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Physical review. B, Condensed matter

Line of continuously varying criticality in the Ashkin-Teller quantum chain

Masanori Yamanaka, Mahito Kohmoto

Published Jul 1, 1995• 5 citations
Publisher

Abstract

We study the line of continuously varying critical exponents found recently in the one-dimensional quantum Ashkin-Teller model using the conformal field theory. On this line, the exponents of the energy density, the polarization, and the magnetization operators are numerically determined by the finite-size scaling method in the conformal field theory. We find that the magnetic exponent is associated with the primary field (1/16, 1) / 16 ), which exists in the coupling constant independent sector of the partition function. Within the existing theory, the numerical results are consistent with the identification of this line with the orbifold line in the Gaussian universality class with the thermal exponent 1\ensuremath{\le}${\mathit{x}}_{\mathit{T}}$\ensuremath{\le}2.

DOI
10.1103/physrevb.52.1138

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We study the line of continuously varying critical exponents found recently in the one-dimensional quantum Ashkin-Teller model using the conformal field theory. On this line, the exponents of the energy density, the polarization, and the magnetization operators are numerically determined by the finite-size scaling method in the conformal field theory. We find that the magnetic exponent is associated with the primary field (1/16, 1) / 16 ), which exists in the coupling constant independent sector of the partition function. Within the existing theory, the numerical results are consistent with the identification of this line with the orbifold line in the Gaussian universality class with the thermal exponent 1\ensuremath{\le}xT{\mathit{x}}_{\mathit{T}}xT​\ensuremath{\le}2.
We use a simple real-space renormalization-group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model in which the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization-group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.
We use the multiscale entanglement renormalization ansatz (MERA) to numerically investigate three critical quantum spin chains with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ on-site symmetry: a staggered XXZ model, a transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a continuous one-parameter family of critical points. Along this critical line, the thermodynamic limit of these models is expected to be described by classes of c=1c=1c=1 conformal field theories (CFTs) of two possible types: the S1{S}^{1}S1 free boson and its Z2{Z}_{2}Z2​ orbifold. Our numerics using MERA with explicitly enforced ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry allow us to extract conformal data for each model, with strong evidence supporting the identification of the staggered XXZ model and critical transverse field cluster model with the S1{S}^{1}S1 boson CFT, and the Ashkin-Teller model with the Z2{Z}_{2}Z2​-orbifold boson CFT. Our first two models describe the phase transitions between symmetry-protected topologically ordered phases and trivial phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry breaking. Our results show that a range of critical theories can arise at the boundary of a single symmetry protected phase.
Renormalization group theory allows continuous variation of critical exponents along a marginal direction (when there is one), keeping the scaling relations invariant. We propose a superuniversality hypothesis (SUH) suggesting that, up to constant scale factors, the scaling functions along the critical line must be identical to that of the base universality class even when all the critical exponents vary continuously. We demonstrate this in the Ashkin-Teller (AT) model on a two-dimensional square lattice where two different phase transitions occur across the self-dual critical line: while magnetic transition obeys the weak-universality hypothesis where exponent ratios remain fixed, the polarization exhibits a continuous variation of all critical exponents. The SUH not only explains both kinds of variations observed in the AT model, it also provides a unified picture of continuous variation of critical exponents observed in several other contexts.
We investigate the boundary critical phenomena of the one-dimensional quantum Ashkin-Teller model using boundary conformal field theory and density matrix renormalization group (DMRG) simulations. Based on the Z2\mathbb{Z}_2Z2​-orbifold of the c=1c=1c=1 compactified boson boundary conformal field theory, we construct microscopic lattice boundary terms that renormalize to the stable conformal boundary conditions, utilizing simple current extensions and the underlying SU(2)\mathrm{SU}(2)SU(2) symmetry to explicitly characterize the four-state Potts point. We validate these theoretical identifications via finite-size spectroscopy of the lattice energy spectra, confirming their consistency with D4D_4D4​ symmetry and Kramers-Wannier duality. Finally, we discuss the boundary renormalization group flows among these identified fixed points to propose a global phase diagram for the boundary criticality.
The commuting row-to-row transfer matrices of magnetic hard squares on the multicritical T manifold are shown to satisfy a special functional equation called an inversion identity.For strip widths up to N=32, these equations are solved numerically for the transfer matrix eigenvalues. The central charge or conformal anomaly c=1 is obtained from 1/N 2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps. The magnetic scaling dimension is x m =1/8 and the sublattice density difference scaling dimension is x e =1/9(2-y), where y=2 lambda / pi and the interaction-dependent crossing parameter lambda varies between 0 and 2 pi /3. Scaling dimensions for further operators fall in the sequence x n =n 2 x e where n=1,2,3,4. The critical exponents also vary continuously along the multicritical line and are given by alpha =(14-9y)/(16-9y), beta m =9(2-y)/16(16-9y), beta e =1/2(16-9y), nu =9(2-y)/2(16-9y) and delta =15. These exponents are simply related to the exponents of the critical eight-vertex and Ashkin-Teller models, which also exhibit Z 2 *Z 2 symmetry, and the authors argue that these models lie in the same universality class.
The eigenenergies for the spin SSS generalization of the critical XXZ\mathrm{XXZ}XXZ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge c=3S(S+1)c=\frac{3S}{(S+1)}c=(S+1)3S​ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.