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We study the evolution and persistence of quantum and classical correlations between spatially separated sites in the disordered XXZ spin chain in its ergodic and many-body-localized phases using exact diagonalization and matrix product state simulations. We show that in the many-body-localized phase, quantum entanglement survives in long-time unitary dynamics of the system, even though it is more fragile than the classical correlations. We study the dependence of the residual entanglement and correlations on the system size and on the disorder strength. Finally, we demonstrate the robustness of entanglement and correlations with respect to local dissipation in the bulk of the system.

An alternative method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent-state basis that leads to the nonlinear $\sigma$ model, we use a basis called the fluctuating matrix product states which embodies intersite entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts the Affleck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadratic effective action derived by gradient expansion around the saddle point is free from constraints that plagued the nonlinear $\sigma$ model and exactly solvable. The obtained excitation modes agree precisely with the single-mode approximation result for the AKLT Hamiltonian. Excitation spectra for other bilinear-biquadratic Hamiltonians are obtained as well by diagonalizing the quadratic action.

Using time-dependent density matrix renormalization group calculations we study the transport properties ($I-V$ curves and shot noise) of the interacting resonant level model (IRLM) in a large range of the interaction parameter $U$, in the scaling limit. We find that these properties can be described remarkably well by those of the boundary sine-Gordon model (BSG), which are known analytically [Fendley et al., Phys. Rev. B 52, 8934 (1995).]. We argue that the two models are nevertheless in different universality classes out-of-equilibrium: this requires a delicate discussion of their infrared (IR) properties (i.e., at low bias), where we prove in particular that the effective tunneling charge is $e$ in the infrared regime of the IRLM (except at the self-dual point where it jumps to $2e$), while it is known to be a continuously varying function of $U$ in the BSG. This behavior is confirmed by careful analysis of the numerical data in the IR. The remarkable agreement of the transport properties, especially in the crossover region, remains however unexplained.

Suppression of electron motion under an alternating current (ac) electric field is examined in a one-dimensional Hubbard model. Utilizing three complementary calculation methods, it is found unambiguously that magnitudes of the kinetic-energy suppressions are influenced sensitively by the Coulomb interaction as well as the electron density. The phase and frequency in the ac field do not bring about major effects. The results are interpreted as a combined effect of the Coulomb interaction and the ac field, and provide a guiding principle for the photocontrol of correlated electron motion.

In this paper we present the exact representation of a fully correlated electronic wavefunction as the single-particle basis approaches completeness. It consists of a half-infinite chain of matrices of exponentially increasing size. The complete basis limit is illustrated numerically using the density-matrix renormalization-group method by computing the core-valence entanglement in the ${\mathrm{C}}_2$ ground state in increasing subsets of cc-pVTZ and pVQZ bases until convergence is reached.

Response functions $\langle {\widehat{A}}_x\left(t\right){\widehat{B}}_y\left(0\right)\rangle$ for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range $|x-y|<\ell$ is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from $\ell$ to $2\sqrt{\ell }$. (b) To go beyond, one can employ infinite MPS (iMPS) such that two evolution runs suffice. To this purpose, iMPS that are heterogeneous only around the causal cone of the perturbation are evolved in time, i.e., the simulation is done with infinite boundary conditions. Computing overlaps of these states, spatially shifted relative to each other, yields the response functions for all distances $|x-y|$. As a specific application, we compute the dynamic structure factor for ground states of bilinear-biquadratic spin-1 chains with very high resolution and explain the underlying low-energy physics. To determine the initial uniform iMPS for such simulations, infinite-system density matrix renormalization group (iDMRG) can be employed. We discuss that, depending on the system and chosen bond dimension, iDMRG with a cell size $n_c$ may converge to a nontrivial limit cycle of length $m$. This then corresponds to an iMPS with an enlarged unit cell of size $m{n}_c$.

In strictly one-dimensional systems, repulsive interactions tend to reduce particle mobility on a lattice. Therefore, the Drude weight, controlling the divergence at zero-frequency of optical conductivities in perfect conductors, is lower than in noninteracting cases. We show that this is not the case when extending to quasi-one-dimensional ladder systems. Relying on bosonization, perturbative and matrix product states (MPS) calculations, we show that nearest-neighbor interactions and magnetic fluxes provide a bias between back- and forward-scattering processes, leading to linear corrections to the Drude weight in the interaction strength. As a consequence, Drude weights counterintuitively increase (decrease) with repulsive (attractive) interactions. Our findings are relevant for the efficient tuning of Drude weights in the framework of ultracold atoms trapped in optical lattices and equally affect topological edge states in condensed matter systems.

We investigate the evolution of string order in a spin-1 chain following a quantum quench. After initializing the chain in the Affleck-Kennedy-Lieb-Tasaki state, we analyze in detail how string order evolves as a function of time at different length scales. The Hamiltonian after the quench is chosen either to preserve or to suddenly break the symmetry which ensures the presence of string order. Depending on which of these two situations arises, string order is either preserved or lost even at infinitesimal times in the thermodynamic limit. The fact that nonlocal order may be abruptly destroyed, what we call string-order melting, makes it qualitatively different from typical order parameters in the manner of Landau. This situation is thoroughly characterized by means of numerical simulations based on matrix product states algorithms and analytical studies based on a short-time expansion for several simplified models.

We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized “analytic clustering properties.” In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration-free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short-range Hamiltonians and MPSs of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin–conformal field theory (CFT) states and their parent Hamiltonians. While focusing on the Laughlin state, generalizations to other CFT-MPSs will become transparent.

We propose a tensor network method for investigating strongly disordered systems that is based on an adaptation of entanglement renormalization [G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)]. This method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse-grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. Benchmark results from the disordered $XXZ$ model demonstrates that this approach accurately captures ground-state entanglement in disordered systems, even at long distances. This approach leads to a new class of efficiently contractible tensor network ansatz for one-dimensional systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.

We investigate the occurrence of exact zero modes in one-dimensional quantum magnets of finite length that possess edge states. Building on conclusions first reached in the context of the spin-1/$2XY$ chain in a field and then for the spin-$1J_1-J_2$ Heisenberg model, we show that the development of incommensurate correlations in the bulk invariably leads to oscillations in the sign of the coupling between edge states, and hence to exact zero energy modes at the crossing points where the coupling between the edge states rigorously vanishes. This is true regardless of the origin of the frustration (e.g., next-nearest-neighbor coupling or biquadratic coupling for the spin-1 chain), of the value of the bulk spin (we report on spin-1/2, spin-1, and spin-2 examples), and of the value of the edge-state emergent spin (spin-1/2 or spin-1).

The Chebyshev expansion offers a numerically efficient and easy-implement algorithm for evaluating dynamic correlation functions using matrix product states (MPS). In this approach, each recursively generated Chebyshev vector is approximately represented by an MPS. However, the recurrence relations of Chebyshev polynomials are broken by the approximation, leading to an error which is accumulated with the increase of the order of expansion. Here we propose a reorthonormalization approach to remove this error introduced in the loss of orthogonality of the Chebyshev polynomials. Our approach, as illustrated by comparison with the exact results for the one-dimensional XY and Heisenberg models, improves significantly the accuracy in the calculation of dynamical correlation functions.

In the Fock representation, we construct matrix product states (MPS) for one-dimensional gapped phases for ${\mathbb{Z}}_p$ parafermions. From the analysis of irreducibility of MPS, we classify all possible gapped phases of ${\mathbb{Z}}_p$ parafermions without extra symmetry other than ${\mathbb{Z}}_p$ charge symmetry, including topological phases, spontaneous symmetry breaking phases, and a trivial phase. For all phases, we find the irreducible forms of local matrices of MPS, which span different kinds of graded algebras. The topological phases are characterized by the nontrivial simple ${\mathbb{Z}}_p$ graded algebras with the characteristic graded centers, yielding the degeneracies of the full transfer matrix spectra uniquely. However, the spontaneous symmetry breaking phases correspond to the trivial semisimple ${\mathbb{Z}}_{p/n}$ graded algebras, which can be further reduced to the trivial simple ${\mathbb{Z}}_{p/n}$ graded algebras, where $n$ is the divisor of $p$. So the present results provide the complete classification of the parafermionic gapped phases and deepen our understanding of topological phases in one dimension.

This paper investigates the temperature dependence of quantum information scrambling in local systems with an energy gap, $m$, above the ground state. We study the speed and shape of growing Heisenberg operators as quantified by out-of-time-order correlators, with particular attention paid to so-called contour dependence, i.e., dependence on the way operators are distributed around the thermal circle. We report large-scale tensor network numerics on a gapped chaotic spin chain down to temperatures comparable to the gap which show that the speed of operator growth is strongly contour-dependent. The numerics also show a characteristic broadening of the operator wavefront at finite temperature $T$. To study the behavior at temperatures much below the gap, we perform a perturbative calculation in the paramagnetic phase of a $2+1\mathrm{D} \mathrm{O}(N$) nonlinear sigma model, which is analytically tractable at large $N$. Using the ladder diagram technique, we find that operators spread at a speed $\sqrt{T/m}$ at low temperatures, $T\ll m$. In contrast to the numerical findings of spin chain, the large $N$ computation is insensitive to the contour dependence and does not show broadening of operator front. We discuss these results in the context of a recently proposed state-dependent bound on scrambling.

We analyze the cluster formation in a nonergodic stochastic system as a result of counterflow, with the aid of an exactly solvable model. To illustrate the clustering, a two species asymmetric simple exclusion process with impurities on a periodic lattice is considered, where the impurity can activate flips between the two nonconserved species. Exact analytical results, supported by Monte Carlo simulations, show two distinct phases, free-flowing phase and clustering phase. The clustering phase is characterized by constant density and vanishing current of the nonconserved species, whereas the free-flowing phase is identified with nonmonotonic density and nonmonotonic finite current of the same. The $n$-point spatial correlation between $n$ consecutive vacancies grows with increasing $n$ in the clustering phase, indicating the formation of two macroscopic clusters in this phase, one of the vacancies and the other consisting of all the particles. We define a rearrangement parameter that permutes the ordering of particles in the initial configuration, keeping all the input parameters fixed. This rearrangement parameter reveals the significant effect of nonergodicity on the onset of clustering. For a special choice of the microscopic dynamics, we connect the present model to a system of run-and-tumble particles used to model active matter, where the two species having opposite net bias manifest the two possible run directions of the run-and-tumble particles, and the impurities act as tumbling reagents that enable the tumbling process.

We introduce a theoretical scheme for the analog quantum simulation of long-range XYZ models using current trapped-ion technology. In order to achieve fully tunable Heisenberg-type interactions, our proposal requires a state-dependent dipole force along a single vibrational axis, together with a combination of standard resonant and detuned carrier drivings. We discuss how this quantum simulator could explore the effect of long-range interactions on the phase diagram by combining an adiabatic protocol with the quasiperiodic drivings, and test the validity of our scheme numerically. At the isotropic Heisenberg point, we show that the long-range Hamiltonian can be mapped onto a nonlinear sigma model with a topological term that is responsible for its low-energy properties, and we benchmark our predictions with matrix-product-state numerical simulations.

We study quench dynamics in an interacting spin chain with a quasiperiodic on-site field, known as the interacting Aubry-André model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to $L=50$. We find that the choice of periodicity $\mathrm{\Phi }$ of the quasiperiodic field influences the dynamics. For $\mathrm{\Phi }=(\sqrt{5}-1)/2$ (the inverse golden ratio) and interaction $\mathrm{\Delta }=1$, the model most frequently considered in the literature, we obtain the critical disorder $W_c=4.8\pm 0.5$ in units where the noninteracting transition is at $W=2$. At the same time, for periodicity $\mathrm{\Phi }=\sqrt{2}/2$ we obtain a considerably higher critical value, $W_c=7.8\pm 0.5$. Finite-size effects on the critical disorder $W_c$ are much weaker than in the purely random case. This supports the enhancement of $W_c$ in the case of a purely random potential by rare “ergodic spots,” which do not occur in the quasiperiodic case. Further, the data suggest that the decay of the antiferromagnetic order in the delocalized phase is faster than a power law.

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying $T^2=1$. Without extra symmetry, it has been found that ${\mathbb{Z}}_p$ parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors $n$ of $p$. In the presence of TR symmetry, however, the enriched classification is characterized by three indices $n$, $\kappa$, and $\mu$, where $\kappa \in {\mathbb{Z}}_2$ denotes the linear or projective TR actions on the edges, and $\mu \in {\mathbb{Z}}_2$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the ${\mathbb{Z}}_r$-symmetric parafermionic ground states, where $r=p$ for trivial or topological phases, and $r=p/n$ for SSB phases, each original gapped phase with odd $r$ is divided into two phases, while each phase with even $r$ is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.

In a many-body quantum system, local operators in the Heisenberg picture $O(t)=e^{iHt}O{e}^{-iHt}$ spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement—the entanglement entropy in operator space—is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time $t$ in chaotic systems, we present evidence that it grows only logarithmically in generic interacting integrable systems. Although this logarithmic growth has been previously established for noninteracting fermions, there has been no progress on interacting integrable systems to date. In this Letter we provide an analytical upper bound on operator entanglement for all local operators in the “Rule 54” qubit chain, a cellular automaton model introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)], and recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically. The possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.