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Strong Zero Modes via Commutant Algebras

Sanjay Moudgalya, Olexei I. Motrunich

Abstract

Strong Zero Modes (SZMs) are (approximately) conserved quantities that result in (approximate) double degeneracies in the entire spectra of certain Hamiltonians, with the Majorana zero mode of the transverse-field Ising chain being a primary example. In this work, we discover via a systematic search that many examples of SZMs can be understood as symmetries in the commutant algebra framework, which reveals novel algebraic structures hidden in Hamiltonians with well-known SZMs, including the transverse-field Ising chain. Our findings unify the understanding of different examples of SZMs in the literature, demystify their connections to ground state phases of matter, and reveal novel symmetries in simple models, such as exact quasilocal $U(1)$ symmetries that sometimes accompany the SZMs such as in the spin-1/2 XY model for certain parameter values. Moreover, while analytically tractable SZMs have mostly been demonstrated only for non-interacting or integrable models, the algebraic structures revealed in this work can be exploited to construct integrability-breaking interactions that exactly preserve these SZMs. Such non-integrable models are expected to show more clear dynamical signatures of SZMs without the interference of other conserved quantities that appear in integrable models, and we discuss many examples, including those of novel hydrodynamic modes associated with such symmetries for some special parameter values. We also show that while this commutant understanding extends to the non-interacting limit of the celebrated Fendley SZM in the spin-1/2 XYZ chain, the SZM in the interacting case cannot be understood in this framework. This suggests that there are two types of SZMs -- those that survive integrability breaking and those that do not. We close by using this commutant understanding to construct an alternate proof of the Fendley SZM, which might be of independent interest.

Strong Zero Modes via Commutant Algebras

Abstract

Strong Zero Modes (SZMs) are (approximately) conserved quantities that result in (approximate) double degeneracies in the entire spectra of certain Hamiltonians, with the Majorana zero mode of the transverse-field Ising chain being a primary example. In this work, we discover via a systematic search that many examples of SZMs can be understood as symmetries in the commutant algebra framework, which reveals novel algebraic structures hidden in Hamiltonians with well-known SZMs, including the transverse-field Ising chain. Our findings unify the understanding of different examples of SZMs in the literature, demystify their connections to ground state phases of matter, and reveal novel symmetries in simple models, such as exact quasilocal symmetries that sometimes accompany the SZMs such as in the spin-1/2 XY model for certain parameter values. Moreover, while analytically tractable SZMs have mostly been demonstrated only for non-interacting or integrable models, the algebraic structures revealed in this work can be exploited to construct integrability-breaking interactions that exactly preserve these SZMs. Such non-integrable models are expected to show more clear dynamical signatures of SZMs without the interference of other conserved quantities that appear in integrable models, and we discuss many examples, including those of novel hydrodynamic modes associated with such symmetries for some special parameter values. We also show that while this commutant understanding extends to the non-interacting limit of the celebrated Fendley SZM in the spin-1/2 XYZ chain, the SZM in the interacting case cannot be understood in this framework. This suggests that there are two types of SZMs -- those that survive integrability breaking and those that do not. We close by using this commutant understanding to construct an alternate proof of the Fendley SZM, which might be of independent interest.
Paper Structure (54 sections, 1 theorem, 153 equations, 3 figures)

This paper contains 54 sections, 1 theorem, 153 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of Key Concepts
  3. Symmetries and Commutant Algebras
  4. Systematic search for symmetries
  5. Strong Zero Modes (SZM)
  6. Ising Strong Zero Mode
  7. Structure of the Commutant
  8. Connection to phases of the Ising Model
  9. Single-particle spectrum
  10. Approximate SZMs
  11. XY Strong Zero Modes and (Hidden) Symmetries
  12. XY Strong Zero Mode
  13. Definition
  14. Connections to the XY model
  15. Conventional Symmetries
  16. ...and 39 more sections

Key Result

Lemma 1

Let $\kappa,\gamma,q\in\mathbb{R}$ with $\kappa+\gamma\neq 0$, $\kappa - \gamma \neq 0$, $q^{-1} \neq 0$, and $q\neq 0$. The MZMs $\Psi^\ell(q)$ and $i Q_Z\Psi^r(s = q^{-1}\frac{\kappa - \gamma}{\kappa + \gamma})$ of Eqs. (eq:SZMMajorana) and (eq:Psirmajorana) are localized on opposite ends of the c

Figures (3)

  • Figure 1: Non-integrable models with the Ising SZM. Data shown for the non-integrable model of Eq. (\ref{['eq:HintIsingSZM']}) that possesses the Ising SZM $\Psi^\ell(q)$ of Eq. (\ref{['eq:ISZMdefns']}) as an exact conserved quantity. (Top) Statistics of energy level differences $\{s_n = E_{n+1} - E_n\}$ (normalized such that the mean spacing $\langle s \rangle = 1$) in the middle-half of the spectrum in the $Q_Z = +1$ sector for parameters $(q, c, r) = (0.55, 0.53, 0.23)$ and system size $L = 14$. It clearly exhibits Wigner-Dyson statistics from the GUE ensemble, characteristic of non-integrable models that break time-reversal symmetry, in contrast to Poisson level statistics exhibited by integrable models. (Bottom) Autocorrelation function $C_{X}(t) = 2^{-L} \text{Tr}[X_1(t) X_1(0)]$ of the operator $X_1$ on the leftmost site for parameters $(c, r) = (0.53, 0.23)$ and different values of $q$ from $0.4$ to $1.2$ and system sizes $L = 8, 10, 12$ (different linestyles---continuous, dashed, dotted). The autocorrelation functions show saturation to a finite value of $\sim 1 - |q|^2$ for $q < 1$, indicating an SZM localized at the left edge, while for $q > 1$ the autocorrelations quickly decay to $0$.
  • Figure 2: Dynamics of autocorrelation functions in Brownian circuits with SZMs. Computations were done using Time-Evolving Block Decimation (TEBD) on Eq. (\ref{['eq:avgcorrfunc']}) setting $g = 1$ and for system size $L = 100$ and maximum bond dimension $\chi = 50$. (Top) $C_{X_1}(t) = 2^{-L} \text{Tr}[X_1(t) X_1(0)]$ of the $X_1$ operator on the left edge spin in a Brownian circuit constructed from the generators of the Ising bond algebra $\mathcal{A}_{\text{I-SZM}}$ of Eq. (\ref{['eq:Isingbond']}) for three values of $q$. This shows saturation to a constant ($q < 1$), diffusive power-law decay ($q = 1$), and exponential decay ($q > 1$), which respectively correspond to the Ising SZM being localized on the left edge of the chain, delocalized and possessing hydrodynamic modes, or localized on the right edge of the chain. (Bottom) $C_{Z_j}(t) = 2^{-L} \text{Tr}[Z_j(t) Z_j(0)]$ for the operator $Z_j$ on the left (site $j = 4$) or middle (site $j = 50$) of the chain in a Brownian circuit constructed from the generators of the bond algebra $\mathcal{A}_{U(1),I}$ of Eq. (\ref{['eq:U1bondI']}) as well as boundary terms $\{Z_1\}$ and $\{Z_L\}$ that break the quasi-local $U(1)$ symmetry. $Z_{j=50}$ in the middle exhibits hydrodynamics similar to a system with regular $U(1)$ conservation with a clear diffusive power-law decay up to large timescales $\sim O(L^2)$. $Z_{j=4}$ close to the boundary of the system senses the breaking of the quasilocal symmetry on the boundary at finite timescales $\sim O(j^2)$, and shows a power-law decay $\sim t^{-\frac{3}{2}}$ corresponding to the physics of a diffusing particle with an absorbing boundary, as discussed in Sec. \ref{['subsubsec:quasilocalhydro']}.
  • Figure 3: Eigenvalues of the Fendley SZM $\boldsymbol{\Psi(x,y)}$. Data shown for $(x, y) = (0.6, 0.4)$ for various system sizes from $L = 4$ to $L = 12$. After scaling by the inverse norm $\lVert\Psi(x,y)\rVert^{-1} = \sqrt{(1 - x^2)(1 - y^2)}$, the eigenvalues $\{E_\Psi\}$ are increasingly clustered around $+1$ and $-1$ for increasing system size $L$, but are in fact completely non-degenerate for any finite system size $L$. This is to be contrasted with other SZMs discussed in this work, where all eigenvalues of the SZM at finite system size are either $+1$ or $-1$.

Theorems & Definitions (2)

  • Lemma 1
  • proof