Journal of Mathematical Physics

Large deviations in a semi-infinite layered ferromagnet with mean-field interactions: Classical and quantum models

N. Angelescu, M. Bundaru, Joseph V. Pulé

Published Sep 1, 1995• 0 citations

Abstract

We consider the probability distribution in a classical semi-infinite layered ferromagnet with mean-field interactions and prove that this satisfies a large deviation principle, giving the rate function explicitly. For an analogous quantum system we show that the limiting states are product states, which are completely characterized by the classical mean-field solutions.

DOI: 10.1063/1.530919

Related papers

A Large Deviation Principle in Many-Body Quantum Dynamics

Kay Kirkpatrick, Simone Rademacher, Benjamin Schlein

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.

arXiv:2010.13754Annales Henri PoincaréApr 8, 202110 citations

Large deviations in mean-field quantum spin systems

Matthias Keller, Christiaan J. F. van de Ven

arXiv:2410.08538Oct 11, 20240 citations

Landau-Zener-Stueckelberg effect in a model of interacting tunneling systems

Dmitry A. Garanin

arXiv:cond-mat/0302107Physical review. B, Condensed matterJul 11, 200324 citations

Quantum Heisenberg model with long-range ferromagnetic interactions: A Green’s function approach

Mohamed Hamedoun, Y. Cherriet, Ahmed Hourmatallah +1 more

Physical review. B, Condensed matterMar 28, 200123 citations

Large Deviations for the Ground State of Weakly Interacting Bose Gases

Simone Rademacher

arXiv:2301.00430Annales Henri PoincaréJul 2, 20244 citations

On the general principle of the mean-field approximation for many-boson dynamics

Clément Rouffort

arXiv:1809.01450Sep 5, 20182 citations

Journal of Mathematical Physics

Large deviations in a semi-infinite layered ferromagnet with mean-field interactions: Classical and quantum models

N. Angelescu, M. Bundaru, Joseph V. Pulé

Published Sep 1, 1995• 0 citations

Publisher

Abstract

DOI: 10.1063/1.530919

Related papers

A Large Deviation Principle in Many-Body Quantum Dynamics

Kay Kirkpatrick, Simone Rademacher, Benjamin Schlein

arXiv:2010.13754Annales Henri PoincaréApr 8, 202110 citations

Large deviations in mean-field quantum spin systems

Matthias Keller, Christiaan J. F. van de Ven

Continuous fields (or bundles) of $C^*$ -algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^*$ -bundle generated by increasing symmetric tensor powers of the complex $\ell\times\ell$ matrices $M_\ell(\mathbb{C})$ , which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.Continuous fields (or bundles) of $C^*$-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^*$-bundle generated by increasing symmetric tensor powers of the complex $\ell\times\ell$ matrices $M_\ell(\mathbb{C})$, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.

arXiv:2410.08538Oct 11, 20240 citations

Landau-Zener-Stueckelberg effect in a model of interacting tunneling systems

Dmitry A. Garanin

The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin $S=N/2.$ The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling $J>0$ tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For $J<0$ LZS transitions are boosted and $P=0$ for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin $S=N/2.$ The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling $J>0$ tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For $J<0$ LZS transitions are boosted and $P=0$ for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.

arXiv:cond-mat/0302107Physical review. B, Condensed matterJul 11, 200324 citations

Quantum Heisenberg model with long-range ferromagnetic interactions: A Green’s function approach

Mohamed Hamedoun, Y. Cherriet, Ahmed Hourmatallah +1 more

A Green-function method is employed within a random phase approximation to study the one-dimensional quantum Heisenberg model with long-range ferromagnetic interaction proportional to ${r}^{\ensuremath{-}p}.$ We have shown that for $1<p<2$ there exists a phase transition at finite temperatures and estimate its critical temperature.A Green-function method is employed within a random phase approximation to study the one-dimensional quantum Heisenberg model with long-range ferromagnetic interaction proportional to ${r}^{\ensuremath{-}p}.$ We have shown that for $1<p<2$ there exists a phase transition at finite temperatures and estimate its critical temperature.

Physical review. B, Condensed matterMar 28, 200123 citations

Large Deviations for the Ground State of Weakly Interacting Bose Gases

Simone Rademacher

Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $N \rightarrow \infty$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.Abstract We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit $$N \rightarrow \infty $$ N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

arXiv:2301.00430Annales Henri PoincaréJul 2, 20244 citations

On the general principle of the mean-field approximation for many-boson dynamics

Clément Rouffort

The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: $\bullet$ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). $\bullet$ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. $\bullet$ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: $\bullet$ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). $\bullet$ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. $\bullet$ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.

arXiv:1809.01450Sep 5, 20182 citations

Continuous fields (or bundles) of

C^*

-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous

C^*

-bundle generated by increasing symmetric tensor powers of the complex

\ell\times\ell

matrices

M_\ell(\mathbb{C})

, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.

S=N/2.

The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling

J>0

tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For

J<0

LZS transitions are boosted and

P=0

for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.

1<p<2

there exists a phase transition at finite temperatures and estimate its critical temperature.

N \rightarrow \infty

N → ∞ bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.

\bullet

The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles).

\bullet

The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part.

\bullet

There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.