Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order

Abstract

Topological orders in 2+1d are spontaneous symmetry-breaking (SSB) phases of 1-form symmetries in pure states. The notion of symmetry is further enriched in the context of mixed states, where a symmetry can be either ``strong" or ``weak". In this work, we apply a Rényi-2 version of the proposed equivalence relation in [Sang, Lessa, Mong, Grover, Wang, & Hsieh, to appear] on density matrices that is slightly finer than two-way channel connectivity. This equivalence relation distinguishes general 1-form strong-to-weak SSB (SW-SSB) states from phases containing pure states, and therefore labels SW-SSB states as ``intrinsically mixed". According to our equivalence relation, two states are equivalent if and only if they are connected to each other by finite Lindbladian evolution that maintains continuously varying, finite Rényi-2 Markov length. We then examine a natural setting for finding such density matrices: disordered ensembles. Specifically, we study the toric code with various types of disorders and show that in each case, the ensemble of ground states corresponding to different disorder realizations form a density matrix with different strong and weak SSB patterns of 1-form symmetries, including SW-SSB. Furthermore we show by perturbative calculations that these disordered ensembles form stable ``phases" in the sense that they exist over a finite parameter range, according to our equivalence relation.

Figures (4)

• Figure 1: The three regions on a plane used to defined the CMI. The radius of $A$ is an unimportant constant. We will take the width of $B$ and $C$, labeled by $r$, to infinity.
• Figure 2: The topological CMI is given by the combination of entanglement entropies in (\ref{['neumanncmi']}), with $A,B,C$ as illustrated above. For pure states, it just gives twice the TEE.
• Figure 3: The phase diagram of toric code with incoherent Pauli $X$ or $Z$ noise (\ref{['Eq: Z dephasing']}).
• Figure 4: Unitary circuit relating $|\rho_B \mathclose{\hbox{$\m@th{\rangle}$}\hbox{$\m@th{\rangle}$}}$ to $|\rho_{\beta} \mathclose{\hbox{$\m@th{\rangle}$}\hbox{$\m@th{\rangle}$}}$ when we take one of the directions (say, $y$) to have open boundary conditions. (a) We can write $e^{\beta A_v}|\uparrow_{m}\rangle$ as $e^{i\tilde{\beta}\tilde{A}_{v,m}}|\uparrow_{m}\rangle$ where $\tilde{A}_{v,m}$ has a single $Y_{e,m}$ operator. (b) We can transform all the $A_{v,m}$ terms in a single row ($x$ direction) simultaneously. (c) Side view of the circuit. The circuit is a sequential circuit in the $y$ direction because the gates in different $y$ coordinates do not commute. The $Y_{e,m}$ operator of the $\tilde{A}_{v,m}$ one row below overlaps with a $X_{e,m}$ operator of the $\tilde{A}_{v,m}$ one row above. The naive construction illustrated here allows periodic boundary conditions in the $x$ direction but not in the $y$ direction; closing up the circuit in the $y$ direction would require some modifications.