25 research outputs found
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
The operator growth hypothesis in open quantum systems
The operator growth hypothesis (OGH) is a technical conjecture about the
behaviour of operators -- specifically, the asymptotic growth of their Lanczos
coefficients -- under repeated action by a Liouvillian. It is expected to hold
for a sufficiently generic closed many-body system. When it holds, it yields
bounds on the high frequency behavior of local correlation functions and
measures of chaos (like OTOCs). It also gives a route to numerically estimating
response functions. Here we investigate the generalisation of OGH to open
quantum systems, where the Liouvillian is replaced by a Lindbladian. For a
quantum system with local Hermitian jump operators, we show that the OGH is
modified: we define a generalisation of the Lanczos coefficient and show that
it initially grows linearly as in the original OGH, but experiences
exponentially growing oscillations on scales determined by the dissipation
strength. We see this behavior manifested in a semi-analytically solvable model
(large-q SYK with dissipation), numerically for an ergodic spin chain, and in a
solvable toy model for operator growth in the presence of dissipation (which
resembles a non-Hermitian single-particle hopping process). Finally, we show
that the modified OGH connects to a fundamental difference between Lindblad and
closed systems: at high frequencies, the spectral functions of the former decay
algebraically, while in the latter they decay exponentially. This is an
experimentally testable statement, which also places limitations on the
applicability of Lindbladians to systems in contact with equilibrium
environments.Comment: 9 pages, 6 figure
Measurement of the entanglement spectrum of a symmetry-protected topological state using the IBM quantum computer
Entanglement properties are routinely used to characterize phases of quantum
matter in theoretical computations. For example the spectrum of the reduced
density matrix, or so-called "entanglement spectrum", has become a widely used
diagnostic for universal topological properties of quantum phases. However,
while being convenient to calculate theoretically, it is notoriously hard to
measure in experiments. Here we use the IBM quantum computer to make the first
ever measurement of the entanglement spectrum of a symmetry-protected
topological state. We are able to distinguish its entanglement spectrum from
those we measure for trivial and long-range ordered states.Comment: 8 pages, 4 figure
Boson Condensation in Topologically Ordered Quantum Liquids
Boson condensation in topological quantum field theories (TQFT) has been
previously investigated through the formalism of Frobenius algebras and the use
of vertex lifting coefficients. While general, this formalism is physically
opaque and computationally arduous: analyses of TQFT condensation are
practically performed on a case by case basis and for very simple theories
only, mostly not using the Frobenius algebra formalism. In this paper we
provide a new way of treating boson condensation that is computationally
efficient. With a minimal set of physical assumptions, such as commutativity of
lifting and the definition of confined particles, we can prove a number of
theorems linking Boson condensation in TQFT with chiral algebra extensions, and
with the factorization of completely positive matrices over the nonnegative
integers. We present numerically efficient ways of obtaining a condensed theory
fusion algebra and S matrices; and we then use our formalism to prove several
theorems for the S and T matrices of simple current condensation and of
theories which upon condensation result in a low number of confined particles.
We also show that our formalism easily reproduces results existent in the
mathematical literature such as the noncondensability of 5 and 10 layers of the
Fibonacci TQFT.Comment: 29 page
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
No-go theorem for boson condensation in topologically ordered quantum liquids
Certain phase transitions between topological quantum field theories (TQFTs) are driven by the condensation of bosonic anyons. However, as bosons in a TQFT are themselves nontrivial collective excitations, there can be topological obstructions that prevent them from condensing. Here we formulate such an obstruction in the form of a no-go theorem. We use it to show that no condensation is possible in SO(3) k TQFTs with odd k. We further show that a 'layered' theory obtained by tensoring SO(3) k TQFT with itself any integer number of times does not admit condensation transitions either. This includes (as the case k = 3) the noncondensability of any number of layers of the Fibonacci TQFT
Observation of discrete time-crystalline order in a disordered dipolar many-body system
Understanding quantum dynamics away from equilibrium is an outstanding
challenge in the modern physical sciences. It is well known that
out-of-equilibrium systems can display a rich array of phenomena, ranging from
self-organized synchronization to dynamical phase transitions. More recently,
advances in the controlled manipulation of isolated many-body systems have
enabled detailed studies of non-equilibrium phases in strongly interacting
quantum matter. As a particularly striking example, the interplay of periodic
driving, disorder, and strong interactions has recently been predicted to
result in exotic "time-crystalline" phases, which spontaneously break the
discrete time-translation symmetry of the underlying drive. Here, we report the
experimental observation of such discrete time-crystalline order in a driven,
disordered ensemble of dipolar spin impurities in diamond at
room-temperature. We observe long-lived temporal correlations at integer
multiples of the fundamental driving period, experimentally identify the phase
boundary and find that the temporal order is protected by strong interactions;
this order is remarkably stable against perturbations, even in the presence of
slow thermalization. Our work opens the door to exploring dynamical phases of
matter and controlling interacting, disordered many-body systems.Comment: 6 + 3 pages, 4 figure
String-net models in condensed matter systems
We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour 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 Ï=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour 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. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk.
Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling. </p
Ballistic to diffusive crossover in a weakly interacting Fermi gas
In the absence of disorder and interactions, fermions move coherently and their associated charge and energy exhibit ballistic spreading, even at finite energy density. In the presence of weak interactions and a finite energy density, fermion-fermion scattering leads to a crossover between early-time ballistic and late-time diffusive transport. The relevant crossover timescales and the transport coefficients are both functions of interaction strength, but the question of determining the precise functional dependence is likely impossible to answer exactly. In this work we develop a numerical method (fDAOE) which is powerful enough to provide an approximate answer to this question, and which is consistent with perturbative arguments in the limit of very weak interactions. Our algorithm, which adapts the existing dissipation-assisted operator evolution (DAOE) to fermions, is applicable to systems of interacting fermions at high temperatures. The algorithm approximates the exact dynamics by systematically discarding information from high n-point functions, and is tailored to capture noninteracting dynamics exactly. Applying our method to a microscopic model of interacting fermions, we numerically determine crossover timescales and diffusion constants for a wide range of interaction strengths. In the limit of weak interaction strength (Δ), we demonstrate that the crossover from ballistic to diffusive transport happens at a time tD∼1/Δ2 and that the diffusion constant similarly scales as D∼1/Δ2. We confirm that these scalings are consistent with a perturbative Fermi's golden rule calculation, and we provide a heuristic operator-spreading picture for the crossover between ballistic and diffusive transport.</p