18 research outputs found
Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk
Quantum walks are promising for information processing tasks because on
regular graphs they spread quadratically faster than random walks. Static
disorder, however, can turn the tables: unlike random walks, quantum walks can
suffer Anderson localization, whereby the spread of the walker stays within a
finite region even in the infinite time limit. It is therefore important to
understand when we can expect a quantum walk to be Anderson localized and when
we can expect it to spread to infinity even in the presence of disorder. In
this work we analyze the response of a generic one-dimensional quantum walk --
the split-step walk -- to different forms of static disorder. We find that
introducing static, symmetry-preserving disorder in the parameters of the walk
leads to Anderson localization. In the completely disordered limit, however, a
delocalization sets in, and the walk spreads subdiffusively. Using an efficient
numerical algorithm, we calculate the bulk topological invariants of the
disordered walk, and interpret the disorder-induced Anderson localization and
delocalization transitions using these invariants.Comment: version 2, submitted to Phys. Rev.
The Physics of (good) LDPC Codes I. Gauging and dualities
Low-depth parity check (LDPC) codes are a paradigm of error correction that
allow for spatially non-local interactions between (qu)bits, while still
enforcing that each (qu)bit interacts only with finitely many others. On
expander graphs, they can give rise to ``good codes'' that combine a finite
encoding rate with an optimal scaling of the code distance, which governs the
code's robustness against noise. Such codes have garnered much recent attention
due to two breakthrough developments: the construction of good quantum LDPC
codes and good locally testable classical LDPC codes, using similar methods.
Here we explore these developments from a physics lens, establishing
connections between LDPC codes and ordered phases of matter defined for systems
with non-local interactions and on non-Euclidean geometries. We generalize the
physical notions of Kramers-Wannier (KW) dualities and gauge theories to this
context, using the notion of chain complexes as an organizing principle. We
discuss gauge theories based on generic classical LDPC codes and make a
distinction between two classes, based on whether their excitations are
point-like or extended. For the former, we describe KW dualities, analogous to
the 1D Ising model and describe the role played by ``boundary conditions''. For
the latter we generalize Wegner's duality to obtain generic quantum LDPC codes
within the deconfined phase of a Z_2 gauge theory. We show that all known
examples of good quantum LDPC codes are obtained by gauging locally testable
classical codes. We also construct cluster Hamiltonians from arbitrary
classical codes, related to the Higgs phase of the gauge theory, and formulate
generalizations of the Kennedy-Tasaki duality transformation. We use the chain
complex language to discuss edge modes and non-local order parameters for these
models, initiating the study of SPT phases in non-Euclidean geometries
Sub-ballistic growth of R\'enyi entropies due to diffusion
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 , in systems with diffusive
transport. We provide strong evidence for this in both a U 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 + supplemen
Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation
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 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 , and apply perturbative arguments to show that
for the ballistic front of information-spreading can only develop at
times exponentially large in -- 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, 1
Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
Thermalization and scrambling are the subject of much recent study from the
perspective of many-body quantum systems with locally bounded Hilbert spaces
(`spin chains'), quantum field theory and holography. We tackle this problem in
1D spin-chains evolving under random local unitary circuits and prove a number
of exact results on the behavior of out-of-time-ordered commutators (OTOCs),
and entanglement growth in this setting. These results follow from the
observation that the spreading of operators in random circuits is described by
a `hydrodynamical' equation of motion, despite the fact that random unitary
circuits do not have locally conserved quantities (e.g., no conserved energy).
In this hydrodynamic picture quantum information travels in a front with a
`butterfly velocity' that is smaller than the light cone
velocity of the system, while the front itself broadens diffusively in time.
The OTOC increases sharply after the arrival of the light cone, but we do
\emph{not} observe a prolonged exponential regime of the form for a fixed Lyapunov exponent .
We find that the diffusive broadening of the front has important consequences
for entanglement growth, leading to an entanglement velocity that can be
significantly smaller than the butterfly velocity. We conjecture that the
hydrodynamical description applies to more generic ergodic systems and support
this by verifying numerically that the diffusive broadening of the operator
wavefront also holds in a more traditional non-random Floquet spin-chain. We
also compare our results to Clifford circuits, which have less rich
hydrodynamics and consequently trivial OTOC behavior, but which can
nevertheless exhibit linear entanglement growth and thermalization.Comment: 11+6 pages, 9 figure
Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians
We show that the combination of charge and dipole
conservation---characteristic of fracton systems---leads to an extensive
fragmentation of the Hilbert space, which in turn can lead to a breakdown of
thermalization. As a concrete example, we investigate the out-of-equilibrium
dynamics of one-dimensional spin-1 models that conserve charge (total )
and its associated dipole moment. First, we consider a minimal model including
only three-site terms and find that the infinite temperature auto-correlation
saturates to a finite value---showcasing non-thermal behavior. The absence of
thermalization is identified as a consequence of the strong fragmentation of
the Hilbert space into exponentially many invariant subspaces in the local
basis, arising from the interplay of dipole conservation and local
interactions. Second, we extend the model by including four-site terms and find
that this perturbation leads to a weak fragmentation: the system still has
exponentially many invariant subspaces, but they are no longer sufficient to
avoid thermalization for typical initial states. More generally, for any finite
range of interactions, the system still exhibits non-thermal eigenstates
appearing throughout the entire spectrum. We compare our results to charge and
dipole moment conserving random unitary circuit models for which we reach
identical conclusions.Comment: close to published version: 10 pages + Appendices. Updated
discussions and conten
Detecting topological invariants in chiral symmetric insulators via losses
We show that the bulk winding number characterizing one-dimensional
topological insulators with chiral symmetry can be detected from the
displacement of a single particle, observed via losses. Losses represent the
effect of repeated weak measurements on one sublattice only, which interrupt
the dynamics periodically. When these do not detect the particle, they realize
negative measurements. Our repeated measurement scheme covers both
time-independent and periodically driven (Floquet) topological insulators, with
or without spatial disorder. In the limit of rapidly repeated, vanishingly weak
measurements, our scheme describes non-Hermitian Hamiltonians, as the lossy
Su-Schrieffer-Heeger model of Phys. Rev. Lett. 102, 065703 (2009). We find,
contrary to intuition, that the time needed to detect the winding number can be
made shorter by decreasing the efficiency of the measurement. We illustrate our
results on a discrete-time quantum walk, and propose ways of testing them
experimentally.Comment: 4.5 pages, 3 figures + 4 pages of Supplemental Materia
Higgs Condensates are Symmetry-Protected Topological Phases: II. Gauge Theory and Superconductors
Classifying Higgs phases within the landscape of gapped and symmetry
preserving states of matter presents a conceptual challenge. We argue that
Higgs phases are symmetry-protected topological (SPT) phases and we
derive their topological response theory and boundary anomaly -- applicable to
superconductors treated with dynamical electromagnetic field. This generalizes
the discussion of discrete gauge theories by Verresen et al., arXiv:2211.01376.
We show that a Higgs phase in spatial dimensions is in a non-trivial SPT
class protected by a global symmetry associated with the Higgs field,
and a form magnetic symmetry, associated with the absence of
magnetic monopoles. In , this gives an SPT with a mixed Hall response
between conventional symmetries, whereas in we obtain a novel SPT
protected by a 0-form and 1-form symmetry whose 2+1d boundary anomaly is
satisfied by a superfluid. The signature properties of superconductors -- Higgs
phases for electromagnetism -- can be reproduced from this SPT response. For
instance, the Josephson effect directly arises from the aforementioned boundary
superfluid. In addition to this minimalist approach being complementary to
Landau-Ginzburg theory, its non-perturbative nature is useful in situations
where fluctuations are significant. We substantiate this by predicting the
stability of the Josephson effect upon introducing monopoles in lattice
gauge theory, where tuning from the charge-1 Higgs phase to the confined phase
leads to a quantum critical point in the junction. Furthermore, this
perspective reveals unexpected connections, such as how persistent currents at
the surface of a superconductor arise from generalized Thouless pumps. We also
treat generalizations to partial-Higgs phases, including "2e" condensates in
electronic superconductors, corresponding to symmetry-enriched topological
orders.Comment: 40 pages + appendices, 7 figure
The ballistic to diffusive crossover in a weakly-interacting Fermi gas
Charge and energy are expected to diffuse in interacting systems of fermions
at finite temperatures, even in the absence of disorder, with the interactions
inducing a crossover from the coherent and ballistic streaming of
quasi-particles at early times, to incoherent diffusive behavior at late times.
The relevant crossover timescales and the transport coefficients are both
controlled by the strength of interactions. In this work we develop a numerical
method to simulate such systems at high temperatures, applicable in a wide
range of interaction strengths, by adapting Dissipation-assisted Operator
Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact
dynamics by systematically discarding information from high -point
functions, is tailored to capture non-interacting dynamics exactly, thus
providing a good starting point for the weakly interacting problem. Applying
our method to a microscopic model of weakly interacting fermions, we
numerically demonstrate that the crossover from ballistic to diffusive
transport happens at a time and that the diffusion
constant similarly scales as , where is the
interaction strength. We substantiate this scaling with a Fermi's golden rule
calculation in the operator spreading picture, interpreting as the
fermion-fermion scattering time and lifetime of the single-particle Green's
function