These notes deal with the mean-field approximation for equilibrium states of N-body systems in classical and quantum statistical mechanics. A general strategy for the justification of effective models based on statistical independence assumptions is presented in details. The main tools are structure theorems {à} la de Finetti, describing the large N limits of admissible states for these systems. These rely on the symmetry under exchange of particles, due to their indiscernability. Emphasis is put on quantum aspects, in particular the mean-field approximation for the ground states of large bosonic systems, in relation with the Bose-Einstein condensation phenomenon. Topics covered in details include: the structure of reduced density matrices for large bosonic systems, Fock-space localization methods, derivation of effective energy functionals of Hartree or non-linear Schr{ö}dinger type, starting from the many-body Schr{ö}dinger Hamiltonian.
These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schrödinger effective energy functionals.
The quantum de Finetti theorem asserts that the |
k|-body density matrices of an |
N|-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when |
N| is large and |
k| fixed. In this note, we review a construction due to Christandl et al. [8] valid for finite-dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella [7], which gives the density matrices of the constructed state as a function of those of the original state.
The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized
SU(M) coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, such as the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semiclassical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyze the depletion and heating of the condensate. A comparison to methods ignoring the fixed particle number shows that in this case artificial number fluctuations lead to ambiguities and large deviations even for quite simple examples.
Quantum versions of de Finetti’s theorem are powerful tools, yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean-field approaches. Such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles. In this work, we derive a de Finetti theorem for finite sized Majorana fermionic systems. It is shown, much reflecting the spirit of other quantum de Finetti theorems, that a state which is invariant under certain permutations of modes loses most of its anti-symmetric character and is locally well described by a mode separable state. We discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the Hartree-Fock approximation of quantum systems. We hint at a link to generalized Pauli principles for one-body reduced density operators. Finally, building upon the obtained de Finetti theorem, we generalize and extend the applicability of Hudson’s fermionic central limit theorem.
Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the product-state ansatz, in which all particles are in the same quantum state. As particle-particle correlations become important, however, the product ansatz begins to fail. We consider a new set of states, which constitute a natural generalization of the product ansatz; the particle-correlated state of
N=l×n identical particles is derived by symmetrizing the
n-fold product of an
l-particle quantum state. The particle-correlated states can be simulated efficiently for large
N, because their parameter spaces, which depend on
l, do not grow with
n. Here we pay special attention to the pure-state case for
l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions for bosons, which we call pair-correlated states (PCS), were introduced by Leggett [Rev. Mod. Phys.
73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves the reduced (marginal) density matrices (RDMs)---these are the correlation functions---associated with PCS in time
O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large
N limit. To test the efficacy of our theory, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-
N approximate form. The relative errors of the ground state energy for both cases are within
10−5 for
N=1000 particles over the entire parameter region from a single condensate to a Mott insulator. Moreover, we present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.