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arXiv:2111.04375

Solvability of spin systems with two body interactions: back to Babylon

Nicola Kistler

Published Nov 8, 2021• 0 citations
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Abstract

We show that spin systems with two body, generic (ferro- or paramagnetic, or random) interactions are 'completely integrable'. The approach is worked out, by way of example, for the Sherrington Kirkpatrick model: we derive an exact, closed formula for the quenched free energy in finite volume which involves an integral over a Gaussian field with correlation structure given by the interaction matrix (with a twist).

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We introduce a novel quantum spin-glass model, a Sherrington-Kirkpatrick model with a transverse mean-field type random magnet. We rigorously derive the exact expression of the free energy of this model at the entire parameter region. The obtained exact solution implies the existence of a 1RSB-fullRSB transition at low temperatures. Our technique can be applied to general classical spin models, telling that any solvable classical spin model has its solvable quantum counterpart.

arXiv:2410.03079•Oct 4, 2024•0 citations
We show that spin systems with two body, generic (ferro- or paramagnetic, or random) interactions are 'completely integrable'. The approach is worked out, by way of example, for the Sherrington Kirkpatrick model: we derive an exact, closed formula for the quenched free energy in finite volume which involves an integral over a Gaussian field with correlation structure given by the interaction matrix (with a twist).
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.
We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, , we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N - 1 integrals of motion, notably, their scaling with N, and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.
Integrability conditions for systems of bosons or fermions with seniority conserving hamiltonians are derived. The conditions are shown to be invariant under a large class of transformations of the interaction matrix elements. Previously published integrable models are shown to satisfy these conditions and the existence of a new class of integrable models is demonstrated. The number of free parameters in the interaction in these models equals the number of single particle levels plus 3 . Equations for the energy eigenstates and eigenvalues are derived and the eigenvalues of the complete set of two-body integrals of the motion are given for the new class. Some two-body correlations in these eigenstates are derived from the integrals of the motion.
Abstract We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, , we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N − 1 integrals of motion, notably, their scaling with N , and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs–Boltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.