# P12: Interaction quenches in closed one-dimensional systems

## Project Description

Ultracold gases in traps and optical lattices form isolated quantum systems which can still be controlled and manipulated externally ^{(1)}. These properties made fundamental questions of statistical physics accessible to experiments, for instance if and how a subsystem of an isolated quantum system prepared in a non-equilibrium state reaches a steady state. Furthermore one might wonder whether the steady state in a subsystem, or equivalently the steady-state expectation values of local observables, can be described by the canonical ensemble as introduced in

equilibrium statistical mechanics. Questions of this type spurred a large number of theoretical studies on the real-time dynamics of bosonic and fermionic model systems ^{(2)}. An often investigated protocol to achieve the non-equilibrium state is an (abrupt or gradual) interaction quench; the system is prepared in an eigenstate or thermal ensemble of an initial Hamiltonian while the propagation is performed with a final Hamiltonian with different interaction strength. In this project we focus on the relaxation dynamics and the (possible) steady states of one-dimensional (1d) correlated Fermi systems. Two issues are particular to these. (i) In equilibrium gapless interacting fermions are known to be strongly correlated even for weak two-particle interactions. The ground-state properties are described by the Luttinger liquid phenomenology characterized e.g. by power-law scaling of correlation functions ^{(3)}. The low-energy fixed point model of any Luttinger liquid is given by the Tomonaga-Luttinger model ^{(4)}. (ii) Several often investigated 1d systems are known to be exactly solvable by the Bethe ansatz ^{(3),(5)} or by mapping them to non-interacting ones ^{(3)}. They are characterized by an exhaustive set of integrals of motion restricting the time evolution.

Within this project we want to investigate two questions of fundamental interest.

Firstly, we plan to study to what extend the steady state of a microscopic 1d model after an interaction quench and the time evolution towards it can be understood in terms of that of the Tomonaga-Luttinger model. One has to distinguish two aspects. The first concerns the complete time evolution of selected observables. Can the one obtained within microscopic models using time-dependent density-matrix renormalization group (DMRG) be fitted by the one derived for the Tomonaga-Luttinger model by bosonization^{(3)}? For certain types of models and specific protocols a surprising agreement was found^{(6)-(8)}. We will extend the DMRG calculations to other than the mostly studied XXZ model and attempt to obtain a better understanding. The second addresses the more fundamental question whether the steady state and/or the asymptotic dynamics towards it is universal in the Luttinger liquid sense. Recently, a non-equilibrium Wilson-like renormalization group (RG) approach applicable to field-theoretical models was suggested ^{(9)-(11)}. Calculations using this approach indicate that terms which spoil Luttinger liquid universality and are RG irrelevant in equilibrium become relevant in the non-equilibrium time evolution. In contrast numerical approaches to microscopic models have found indications of Luttinger liquid universality ^{(12)-(14)}. Applying state of the art matrix-product state based DMRG (see below) to different types of 1d lattice models (not only the XXZ model mostly studied so far) as well as RG based approaches (Wilson-like RG and functional RG ^{(15)}) we aim at a more detailed understanding of the relation between field-theoretical models and microscopic ones in non-equilibrium. In equilibrium this is essentially understood ^{(3),(4)}. This way we hope to be able to conciliate the conflicting results. Possible solutions are: large time scales in microscopic models beyond the ones reached in the previous DMRG studies or weaknesses of the suggested RG approach.

The second fundamental question to be investigated concerns the possible thermalization of correlated 1d systems. A system is considered as thermalized if the steady state reduced density matrix of a subsystem is given by the canonical density matrix. It is generally believed that systems with many integrals of motion (``integrable'' models), that is the Bethe ansatz solvable ones as well as effectively non-interacting ones do not thermalize, while generic systems do so ^{(16),(2),(17)-(19)}. Analytical as well as numerical results supporting this view are still ambiguous. Systematically adding terms breaking ``integrability'' to ``integrable'' models (XXZ model, Hubbard model) we plan to investigate this question using DMRG.

To tackle the above questions numerically we employ a time dependent DMRG approach set up in matrix product states ^{(20)}. In order to optimize the numerical cost, we will make use of the fact that the bond dimension (governing the numerical effort) usually grows non-uniformly within the chain and flexibly adapt the bond dimensions across the chain independently according to this growth, combined with methods such as the folding trick ^{(21)} to reduce the correlations in the system. Furthermore, we will study infinite and finite systems and also utilize the speedups stemming from additive Abelian symmetries. To calculate the dynamics at finite temperature very efficiently we make use of the disentangler proposed recently in Ref. (12).

This project is linked to P8 in which correlated quantum wires coupled to leads are studied as well as to P13 in which the real-time dynamics of dots with time-dependent parameters are considered.

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(4) J. Sólyom, Adv. Phys. **28**, 209 (1979)

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* The One-dimensional Hubbard Model* (Cambridge University Press, Cambridge, 2010)

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(20) U. Schollwöck, Ann. of Phys. **326**, 96 (2011)

(21) M. C. Bañuls, M. B. Hastings, F. Verstraete and J.I. Cirac, Phys. Rev. Lett. **102**, 240603 (2009)