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Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions

Anthony Rey, Ömer M. Aksoy, Daniel P. Arovas, Claudio Chamon, Christopher Mudry

TL;DR

This work constructs a one-dimensional, $ ext{Z}_2^X imes ext{Z}_2^Y imes ext{Z}_2^T$-symmetric spin-1/2 lattice model with two tunable couplings $( heta, extlambda)$ that realizes four gapped AFM phases and an extended gapless FM$_z$ phase. Numerics (ED and DMRG) map a phase diagram in which each corner hosts a stable gapped phase and each midpoint hosts a DQCP; remarkably, all four DQCPs connect to a central gapless FM$_z$ hub through tricritical points. The FM$_z$ phase exhibits incommensurate long-range order along $Z$, algebraic decay of transverse spin correlations, and a conformal field theory with central charge $ ext{c}=1$, indicating rich critical behavior coexisting with FM order. The phase boundaries display a mix of conventional DQCP physics and quantum topological transitions (QTPTs) between distinct SPT and AFM states, with boundary degeneracies and anomalous symmetry fractionalization depending on the boundary conditions. Collectively, the results illustrate how competition among symmetry-enriched DQCPs can stabilize an exotic gapless phase that bridges multiple criticalities and topological sectors in one dimension, with potential extensions to higher dimensions and sign-problem-free Monte Carlo studies.

Abstract

We present a scenario, in which a gapless extended phase serves as a "hub" connecting multiple symmetry-enriched deconfined quantum critical points. As a concrete example, we construct a lattice model with $\mathbb{Z}^{\,}_{2}\times \mathbb{Z}^{\,}_{2}\times \mathbb{Z}^{\,}_{2}$ symmetry for quantum spin-1/2 degrees of freedom that realizes four distinct gapful phases supporting antiferromagnetic long-range order and one extended incommensurate gapless ferromagnetic phase. The quantum phase transition between any two of the four gapped and antiferromagnetic phases goes through either a (deconfined) quantum critical point, a quantum tricritical point, or the incommensurate gapless ferromagnetic phase. In this phase diagram, it is possible to interpolate between four deconfined quantum critical points by passing through the extended gapless ferromagnetic phase. We identify the phases in the model and the nature of the transitions between them through a combination of analytical arguments and density matrix renormalization group studies.

Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions

TL;DR

This work constructs a one-dimensional, -symmetric spin-1/2 lattice model with two tunable couplings that realizes four gapped AFM phases and an extended gapless FM phase. Numerics (ED and DMRG) map a phase diagram in which each corner hosts a stable gapped phase and each midpoint hosts a DQCP; remarkably, all four DQCPs connect to a central gapless FM hub through tricritical points. The FM phase exhibits incommensurate long-range order along , algebraic decay of transverse spin correlations, and a conformal field theory with central charge , indicating rich critical behavior coexisting with FM order. The phase boundaries display a mix of conventional DQCP physics and quantum topological transitions (QTPTs) between distinct SPT and AFM states, with boundary degeneracies and anomalous symmetry fractionalization depending on the boundary conditions. Collectively, the results illustrate how competition among symmetry-enriched DQCPs can stabilize an exotic gapless phase that bridges multiple criticalities and topological sectors in one dimension, with potential extensions to higher dimensions and sign-problem-free Monte Carlo studies.

Abstract

We present a scenario, in which a gapless extended phase serves as a "hub" connecting multiple symmetry-enriched deconfined quantum critical points. As a concrete example, we construct a lattice model with symmetry for quantum spin-1/2 degrees of freedom that realizes four distinct gapful phases supporting antiferromagnetic long-range order and one extended incommensurate gapless ferromagnetic phase. The quantum phase transition between any two of the four gapped and antiferromagnetic phases goes through either a (deconfined) quantum critical point, a quantum tricritical point, or the incommensurate gapless ferromagnetic phase. In this phase diagram, it is possible to interpolate between four deconfined quantum critical points by passing through the extended gapless ferromagnetic phase. We identify the phases in the model and the nature of the transitions between them through a combination of analytical arguments and density matrix renormalization group studies.

Paper Structure

This paper contains 41 sections, 227 equations, 50 figures.

Table of Contents

  1. Definitions
  2. Symmetries
  3. Gapped and gapless fixed-points
  4. Gapped stable fixed-points
  5. Gapless unstable fixed-points
  6. Numerics
  7. Five phases
  8. Order parameters and correlation functions
  9. The $\mathrm{\hbox{N'eel}}^{\,}_{x}$ phase
  10. The $\mathrm{\hbox{N'eel}}^{\,}_{y}$ phase
  11. The $\mathrm{\hbox{N'eel}}^{\mathrm{SPT}}_{x}$ phase
  12. The $\mathrm{\hbox{N'eel}}^{\mathrm{SPT}}_{y}$ phase
  13. The $\mathrm{FM}^{\,}_{z}$ phase
  14. Criticality and incommensuration of the $\mathrm{FM}^{\,}_{z}$ phase
  15. Criticality
  16. ...and 26 more sections

Figures (50)

  • Figure 1: (Color online) Phase diagram of Hamiltonian \ref{['eq:def H']}. Squares (circles) on the boundary denote stable (unstable) fixed points (DQCPs).
  • Figure 2:
  • Figure 3: (Color online) Magnitudes of the staggered magnetization per site along the $X$ axis \ref{['eq:mx_sta']} and the uniform magnetization per site along the $Z$ axis \ref{['eq:mz_uni']}, obtained along the line $\lambda=0.4$ using DMRG with OBCs for $2N=64,128$, and $256$ sites. The sharp suppression of the order parameters on both sides of the quantum phase transition around $\theta=0.57$ indicates that the correlation length is larger than the length of the chain.
  • Figure 4: (Color online) Spin-spin correlation functions $C^{\alpha}$ defined in Eq. \ref{['eq:correlations']} computed using DMRG with OBCs, for $2N=128$, $\lambda=0.4$, and $\theta=\pi/4-0.1$ with (a) $C^{x}$ and $C^{y}$ and (b) $C^{z}$. Fits used are labeled "p" for power law, "e" for exponential, and "q" for cosine form. We find $q^{z}\simeq2\pi\times3/10$. The goodness of fit for p is always greater than that for e. The insets show the Fourier transforms of the respective correlations, where the abscissa is the frequency $f^{\alpha}\equiv q^{\alpha}/(2\pi)$ and the ordinate is the expansion coefficient $A(f^{\alpha})$, $\alpha=x,y,$ and $z$. (c) Dependence on $\lambda$ of the uniform magnetization $m^{z}_{\mathrm{uni}}$ defined in Eq. \ref{['eq:mz_uni']} and the frequency $f^{z}$ of the Fourier transform of the spin-spin correlations $C^{z}$ in the bulk of a chain with $2N=512$ sites obtained using DMRG with OBCs at $\theta=\pi/4$. The inset shows the fit$^{\,}_{\mathrm{log}}$ ansatz $\ln|m^{z}_{\mathrm{uni}}(\lambda)|\sim 1/6\,\ln|\lambda-\lambda^{\,}_{\mathrm{tri}}|$, where $\lambda^{\,}_{\mathrm{tri}}\approx0.25$. (d) Proportionality between $f^{z}(\lambda)$ and $m^{z}_{\mathrm{uni}}(\lambda)$ on a chain of $2N=512$ sites obtained using DMRG with OBCs at $\theta=\pi/4$. The different values of $\lambda$ are specified by different colors.
  • Figure 5: (Color online) Phase diagram of Hamiltonian \ref{['suppeq:def H']}. Squares (circles) on the boundary denote stable (unstable) fixed points (DQCP).
  • ...and 45 more figures