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

Condensed Matter Physics

QUANTUM ANHARMONIC CRYSTAL IN FUNCTIONAL INTEGRAL APPROACH

Kondratiev, Kozitsky, Rockner

Published Jan 1, 2003• 1 citation
Publisher

Abstract

A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals.The main aspects of this approach are described on an introductory level.Then a mechanism of the stabilization of this model by quantum effects is suggested.In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given.It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough.It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.

DOI
10.5488/cmp.6.4.647

Related papers

QUANTUM EFFECTS IN AN ANHARMONIC CRYSTAL

Kozitsky

A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.

Condensed Matter Physics•Jan 1, 2002•2 citations

PHASE TRANSITIONS AND QUANTUM EFFECTS IN ANHARMONIC CRYSTALS

Sergio A Albeverio, YURI KONDRATIEV, YURI KOZITSKY +1 more

The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the nondifferentiability of the free energy density and on the appearance of ordering, are also discussed.

arXiv:1204.6279•International Journal of Modern Physics B•Apr 17, 2012•4 citations

Quantum effects in an anharmonic crystal

Yuri V. Kozitsky

A model of quantum particles performing D -dimensional anharmonic oscillations \naround their equilibrium positions which form the d -dimensional \nsimple cubic lattice Zd is considered. The model undergoes a structural \nphase transition when the fluctuations of displacements of particles become \nmacroscopic. This phenomenon is described by susceptibilities depending \non Matsubara frequencies ωn , n ∈ Z . We prove two theorems \nconcerning the thermodynamic limits of these susceptibilities. The first theorem \nstates that the susceptibilities with nonzero ωn remain bounded at all \ntemperatures, which means that the macroscopic fluctuations in the model \nare always non-quantum. The second theorem gives a sufficient condition \nfor the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded \nat all temperatures. This condition involves the particle mass, the anharmonicity \nparameters and the interaction intensity. The physical meaning of \nthis result is that, for all D and all values of the temperature, strong quantum \neffects suppress critical points and the long range order. The proof \nis performed in the approach where the susceptibilities are represented as \nfunctional integrals. A brief description of the main features of this approach \nis delivered.

•Jan 1, 2002•0 citations

Toward extracting the scattering phase shift from integrated correlation functions. II. A relativistic lattice field theory model

Peng Guo

In the present work, a relativistic relation that connects the difference of interacting and noninteracting integrated two-particle correlation functions in finite volume to infinite volume scattering phase shift through an integral is derived. We show that the difference of integrated finite volume correlation functions converges rapidly to its infinite volume limit as the size of the periodic box is increased. The fast convergence of our proposed formalism is illustrated by analytic solutions of a contact interaction model, the perturbation theory calculation, and also the Monte Carlo simulation of a complex ϕ 4 lattice field theory model. Published by the American Physical Society 2024

arXiv:2402.15628•Physical review. D/Physical review. D.•Jul 9, 2024•7 citations

ON THE EXISTENCE OF THE DYNAMICS FOR ANHARMONIC QUANTUM OSCILLATOR SYSTEMS

Bruno Nachtergaele, Benjamin Schlein, Robert Sims +2 more

We construct a W*-dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved Lieb–Robinson bounds for such systems on finite lattices [19].

arXiv:0909.2249•Reviews in Mathematical Physics•Mar 1, 2010•43 citations

Toward extracting the scattering phase shift from integrated correlation functions

Peng Guo, V. Gasparian

In present work, a relation that connects the integrated correlation function of a trapped two-particle system to infinite volume particles scattering phase shift is derived. It has the potential to provide an alternative approach for extracting two-particle scattering phase shift from integrated correlation function in lattice simulation at small Euclidean time region. Both (i) perturbation calculation of ($1+1$)-dimensional lattice Euclidean field theory model of fermions interacting with a contact interaction and (ii) Monte Carlo simulation of a 1D exactly solvable quantum mechanics model are carried out to test the proposed relation. In contrast to conventional two-step approach of extracting energy levels from temporal correlation function in lattice simulation at large Euclidean time first and then applying L\"uscher formula to convert energy levels into scattering phase shifts, we show that the difference of integrated correlation functions between interacting and noninteracting trapped systems converges rapidly to infinite volume limit that is given in terms of scattering phase shifts at small Euclidean time region.

Physical review. D/Physical review. D.•Oct 10, 2023•10 citations
A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals.The main aspects of this approach are described on an introductory level.Then a mechanism of the stabilization of this model by quantum effects is suggested.In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given.It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough.It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.
A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.
The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the nondifferentiability of the free energy density and on the appearance of ordering, are also discussed.
A model of quantum particles performing D -dimensional anharmonic oscillations \naround their equilibrium positions which form the d -dimensional \nsimple cubic lattice Zd is considered. The model undergoes a structural \nphase transition when the fluctuations of displacements of particles become \nmacroscopic. This phenomenon is described by susceptibilities depending \non Matsubara frequencies ωn , n ∈ Z . We prove two theorems \nconcerning the thermodynamic limits of these susceptibilities. The first theorem \nstates that the susceptibilities with nonzero ωn remain bounded at all \ntemperatures, which means that the macroscopic fluctuations in the model \nare always non-quantum. The second theorem gives a sufficient condition \nfor the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded \nat all temperatures. This condition involves the particle mass, the anharmonicity \nparameters and the interaction intensity. The physical meaning of \nthis result is that, for all D and all values of the temperature, strong quantum \neffects suppress critical points and the long range order. The proof \nis performed in the approach where the susceptibilities are represented as \nfunctional integrals. A brief description of the main features of this approach \nis delivered.
In the present work, a relativistic relation that connects the difference of interacting and noninteracting integrated two-particle correlation functions in finite volume to infinite volume scattering phase shift through an integral is derived. We show that the difference of integrated finite volume correlation functions converges rapidly to its infinite volume limit as the size of the periodic box is increased. The fast convergence of our proposed formalism is illustrated by analytic solutions of a contact interaction model, the perturbation theory calculation, and also the Monte Carlo simulation of a complex ϕ 4 lattice field theory model. Published by the American Physical Society 2024
In present work, a relation that connects the integrated correlation function of a trapped two-particle system to infinite volume particles scattering phase shift is derived. It has the potential to provide an alternative approach for extracting two-particle scattering phase shift from integrated correlation function in lattice simulation at small Euclidean time region. Both (i) perturbation calculation of (1+11+11+1)-dimensional lattice Euclidean field theory model of fermions interacting with a contact interaction and (ii) Monte Carlo simulation of a 1D exactly solvable quantum mechanics model are carried out to test the proposed relation. In contrast to conventional two-step approach of extracting energy levels from temporal correlation function in lattice simulation at large Euclidean time first and then applying L\"uscher formula to convert energy levels into scattering phase shifts, we show that the difference of integrated correlation functions between interacting and noninteracting trapped systems converges rapidly to infinite volume limit that is given in terms of scattering phase shifts at small Euclidean time region.