80 research outputs found
Efficient representation of fully many-body localized systems using tensor networks
We propose a tensor network encoding the set of all eigenstates of a fully
many-body localized system in one dimension. Our construction, conceptually
based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built
from two layers of unitary matrices which act on blocks of contiguous
sites.
We argue this yields an exponential reduction in computational time and
memory requirement as compared to all previous approaches for finding a
representation of the complete eigenspectrum of large many-body localized
systems with a given accuracy. Concretely, we optimize the unitaries by
minimizing the magnitude of the commutator of the approximate integrals of
motion and the Hamiltonian, which can be done in a local fashion. This further
reduces the computational complexity of the tensor networks arising in the
minimization process compared to previous work. We test the accuracy of our
method by comparing the approximate energy spectrum to exact diagonalization
results for the random field Heisenberg model on 16 sites. We find that the
technique is highly accurate deep in the localized regime and maintains a
surprising degree of accuracy in predicting certain local quantities even in
the vicinity of the predicted dynamical phase transition. To demonstrate the
power of our technique, we study a system of 72 sites and we are able to see
clear signatures of the phase transition. Our work opens a new avenue to study
properties of the many-body localization transition in large systems.Comment: Version 2, 16 pages, 16 figures. Larger systems and greater
efficienc
Simulating quantum circuits using efficient tensor network contraction algorithms with subexponential upper bound
We derive a rigorous upper bound on the classical computation time of
finite-ranged tensor network contractions in dimensions. By means of
the Sphere Separator Theorem, we are able to take advantage of the structure of
quantum circuits to speed up contractions to show that quantum circuits of
single-qubit and finite-ranged two-qubit gates can be classically simulated in
subexponential time in the number of gates. In many practically relevant cases
this beats standard simulation schemes. Moreover, our algorithm leads to
speedups of several orders of magnitude over naive contraction schemes for
two-dimensional quantum circuits on as little as an lattice. We
obtain similarly efficient contraction schemes for Google's Sycamore-type
quantum circuits, instantaneous quantum polynomial-time circuits and
non-homogeneous (2+1)-dimensional random quantum circuits.Comment: 8 pages, 6 figure
Fermionic Projected Entangled Pair States and Local U(1) Gauge Theories
Tensor networks, and in particular Projected Entangled Pair States (PEPS),
are a powerful tool for the study of quantum many body physics, thanks to both
their built-in ability of classifying and studying symmetries, and the
efficient numerical calculations they allow. In this work, we introduce a way
to extend the set of symmetric PEPS in order to include local gauge invariance
and investigate lattice gauge theories with fermionic matter. To this purpose,
we provide as a case study and first example, the construction of a fermionic
PEPS, based on Gaussian schemes, invariant under both global and local U(1)
gauge transformations. The obtained states correspond to a truncated U(1)
lattice gauge theory in 2 + 1 dimensions, involving both the gauge field and
fermionic matter. For the global symmetry (pure fermionic) case, these PEPS can
be studied in terms of spinless fermions subject to a p-wave superconducting
pairing. For the local symmetry (fermions and gauge fields) case, we find
confined and deconfined phases in the pure gauge limit, and we discuss the
screening properties of the phases arising in the presence of dynamical matter
Classification of symmetry-protected topological many-body localized phases in one dimension.
We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians
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