Journal of Physics A Mathematical and General

Construction of quantum systems with soluble ground-state properties

T. Schneider, R. Badii

Published Mar 11, 1985• 6 citations

Abstract

The authors propose a transformation of the Langevin equation into an eigenvalue problem as a method to construct systems with soluble ground-state properties. As examples, they discuss quantum systems resulting from the Toda and sine-Gordon chains evolving according to the Langevin equation. Some ground-state properties are then evaluated with methods originally devised to calculate the partition function of one-dimensional classical systems. They also present numerical results for the two-phonon bound-state frequency and its coupling constant dependence by simulating a generalised quantum sine-Gordon system.

DOI: 10.1088/0305-4470/18/4/001

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Journal of Physics A Mathematical and General

Construction of quantum systems with soluble ground-state properties

T. Schneider, R. Badii

Published Mar 11, 1985• 6 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/18/4/001

Related papers

Langevin simulations of quantum systems

R. Badii, T. Schneider, Mads Peter Soerensen

We explore the potential of the Langevin simulation of quantum systems. A generalized quantum sine-Gordon chain, a generalized Toda chain, and impenetrable bosons on a ring are treated specifically to estimate low-lying excitation energies from the long-time behavior of the Langevin process. In the sine-Gordon case, we investigated the dependence of the bound-state frequency on the coupling constant, while in the Toda chain the phonon dispersion curves are obtained for different coupling constants. Finally, for impenetrable bosons the estimated dynamic form factor for density fluctuations is compared with exact results.We explore the potential of the Langevin simulation of quantum systems. A generalized quantum sine-Gordon chain, a generalized Toda chain, and impenetrable bosons on a ring are treated specifically to estimate low-lying excitation energies from the long-time behavior of the Langevin process. In the sine-Gordon case, we investigated the dependence of the bound-state frequency on the coupling constant, while in the Toda chain the phonon dispersion curves are obtained for different coupling constants. Finally, for impenetrable bosons the estimated dynamic form factor for density fluctuations is compared with exact results.

Physical review. B, Condensed matterJan 1, 19875 citations

Stochastic simulation of quantum systems and critical dynamics

T. Schneider, Marco Zannetti, R. Badii

We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.

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We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical $d$ -dimensional system is mapped onto the statics of an associated quantum system and of its classical ( $d+1$ )-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical $d$-dimensional system is mapped onto the statics of an associated quantum system and of its classical ($d+1$)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.

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Tai Kai Ng

Physical review. B, Condensed matterAug 1, 19913 citations

ODE/IM Correspondence in Toda Field Theories and Fermionic Basis in sin(h)-Gordon Model

Stefano Negro

The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Kač-Moody algebra $\hat{\mathfrak g}$ . In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called $ψ$ -system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$ . Finally, from the $ψ$ -system for the algebras $\mathfrak a$ and $\mathfrak d$ , we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Kač-Moody algebra $\hat{\mathfrak g}$. In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called $ψ$-system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$. Finally, from the $ψ$-system for the algebras $\mathfrak a$ and $\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.

arXiv:1702.06657Feb 22, 20175 citations

d

-dimensional system is mapped onto the statics of an associated quantum system and of its classical (

d+1

)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.

\hat{\mathfrak a}_1

to a general affine Lie-Kač-Moody algebra

\hat{\mathfrak g}

. In order to achieve our goal, we investigate the structure of evaluation representations of

\hat{\mathfrak g}

and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called

ψ

-system, for the solutions to complex differential equations associated to

\hat{\mathfrak g}

. Finally, from the

ψ

-system for the algebras

\mathfrak a

and

\mathfrak d

, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.