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 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.). AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan ID - 554 IS - 1 JF - Communications in Mathematical Physics SN - 00103616 TI - The Bogoliubov free energy functional II: The dilute Limit VL - 360 ER - TY - JOUR AB - 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. AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan ID - 399 IS - 1 JF - EPL TI - Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation VL - 121 ER - TY - JOUR AB - 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. AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan Philip ID - 6002 IS - 3 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - The Bogoliubov free energy functional I: Existence of minimizers and phase diagram VL - 229 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M ID - 484 IS - 3 JF - Advances in Theoretical and Mathematical Physics SN - 10950761 TI - Bogoliubov correction to the mean-field dynamics of interacting bosons VL - 21 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M ID - 739 IS - 5 JF - Journal de Mathématiques Pures et Appliquées SN - 00217824 TI - A note on the validity of Bogoliubov correction to mean field dynamics VL - 108 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M AU - Solovej, Jan ID - 1545 IS - 11 JF - Journal of Functional Analysis TI - Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations VL - 270 ER - TY - JOUR AB - We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys.306:565–578, 2011) to large volumes. AU - Dereziński, Jan AU - Napiórkowski, Marcin M ID - 5813 IS - 12 JF - Annales Henri Poincaré SN - 1424-0637 TI - Excitation spectrum of interacting bosons in the Mean-Field Infinite-Volume limit VL - 15 ER -