The topic of this article are transport properties of many-body quantum chains accompanying DMRG-type simulations of the Bose–Hubbard model. To set the stage, we first provide a very brief introduction to many-body quantum theory and tensor network approximations. Transport properties are studied via dynamical density correlation functions, in line with linear response theory. We observe diffusive behavior at “infinite temperature” T → ∞ . Finally, we mention other approaches to study transport, e.g., explicitly imposing a temperature gradient via thermal reservoirs at the boundary.
The topic of this article are transport properties of many-body quantum chains accompanying DMRG-type simulations of the Bose–Hubbard model. To set the stage, we first provide a very brief introduction to many-body quantum theory and tensor network approximations. Transport properties are studied via dynamical density correlation functions, in line with linear response theory. We observe diffusive behavior at “infinite temperature” T → ∞ . Finally, we mention other approaches to study transport, e.g., explicitly imposing a temperature gradient via thermal reservoirs at the boundary.
One of the most straightforward ways to study thermal properties beyond linear response is to monitor the relaxation of an arbitrarily large left-right temperature gradient ${T}_{L}\ensuremath{-}{T}_{R}$. In one-dimensional systems which support ballistic thermal transport, the local energy currents $\ensuremath{\langle}j(t)\ensuremath{\rangle}$ acquire a nonzero value at long times, and it was recently investigated whether or not this steady state fulfills a simple additive relation $\ensuremath{\langle}j(t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\rangle}=f({T}_{L})\ensuremath{-}f({T}_{R})$ in integrable models. In this paper, we probe the nonequilibrium dynamics of the Hubbard chain using density matrix renormalization group (DMRG) numerics. We show that the above form provides an effective description of thermal transport in this model; violations are below the finite-time accuracy of the DMRG. As a second setup, we study how an initially equilibrated system radiates into different nonthermal states (such as the vacuum).
In this thesis, quantum many-body systems are studied by means of tensor network methods. After introducing the quantum many-body problem and discussing paradigmatic models as well as established methods for their solution, the focus is set on a practical approach to tensor network methods, always keeping in mind the realistic numerical applicability of the tools devised. In order to achieve this objective, we focus on loop-free network geometries, which allow to boost the computational performance to the greatest possible extent. The developed algorithms are described and benchmarked in detail, starting from their elementary constituents, namely the multidimensional arrays called tensors which are at the heart of any tensor network algorithm. We then apply the developed simulation techniques to a diverse collection of quantum many-body models of current research interest. We are interested, in particular, in a characterization of the ground-state properties of these models, but we also investigate problems beyond the zero-temperature equilibrium setting, such as time evolution or finite-temperature behavior. Specifically, novel findings for the phase diagrams of the disordered Bose-Hubbard chain and of the Heisenberg chain with additional biquadratic interactions are presented, as well as compelling new evidence for the existence of fractional quantum Hall states in the interacting Harper-Hofstadter model.
Technological and scientific advances have given rise to an era in which coherent quantum-mechanical phenomena can be probed and experimentally-realised over unprecedented timescales in condensed matter physics. In turn, scientific interest in non-equilibrium dynamics and irreversibility signatures of thermodynamics has taken place in recent decades, particularly in relation to cold-atom platforms and thermoelectric devices. In this PhD thesis I summarise some of the most important results obtained over the duration of my PhD, on the topic of thermodynamics involving interacting many-body quantum systems. The topics of discussion encompass three main themes: spin/particle transport in non-integrable systems, explorations in eigenstate thermalisation and finite-temperature transport in autonomous thermal machines. By questioning the effect of local integrability-breaking perturbations, I describe the subtle effects that may rise from conservation laws and their connection to linear response transport using microscopic models. Then, I describe thermalisation in the context of the eigenstate thermalisation hypothesis and its consequence to multipartite entanglement and high-order correlation functions. Finally, motivated by the necessity of describing quantum thermal machines in the finite-temperature regime, I introduce a novel tensor-network based method to investigate thermodynamic properties of autonomous machines that employ interacting many-body systems as working media.
We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction U. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
Thermalization in quantum many-body systems is a central problem of quantum mechanics. Despite the unitary nature of quantum evolution, isolated many-body systems often appear to relax toward states that are locally described by statistical ensembles. However, phenomena such as integrability and quantum many-body scars demonstrate that thermalization is not universal and can fail in unexpected ways, leading to nonergodic dynamics and long-lived nonequilibrium behavior. In this thesis, the problem is addressed from multiple complementary perspectives, including thermalization, integrability breaking, constrained dynamics, and quantum many-body scars, together with the development of new tensor-network methods for nonequilibrium quantum systems and quantum metrological applications. First, we study energy-filtering protocols for the preparation of thermal states, analyzing the possibility of simulating them classically through the behavior of entanglement and correlations. The effects of weak integrability breaking are then investigated in models relevant to Rydberg-atom experiments, leading to the identification of novel scattering mechanisms and the introduction of subsystem variance as a sensitive probe of weak integrability breaking. A part of the thesis is devoted to quantum many-body scars and their consequences for relaxation and transport phenomena. Their dynamical signatures are studied in both closed and open quantum systems, revealing new mechanisms that sustain long-lived nonequilibrium behavior. The thermalization processes arising during the quasi-adiabatic preparation of the ground state in the trapped Bose-Hubbard model are also examined, showing how transport constraints can stabilize low-entropy states under experimentally realistic conditions. Finally, a novel tensor-network approach based on Tree Tensor Operators is introduced for the efficient computation of the Quantum Fisher Information of generic mixed states. Applicable to both closed and open quantum systems, the method provides a promising numerical tool for quantum sensing and metrological applications.