27 research outputs found
Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines
Using the framework of infinite Matrix Product States, the existence of an
\textit{anomalous} dynamical phase for the transverse-field Ising chain with
sufficiently long-range interactions was first reported in [J.~C.~Halimeh and
V.~Zauner-Stauber, arXiv:1610:02019], where it was shown that
\textit{anomalous} cusps arise in the Loschmidt-echo return rate for
sufficiently small quenches within the ferromagnetic phase. In this work we
further probe the nature of the anomalous phase through calculating the
corresponding Fisher-zero lines in the complex time plane. We find that these
Fisher-zero lines exhibit a qualitative difference in their behavior, where,
unlike in the case of the regular phase, some of them terminate before
intersecting the imaginary axis, indicating the existence of smooth peaks in
the return rate preceding the cusps. Additionally, we discuss in detail the
infinite Matrix Product State time-evolution method used to calculate Fisher
zeros and the Loschmidt-echo return rate using the Matrix Product State
transfer matrix. Our work sheds further light on the nature of the anomalous
phase in the long-range transverse-field Ising chain, while the numerical
treatment presented can be applied to more general quantum spin chains.Comment: Journal article. 9 pages and 6 figures. Includes in part what used to
be supplemental material in arXiv:1610:0201
Quasiparticle origin of dynamical quantum phase transitions
Considering nonintegrable quantum Ising chains with exponentially decaying
interactions, we present matrix product state results that establish a
connection between low-energy quasiparticle excitations and the kind of
nonanalyticities in the Loschmidt return rate. When domain walls in the
spectrum of the quench Hamiltonian are energetically favored to be bound rather
than freely propagating, anomalous cusps appear in the return rate regardless
of the initial state. In the nearest-neighbor limit, domain walls are always
freely propagating, and anomalous cusps never appear. As a consequence, our
work illustrates that models in the same equilibrium universality class can
still exhibit fundamentally distinct out-of-equilibrium criticality. Our
results are accessible to current ultracold-atom and ion-trap experiments.Comment: 9 pages, 8 figures, accepted versio
Matrix product state renormalization
The truncation or compression of the spectrum of Schmidt values is inherent
to the matrix product state (MPS) approximation of one-dimensional quantum
ground states. We provide a renormalization group picture by interpreting this
compression as an application of Wilson's numerical renormalization group along
the imaginary time direction appearing in the path integral representation of
the state. The location of the physical index is considered as an impurity in
the transfer matrix and static MPS correlation functions are reinterpreted as
dynamical impurity correlations. Coarse-graining the transfer matrix is
performed using a hybrid variational ansatz based on matrix product operators,
combining ideas of MPS and the multi-scale entanglement renormalization ansatz.
Through numerical comparison with conventional MPS algorithms, we explicitly
verify the impurity interpretation of MPS compression, as put forward by [V.
Zauner et al., New J. Phys. 17, 053002 (2015)] for the transverse-field Ising
model. Additionally, we motivate the conceptual usefulness of endowing MPS with
an internal layered structure by studying restricted variational subspaces to
describe elementary excitations on top of the ground state, which serves to
elucidate a transparent renormalization group structure ingrained in MPS
descriptions of ground states.Comment: 15 pages, 10 figures, published versio
Shadows of Anyons
The eigenvalue structure of the quantum transfer matrix is known to encode
essential information about the elementary excitations. Here we study transfer
matrices of quantum states in a topological phase using the tensor network
formalism. We demonstrate that topological quantum order requires a particular
type of `symmetry breaking' for the fixed point subspace of the transfer
matrix, and relate physical anyon excitations to domain wall excitations at the
level of the transfer matrix. A topological phase transition to a trivial phase
triggers a change in the fixed point subspace to either a larger or smaller
symmetry and we explain how this relates to a condensation or confinement of
the corresponding anyon sectors. The tensor network formalism enables us to
determine the structure of the topological sectors in two-dimensional gapped
phases very efficiently, therefore opening novel avenues for studying
fundamental open questions related to anyon condensation.Comment: updated version, extended discussio
Prethermalization and Persistent Order in the Absence of a Thermal Phase Transition
We numerically study the dynamics after a parameter quench in the
one-dimensional transverse-field Ising model with long-range interactions
( with distance ), for finite chains and also directly
in the thermodynamic limit. In nonequilibrium, i.e., before the system settles
into a thermal state, we find a long-lived regime that is characterized by a
prethermal value of the magnetization, which in general differs from its
thermal value. We find that the ferromagnetic phase is stabilized dynamically:
as a function of the quench parameter, the prethermal magnetization shows a
transition between a symmetry-broken and a symmetric phase, even for those
values of for which no finite-temperature transition occurs in
equilibrium. The dynamical critical point is shifted with respect to the
equilibrium one, and the shift is found to depend on as well as on the
quench parameters.Comment: 6 pages, 4 figure
Transfer matrices and excitations with matrix product states
We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)—a central object in the computation of static correlation functions—provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system
Truncating an exact Matrix Product State for the XY model: transfer matrix and its renormalisation
We discuss how to analytically obtain an -- essentially infinite -- Matrix
Product State (MPS) representation of the ground state of the XY model. On the
one hand this allows to illustrate how the Ornstein-Zernike form of the
correlation function emerges in the exact case using standard MPS language. On
the other hand we study the consequences of truncating the bond dimension of
the exact MPS, which is also part of many tensor network algorithms, and
analyze how the truncated MPS transfer matrix is representing the dominant part
of the exact quantum transfer matrix. In the gapped phase we observe that the
correlation length obtained from a truncated MPS approaches the exact value
following a power law in effective bond dimension. In the gapless phase we find
a good match between a state obtained numerically from standard MPS techniques
with finite bond dimension, and a state obtained by effective finite imaginary
time evolution in our framework. This provides a direct hint for a geometric
interpretation of Finite Entanglement Scaling at the critical point in this
case. Finally, by analyzing the spectra of transfer matrices, we support the
interpretation put forward by [V. Zauner at. al., New J. Phys. 17, 053002
(2015)] that the MPS transfer matrix emerges from the quantum transfer matrix
though the application of Wilson's Numerical Renormalisation Group along the
imaginary-time direction.Comment: 14 pages, 9 figures, significantly extended, comments welcom