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Journal of Physics A Mathematical and General

A critical-amplitude relation for one-dimensional quantum transitions and determination of the exponent eta

Naomichi Hatano, Yoshihiro Nishiyama, Masuo Suzuki

Published Sep 21, 1994• 7 citations
Publisher

Abstract

A new critical-amplitude relation which holds for one-dimensional quantum ground-state transitions is presented. The relation yields an estimate of the sound velocity which appears in the conformal field theory as an a priori unknown parameter. With the use of this relation, the exponent eta can be explicitly determined from only the scaling fit of the off-critical energy gap. The relation is confirmed in the transverse Ising model, the S= 1/2 anisotropic XY model, and the three-state Potts model.

DOI
10.1088/0305-4470/27/18/017

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A new critical-amplitude relation which holds for one-dimensional quantum ground-state transitions is presented. The relation yields an estimate of the sound velocity which appears in the conformal field theory as an a priori unknown parameter. With the use of this relation, the exponent eta can be explicitly determined from only the scaling fit of the off-critical energy gap. The relation is confirmed in the transverse Ising model, the S= 1/2 anisotropic XY model, and the three-state Potts model.
The low-energy excitation spectrum of an Ising quantum chain in a transverse field which is critical (h=J=1) for 1<or=n<or=L/2 and off-critical (h not=J) for L/2(n<or=L is calculated up to O(L -1 ). Interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite-size scaling. One gets an ordinary surface transition when h)J and an extraordinary surface transition when h(J. The gap-exponent relation is verified, the mass gaps and the level degeneracy are in agreement with conformal invariance.
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.
This manuscript develops a geometric theory of quantum phase transitions that does not begin from a preassigned order parameter, symmetry-breaking pattern, or topological invariant. The basic object is a phase-sensitive state kernel extracted from a ground state, thermal state, band projector, correlation matrix, or quantum geometric tensor. A kernel-resolved phase is defined operationally as a maximal parameter region on which the principal phase-sensitive direction of this kernel can be transported continuously with finite accumulated projective variation. The central diagnostic is the rupture index\kappa=\alpha(1-\beta)=\alpha-\gamma,where \alpha is the local closing exponent of the relevant spectral or kernel separation, \gamma is the independently measured exponent of the transverse carrier amplitude, and \beta=\gamma/\alpha is the relative carrier-amplitude exponent. The theorem-bearing class treated in this paper is the finite-dimensional free-fermion / two-level Dirac-mass class. The transverse-field Ising chain, the SSH chain, and the Kitaev chain are developed as the first model realizations. In all three cases the local channel is a nondegenerate Dirac mass closure with\alpha=1,\qquad \gamma=0,\qquad \beta=0,\qquad \kappa=1.The numerical section is presented as an implementation check of the common local theorem, not as an independent proof of three unrelated mechanisms. A next-harmonic deformation of the SSH model verifies stability under nonuniversal velocity and curvature corrections, while a gapped noncritical control confirms saturation rather than an artificial rupture peak. The paper explicitly delimits BKT, quadratic-touching, multicritical, infinite-dimensional-kernel, interacting-kernel, and non-gap-controlled cases as frontier problems rather than classifying them under the present theorem.