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Detecting two dimensional symmetry protected topological order in a ground state wave function

Michael P. Zaletel

TL;DR

This paper presents a numerical framework to detect two-dimensional symmetry-protected topological (SPT) order by threading g-flux and measuring two bulk responses: the spin s_g via momentum polarization and the projective representation [χ_g] of g-flux ends. By relating these invariants to the slant product i_g ω of the 3rd cohomology class [ω], the authors show that, for finite Abelian symmetry groups G, s_g and [χ_g] suffice to reconstruct [ω], providing a complete 2D SPT characterization. The method is validated numerically on a Z2 SPT model using infinite DMRG, recovering the expected nontrivial s_g and [χ_g] consistent with the known ω and demonstrating robustness. Overall, the work offers a practical, cohomology-grounded approach to identify and classify interacting 2D SPT phases in microscopic models.

Abstract

Symmetry protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two dimensional ground state has symmetry protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite abelian onsite symmetry, the measurement completely determines the 3rd cohomology class that characterizes the order. The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with $\mathbb{Z}_2$ symmetry protected order.

Detecting two dimensional symmetry protected topological order in a ground state wave function

TL;DR

This paper presents a numerical framework to detect two-dimensional symmetry-protected topological (SPT) order by threading g-flux and measuring two bulk responses: the spin s_g via momentum polarization and the projective representation [χ_g] of g-flux ends. By relating these invariants to the slant product i_g ω of the 3rd cohomology class [ω], the authors show that, for finite Abelian symmetry groups G, s_g and [χ_g] suffice to reconstruct [ω], providing a complete 2D SPT characterization. The method is validated numerically on a Z2 SPT model using infinite DMRG, recovering the expected nontrivial s_g and [χ_g] consistent with the known ω and demonstrating robustness. Overall, the work offers a practical, cohomology-grounded approach to identify and classify interacting 2D SPT phases in microscopic models.

Abstract

Symmetry protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two dimensional ground state has symmetry protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite abelian onsite symmetry, the measurement completely determines the 3rd cohomology class that characterizes the order. The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with symmetry protected order.

Paper Structure

This paper contains 15 sections, 44 equations, 2 figures.

Table of Contents

  1. Physical responses of 2d SPT states
  2. Detecting the spin $s_g$ in numerics
  3. Detecting $[\chi_g]$ in numerics
  4. Calculating $s_g$ and $[\chi_g]$ from the cohomology classification
  5. Application to a model with $\mathbb{Z}_2$ SPT order
  6. Completeness of the characterization for finite Abelian $G$.
  7. Type I
  8. Type II
  9. Type III
  10. Ground state degeneracy of an open cylinder and 'decorated domain walls.'
  11. The spin of a dyon and constraints on $s_g$.
  12. Uniqueness of $\alpha_g(1)^{N_g}$ and $\lambda_g^{N_g L_x}$
  13. $\alpha_g(1)^{N_g}$.
  14. $\lambda_g^{N_g L_x}$.
  15. The cohomology classification of 1d SPT order and its numerical detection

Figures (2)

  • Figure 1: a) The cylinder with a defect line '$g$' has interactions modified on the thick red links, with ground state $\ket{g}$. b) Hamiltonian after one step of the momentum polarization Berry cycle, with ground state $\hat{T}_L \ket{g}$. After applying $\hat{T}_L^{L_x}$, the Hamiltonian will have a $g$-defect at the cut $y = y_0$ in addition to the starting $g$-defect at $x = x_0$. c) We define $U_a^{(g)}$ to be the operation of locally nucleating an $a$-defect line and adiabatically extending it to encircle a $g$-flux. d) Effect of a $2 \pi$-rotation on the end of a $g$-defect: an additional $g$-defect encloses the endpoint, equivalent to $U_g^{(g)}$
  • Figure 2: Entanglement spectrum of the trivial/SPT phase (column) both with with/without a $\mathbb{Z}_2$ defect line (row) at circumference $L_x = 6$. Red/yellow coloring denotes the $\mathbb{Z}_2$ charge of the Schmidt state. The SPT phase shows a distinctive response to the defect; the entanglement spectrum is symmetric about quarter-integer momenta. All $|s|<10^{-15}$ except for $2 s_g = 0.50000055$ in the SPT phase.