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Subsystem quantum error-correcting codes typically involve measuring a sequence of non-commuting parity check operators. They can sometimes exhibit greater fault-tolerance than conventional subspace codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes -- in particular their advantages -- can be understood in terms of ground state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [Nat. Comm. 13, 6272(2022)], which exhibits single-shot error correction (SSEC). Motivated by a conjectured relation between SSEC and thermal stability, we study the zero and finite temperature phases of an associated non-commuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D Z_2 gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: there are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the SSEC in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of SSEC can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective which inspires alternative measurement sequences enabling SSEC. Moreover, since none of the above-mentioned phases survives at nonzero temperature, SSEC of the code does not imply thermal stability of the associated Hamiltonian phase., Comment: 23 pages, 22 figures. abstract shortened to meet arxiv requirements, see pdf for full abstract. v2: 25 pages, 22 figures. new background section. accepted version

Classical shadows are a computationally efficient approach to storing quantum states on a classical computer for the purposes of estimating expectation values of local observables, obtained by performing repeated random measurements. In this note we offer some comments on this approach. We note that the resources needed to form classical shadows with bounded relative error depend strongly on the target state. We then comment on the advantages and limitations of using classical shadows to simulate many-body dynamics. In addition, we introduce the notion of a hybrid shadow, constructed from measurements on a part of the system instead of the entirety, which provides a framework to gain more insight into the nature of shadow states as one reduces the size of the subsystem measured, and a potential alternative to compressing quantum states., Comment: 8+3 pages, 4+1 figures

Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the \emph{dynamics} of many-body systems is less clear: the memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wavefunction which are expected to have little effect on the hydrodynamic observables typically of interest: for example, some methods discard fine-grained correlations associated with $n$-point functions, with $n$ exceeding some cutoff $\ell_*$. In this work, we present a theory for the sizes of `backflow corrections', i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff $\ell_*$. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin-chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision $\epsilon$ with an amount of memory scaling as $\exp[\mathcal{O}(\log(\epsilon)^2)]$, significantly better than the naive estimate of memory $\exp[\mathcal{O}(\mathrm{poly}(\epsilon^{-1}))]$ required by more brute-force methods., Comment: 18 pages, 7 figures

The spread and scrambling of quantum information is a topic of considerable current interest. Numerous studies suggest that quantum information evolves according to hydrodynamical equations of motion, even though it is a starkly different quantity to better-known hydrodynamical variables such as charge and energy. In this work we show that the well-known memory matrix formalism for traditional hydrodynamics can be applied, with relatively little modification, to the question of operator growth in many-body quantum systems. On a conceptual level, this shores up the connection between information scrambling and hydrodynamics. At a practical level, it provides a framework for calculating quantities related to operator growth like the butterfly velocity and front diffusion constant, and for understanding how these quantities are constrained by microscopic symmetries. We apply this formalism to calculate operator-hydrodynamical coefficients perturbatively in a family of Floquet models. Our formalism allows us to identify the processes affecting information transport that arise from the spatiotemporal symmetries of the model., Comment: 58 pages, 3 figures

In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces., Comment: 2 pages; Comment on Commun. Phys. 3, 100

We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision., Comment: 4 + 4 pages, 3 + 4 figures

We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of \textit{Jonay et al.} in such a way that the `line tension' of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by a Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as $\propto\sqrt{t}$. In particular, for a domain wall quench, diffusion restricts entanglement growth to be $S_\text{vN} \lesssim \sqrt{t}$. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains., Comment: 11 pages, 9 figures

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher R\'enyi entropies. We argue that the latter generically grow \emph{sub-ballistically}, as $\propto\sqrt{t}$, in systems with diffusive transport. We provide strong evidence for this in both a U$(1)$ symmetric random circuit model and in a paradigmatic non-integrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second R\'enyi entropy and show that it exhibits hydrodynamic tails with \emph{three distinct power laws} occurring for different classes of initial states., Comment: close to published version: 4 + epsilon pages, 3 figures + supplement

Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of "operator spreading" in quantum maps on a torus, systems which do not have a tensor-product Hilbert space or a notion of spatial locality. Using the perturbed Arnold cat map as an example, we analytically compare and contrast the evolutions of functions on classical phase space and quantum operator evolutions, and identify distinct timescales that characterize the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the quantum system exhibits classical chaos, i.e. it mimics the behavior of the corresponding classical system. After an operator scrambling time, the operator looks "random" in the initial basis, a characteristic feature of quantum chaos. These timescales can be related to the quasi-energy spectrum of the unitary via the spectral form factor. Furthermore, we show examples of "emergent classicality" in quantum problems far away from the classical limit. Finally, we study operator evolution in non-chaotic and mixed quantum maps using the Chirikov standard map as an example., Comment: 21 pages, 7 figures v2: References added

The scrambling of quantum information in closed many-body systems, as measured by out-of-time-ordered correlation functions (OTOCs), has lately received considerable attention. Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits, aspects of which are conjectured to be universal to ergodic many-body systems, even without randomness. Here we extend this approach to systems with locally conserved quantities (e.g., energy). We do this by considering local random unitary circuits with a conserved U$(1)$ charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time ordered {\textit{and}} out-of-time-ordered correlation functions, both can have a diffusively relaxing component or "hydrodynamic tail" at late times. We verify the presence of such tails also in a deterministic, peridocially driven system. We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific, asymmetric shape, decaying as a power law behind the front. These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation. Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential $\mu$, and apply perturbative arguments to show that for $\mu\gg 1$ the ballistic front of information-spreading can only develop at times exponentially large in $\mu$ -- with the information traveling diffusively at earlier times. We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators., Comment: Close to published version: 17 + 9.5 pages. Improved presentation. Contains new section on clean Floquet spin chain. New and/or improved numerical data in Figures 4-7, 11, 14

Time crystals are proposed states of matter which spontaneously break time translation symmetry. There is no settled definition of such states. We offer a new definition which follows the traditional recipe for Wigner symmetries and order parameters. Supplementing our definition with a few plausible assumptions we find that a) systems with time independent Hamiltonians should not exhibit TTSB while b) the recently studied $\pi$ spin glass/Floquet time crystal can be viewed as breaking a global internal symmetry and as breaking time translation symmetry as befits its two names.

Recent work has shown that a variety of novel phases of matter arise in periodically driven Floquet systems. Among these are many-body localized phases which spontaneously break global symmetries and exhibit novel multiplets of Floquet eigenstates separated by quantized quasienergies. Here we show that these properties are stable to all weak local deformations of the underlying Floquet drives -- including those that explicitly break the defining symmetries -- and that the models considered until now occupy sub-manifolds within these larger "absolutely stable" phases. While these absolutely stable phases have no explicit global symmetries, they spontaneously break Hamiltonian dependent emergent symmetries, and thus continue to exhibit the novel multiplet structure. The multiplet structure in turn encodes characteristic oscillations of the emergent order parameter at multiples of the fundamental period. Altogether these phases exhibit a form of simultaneous long-range order in space and time which is new to quantum systems. We describe how this spatiotemporal order can be detected in experiments involving quenches from a broad class of initial states., Comment: Published version. Minor typos corrected, some discussions expanded

Recent work suggests that a sharp definition of `phase of matter' can be given for periodically driven `Floquet' quantum systems exhibiting many-body localization. In this work we propose a classification of the phases of interacting Floquet localized systems with (completely) spontaneously broken symmetries -- we focus on the one dimensional case, but our results appear to generalize to higher dimensions. We find that the different Floquet phases correspond to elements of $Z(G)$, the centre of the symmetry group in question. In a previous paper we offered a companion classification of unbroken, i.e., paramagnetic phases., Comment: Published version

Recent work suggests that a sharp definition of `phase of matter' can be given for some quantum systems out of equilibrium---first for many-body localized systems with time independent Hamiltonians and more recently for periodically driven or Floquet localized systems. In this work we propose a classification of the finite abelian symmetry protected phases of interacting Floquet localized systems in one dimension. We find that the different Floquet phases correspond to elements of $\text{Cl}\times\mathcal{A}_G$, where $\text{Cl}$ is the undriven interacting classification, and $\mathcal{A}_G$ is a set of (twisted) 1d representations of $G$. We will address symmetry broken phases in a subsequent paper., Comment: 21 pages. Explained connection to the classification schemes in other recent work. Close to published version

We study the effects of bulk-edge Coulomb coupling on quantum Hall Fabry-Perot interferometers. We find that these effects can be appreciable in devices which would not usually be associated with strong bulk-edge Coulomb coupling, provided the devices in question exhibit certain fractional plateaus. With this in mind, we analyze recent experiments at $\nu=5/2$ by taking into account a tunnel coupling between localized bulk Majorana states and Majorana edge states. We find that these experimental data are consistent with the widely held view that the $\nu=5/2$ state harbors Moore-Read topological order. However, experiments may have measured Coulomb effects rather than an `even-odd effect' due to non-abelian braiding.

We connect a family of gauge theories (Maxwell theories with a magnetoelectric coupling $\theta = 2 \pi k, k \in \mathbb{Z}$) to the family of 3D topological lattice models introduced by Walker and Wang. In particular, we show that the lattice Hamiltonians capture a certain strong-coupling limit of these gauge theories, in which the system enters a gapped (confined) phase. We discuss the relationship between the topological order exhibited by certain of these lattice Hamiltonians and the characteristic electromagnetic response of the symmetry-protected bosonic topological insulator., Comment: Added supplementary material on surface behaviour of the models

We study the phase diagrams of a family of 3D "Walker-Wang" type lattice models, which are not topologically ordered but have deconfined anyonic excitations confined to their surfaces. We add a perturbation (analogous to that which drives the confining transition in Z_p lattice gauge theories) to the Walker-Wang Hamiltonians, driving a transition in which all or some of the variables associated with the loop gas or string-net ground states of these models become confined. We show that in many cases the location and nature of the phase transitions involved is exactly that of a generalized Z_p lattice gauge theory, and use this to deduce the basic structure of the phase diagram. We further show that the relationship between the phases on opposite sides of the transition is fundamentally different than in conventional gauge theories: in the Walker-Wang case, the number of species of excitations that are deconfined in the bulk can increase across a transition that confines only some of the species of loops or string-nets. The analogue of the confining transition in the Walker-Wang models can therefore lead to bulk deconfinement and topological order.

Quantum fluctuations are of central importance in itinerant ferromagnets; in the case of the Stoner Hamiltonian, with contact interactions, they deliver a rich phase diagram featuring a first order ferromagnetic transition preempted by a spin spiral and a paired density wave. However, to date all analyses of fluctuation corrections neglect the finite ranged nature of the Coulomb interaction. We develop the formalism to consider the effects of fluctuations with a realistic screened Coulomb potential. The finite ranged interaction suppresses the tricritical point temperature of the first order ferromagnetic transition, bringing theory into line with experiment, whilst retaining the exotic spin spiral and paired density wave. In an ultracold atomic gas a finite ranged interaction damps the competing molecular instability, permitting the observation of ferromagnetic correlations.

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang [arXiv:1104.2632v2]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a $\nu=1/2$ bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of $bF$ theories. Our special example of the 3D semion model captures much of the behavior of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum hall effect. We find that in these models a fermion is always deconfined in the three dimensional bulk.

We study the emergence of itinerant ferromagnetism in an ultra-cold atomic gas with a variable mass ratio between the up and down spin species. Mass imbalance breaks the SU(2) spin symmetry leading to a modified Stoner criterion. We first elucidate the phase behavior in both the grand canonical and canonical ensembles. Secondly, we apply the formalism to a harmonic trap to demonstrate how a mass imbalance delivers unique experimental signatures of ferromagnetism. These could help future experiments to better identify the putative ferromagnetic state. Furthermore, we highlight how a mass imbalance suppresses the three-body loss processes that handicap the formation of a ferromagnetic state. Finally, we study the time dependent formation of the ferromagnetic phase following a quench in the interaction strength.