TY - JOUR AB - 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. AU - Napiórkowski, Marcin M AU - Seiringer, Robert ID - 9256 IS - 2 JF - Letters in Mathematical Physics SN - 03779017 TI - Free energy asymptotics of the quantum Heisenberg spin chain VL - 111 ER - TY - JOUR AB - 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. AU - Bossmann, Lea AU - Petrat, Sören P AU - Seiringer, Robert ID - 9318 JF - Forum of Mathematics, Sigma TI - Asymptotic expansion of low-energy excitations for weakly interacting bosons VL - 9 ER - TY - JOUR AB - 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. AU - Mitrouskas, David Johannes ID - 9333 JF - Letters in Mathematical Physics SN - 03779017 TI - A note on the Fröhlich dynamics in the strong coupling limit VL - 111 ER - TY - JOUR AB - 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. AU - Kirkpatrick, Kay AU - Rademacher, Simone Anna Elvira AU - Schlein, Benjamin ID - 9351 JF - Annales Henri Poincare SN - 1424-0637 TI - A large deviation principle in many-body quantum dynamics VL - 22 ER - TY - JOUR AB - 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. AU - Brooks, Morris AU - Di Gesù, Giacomo ID - 9348 IS - 3 JF - Journal of Functional Analysis SN - 0022-1236 TI - Sharp tunneling estimates for a double-well model in infinite dimension VL - 281 ER - TY - JOUR AB - 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. AU - Deuchert, Andreas AU - Seiringer, Robert ID - 9462 IS - 6 JF - Journal of Functional Analysis SN - 0022-1236 TI - Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons VL - 281 ER - TY - JOUR AB - 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. AU - Lauritsen, Asbjørn Bækgaard ID - 9891 IS - 8 JF - Journal of Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-2488 TI - Floating Wigner crystal and periodic jellium configurations VL - 62 ER - TY - JOUR AB - 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. AU - Feliciangeli, Dario AU - Seiringer, Robert ID - 10224 IS - 3 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics VL - 242 ER - TY - JOUR AB - 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. AU - Benedikter, Niels P AU - Nam, Phan Thành AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 10537 JF - Annales Henri Poincaré SN - 1424-0637 TI - Bosonization of fermionic many-body dynamics ER - TY - JOUR AB - 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. AU - Benedikter, Niels P AU - Nam, Phan Thành AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 7901 JF - Inventiones Mathematicae SN - 0020-9910 TI - Correlation energy of a weakly interacting Fermi gas VL - 225 ER -