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Register a new userMany-body localisation implies that eigenvectors are matrix-product states
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The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties - the absence of a group velocity and transport - with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its many-body eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.
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We compare accuracy of two prime time evolution algorithms involving Matrix Product States - tDMRG (time-dependent density matrix renormalization group) and TDVP (time-dependent variational principle). The latter is supposed to be superior within a limited and fixed auxiliary space dimension. Surprisingly, we find that the performance of algorithms depends on the model considered. In particular, many-body localized systems as well as the crossover regions between localized and delocalized phases are better described by tDMRG, contrary to the delocalized regime where TDVP indeed outperforms tDMRG in terms of accuracy and reliability. As an example, we study many-body localization transition in a large size Heisenberg chain. We discuss drawbacks of previous estimates [Phys. Rev. B 98, 174202 (2018)] of the critical disorder strength for large systems.
Matrix Product States form the basis of powerful simulation methods for ground state problems in one dimension. Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the area law. In this work, we establish the mixed state analogue of this result: We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by Matrix Product Density Operators (MPDOs). In combination with results establishing area laws for thermal states, this helps to put the use of MPDOs in the simulation of thermal states on a formal footing.
We introduce reinforcement learning (RL) formulations of the problem of finding the ground state of a many-body quantum mechanical model defined on a lattice. We show that stoquastic Hamiltonians - those without a sign problem - have a natural decomposition into stochastic dynamics and a potential representing a reward function. The mapping to RL is developed for both continuous and discrete time, based on a generalized Feynman-Kac formula in the former case and a stochastic representation of the Schrodinger equation in the latter. We discuss the application of this mapping to the neural representation of quantum states, spelling out the advantages over approaches based on direct representation of the wavefunction of the system.
Any quantum process is represented by a sequence of quantum channels. We consider ergodic processes, obtained by sampling channel valued random variables along the trajectories of an ergodic dynamical system. Examples of such processes include the effect of repeated application of a fixed quantum channel perturbed by arbitrary correlated noise, or a sequence of channels drawn independently and identically from an ensemble. Under natural irreducibility conditions, we obtain a theorem showing that the state of a system evolving by such a process converges exponentially fast to an ergodic sequence of states depending on the process, but independent of the initial state of the system. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk. Further applications and physical implications of our results are discussed in the companion paper [11].
The phenomenon of many-body localised (MBL) systems has attracted significant interest in recent years, for its intriguing implications from a perspective of both condensed-matter and statistical physics: they are insulators even at non-zero temperature and fail to thermalise, violating expectations from quantum statistical mechanics. What is more, recent seminal experimental developments with ultra-cold atoms in optical lattices constituting analog quantum simulators have pushed many-body localised systems into the realm of physical systems that can be measured with high accuracy. In this work, we introduce experimentally accessible witnesses that directly probe distinct features of MBL, distinguishing it from its Anderson counterpart. We insist on building our toolbox from techniques available in the laboratory, including on-site addressing, super-lattices, and time-of-flight measurements, identifying witnesses based on fluctuations, density-density correlators, densities, and entanglement. We build upon the theory of out of equilibrium quantum systems, in conjunction with tensor network and exact simulations, showing the effectiveness of the tools for realistic models.
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