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arXiv:1501.02817 · Physical Review B

Multiscale entanglement renormalization ansatz for spin chains with continuously varying criticality

Jacob C. Bridgeman, Aroon O’Brien, Stephen D. Bartlett, Andrew C. Doherty

Published Apr 24, 2015• 35 citations
arXivHTMLPDF

Abstract

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 $c=1$ conformal field theories (CFTs) of two possible types: the ${S}^{1}$ free boson and its ${Z}_{2}$ 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 ${S}^{1}$ boson CFT, and the Ashkin-Teller model with the ${Z}_{2}$-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.

DOI
10.1103/physrevb.91.165129

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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.
We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.
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 employ the Multiscale Entanglement Renormalization Ansatz (MERA) tensor network to investigate a critical line of continuous quantum phase transitions of the Z3\mathbb{Z}_3Z3​ chiral clock model. This critical line is believed to be described by a slow renormalization group flow from the 3-state Potts fixed point to another fixed point that features anisotropic scaling of space and time. We use the variational principle to construct a MERA representation of the model's ground state, from which we obtain the ground state energy and the set of scaling operators and their scaling dimensions. These scaling dimensions determine the critical exponents of the model, and we study these critical exponents and other scaling data as a function of the model's chiral parameter. We find a set of effective scaling data that smoothly varies starting from the Potts data as the chiral parameter is increased. Within the context of our approach, we discuss how this result may nevertheless be consistent with the two fixed point hypothesis provided the renormalization group flow is sufficiently slow. Our findings demonstrate MERA's effectiveness in capturing the complex low-energy physics of the chiral clock model and in extracting field theory data for an anisotropic continuum theory.
The multiscale entanglement renormalization ansatz (MERA) can be used, in its scale invariant version, to describe the ground state of a lattice system at a quantum-critical point. From the scale invariant MERA one can determine the local scaling operators of the model. Here we show that, in the presence of a global symmetry G\mathcal{G}G, it is also possible to determine a class of nonlocal scaling operators. Each operator consists, for a given group element g∊Gg∊\mathcal{G}g∊G, of a semi-infinite string ${\ensuremath{\Gamma}}_{g}^{\ensuremath{\vartriangleleft}}$ with a local operator $\ensuremath{\varphi}$ attached to its open end. In the case of the quantum Ising model, G=Z2\mathcal{G}={\mathbb{Z}}_{2}G=Z2​, they correspond to the disorder operator $\ensuremath{\mu}$, the fermionic operators $\ensuremath{\psi}$ and $\overline{\ensuremath{\psi}}$, and all their descendants. Together with the local scaling operators identity I\mathbb{I}I, spin $\ensuremath{\sigma}$, and energy ϵϵϵ, the fermionic and disorder scaling operators $\ensuremath{\psi}$, $\overline{\ensuremath{\psi}}$, and $\ensuremath{\mu}$ are the complete list of primary fields of the Ising CFT. Therefore the scale invariant MERA allows us to characterize all the conformal towers of this CFT.