The Bethe-ansatz (BA) equation for the classical sine-Gordon (SG) thermodynamics is shown to reduce to the one for the classical Toda lattice within two approximations; the phonon-phonon interactions are neglected (the harmonic phonon) and solitons are treated nonrelativistically. Within these approximations the solitons in the SG model correspond to the particles in the Toda lattice. Making use of an analytical solution of the BA equation for the classical Toda lattice due to Opper, we obtain an analytical solution of the approximate SG equation. We propose also a similar approximate BA equation for the ${\ensuremath{\varphi}}^{4}$ model, although the ${\ensuremath{\varphi}}^{4}$ model does not have the BA equation for the classical thermodynamics.The Bethe-ansatz (BA) equation for the classical sine-Gordon (SG) thermodynamics is shown to reduce to the one for the classical Toda lattice within two approximations; the phonon-phonon interactions are neglected (the harmonic phonon) and solitons are treated nonrelativistically. Within these approximations the solitons in the SG model correspond to the particles in the Toda lattice. Making use of an analytical solution of the BA equation for the classical Toda lattice due to Opper, we obtain an analytical solution of the approximate SG equation. We propose also a similar approximate BA equation for the ${\ensuremath{\varphi}}^{4}$ model, although the ${\ensuremath{\varphi}}^{4}$ model does not have the BA equation for the classical thermodynamics.

Physical review. B, Condensed matterMar 1, 19873 citations

Spreading of Perturbations in Long-Range Interacting Classical Lattice Models

David Métivier, Romain Bachelard, Michael Kästner

Lieb-Robinson-type bounds are reported for a large class of classical Hamiltonian lattice models. By a suitable rescaling of energy or time, such bounds can be constructed for interactions of arbitrarily long range. The bound quantifies the dependence of the system's dynamics on a perturbation of the initial state. The effect of the perturbation is found to be effectively restricted to the interior of a causal region of logarithmic shape, with only small, algebraically decaying effects in the exterior. A refined bound, sharper than conventional Lieb-Robinson bounds, is required to correctly capture the shape of the causal region, as confirmed by numerical results for classical long-range $XY$ chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.Lieb-Robinson-type bounds are reported for a large class of classical Hamiltonian lattice models. By a suitable rescaling of energy or time, such bounds can be constructed for interactions of arbitrarily long range. The bound quantifies the dependence of the system's dynamics on a perturbation of the initial state. The effect of the perturbation is found to be effectively restricted to the interior of a causal region of logarithmic shape, with only small, algebraically decaying effects in the exterior. A refined bound, sharper than conventional Lieb-Robinson bounds, is required to correctly capture the shape of the causal region, as confirmed by numerical results for classical long-range $XY$ chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.

arXiv:1405.7556Physical Review LettersMay 28, 201443 citations

A probabilistic view of the quantum Toda lattice

Neil O’Connell

Jan 1, 20100 citations

Exact continuum representation of long-range interacting systems and emerging exotic phases in unconventional superconductors

Andreas Alexander Buchheit, Torsten Keßler, Peter K. Schuhmacher +1 more

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this paper, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, for any lattice, for any power-law interaction, and for linear, nonlinear, and multiatomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely, the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize nonequilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization. By providing a unifying framework for long-range interactions on a lattice, both classical and quantum, our research can guide the search for exotic phases of matter across different fields.Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this paper, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, for any lattice, for any power-law interaction, and for linear, nonlinear, and multiatomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely, the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize nonequilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization. By providing a unifying framework for long-range interactions on a lattice, both classical and quantum, our research can guide the search for exotic phases of matter across different fields.

arXiv:2201.11101Physical Review ResearchOct 20, 20236 citations

XY

chains. We discuss the relevance of our findings for the relaxation to equilibrium of long-range interacting lattice models.