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International Journal of Modern Physics A

TRUNCATED-FERMIONIC-SPACE APPROACH TO THE CRITICAL 2D ISING MODEL WITH MAGNETIC FIELD

V.P. Yurov, Al.B. Zamolodchikov

Published Oct 20, 1991• 156 citations
Publisher

Abstract

The critical Ising model perturbed by the spin field (conjugated to the magnetic field) is studied numerically by the method of truncated fermionic space of states. The matrix elements of the perturbed Hamiltonian are found between the states of free Majorana fermions living on a finite-length circle. The Hamiltonian spectrum is studied numerically in the level-5 truncated space. Reasonable estimations are obtained for the vacuum energy, a part of the mass spectrum and the simplest scattering amplitude of the perturbed system.

DOI
10.1142/s0217751x91002161

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The critical Ising model perturbed by the spin field (conjugated to the magnetic field) is studied numerically by the method of truncated fermionic space of states. The matrix elements of the perturbed Hamiltonian are found between the states of free Majorana fermions living on a finite-length circle. The Hamiltonian spectrum is studied numerically in the level-5 truncated space. Reasonable estimations are obtained for the vacuum energy, a part of the mass spectrum and the simplest scattering amplitude of the perturbed system.
A numerical approach to 2-D relativistic field theories is suggested. Considering a field theory model as an ultraviolet conformal field theory perturbed by a suitable relevant scalar operator one studies it in finite volume (on a circle). The perturbed Hamiltonian acts in the conformal field theory space of states and its matrix elements can be extracted from the conformal field theory. Truncation of the space at a reasonable level results in a finite dimensional problem for numerical analyses. The nonunitary field theory with the ultraviolet region controlled by the minimal conformal theory {ital M}(2/5) is studied in detail.
A numerical approach to 2-D relativistic field theories is suggested. Considering a field theory model as an ultraviolet conformal field theory perturbed by a suitable relevant scalar operator one studies it in finite volume (on a circle). The perturbed Hamiltonian acts in the conformal field theory space of states and its matrix elements can be extracted from the conformal field theory. Truncation of the space at a reasonable level results in a finite dimensional problem for numerical analyses. The nonunitary field theory with the ultraviolet region controlled by the minimal conformal theory [Formula: see text] is studied in detail.
In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1 + 1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theory, we study the effect of an integrability breaking perturbation by the longitudinal magnetic field. In both the ferromagnetic and paramagnetic phases of the model we find persistent oscillations with frequencies set by the low-lying particle excitations not only for small, but even for moderate size quenches. In the ferromagnetic phase these particles are the various non-perturbative confined bound states of the domain wall excitations, while in the paramagnetic phase the single magnon excitation governs the dynamics, allowing us to capture the time evolution of the magnetisation using a combination of known results from perturbation theory and form factor based methods. We point out that the dominance of low lying excitations allows for the numerical or experimental determination of the mass spectra through the study of the quench dynamics.
We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.
We apply the massive analogue of the truncated conformal space approach to study the two dimensional φ4 theory in finite volume. We focus on the broken phase and determine the finite size spectrum of the model numerically. We interpret the results in terms of the Bethe- Yang spectrum, from which we extract the infinite volume masses and scattering matrices for various couplings. We compare these results against semiclassical analysis and perturbation theory. We also analyze the critical point of the model and confirm that it is in the Ising universality class.
We investigate the idea that the effect of the truncation applied in the truncated conformal space approach method to the spectrum coincides with the effect of a suitable changing of the coefficients of the terms in the Hamiltonian operator. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. A detailed quantum field theoretical description of this model is also given, and we propose a description as a perturbation of the infinite coupling limit. The investigation is also carried out for a truncation method which preserves the solvability of the model. The results of perturbative and numerical calculations presented support the above idea and show that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method.