From Linear to Nonlinear Responses of Thermal Pure Quantum States

Abstract

We propose a self-validating scheme to calculate the unbiased responses of quantum many-body systems to external fields of arbibraty strength at any temperature. By switching on a specified field to a thermal pure quantum state of an isolated system, and tracking its time evolution, one can observe an intrinsic thermalization process driven solely by many-body effects. The transient behavior before thermalization contains rich information on excited states, giving the linear and nonlinear response functions at all frequencies. We uncover the necessary conditions to clarify the applicability of this formalism, supported by a proper definition of the nonlinear response function. The accuracy of the protocol is guaranteed by a rigorous upper bound of error exponentially decreasing with system size, and is well implemented in the simple ferromagnetic Heisenberg chain, whose response at high fields exhibits a nonlinear band deformation. We further extract the characteristic features of excitation of the spin-1/2 kagome antiferromagnet; the wavenumber-insensitive linear responses from the possible spin liquid ground state, and the significantly broad nonlinear peaks which should be generated from numerous collisions of quasi-particles, that are beyond the perturbative description.

Finite-temperature transport in one-dimensional quantum lattice models

“…( 138) is a mathematically rigorous statement if |ψ is essentially drawn at random from a sufficiently large Hilbert space (Bartsch and Gemmer, 2009;Reimann, 2007). In fact, the idea of using random states |ψ has a long history (Alben et al, 1975;De Raedt and De Vries, 1989;Jaklič and Prelovšek, 1994) and is at the basis of various numerical approaches to the density of states (Hams and De Raedt, 2000), thermodynamic quantities (De Vries and De Raedt, 1993;Shimizu, 2012, 2013;Wietek et al, 2019), equilibrium correlation functions (Elsayed and Fine, 2013;Iitaka and Ebisuzaki, 2003;Rousochatzakis et al, 2019;Steinigeweg et al, 2014aSteinigeweg et al, , 2016b, non-equilibrium processes (Endo et al, 2018;Monnai and Sugita, 2014;Richter et al, 2019c), as well as ETH (Steinigeweg et al, 2014c). In this review, we focus on the case of equilibrium correlation functions.…”

Thermally enhanced Majorana-mediated spin transport in the Kitaev model

“…Thus, we use the TPQ state method [30,31], where local quantities are efficiently evaluated without the trace calculations [36][37][38][39][40][41][42]. An important point is that this numerical method takes several energy scales into account on equal footing, and thereby has been successfully used in several systems such as the Heisenberg model on frustrated lattices [30][31][32][43][44][45][46] and the Kitaev models [47][48][49][50][51][52][53].…”

“…where |Ψ T is the TPQ state at the temperature T and |Ψ T (t) = U (t)|Ψ T . The time-evolution of the physical quantities can be evaluated by the time-evolution of the TPQ state [32].…”

“…To answer this question, we deal with the Kitaev model with edges and consider the spin transport at finite tem-peratures. By means of the time-dependent thermal pure quantum (TPQ) state method [30][31][32], we examine the dynamics of the system after the magnetic pulse is introduced at one of the edges. Then, we discuss how thermal fluctuations affect the Majorana-mediated spin transport.…”

Quantum hydrodynamics from local thermal pure states

“…We introduce a special class of random pure quantum states describing systems in local thermal equilibrium, which we refer to as local thermal pure quantum ( TPQ) states. This is a generalization of thermal pure quantum (TPQ) states [52][53][54][55][56][57][58][59] randomly sampled from the Hilbert space to reproduce thermal equilibrium behaviors. We show that TPQ states have a numerical advantage in computing hydrodynamic expectation values [55,56,58,59] and give the equivalent results with the conventional method based on the LG ensemble in the large fluid-cell limit.…”