arXiv subjects

Thorsten B. Wahl

Publications and source records attributed to Thorsten B. Wahl.

At least 19 recordsLinked to original sources

arXiv2025

Gapped spinful phases obtained via Gutzwiller projections of Euler states

Gutzwiller projections of non-interacting chiral topological phases are known to give rise to fractional, topologically ordered chiral phases. Here, we carry out a similar construction using two copies of non-interacting Euler insulators to produce a class of spinful interacting Euler models. To that end, we take advantage of the recently discovered exact representation of certain Euler insulators by a projected entangled pair state (PEPS) of bond dimension $D = 2$. The Gutzwiller projection can be implemented within the tensor network formalism, giving rise to a new PEPS of bond dimension $D = 4$. We, moreover, apply very recent state-of-the-art tensor network tools to evaluate these phases. In particular, we analyze its entanglement entropy scaling and find no topological correction to the area law, indicating that the state is not intrinsically topologically ordered. Its entanglement spectrum shows a clear cusp at momentum zero, similar to non-interacting Euler insulators, and the spectrum of the transfer operator indicates that the state is gapped, which could imply non-intrinsic topological features. Finally, the static structure factor displays Bragg peaks, indicating the simultaneous presence of local order.

cond-mat.str-el↗

arXiv2025

Two-dimensional many-body localized systems coupled to a heat bath

We numerically investigate the effect of coupling a two-dimensional many-body localized system to a finite heat bath, using shallow quantum circuits as a variational ansatz. Specifically, we simulate optical lattice experiments with two components of ultracold bosons, where only one species is subject to a random disorder potential and the other acts as a heat bath. We obtain a filling fraction dependent phase diagram with a critical filling consistent with experiments. We also calculate two-point correlation functions and the quantum mutual information between sites. We observe a distribution of the quantum mutual information in the many-body localized regime which is consistent with the presence of resonances, similar to those observed in one-dimensional many-body localized systems.

cond-mat.dis-nn↗

arXiv2025

Exact projected entangled pair ground states with topological Euler invariant

We report on a class of gapped projected entangled pair states (PEPS) with non-trivial Euler topology motivated by recent progress in band geometry. In the non-interacting limit, these systems have optimal conditions relating to saturation of quantum geometrical bounds, allowing for parent Hamiltonians whose lowest bands are completely flat and which have the PEPS as unique ground states. Protected by crystalline symmetries, these states evade restrictions on capturing tenfold-way topological features with gapped PEPS. These PEPS thus form the first tensor network representative of a non-interacting, gapped two-dimensional topological phase, similar to the Kitaev chain in one dimension. Using unitary circuits, we then formulate interacting variants of these PEPS and corresponding gapped parent Hamiltonians. We reveal characteristic entanglement features shared between the free-fermionic and interacting states with Euler topology. Our results hence provide a rich platform of PEPS models that have, unexpectedly, a finite topological invariant, forming the basis for new spin liquids, quantum Hall physics, and quantum information pursuits.

quant-ph↗

arXiv2024

Many-body-localization protection of eigenstate topological order in two dimensions

Many-body localization (MBL) has been proposed to enable and protect topological order in all eigenstates, vastly expanding the traditional ground-state setting. However, for the most intriguing case of two-dimensional (2D) systems with anyons and topology-dependent degeneracies, the dense many-body spectrum challenges studying this MBL protection numerically. Here we use large-scale full-spectrum variational ansätze to demonstrate MBL-protected topological order in the disordered 2D toric code perturbed by magnetic fields. We show that the system has topological local integrals of motion (tLIOMs) for magnetic field strengths below $h_c\approx0.1$ times the toric code coupling scale. Combining tLIOMs with exact diagonalization, we also identify high-energy topological multiplets in the dense many-body spectrum. The phase diagram we find is consistent with toric-code and trivial MBL phases being separated by an intervening thermal phase.

cond-mat.dis-nn↗

arXiv2024

Quantum Circuits Reproduce Experimental Two-dimensional Many-body Localization Transition Point

While many studies point towards the existence of many-body localization (MBL) in one dimension, the fate of higher-dimensional strongly disordered systems is a topic of current debate. The latest experiments as well as several recent numerical studies indicate that such systems behave many-body localized -- at least on practically relevant time scales. However, thus far, theoretical approaches have been unable to quantitatively reproduce experimentally measured MBL features -- an important requirement to demonstrate their validity. In this work, we use fermionic quantum circuits as a variational method to approximate the full set of eigenstates of two-dimensional MBL systems realized in fermionic optical lattice experiments. Using entanglement-based features, we obtain a phase transition point in excellent agreement with the experimentally measured value. Moreover, we calculate, the filling fraction-dependent MBL phase diagram, an important feature which has not been addressed in previous literature. We argue that our approach best captures the underlying charge-density-wave experiments and compute the mean localization lengths, which can be compared to future experiments.

cond-mat.dis-nn↗

arXiv2023

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 $d \geq 2$ dimensions. Consequently, we 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. Moreover, we present and implement an algorithm guaranteed to meet our bound and which finds contraction orders with vastly lower computational times in practice. In many practically relevant cases this beats standard simulation schemes and, for certain quantum circuits, also a state-of-the-art method. Specifically, our algorithm leads to speedups of several orders of magnitude over naive contraction schemes for two-dimensional quantum circuits on as little as an $8 \times 8$ 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.

quant-ph↗

arXiv2023

Topologically ordered time crystals

We define topological time crystals, a dynamical phase of periodically driven quantum many-body systems capturing the coexistence of intrinsic topological order with the spontaneous breaking of discrete time-translation symmetry. We show that many-body localization can stabilize this phase against generic perturbations and establish some of its key features and signatures, including a dynamical, time-crystal form of the perimeter law for topological order. We link topological and ordinary time crystals through three complementary perspectives: higher-form symmetries, quantum error-correcting codes, and a holographic correspondence. We also propose an experimental realization of a surface-code-based topological time crystal for the Google Sycamore processor.

cond-mat.dis-nn↗

arXiv2022

Local integrals of motion detection of localization-protected topological order

Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the DMRG-X algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.

cond-mat.dis-nn↗

arXiv2021

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 $\mathbb{Z}_2$ classification (cf. [Phys. Rev. B 98, 054204 (2018)]). 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 $\mathbb{Z}_2$ topological index expected from the classification of fermionic SPT ground states. Furthermore, 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.

cond-mat.dis-nn↗

arXiv2020

Local Integrals of Motion for Topologically Ordered Many-Body Localized Systems

Many-body localized (MBL) systems are often described using their local integrals of motion, which, for spin systems, are commonly assumed to be a local unitary transform of the set of on-site spin-z operators. We show that this assumption cannot hold for topologically ordered MBL systems. Using a suitable definition to capture such systems in any spatial dimension, we demonstrate a number of features, including that MBL topological order, if present: (i) is the same for all eigenstates; (ii) is robust in character against any perturbation preserving MBL; (iii) implies that on topologically nontrivial manifolds a complete set of integrals of motion must include nonlocal ones in the form of local-unitary-dressed noncontractible Wilson loops. Our approach is well suited for tensor-network methods, and is expected to allow these to resolve highly-excited finite-size-split topological eigenspaces despite their overlap in energy. We illustrate our approach on the disordered Kitaev chain, toric code, and X-cube model.

cond-mat.dis-nn↗

arXiv2020

Classification of symmetry-protected topological phases in two-dimensional many body-localized systems

We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we show that the (generalized) third cohomology class of the symmetry group is a topological invariant; however our approach leaves room for the existence of additional topological indices. We argue that our classification applies to quasi-periodic systems in two dimensions and systems with true random disorder within times which scale superexponentially with the inverse interaction strength. Our technique might be adapted to supply arguments suggesting the same classification for two-dimensional symmetry-protected topological ground states with a rigorous proof.

cond-mat.dis-nn↗

arXiv2020

Signatures of the Many-body Localized Regime in Two Dimensions

Lessons from Anderson localization highlight the importance of dimensionality of real space for localization due to disorder. More recently, studies of many-body localization have focussed on the phenomenon in one dimension using techniques of exact diagonalization and tensor networks. On the other hand, experiments in two dimensions have provided concrete results going beyond the previously numerically accessible limits while posing several challenging questions. We present the first large-scale numerical examination of a disordered Bose-Hubbard model in two dimensions realized in cold atoms, which shows entanglement based signatures of many-body localization. By generalizing a low-depth quantum circuit to two dimensions we approximate eigenstates in the experimental parameter regimes for large systems, which is beyond the scope of exact diagonalization. A careful analysis of the eigenstate entanglement structure provides an indication of the putative phase transition marked by a peak in the fluctuations of entanglement entropy in a parameter range consistent with experiments.

cond-mat.dis-nn↗

arXiv2020

Tensor networks demonstrate the robustness of localization and symmetry protected topological phases

We prove that all eigenstates of many-body localized symmetry protected topological systems with time reversal symmetry have four-fold degenerate entanglement spectra in the thermodynamic limit. To that end, we employ unitary quantum circuits where the number of sites the gates act on grows linearly with the system size. We find that the corresponding matrix product operator representation has similar local symmetries as matrix product ground states of symmetry protected topological phases. Those local symmetries give rise to a $\mathbb{Z}_2$ topological index, which is robust against arbitrary perturbations so long as they do not break time reversal symmetry or drive the system out of the fully many-body localized phase.

cond-mat.dis-nn↗

arXiv2020

Behavior of l-bits near the many-body localization transition

Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators $τ_i^z$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators $τ_i^z$ and $τ_i^x$ associated with l-bits ($\boldsymbolτ_i$) completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a non-perturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli-algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.

cond-mat.dis-nn↗

arXiv2020

Approximating observables on eigenstates of large many-body localized systems

Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called l-bits. These spin algebras define quasilocal l-bit measurement ($τ^z_i$) and l-bit flip ($τ^x_i$) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate l-bit flip operators by finding them exactly on small windows of systems and extending them onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates. We then describe a method to calculate products of local observables on these eigenstates for systems of size $L$ in $O(L^2)$ time. This algorithm is used to compute the error of the approximate eigenstates.

cond-mat.dis-nn↗

arXiv2017

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 $\ell$ 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.

cond-mat.dis-nn↗

arXiv2016

Projected Entangled Pair States with non-Abelian gauge symmetries: an SU(2) study

Over the last years, Projected Entangled Pair States have demonstrated great power for the study of many body systems, as they naturally describe ground states of gapped many body Hamiltonians, and suggest a constructive way to encode and classify their symmetries. The PEPS study is not only limited to global symmetries, but has also been extended and applied for local symmetries, allowing to use them for the description of states in lattice gauge theories. In this paper we discuss PEPS with a local, SU(2) gauge symmetry, and demonstrate the use of PEPS features and techniques for the study of a simple family of many body states with a non-Abelian gauge symmetry. We present, in particular, the construction of fermionic PEPS able to describe both two-color fermionic matter and the degrees of freedom of an SU(2) gauge field with a suitable truncation.

quant-ph↗

arXiv2015

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.

quant-ph↗