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Non-Existence of Local Integrals of Motion in the Multi-Deformed Ising Model

Fonseca, P D

Published Apr 22, 1998• 3 citations
OpenAlex

Abstract

We confirm the non-integrability of the multi-deformed Ising Model, an already expected result. After deforming with the energy operator $\\phi_{1,3}$ we use the Majorana free fermionic representation for the massive theory to show that, besides the trivial one, no local integrals of motion can be built in the theory arising from perturbing with both energy and spin operators.

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NON-EXISTENCE OF LOCAL INTEGRALS OF MOTION IN THE MULTI-DEFORMED ISING MODEL

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We confirm the non-integrability of the multi-deformed Ising model — an already expected result. After deforming with the energy operator ϕ1,3, we use the Majorana free fermionic representation for the massive theory to show that, besides the trivial one, no local integrals of motion can be built in the theory arising from perturbing with both energy and spin operators.

arXiv:hep-th/9804144•Modern Physics Letters A•Aug 10, 1998•2 citations

On the integrable structure of the Ising model

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Starting from the lattice A 3 realization of the Ising model defined on a strip with integrable boundary conditions, the exact spectrum (including excited states) of all the local integrals of motion is derived in the continuum limit by means of thermodynamic Bethe ansatz techniques. It is also possible to follow the massive flow of this spectrum between the UV c = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates of such integrals of motion in terms of Virasoro modes is found to have only rational coefficients and their fermionic representation turns out to be simply related to the quantum numbers describing the spectrum.

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Free Fermions and Two-Dimensional Ising Model

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On the S-matrix of Ising field theory in two dimensions

Barak Gabai, Xi Yin

Abstract We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 → 2 S-matrix element to the complex s-plane. Focusing primarily on the “high temperature” regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.

Journal of High Energy Physics•Oct 26, 2022•22 citations

On The S-Matrix of Ising Field Theory in Two Dimensions

Barak Gabai, Xi Yin

We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.

arXiv:1905.00710•May 2, 2019•2 citations

Identification of Majorana Modes in Interacting Systems by Local Integrals of Motion

Andrzej Więckowski, Maciej M. Maśka, Marcin Mierzejewski

Recently, there has been substantial progress in methods of identifying local integrals of motion in interacting integrable models or in systems with many-body localization. We show that one of these approaches can be utilized for constructing local, conserved, Majorana fermions in systems with an arbitrary many-body interaction. As a test case, we first investigate a noninteracting Kitaev model and demonstrate that this approach perfectly reproduces the standard results. Then, we discuss how the many-body interactions influence the spatial structure and the lifetime of the Majorana modes. Finally, we determine the regime for which the information stored in the Majorana correlators is also retained for arbitrarily long times at high temperatures. We show that it is included in the regime with topologically protected soft Majorana modes, but in some cases is significantly smaller.

arXiv:1707.08125•Physical Review Letters•Jan 26, 2018•20 citations
We confirm the non-integrability of the multi-deformed Ising Model, an already expected result. After deforming with the energy operator phi1,3\\phi_{1,3}phi1,3​ we use the Majorana free fermionic representation for the massive theory to show that, besides the trivial one, no local integrals of motion can be built in the theory arising from perturbing with both energy and spin operators.
Starting from the lattice A 3 realization of the Ising model defined on a strip with integrable boundary conditions, the exact spectrum (including excited states) of all the local integrals of motion is derived in the continuum limit by means of thermodynamic Bethe ansatz techniques. It is also possible to follow the massive flow of this spectrum between the UV c = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates of such integrals of motion in terms of Virasoro modes is found to have only rational coefficients and their fermionic representation turns out to be simply related to the quantum numbers describing the spectrum.
Abstract We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 → 2 S-matrix element to the complex s-plane. Focusing primarily on the “high temperature” regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.
We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.
Recently, there has been substantial progress in methods of identifying local integrals of motion in interacting integrable models or in systems with many-body localization. We show that one of these approaches can be utilized for constructing local, conserved, Majorana fermions in systems with an arbitrary many-body interaction. As a test case, we first investigate a noninteracting Kitaev model and demonstrate that this approach perfectly reproduces the standard results. Then, we discuss how the many-body interactions influence the spatial structure and the lifetime of the Majorana modes. Finally, we determine the regime for which the information stored in the Majorana correlators is also retained for arbitrarily long times at high temperatures. We show that it is included in the regime with topologically protected soft Majorana modes, but in some cases is significantly smaller.