Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Theoretical and Mathematical Physics

Mathematical description of the states of bose and fermi systems by the method of partial density matrices of the canonical ensemble

К. С. Матвийчук

Published Dec 1, 1979• 3 citations
Publisher
DOI
10.1007/bf01019377

Related papers

Mathematical description of quantum systems by the method of reduced density matrices of a canonical ensemble

К. С. Матвийчук

Доклады Академии наук•Mar 1, 1976•0 citations

Method of m-particle density matrices of the canonical ensemble in the description of states of quantum systems

К. С. Матвийчук

Theoretical and Mathematical Physics•Jun 1, 1976•5 citations

The mathematical description of quantum systems by reduced density matrices of a canonical ensemble

К. С. Матвийчук

SPhD•Mar 1, 1976•0 citations

Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

Kazumasa Tsutsui, Takafumi Kita

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hatψ({\bf r})\rangle$ of the field operator $\hatψ({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hatψ({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.

arXiv:1610.03617•Journal of the Physical Society of Japan•Oct 12, 2016•4 citations

Bose-Einstein and Fermi-Dirac systems and reduced density matrices with nonzero boundary conditions

К. С. Матвийчук

Ukrainian Mathematical Journal•Jan 1, 1974•0 citations

Stable recursive auxiliary field quantum Monte Carlo algorithm in the canonical ensemble: Applications to thermometry and the Hubbard model

Tong Shen, Hatem Barghathi, Jiangyong Yu +2 more

Many experimentally accessible, finite-sized interacting quantum systems are most appropriately described by the canonical ensemble of statistical mechanics. Conventional numerical simulation methods either approximate them as being coupled to a particle bath or use projective algorithms which may suffer from nonoptimal scaling with system size or large algorithmic prefactors. In this paper, we introduce a highly stable, recursive auxiliary field quantum Monte Carlo approach that can directly simulate systems in the canonical ensemble. We apply the method to the fermion Hubbard model in one and two spatial dimensions in a regime known to exhibit a significant "sign" problem and find improved performance over existing approaches including rapid convergence to ground-state expectation values. The effects of excitations above the ground state are quantified using an estimator-agnostic approach including studying the temperature dependence of the purity and overlap fidelity of the canonical and grand canonical density matrices. As an important application, we show that thermometry approaches often exploited in ultracold atoms that employ an analysis of the velocity distribution in the grand canonical ensemble may be subject to errors leading to an underestimation of extracted temperatures with respect to the Fermi temperature.

arXiv:2212.08654•Physical review. E•May 9, 2023•9 citations
We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with NNN particles. The pair distribution function g(2)(r)g^{(2)}(r)g(2)(r) of fermions clearly exhibits antibunching with g(2)(0)=0g^{(2)}(0)=0g(2)(0)=0 due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with g(2)(0)≈2g^{(2)}(0)\approx 2g(2)(0)≈2, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature T0T_0T0​, an off-diagonal long-range order develops in the one-particle density matrix to reach g(1)(r)=1g^{(1)}(r)=1g(1)(r)=1 at T=0T=0T=0, and the pair correlation starts to decrease towards g(2)(r)≈1g^{(2)}(r)\approx 1g(2)(r)≈1 at T=0T=0T=0. The results for N→∞N\rightarrow \inftyN→∞ are seen to converge to those of the grand canonical ensemble obtained by assuming the average ⟨ψ^(r)⟩\langle\hatψ({\bf r})\rangle⟨ψ^​(r)⟩ of the field operator ψ^(r)\hatψ({\bf r})ψ^​(r) below T0T_0T0​. This fact justifies the introduction of the "anomalous" average ⟨ψ^(r)⟩≠0\langle\hatψ({\bf r})\rangle\neq 0⟨ψ^​(r)⟩=0 below T0T_0T0​ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.
Many experimentally accessible, finite-sized interacting quantum systems are most appropriately described by the canonical ensemble of statistical mechanics. Conventional numerical simulation methods either approximate them as being coupled to a particle bath or use projective algorithms which may suffer from nonoptimal scaling with system size or large algorithmic prefactors. In this paper, we introduce a highly stable, recursive auxiliary field quantum Monte Carlo approach that can directly simulate systems in the canonical ensemble. We apply the method to the fermion Hubbard model in one and two spatial dimensions in a regime known to exhibit a significant "sign" problem and find improved performance over existing approaches including rapid convergence to ground-state expectation values. The effects of excitations above the ground state are quantified using an estimator-agnostic approach including studying the temperature dependence of the purity and overlap fidelity of the canonical and grand canonical density matrices. As an important application, we show that thermometry approaches often exploited in ultracold atoms that employ an analysis of the velocity distribution in the grand canonical ensemble may be subject to errors leading to an underestimation of extracted temperatures with respect to the Fermi temperature.