (2017)). Here we demonstrate that in the experimentally realized regime the angulon can be seen as a point charge on a 2-sphere interacting with a gauge field of a non-abelian magnetic monopole. Unlike in several other settings, the gauge fields of the angulon problem emerge in the real coordinate space, as opposed to the momentum space or some effective parameter space. Furthermore, we find a topological transition associated with making the monopole abelian, which takes place in the vicinity of the previously reported angulon instabilities. These results pave the way for studying topological phenomena in experiments on molecules trapped in superfluid helium nanodroplets, as well as on other realizations of orbital impurity problems.},
author = {Yakaboylu, Enderalp and Deuchert, Andreas and Lemeshko, Mikhail},
issn = {00319007},
journal = {APS Physics, Physical Review Letters},
number = {23},
publisher = {American Physiological Society},
title = {{Emergence of non-abelian magnetic monopoles in a quantum impurity problem}},
doi = {10.1103/PhysRevLett.119.235301},
volume = {119},
year = {2017},
}
@article{1143,
abstract = {We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive nonlinear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson\'s lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.},
author = {Nam, Phan and Rougerie, Nicolas and Seiringer, Robert},
journal = {Analysis and PDE},
number = {2},
pages = {459 -- 485},
publisher = {Mathematical Sciences Publishers},
title = {{Ground states of large bosonic systems: The gross Pitaevskii limit revisited}},
doi = {10.2140/apde.2016.9.459},
volume = {9},
year = {2016},
}
@article{1545,
abstract = {We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.},
author = {Nam, Phan and Napiórkowski, Marcin M and Solovej, Jan},
journal = {Journal of Functional Analysis},
number = {11},
pages = {4340 -- 4368},
publisher = {Academic Press},
title = {{Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations}},
doi = {10.1016/j.jfa.2015.12.007},
volume = {270},
year = {2016},
}
@article{1620,
abstract = {We consider the Bardeen–Cooper–Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg–Landau equation.},
author = {Frank, Rupert and Hainzl, Christian and Seiringer, Robert and Solovej, Jan},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {189 -- 216},
publisher = {Springer},
title = {{The external field dependence of the BCS critical temperature}},
doi = {10.1007/s00220-015-2526-2},
volume = {342},
year = {2016},
}
@article{1622,
abstract = {We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.},
author = {Lundholm, Douglas and Nam, Phan and Portmann, Fabian},
journal = {Archive for Rational Mechanics and Analysis},
number = {3},
pages = {1343 -- 1382},
publisher = {Springer},
title = {{Fractional Hardy–Lieb–Thirring and related Inequalities for interacting systems}},
doi = {10.1007/s00205-015-0923-5},
volume = {219},
year = {2016},
}
@article{1259,
abstract = {We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional.},
author = {Bräunlich, Gerhard and Hainzl, Christian and Seiringer, Robert},
journal = {Mathematical Physics, Analysis and Geometry},
number = {2},
publisher = {Springer},
title = {{Bogolubov–Hartree–Fock theory for strongly interacting fermions in the low density limit}},
doi = {10.1007/s11040-016-9209-x},
volume = {19},
year = {2016},
}
@article{1267,
abstract = {We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result.},
author = {Frank, Rupert and Killip, Rowan and Nam, Phan},
journal = {Letters in Mathematical Physics},
number = {8},
pages = {1033 -- 1036},
publisher = {Springer},
title = {{Nonexistence of large nuclei in the liquid drop model}},
doi = {10.1007/s11005-016-0860-8},
volume = {106},
year = {2016},
}
@article{1291,
abstract = {We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than dÂ +Â 1, with d the space dimension, this happens for all values of J smaller than a critical value Jc(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for pÂ >Â 2d and J in a left neighborhood of Jc(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (dÂ =Â 2) or slabs (dÂ =Â 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.},
author = {Giuliani, Alessandro and Seiringer, Robert},
journal = {Communications in Mathematical Physics},
number = {3},
pages = {983 -- 1007},
publisher = {Springer},
title = {{Periodic striped ground states in Ising models with competing interactions}},
doi = {10.1007/s00220-016-2665-0},
volume = {347},
year = {2016},
}
@article{1422,
abstract = {We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.},
author = {Frank, Rupert and Hainzl, Christian and Schlein, Benjamin and Seiringer, Robert},
journal = {Letters in Mathematical Physics},
number = {7},
pages = {913 -- 923},
publisher = {Springer},
title = {{Incompatibility of time-dependent Bogoliubov–de-Gennes and Ginzburg–Landau equations}},
doi = {10.1007/s11005-016-0847-5},
volume = {106},
year = {2016},
}
@inproceedings{1428,
abstract = {We report on a mathematically rigorous analysis of the superfluid properties of a Bose- Einstein condensate in the many-body ground state of a one-dimensional model of interacting bosons in a random potential.},
author = {Könenberg, Martin and Moser, Thomas and Seiringer, Robert and Yngvason, Jakob},
booktitle = {Journal of Physics: Conference Series},
location = {Shanghai, China},
number = {1},
publisher = {IOP Publishing Ltd.},
title = {{Superfluidity and BEC in a Model of Interacting Bosons in a Random Potential}},
doi = {10.1088/1742-6596/691/1/012016},
volume = {691},
year = {2016},
}
@article{1436,
abstract = {We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.},
author = {Bach, Volker and Breteaux, Sébastien and Petrat, Sören P and Pickl, Peter and Tzaneteas, Tim},
journal = {Journal de Mathématiques Pures et Appliquées},
number = {1},
pages = {1 -- 30},
publisher = {Elsevier},
title = {{Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction}},
doi = {10.1016/j.matpur.2015.09.003},
volume = {105},
year = {2016},
}
@article{1478,
abstract = {We consider the Tonks-Girardeau gas subject to a random external potential. If the disorder is such that the underlying one-particle Hamiltonian displays localization (which is known to be generically the case), we show that there is exponential decay of correlations in the many-body eigenstates. Moreover, there is no Bose-Einstein condensation and no superfluidity, even at zero temperature.},
author = {Seiringer, Robert and Warzel, Simone},
journal = {New Journal of Physics},
number = {3},
publisher = {IOP Publishing Ltd.},
title = {{Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas}},
doi = {10.1088/1367-2630/18/3/035002},
volume = {18},
year = {2016},
}
@article{1486,
abstract = {We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime.},
author = {Hainzl, Christian and Seiringer, Robert},
journal = {Journal of Mathematical Physics},
number = {2},
publisher = {American Institute of Physics},
title = {{The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties}},
doi = {10.1063/1.4941723},
volume = {57},
year = {2016},
}
@article{1491,
abstract = {We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.},
author = {Lewin, Mathieu and Nam, Phan and Rougerie, Nicolas},
journal = {Transactions of the American Mathematical Society},
number = {9},
pages = {6131 -- 6157},
publisher = {American Mathematical Society},
title = {{The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases}},
doi = {10.1090/tran/6537},
volume = {368},
year = {2016},
}
@article{1493,
abstract = {We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.},
author = {Petrat, Sören P and Pickl, Peter},
journal = {Mathematical Physics, Analysis and Geometry},
number = {1},
publisher = {Springer},
title = {{A new method and a new scaling for deriving fermionic mean-field dynamics}},
doi = {10.1007/s11040-016-9204-2},
volume = {19},
year = {2016},
}
@article{473,
abstract = {We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction strength behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions d 2.},
author = {Lewin, Mathieu and Phan Thanh, Nam and Rougerie, Nicolas},
journal = {Journal de l'Ecole Polytechnique - Mathematiques},
pages = {65 -- 115},
publisher = {Ecole Polytechnique},
title = {{Derivation of nonlinear gibbs measures from many-body quantum mechanics}},
doi = {10.5802/jep.18},
volume = {2},
year = {2015},
}
@article{1572,
abstract = {We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein-Primakoff representation. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice.
},
author = {Correggi, Michele and Giuliani, Alessandro and Seiringer, Robert},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {279 -- 307},
publisher = {Springer},
title = {{Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet}},
doi = {10.1007/s00220-015-2402-0},
volume = {339},
year = {2015},
}
@article{1573,
abstract = {We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in ℝ3. One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdos, Schlein, and Yau.},
author = {Chen, Thomas and Hainzl, Christian and Pavlović, Nataša and Seiringer, Robert},
journal = {Communications on Pure and Applied Mathematics},
number = {10},
pages = {1845 -- 1884},
publisher = {Wiley},
title = {{Unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti}},
doi = {10.1002/cpa.21552},
volume = {68},
year = {2015},
}
@article{1704,
abstract = {Given a convex function (Formula presented.) and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by (Formula presented.). Among other things, they prove that the so-defined quantity is monotone if and only if (Formula presented.) is operator monotone. The monotonicity is then used to properly define (Formula presented.) for bounded self-adjoint operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections (Formula presented.) with (Formula presented.) strongly, the limit (Formula presented.) is shown to exist and to be independent of the sequence of projections (Formula presented.). The question whether this sequence converges to its "obvious" limit, namely (Formula presented.), has been left open. We answer this question in principle affirmatively and show that (Formula presented.). If the operators A and B are regular enough, that is (A − B), (Formula presented.) and (Formula presented.) are trace-class, the identity (Formula presented.) holds.},
author = {Deuchert, Andreas and Hainzl, Christian and Seiringer, Robert},
journal = {Letters in Mathematical Physics},
number = {10},
pages = {1449 -- 1466},
publisher = {Springer},
title = {{Note on a family of monotone quantum relative entropies}},
doi = {10.1007/s11005-015-0787-5},
volume = {105},
year = {2015},
}
@article{1807,
abstract = {We study a double Cahn-Hilliard type functional related to the Gross-Pitaevskii energy of two-components Bose-Einstein condensates. In the case of large but same order intercomponent and intracomponent coupling strengths, we prove Γ-convergence to a perimeter minimisation functional with an inhomogeneous surface tension. We study the asymptotic behavior of the surface tension as the ratio between the intercomponent and intracomponent coupling strengths becomes very small or very large and obtain good agreement with the physical literature. We obtain as a consequence, symmetry breaking of the minimisers for the harmonic potential.},
author = {Goldman, Michael and Royo-Letelier, Jimena},
journal = {ESAIM - Control, Optimisation and Calculus of Variations},
number = {3},
pages = {603 -- 624},
publisher = {EDP Sciences},
title = {{Sharp interface limit for two components Bose-Einstein condensates}},
doi = {10.1051/cocv/2014040},
volume = {21},
year = {2015},
}