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

Physical review. B, Condensed matter

Path-integral methods for treating quantal behavior in solids: Mean-field theory and the effects of fluctuations

Richard M. Stratt

Published Feb 1, 1986• 59 citations
Publisher

Abstract

Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.

DOI
10.1103/physrevb.33.1921

Related papers

Quantum fluctuations in the transverse Ising spin glass model: A field theory of random quantum spin systems

Kazutaka Takahashi

We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at $T=0$.

arXiv:0708.2966•Physical Review B•Nov 19, 2007•22 citations

Quantal Ising Liquid

Richard M. Stratt

An example is presented of a model of an amorphous quantum mechanical system, a liquid of quantal Ising spins, which can be solved exactly within certain many-body theories. Analytical solutions of the model in mean-field theory are shown to reveal a decrease in the extent of the ferromagnetic region (compared to an equivalent classical system) and the occurrence of some degree of quantum localization. Both phenomena are analyzed as a competition between quantum mechanics and the condensed phase.

Physical Review Letters•Oct 1, 1984•32 citations

Quantum mean-field approximation for lattice quantum models: Truncating quantum correlations and retaining classical ones

Daniele Malpetti, Tommaso Roscilde

The mean-field approximation is at the heart of our understanding of complex systems, despite its fundamental limitation of completely neglecting correlations between the elementary constituents. In a recent work [Phys. Rev. Lett. 117, 130401 (2016)], we have shown that in quantum many-body systems at finite temperature, two-point correlations can be formally separated into a thermal part and a quantum part and that quantum correlations are generically found to decay exponentially at finite temperature, with a characteristic, temperature-dependent quantum coherence length. The existence of these two different forms of correlation in quantum many-body systems suggests the possibility of formulating an approximation, which affects quantum correlations only, without preventing the correct description of classical fluctuations at all length scales. Focusing on lattice boson and quantum Ising models, we make use of the path-integral formulation of quantum statistical mechanics to introduce such an approximation, which we dub quantum mean-field (QMF) approach, and which can be readily generalized to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to cluster mean-field theory at $T=0$, while at any finite temperature it produces a family of systematically improved, semi-classical approximations to the quantum statistical mechanics of the lattice theory at hand. Contrary to standard MF approximations, the correct nature of thermal critical phenomena is captured by any cluster size. In the two exemplary cases of the two-dimensional quantum Ising model and of two-dimensional quantum rotors, we study systematically the convergence of the cQMF approximation towards the exact result, and show that the convergence is typically linear or sublinear in the boundary-to-bulk ratio of the clusters as $T\ensuremath{\rightarrow}0$, while it becomes faster than linear as $T$ grows. These results pave the way towards the development of semiclassical numerical approaches based on an approximate, yet systematically improved account of quantum correlations.

arXiv:1608.01205•Physical review. B./Physical review. B•Feb 6, 2017•4 citations

A Study of the Transverse Ising Model

Jialin Zhong, Chuan‐Zhang Yang, Xuefan Jiang

Abstract A new method which combines the discretized path‐integral with variational approximation is proposed and used to derive the free energy of the transverse Ising model in a quantum spin system. The transition temperature dependence of both, the transverse field and the coordination number is obtained analytically. The numerical results are compared with those of other methods.

physica status solidi (b)•Oct 1, 1991•1 citation

Path Integrals in Quantum Physics

Rosenfelder, R.

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.

arXiv:1209.1315•Sep 6, 2012•3 citations

Effects of dynamical paths on the energy gap and the corrections to the free energy in path integrals of mean-field quantum spin systems

Yang Wei Koh

In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter, which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. 7, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and nonextensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.

arXiv:1803.08632•Physical review. B./Physical review. B•Mar 16, 2018•1 citation
Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.
We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at T=0T=0T=0.
An example is presented of a model of an amorphous quantum mechanical system, a liquid of quantal Ising spins, which can be solved exactly within certain many-body theories. Analytical solutions of the model in mean-field theory are shown to reveal a decrease in the extent of the ferromagnetic region (compared to an equivalent classical system) and the occurrence of some degree of quantum localization. Both phenomena are analyzed as a competition between quantum mechanics and the condensed phase.
The mean-field approximation is at the heart of our understanding of complex systems, despite its fundamental limitation of completely neglecting correlations between the elementary constituents. In a recent work [Phys. Rev. Lett. 117, 130401 (2016)], we have shown that in quantum many-body systems at finite temperature, two-point correlations can be formally separated into a thermal part and a quantum part and that quantum correlations are generically found to decay exponentially at finite temperature, with a characteristic, temperature-dependent quantum coherence length. The existence of these two different forms of correlation in quantum many-body systems suggests the possibility of formulating an approximation, which affects quantum correlations only, without preventing the correct description of classical fluctuations at all length scales. Focusing on lattice boson and quantum Ising models, we make use of the path-integral formulation of quantum statistical mechanics to introduce such an approximation, which we dub quantum mean-field (QMF) approach, and which can be readily generalized to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to cluster mean-field theory at T=0T=0T=0, while at any finite temperature it produces a family of systematically improved, semi-classical approximations to the quantum statistical mechanics of the lattice theory at hand. Contrary to standard MF approximations, the correct nature of thermal critical phenomena is captured by any cluster size. In the two exemplary cases of the two-dimensional quantum Ising model and of two-dimensional quantum rotors, we study systematically the convergence of the cQMF approximation towards the exact result, and show that the convergence is typically linear or sublinear in the boundary-to-bulk ratio of the clusters as $T\ensuremath{\rightarrow}0$, while it becomes faster than linear as TTT grows. These results pave the way towards the development of semiclassical numerical approaches based on an approximate, yet systematically improved account of quantum correlations.
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.
In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter, which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. 7, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and nonextensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.