Topological Phases of Many-Body Localized Systems: Beyond Eigenstate Order

TL;DR

This work develops a comprehensive framework connecting anomalous localized topological (ALT) phases in many-body localized (MBL) systems to quantum cellular automata (QCA). By positing that ALT phases can be captured by short-ranged entangled MBL Hamiltonians that are related by locality-preserving unitaries, it introduces an Omega-spectrum structure that links driven ALT phases across dimensions via edge pumping (swindle map). The authors classify static and driven ALT phases in low dimensions, extend the framework to symmetry-enriched ALT phases (SALT), and provide explicit solvable models (including a quantum quasiperiodic pump) that realize anomalous edge dynamics. This approach clarifies why some localized topological phases lack eigenstate order and offers practical pathways to realize these phases in quantum simulators. Overall, the Omega-spectrum QCA viewpoint advances the systematic classification of interacting nonequilibrium localization phenomena and their symmetry enrichments, with implications for experimental realization and theory of generalized cohomology classifications.

Abstract

Many-body localization (MBL) lends remarkable robustness to nonequilibrium phases of matter. Such phases can show topological and symmetry breaking order in their ground and excited states, but they may also belong to an anomalous localized topological phase (ALT phase). All eigenstates in an ALT phase are trivial, in that they can be deformed to product states, but the entire Hamiltonian cannot be deformed to a trivial localized model without going through a delocalization transition. Using a correspondence between MBL phases with short-ranged entanglement and locality preserving unitaries - called quantum cellular automata (QCA) - we reduce the classification of ALT phases to that of QCA. This method extends to periodically (Floquet) and quasiperiodically driven ALT phases, and captures anomalous Floquet phases within the same framework as static phases. We considerably develop the study of the topology of QCA, allowing us to classify static and driven ALT phases in low dimensions. The QCA framework further generalizes to include symmetry-enriched ALT phases (SALT phases) - which we also classify in low dimensions - and provides a large class of soluble models suitable for realization in quantum simulators. In systematizing the study of ALT phases, we both greatly extend the classification of interacting nonequilibrium systems and clarify a confusion in the literature which implicitly equates nontrivial Hamiltonians with nontrivial ground states.

Figures (6)

• Figure 1: The topological classification of gapped or MBL Hamiltonians aims to identify the path connected components of gapped or MBL models (respectively), viewed as a subspace of all local Hamiltonians. MBL Hamiltonians are a subspace of gapped Hamiltonians. The path component (in either classification) containing a specified trivial model is called the trivial phase. An anomalous localized topological phase (ALT phase) is a nontrivial MBL phase with no eigenstate order, and so in particular no ground state order, and thus is contained inside the trivial gapped phase. There are also nontrivial gapped phases with no MBL Hamiltonians.
• Figure 2: (a) Many-body localization is characterized by commuting LIOMs, $\tau^z_j$Huse2014fullyMBL. The LIOMs of an MBL model with short-ranged entanglement can be prepared from single-site operators $\sigma^z_j$ by a QCA, $\tau^z_j = V \sigma^z_j V^\dagger$. ALT phases cannot be deformed back to a trivial model without delocalizing, even though they have no eigenstate order. (b) Multi-tone driven ALT phases correspond to loops or tori of QCA. The $\Omega$-spectrum property $\mathrm{QCA}_{d-1} \simeq \Omega \mathrm{QCA}_{d}$\ref{['eqn:Omega_spectrum']} implies that $n$-tone-driven ALT phases in $d$ dimensions correspond to $(d-1)$-dimensional phases with $n-1$ drives. The translation QCA in $d=1$Gross2012QCA corresponds with the anomalous Floquet phase in $d=2$Po2016QCA, which in turn corresponds with a two-tone-driven phase in $d=3$ with circulating surface states.
• Figure 3: The swindle map $S: \mathrm{QCA}_{d-1} \to \Omega \mathrm{QCA}_{d}$ maps a QCA $v$ to a loop of QCA $S(v,t)$ in one higher dimension. $S(v,t)$ is a finite depth circuit built from $v$, $v^\dagger$ (squares), and a path from the identity to a swap gate, $\mathrm{SWAP}(t)$ (line with crosses).
• Figure 4: Sketch of the pulse schedule for the models defined by the modified swindle map $\tilde{S}$. Sites are coupled along even bonds (red, $t \in [0,\tfrac{1}{3})$) and then along odd bonds (blue, $t \in [\tfrac{1}{3},\tfrac{2}{3})$). Finally, an on-site disorder potential may be applied to promote localization (gray, $t \in [\tfrac{2}{3},1]$).
• Figure 5: (a) The power pumped into the $\theta$ drive in the state $\rho_{0-,(L-1)+}$\ref{['eqn:pump_state']}. The average value is quantized to $\omega W$, where $W$ is the winding number of the unitary used to construct the QP pump model and $\omega$ is the frequency of the $\theta$ drive. The power is only nonzero when $t \bmod 1 \in [0,\tfrac{1}{3})$. (b) The quantized power appears as an average slope (red dashed) in the energy absorbed by the $\theta$ drive (blue). In fact, the modified swindle model has $E_\pm(t) = \mp \omega W t$ at integer times. Parameters:$\omega = (1+\sqrt{5})/2$, $W=1$, $\eta_1 = \sin^2(3\pi t/2)$.

Theorems & Definitions (9)

• Definition A.1
• Definition A.2
• Definition A.3
• Theorem A.1
• Lemma A.2
• proof
• Lemma A.3
• proof
• proof : Proof of Theorem \ref{['thm:QCA_exists']}