Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems

TL;DR

This work builds an exactly solvable open quantum-system framework of interacting spins in two dimensions by introducing gamma-matrix spin models on arbitrary graphs coupled to a Markovian bath. Through a careful mapping to bilayer, non-Hermitian free Majorana fermions living on a background $\,\mathbb{Z}_2$ gauge field, the authors obtain an exponentially large steady-state manifold labeled by conserved fluxes, and demonstrate that the steady states encode a mixed-state topological order via one-form strong-to-weak spontaneous symmetry breaking. The analysis combines an exact Majorana/vectorized formalism with gauge fixing and Pfaffian criteria to classify excitations and derive both analytic bounds and numerical results for relaxation rates, revealing phenomena such as anomalous relaxation and quantum Zeno effects depending on symmetry sectors. The findings establish a tractable platform to study nonequilibrium quantum phases and relaxation pathways toward steady states in topologically structured open systems, with potential implications for robust quantum memories in dissipative environments.

Abstract

We study the dissipative dynamics of a class of interacting ``gamma-matrix'' spin models coupled to a Markovian environment. For spins on an arbitrary graph, we construct a Lindbladian that maps to a non-Hermitian model of free Majorana fermions hopping on the graph with a background classical $\mathbb{Z}_2$ gauge field. We show, analytically and numerically, that the steady states and relaxation dynamics are qualitatively independent of the choice of the underlying graph, in stark contrast to the Hamiltonian case. We also show that the exponentially many steady states provide a concrete example of mixed-state topological order, in the sense of strong-to-weak spontaneous symmetry breaking of a one-form symmetry. While encoding only classical information, the steady states still exhibit long-range quantum correlations. Afterward, we examine the relaxation processes toward the steady state by numerically computing decay rates, which we generically find to be finite, even in the dissipationless limit. However, we identify symmetry sectors where fermion-parity conservation is enhanced to fermion-number conservation, where we can analytically bound the decay rates and prove that they vanish in the limits of both infinitely weak and infinitely strong dissipation. Finally, we show that while the choice of coherent dynamics is very flexible, exact solvability strongly constrains the allowed form of dissipation. Our work establishes an analytically tractable framework to explore nonequilibrium quantum phases of matter and the relaxation mechanisms toward them.

Figures (8)

• Figure 1: Schematic representation of the main results. (a) A "proper edge coloring" of a graph (i.e., such that no two edges of the same color meet at any vertex) defines a gamma-matrix Hamiltonian with Heisenberg interactions set by the edge colors. The gamma-matrix "spins" and itinerant Majorana fermions (resulting from the gamma-matrix-to-Majorana mapping of Sec. \ref{['sec:free-fermion']}) live on the vertices and gauge fields $u_{ij}$ (resulting from the same mapping) live on the edges. The gauge fields give rise to fluxes $W_p$ through elementary plaquettes $p$. (b) When the system is realized on a topologically nontrivial surface like a torus, fluxes can thread noncontractible loops. One can define logical operators $\overline{X}_a$ and $\overline{Z}_a$ for each noncontractible loop. In the steady state, the product of two $\overline{Z}$ operators acquires a nonzero correlation even when the two operators are arbitrarily far apart. Consequently, the $\overline{X}$ operators are spontaneously broken from a strong to a weak one-form symmetry. (c) Vectorization: doubling of degrees of freedom to represent the open system. In the free-fermion solvable case, two copies (layers) of the graph are connected by interlayer gauge fields $v_j$ (black vertical lines) arising from dephasing. The symmetry sectors of the Lindbladian are labeled by $W_p$, $\overline{X}_a$, and interlayer fluxes $\mathcal{W}_q$, and each has its own fermionic vacuum. The steady states are the fermionic vacua in sectors with $\mathcal{W}_q=1$ for all interlayer plaquettes $q$, i.e., without any gauge flips ($\mathcal{W}_q=-1)$, and they are labeled by their flux patterns. The decaying states of the Lindbladian are the vacua with gauge flips and fermionic excitations. (d) When intralayer gauge fields are flipped, the excitations conserve only fermionic parity, which is determined by a Pfaffian method we describe. For weak dissipation, their decay rate is proportional to the Hamiltonian coupling strength $J$ when the thermodynamic limit is taken first (anomalous relaxation). (d) If only interlayer gauge fields $v_j$ are flipped, the excitations are labeled by a conserved fermion number, and their decay rates can be analytically bounded with Bendixson inequalities (shaded region). At weak dissipation, the decay rate of all excitations is linear in the dissipation strength $\gamma$.
• Figure 2: Spin Hamiltonians defined by the color rule. Each edge of the graph is assigned a color, and two gamma matrices with that color at the two endpoints of the edge interact in a term of the Hamiltonian. We illustrate the general construction for a vertex $i$ of valence $z(i)=5$.
• Figure 3: Logical operators of the Lindbladian gamma-matrix model. (a) Two noncontractible loops $\ell_1$ and $\ell_2$ for a random graph on the torus and two noncontractible loops $\tilde{\ell}_1$ and $\tilde{\ell}_2$ on the dual graph. $\ell_a$ and $\tilde{\ell}_b$ intersect and odd (even) number of times if $a=b$ ($a\neq b$). The same construction holds for higher-genus surfaces and any graph. (b) Construction of the logical operations in terms of gamma matrices.
• Figure 4: Fractionalized representation of string excitations of the gamma-matrix model. Schematic representation of strings of gamma matrices on an open path $\mathcal{P}$, defined in Eqs. (\ref{['eq:def_W']})--(\ref{['eq:Wch']}) (only one endpoint $j$ depicted). The body of the string, $\prod_{\langle ij \rangle\in\mathcal{P}}K_{ij}$, is mapped to a product of $\mathbb{Z}_2$ gauge fields, $\prod_{\langle ij \rangle\in\mathcal{P}}(-\mathrm{i} u_{ij})$, with Majorana fermions at each end. (a) The string $W_\mathcal{P}^{\mu\nu}$ with $\mu,\nu\in\{1,\dots,2k\}$ carries a Majorana fermion $b_j^\mu$ at its endpoint, which excites intralayer gauge flips on the edge with color $\mu$ that emanates from $j$ (represented by a bold colored edge) and, for the adjoint action, two vortices adjacent to that edge. (b) The string $W_\mathcal{P}^\mathrm{ch}$ carries a chiral Majorana fermion $b_j^{\mathrm{ch}}$ at its endpoint that effects an interlayer gauge flip at $j$ (which is gauge-equivalent to flipping all intralayer gauge fields emanating from $j$, as depicted by the colored edges). (c) The string $W_\mathcal{P}$ carries an itinerant Majorana fermion $c_j$ at its endpoint, introducing a superposition of fermionic excitations at $j$.
• Figure 5: Spectral gap as a function of dissipation strength in fermion-parity-conserving sectors. The gap was computed for the honeycomb lattice with $N=128$--$512$ and periodic boundary conditions on the torus (with the same number of sites along each direction) using the prescription of Sec. \ref{['sec:parity_gap']}, with the single-particle eigenvalues $\beta_k$ obtained from exact diagonalization of Eq. (\ref{['eq:def_A']}). We set uniform Hamiltonian couplings and gauge fields $J_{ij}=u_{ij}=1$ (which gives uniform 0-flux), uniform dissipation $\gamma_i=\gamma$ for all $i$, and fixed the gauge to $v_i=-1$ for all $i$ (i.e., $\mathfrak{V}=\emptyset$). Each panel shows a different type of intralayer gauge flips $\tilde{u}_{ij}=-1$ (i.e., choice of $\mathfrak{U}$): (a) a single arbitrary edge; (b) all edges of a given color; (c) all edges on a noncontractible loop; and (d) all edges emanating from an arbitrary vertex (which is gauge-equivalent to a single interlayer flip on that vertex and $\mathfrak{U}=\emptyset$, with conserved fermion number, cf. upper left panel of Fig. \ref{['fig:intergapUniform']}).

Theorems & Definitions (4)

• Lemma
• proof
• Theorem : Bendixson inequalities bendixson1902
• proof