Condensed Matter Theory

Abstract:

SciPost Journals Publication Detail SciPost Phys. 19, 086 (2025) Long-time divergences in the nonlinear response of gapped one-dimensional many-particle systems

Abstract:

We investigate the lifted TASEP and its generalization, the GL-TASEP. We analyze the spectral properties of the transition matrix of the lifted TASEP using its Bethe ansatz solution, and use them to determine the scaling of the relaxation time (the inverse spectral gap) with particle number. The observed scaling with particle number was previously found to disagree with Monte Carlo simulations of the equilibrium autocorrelation times of the structure factor and of other large-scale density correlators for a particular value of the pullback \alpha_{\rm crit} . We explain this discrepancy. We then construct the continuum limit of the lifted TASEP, which remains integrable, and connect it to the event-chain Monte Carlo algorithm. The critical pullback \alpha_{\rm crit} then equals the system pressure. We generalize the lifted TASEP to a large class of nearest-neighbour interactions, which lead to stationary states characterized by non-trivial Boltzmann distributions. By tuning the pullback parameter in the GL-TASEP to a particular value we can again achieve a polynomial speedup in the time required to converge to the steady state. We comment on the possible integrability of the GL-TASEP.

Abstract:

Half a century ago, two theoretical papers were published that together sparked major new directions — conceptual, mathematical and practically applicable — in several previously disparate fields of science. In this Comment, the authors of one of those papers expose key aspects of the thinking behind them, their implementations and implications, along with sketches of several subsequent and consequential developments.

Abstract:

We consider the universal aspects of two problems: (i) the singular value structure of a product $M_t=m_t{m}_{t-1}\dots m_1$ of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices $m_i$ , representing time steps in the evolution, but the $m_i$ are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.