Classifying one-dimensional Floquet phases through two-dimensional topological order

TL;DR

This work develops a symmetry topological field theory (SymTFT) framework to classify one-dimensional Floquet many-body localized phases with finite Abelian on-site symmetry G by treating the 1D system as the boundary of a two-dimensional G-topological order. The classification arises from boundary data of quantum double models, specifically Lagrangian subgroups and boundary algebras, unifying static SSB/SPT and Floquet SSB/SPT into a single scheme and revealing dual time-crystal phases and time-translation symmetry breaking signatures. It extends to twisted quantum doubles to capture non-onsite symmetries, and demonstrates a comprehensive drive classification with observable bulk signatures, g-twisted boundaries, and potential experimental realizations on programmable quantum devices. The results provide a conceptual and practical toolkit for understanding and probing Floquet phases via topological boundary conditions and anyonBraiding data, offering new avenues for both theory and experiment in driven quantum matter.

Abstract

Floquet systems display rich phenomena, such as time crystals, with many-body localisation (MBL) protecting the phases from heating. While several types of Floquet phases have been classified, a unified picture of Floquet MBL is still emerging. Static phases have been fruitfully studied via "symmetry topological field theory" (SymTFT), wherein the universal features of $G$-symmetric systems are elucidated by placing them on the boundary of a topological order of one dimension higher. In this work, we provide a SymTFT approach to classifying $G$-symmetric Floquet MBL phases in 1D, for $G$ a finite Abelian group with on-site unitary action. In the SymTFT, these 1D systems correspond to the boundaries of the quantum double associated to $G$, and the classification naturally arises from considering the Lagrangian subgroups and boundary excitations of the quantum double. The classification covers all known Floquet phases while uncovering others previously unexplored, along with bulk features of phases thought to have only boundary signatures. We refer to the latter phases as "dual" time crystals. For static phases, we show how anyons of the quantum double and (string) order parameters provide a natural and simple interpretation of known classification schemes. By extending our framework to the boundaries of twisted quantum doubles, we uncover a new time-crystalline phase with non-onsite symmetry, which cannot be obtained through local, symmetric Hamiltonian drives. We numerically demonstrate evidence for the absolute stability of this phase, and observe that for open boundary conditions it has greater stability to symmetric perturbations. We finally discuss perspectives on using programmable quantum devices to realise and probe the phases we discuss. Our results show that SymTFT provides a powerful approach to unifying phases and features of Floquet systems.

Figures (5)

• Figure 1: $\mathbb{Z}_2$-symmetric Floquet phases on the boundary of the toric code TO. (a) Toric code with left and right boundaries identified and top and bottom boundaries $e$-condensing. An example of a vertex operator ($A_v$) and a plaquette operator ($B_p$) are shown. String operators $W^{(m)}$ and $W^{(e)}$, and some members of the boundary algebra are shown. (b) Schematic of the $\mathbb{Z}_2$-TO on the cylinder with the top (bottom) boundary, $B_\text{phys}$ ($B_\text{ref}$), gapped according to Lagrangian subgroup $\mathcal{M}_a$ ($\mathcal{M}_e$), for anyon $a\in \lbrace e,m\rbrace$. Time-translation symmetry breaking (TTSB) is generated by the inclusion of string operator $\overline{W}^{(b)}$, for $b\notin \mathcal{M}_a$, in the drive. For strong disorder, local integrals of motion pick up exponentially decaying tails. This is schematically illustrated for LIOMs $D_i$, with support shown in red. (c) The phase diagram of the driven Ising model. $\bar{J}$ and $\bar{g}$ are the means of disordered parameters $J_i$ and $g_i$ (see Eqs. \ref{['eqn:H_0_ZZ']} and \ref{['eqn:H_1_X']}). We identify two TTSB fixed-point drives, relating them to drives from the model in (b) (see Eqs. \ref{['eqn:piSG_U']} and \ref{['eqn:0piPM_U']}).
• Figure 2: Schematic illustration of $G$-symmetric 1D static phase constructions in SymTFT, and of the boundary algebra (BA). (a) We illustrate general members of the BA. Local operator $W_{il}^{(h,\alpha)}$, shown in blue, corresponds to the process of anyons $(h,\alpha)$ and $(h^{-1},\alpha^{-1})$ being created close to the boundary and condensed at that boundary. Local order parameters $W_j^{(1,\beta)}$, correspond to strings running between top and bottom boundaries, and global symmetry/dual symmetry transformations $\overline{W}^{(g,\gamma)}$ (only a symmetry transformation shown in green here) correspond to strings wrapping the cylinder. The example shown is for a partially-SSB phase, with symmetry broken to subgroup $H<G$. The Lagrangian subgroup $\mathcal{M}$ of the physical boundary (top boundary) includes anyons $(h,\alpha)$ for $h\in H$ and some $\alpha\in\text{Rep}\, G$, such that $\alpha(H) = 1$. Local order parameters $W_k^{(1,\beta)\dagger}$ and $W_j^{(1,\beta)}$ (with $\beta$ satisfying $\beta(H) = 1$) signal long-range order on $B_\text{phys}$. These do not commute with symmetry operator $\overline{W}^{(g,1)}$ for $g\notin H$ shown in green. (b) An example of SPT order for $G = \mathbb{Z}_4\times \mathbb{Z}_6\times \mathbb{Z}_3$, with Lagrangian subgroup $\mathcal{M}_1 = \langle m_1 e_2^3, e_1^2m_2, m_3\rangle$. Only a short section of the $G$-TO and $B_\text{phys}$ is shown. The three layers represent $\mathbb{Z}_4$, $\mathbb{Z}_6$, and $\mathbb{Z}_3$ toric code layers, respectively. String operators corresponding to the three anyons generating $\mathcal{M}_1$ are shown. The $\mathbb{Z}_4$ and $\mathbb{Z}_6$ layers are folded together along the boundary (dashed line represents a domain wall) to indicate the multi-layer anyons that appear in $\mathcal{M}_1$.
• Figure 3: Reconnecting semion strings. The reconnection sign Levin-Wen, in panel (a), implies that starting from a $(1s1)$ configuration on three adjacent sites, hopping the $s$ to the right and then creating an $s$ pair to get $(sss)$ gives $(-1)$ times the state we get by moving the $s$ to the left and then creating the $s$ pair at the other two sites JiWen2020categorical. [The same holds for the reverse process starting from $(sss)$.] It also implies, as shown in (b), that the action of the $\mathbb{Z}_2$ symmetry $P=\overline{W}^{(s)}$ on a computational basis state $\ket{\{z_j\}}$ of $B_\text{phys}$ flips all spins and multiplies by $(-1)^{(n/2)}$ where $n$ is the number of domain walls JiWen2020categorical, created in pairs by the $X_i$ in $W^{(s)}_{i,i+1}$ (for various $i$) acting on a state with all spins aligned.
• Figure 4: Average eigenvalue splitting/$\pi$-splitting in the spectrum of (a) $U_\text{DS}^\text{static}$, (b) $U_\text{DS}^\text{TTSB}$, and (c) $U_\text{DS}^\text{TTSB}$ with a noisy implementation of $P$. $L$ denotes the length of the spin chain. $J_j$ were chosen uniformly at random from range $[\frac{1}{2}\bar{J}, \frac{3}{2}\bar{J}]$ for $\bar{J} = 3\pi/8$. Averages were taken over 1000 disorder realisations. (a) $\Delta_0$ is the average splitting between minimally-separated eigenvalues of the Floquet unitary. Perturbation couplings $h_j^{(i)}$ are chosen uniformly at random from $[-\lambda \bar{J}, \lambda \bar{J}]$. (b) $\Delta_\pi$ is the average $\pi$-splitting in the spectrum. Both perturbation couplings $h_j^{(i)}$ and local field strengths for $X$ and $Z$ fields are chosen at random from the same distribution as above. The results demonstrate absolute stability in the presence of non-symmetric perturbations. (c) Perturbation couplings $h_j^{(i)}$ are chosen as above, for $\lambda = 0.005$. Noisy $P$ is chosen at random with noise strength $\lambda_P$, as explained in the main text.
• Figure 5: Average splitting between ($\pi/2$)-paired eigenvalues of $U^{\text{TTSB}}_{\text{DS,OBC}}$ with all terms in $H_\text{DS,OBC}$ that straddle bond $[\lfloor L/2\rfloor, \lfloor L/2\rfloor + 1]$ being omitted. $\lambda$ is the strength of both symmetric perturbations (not including those that straddle the bond) and non-symmetric local $X$ and $Z$ fields. The $(\pi/2)$-splitting is averaged over 1000 disorder realisations. The $L=8$ data points with $\lambda \leq 0.1$ are used to produce the fit shown, which is $\Delta_{\pi/2} \sim \lambda^{1.5}$.

Theorems & Definitions (2)

• Lemma 1
• proof