We investigate the equilibration dynamics of string order in one-dimensional quantum systems. After initializing a spin-1 chain in the Haldane phase, the time evolution of nonlocal correlations following a sudden quench is studied by means of matrix-product-state-based algorithms. Thermalization occurs only for scales up to a horizon growing at a well defined speed, due to the finite maximal velocity at which string correlations can propagate, related to a Lieb-Robinson bound. The persistence of string ordering at finite times is nontrivially related to symmetries of the quenched Hamiltonian. A qualitatively similar behavior is found for the string order of the Mott insulating phase in the Bose-Hubbard chain. This paves the way towards an experimental testing of our results in present cold-atom setups.
We revisit the question of string order and hidden symmetry breaking in the q q -deformed AKLT model, an example of a spin chain that possesses generalized symmetry. We first argue that the non-local Kennedy-Tasaki duality transformation that was previously proposed to relate the string order to a local order parameter leads to a non-local Hamiltonian and thus does not provide a physically adequate description of the symmetry breaking. We then present a modified non-local transformation which is based on a recently developed generalization of Witten’s Conjugation to frustration-free lattice models and capable of resolving this issue.
We study the robustness of nonlocal string order in two paradigmatic disordered spin-chain models, a spin-1/2 cluster-Ising and a spin-1 XXZ Heisenberg chain. In the clean case, they both display a transition from antiferromagnetic to string order. Applying a disorder, which preserves the Hamiltonian symmetries, we find that the transition persists in both models. In the disordered cluster-Ising model, we can study the transition analytically---by applying the strongest coupling renormalization group ---and numerically---by exploiting integrability to study the antiferromagnetic and string order parameters. We map the model into a quadratic fermion chain, where the transition appears as a change in the number of zero-energy edge modes. We also explore its zero-temperature-singularity behavior and find a transition from a nonsingular to a singular region, at a point that is different from the one separating nonlocal and local ordering. The disordered Heisenberg chain can be treated only numerically: by means of MPS-based simulations, we are able to locate the existence of a transition between antiferromagnetic and string-ordered phase, through the study of order parameters. Finally, we discuss possible connections of our findings with many-body localization.
The global symmetry, either invertible or noninvertible, has been extensively studied in two-dimensional conformal field theories in recent years. When the theory is defined on a manifold with open boundaries, however, many interesting conformal boundary conditions will fully or partially break such global symmetry. In this work, we study the effect of symmetry-breaking boundaries or interfaces when the system is out of equilibrium. We show that the boundary or interface symmetry breaking can be detected by the time evolution of string order parameters, which are constructed from the symmetry operators that implement the symmetry transformations. While the string order parameters are independent of time if the symmetry is preserved over the whole system, they evolve in time in a universal way if the boundary or interface breaks the symmetry. More explicitly, in the presence of boundary or interface symmetry breaking, the string order parameters decay exponentially in time after a global quantum quench, and decay as a power law in time after a local quantum quench. We also generalize our study to the case when the string order parameters are defined in a subsystem, which are related to the full counting statistics. It is found there are also universal features in the time evolution of string order parameters in this case. We verify our field-theory results by studying the time evolution of these two different types of string order parameters in lattice models.
We study the anisotropic XY chain in a transverse field by introducing a global string order parameter and examining its equilibrium behavior and its quench dynamics from a fully polarized initial state. The corresponding string operator is defined as the product of local projected spin operators over the entire chain. The nonequilibrium order parameter is the long-time average of the logarithm of the string-operator expectation value per site after the quench. For the ground state, its field derivative is singular on the Ising critical lines; after the quench, the steady-state response also records the field boundaries and the anisotropy transition. We derive analytically the nonanalytic behavior of the order parameter at the quantum critical lines. Our results reveal a hidden nonlocal order underlying this paradigmatic model of conventional quantum phase transitions. They also connect measurable spin populations with equilibrium and dynamical topological structure.
In this paper we study an antiferromagnetic Heisenberg
S=2 quantum spin chain at
T=0 with both interaction and on-site anisotropy, $H={\ensuremath{\sum}}_{i}\frac{1}{2}{(S}_{i}^{+}{S}_{i+1}^{\ensuremath{-}}{+S}_{i}^{\ensuremath{-}}{S}_{i+1}^{+}{)+J}^{z}{S}_{i}^{z}{S}_{i+1}^{z}{+D(S}_{i}^{z}{)}^{2}$. While the phase diagram of this chain for
S=1 and the isotropic point for
S=2 are by now excellently understood, contradictory scenarios exist for the
S=2 anisotropic phase diagram, which imply completely different mechanisms of the emergence of the classical $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ limit, hence the importance of the question. One main scenario predicts the emergence of a cascade of phase transitions not seen in the
S=1 case, based on an analysis using the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Another scenario is in favor of an almost classical phase diagram for
S=2, from which the $S=\ensuremath{\infty}$ phase diagram evolves quite trivially; the
S=1 case then emerges as very special with its dominant quantum effects. Numerical studies have so far not been conclusive. Using the density matrix renormalization group, we study gaps and correlation functions in the anisotropic
S=2 chain. The question of the existence of hidden topological order in the anisotropic
S=2 chain is emphasized, as it distinguishes between the proposed scenarios. Careful extrapolation allows us to show that the topological order is zero in the thermodynamical limit in all disordered phases, in particular in the new phase interposed between the Haldane and large-
D phases. The decay of the hidden order is substantially faster than that of ${S}^{+}{S}^{\ensuremath{-}}$ correlations and excellently fitted by power laws typical of massless phases. This excludes the AKLT-model-based scenario in favor of an almost classical phase diagram for the
S=2 and thus higher spin chains.