© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field

G von Gehlen

Published Nov 21, 1991• 79 citations

Abstract

The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.

DOI: 10.1088/0305-4470/24/22/021

Related papers

NON-HERMITIAN TRICRITICALITY IN THE BLUME-CAPEL MODEL WITH IMAGINARY FIELD

G von Gehlen

Using finite-size-scaling methods, we study the quantum chain version of the spin-1-Blume-Capel model coupled to an imaginary field. The aim is to realize simple non-unitary minimal conformal field theories in an Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the three lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line realizes the universality class of the ℳ2,7-conformal theory with c=−68/7.

International Journal of Modern Physics B•Nov 1, 1994•24 citations

Non-hermitian tricriticality in the Blume-Capel model with imaginary field

G von Gehlen

Using finite-size-scaling methods, we study the quantum chain version of the spin-$1$-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge $c=-22/5$). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the ${\cal M}_{2,7}$-conformal theory with $c=-68/7$.

arXiv:hep-th/9402143•Feb 25, 1994•0 citations

Universal finite-size gap scaling of the quantum Ising chain

Masaki Oshikawa

I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in $1+1$ dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.

arXiv:1910.06353•Oct 14, 2019•5 citations

Critical and off-critical properties of the XXZ chain in external homogeneous and staggered magnetic fields

Francisco Castilho Alcaraz, André Luiz Malvezzi

The phase diagram of the XXZ chain is calculated under the influence of homogeneous and staggered magnetic fields. The model has a rich phase structure with three types of phases. A fully ferromagnetic phase, an antiferromagnetic phase and a massless phase with partial ferromagnetic and antiferromagnetic order. This massless phase's critical fluctuations are governed by a conformal field theory with central charge c=1. When the sigma z -anisotropy Delta is zero the model is exactly integrable through a Jordan-Wigner fermionization and our results are analytic. For Delta not=0 our analysis is done numerically using lattice sizes up to M=20. Our results show that for -1<or= Delta < square root (2/2) the staggered magnetic field perturbation is relevant, whereas for square root (2/2)< Delta <or=1 it is irrelevant. In the region of relevant perturbations, we show that the massive continuum field theory associated with the model, has the same mass spectrum as the sine-Gordon model.

arXiv:cond-mat/9407050•Journal of Physics A Mathematical and General•Mar 23, 1995•66 citations

Excitation Spectrum at the Yang-Lee Edge Singularity of the 2D Ising model

Tomasz Wydro, McCabe, John F.

This paper studies the Yang-Lee edge singularity of 2-dimensional 2D Ising model through a quantum spin chain. In particular, finite-size scaling measurements on the quantum spin chain are used to determine the low-lying excitation spectrum and central charge at the Yang-Lee edge singularity. The measured values are consistent with predictions for the (A4, A1) minimal conformal field theory.

arXiv:cond-mat/0510190•Oct 7, 2005•0 citations

Finite-size scaling and universality in the spectrum of the quantum ising chain. I. Periodic and antiperiodic boundary condition

Malte Henkel

The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.

Journal of Physics A Mathematical and General•Mar 11, 1987•47 citations

The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.

Using finite-size-scaling methods, we study the quantum chain version of the spin-1-Blume-Capel model coupled to an imaginary field. The aim is to realize simple non-unitary minimal conformal field theories in an Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the three lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line realizes the universality class of the ℳ2,7-conformal theory with c=−68/7.

Using finite-size-scaling methods, we study the quantum chain version of the spin-

1

-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge

c=-22/5

). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the

{\cal M}_{2,7}

-conformal theory with

c=-68/7

.

I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in

1+1

dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.

The phase diagram of the XXZ chain is calculated under the influence of homogeneous and staggered magnetic fields. The model has a rich phase structure with three types of phases. A fully ferromagnetic phase, an antiferromagnetic phase and a massless phase with partial ferromagnetic and antiferromagnetic order. This massless phase's critical fluctuations are governed by a conformal field theory with central charge c=1. When the sigma z -anisotropy Delta is zero the model is exactly integrable through a Jordan-Wigner fermionization and our results are analytic. For Delta not=0 our analysis is done numerically using lattice sizes up to M=20. Our results show that for -1<or= Delta < square root (2/2) the staggered magnetic field perturbation is relevant, whereas for square root (2/2)< Delta <or=1 it is irrelevant. In the region of relevant perturbations, we show that the massive continuum field theory associated with the model, has the same mass spectrum as the sine-Gordon model.

The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.