## Abstract

Loosely speaking, the concept of quantum typicality refers to the fact that a single pure state can imitate the full statistical ensemble. This fact has given rise to a rather simple but remarkably useful numerical approach to simulate the dynamics of quantum many-body systems, called *dynamical quantum typicality* (DQT). In this paper, we give a brief overview of selected applications of DQT, where particular emphasis is given to questions on transport and thermalization in low-dimensional lattice systems like chains or ladders of interacting spins or fermions. For these systems, we discuss that DQT provides an efficient means to obtain time-dependent equilibrium correlation functions for comparatively large Hilbert-space dimensions and long time scales, allowing the quantitative extraction of transport coefficients within the framework of, e. g., linear response theory (LRT). Furthermore, it is discussed that DQT can also be used to study the far-from-equilibrium dynamics resulting from sudden quench scenarios, where the initial state is a thermal Gibbs state of the pre-quench Hamiltonian. Eventually, we summarize a few combinations of DQT with other approaches such as numerical linked cluster expansions or projection operator techniques. In this way, we demonstrate the versatility of DQT.

## 1 Introduction

Unraveling the dynamics of isolated quantum many-body systems is a central objective of modern experimental and theoretical physics. On the one hand, new experimental platforms composed of cold atoms or trapped ions have opened the door to perform quantum simulations with a high amount of control over Hamiltonian parameters and initial conditions [1], [2]. On the other hand, there has been substantial progress from the theoretical side to understand (i) experimental observations and (ii) long-standing questions about the fundamentals of statistical mechanics [3], [4], [5], [6], [7]. One such question is how to reconcile the emergence of thermodynamic behavior with the unitary time evolution of isolated quantum systems, i. e., to explain whether and in which way an isolated system relaxes towards a stationary long-time state which agrees with the predictions from standard statistical mechanics. Another similarly intriguing question in this context is to explain the onset of conventional hydrodynamic transport, i. e., diffusion, from truly microscopic principles [8], [9], [10]. The numerical analysis of thermalization and transport in isolated quantum many-body systems is at the heart of this paper.

Generally, the theoretical analysis of quantum many-body dynamics is notoriously difficult. Given a quantum system

While the presence of strong interactions often prohibits any analytical solution, numerical studies of Eq. (2) are plagued by the exponential growth of the Hilbert space upon increasing the number of degrees of freedom. Moreover, since thermalization and transport can potentially be very slow processes, the necessity to study long time scales adds another layer of complexity.

Of course, for situations close to equilibrium, e. g., a system being weakly perturbed by an external force, linear response theory (LRT) provides a successful framework to describe the system's response in terms of dynamical correlation functions evaluated exactly at equilibrium [14]. However, analogous to Eqs. (1) and (2), the calculation of such time-dependent correlation functions for large system sizes and long time scales is a severe challenge in practice.

Despite these difficulties, significant progress has been made over the years thanks to the augmented availability of computational resources and the development of sophisticated numerical techniques. Especially for one-dimensional systems the time-dependent density matrix renormalization group (tDMRG), including related methods based on matrix product states, provides a powerful approach to dynamical properties in the thermodynamic limit (for reviews, see [15], [16]). However, due to the inevitable build-up of entanglement, this approach is limited in the time scales which can be reached in simulations.

In the present paper, the focus is on another useful numerical approach to the dynamics of quantum many-body systems, which is based on the concept of dynamical quantum typicality (DQT) [17], [18]. In a nutshell, DQT means that “the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time” [17]. In fact, the idea of using random vectors has a long and fruitful history [19], [20], [21], [22], [23], [24], [25], [26]. By virtue of an iterative forward propagation of these vectors in real or imaginary time, dependencies on time and temperature can be obtained. Since DQT can be implemented rather memory efficiently, it is possible to study dynamical properties of quantum many-body systems with Hilbert-space dimensions significantly larger compared to standard exact diagonalization (ED). Moreover, there are no conceptual limitations on the reachable time scales.

It is worth pointing out that DQT can not only be used to obtain time-dependent properties [27], [28], [29] or spectral functions [22], [30], [31], [32] but also static properties such as the density of states [33] or thermodynamic quantities [34], [35], [36], [37]. However, it is the aim of this paper to discuss the usefulness and versatility of DQT especially in the context of thermalization and transport.

This paper is structured as follows. In Sec. 2, we give a brief introduction to the concept of typicality and also elaborate on the differences between typicality and the eigenstate thermalization hypothesis (ETH). In Sec. 3, we discuss various applications of typicality to the dynamics of quantum many-body systems. Finally, we summarize and conclude in Sec. 4, where we also provide an outlook on further applications of DQT.

## 2 What is typicality?

Loosely speaking, the notion of typicality means that even a single pure quantum state can imitate the full statistical ensemble, or, more precisely, expectation values of typical pure states are close to the expectation value of the statistical ensemble [20], [23], [24], [25], [26]. While typicality has been put forward as an important insight to explain the emergence of thermodynamic behavior (see e. g., Ref. [23] for an overview), let us here focus on the practical consequences of typicality. In particular, let us consider the, e. g., canonical equilibrium expectation value *A* defined as

where

where we have introduced the abbreviation *d* according to the unitary invariant Haar measure [17], i. e.,

where the coefficients *A* is a local operator (or a low-degree polynomial in system size), which applies to all examples discussed in this paper. For more details on error bounds see, e. g., Refs. [18], [36]. For empirical estimates, see, e. g., Ref. [41]. Thus, *L*, leads to an exponential improvement of the accuracy (the higher the temperature, the faster), and Eq. (4) becomes exact in the thermodynamic limit

The typicality approximation (4) has proven to be very useful to calculate equilibrium quantities of quantum many-body systems such as the specific heat, entropy, or magnetic susceptibility [34], [35], [36], [37], [41]. For the purpose of this review, however, it is most important to note that typicality is not just restricted to equilibrium properties, but also extends to the real-time dynamics of quantum expectation values [17], [27], [28], [29], [42], [43], [44]. This dynamical version of typicality forms the basis of the numerical approach to time-dependent correlation functions and out-of-equilibrium dynamics more generally, which is discussed in Sec. 3.

Let us briefly discuss the relationship between typicality and the ETH [45], [46], [47]. The ETH states that the expectation values of local observables evaluated within individual eigenstates

While this fact (i. e., pure states can approximate ensemble expectation values) appears similar to our discussion of typicality in the context of Eq. (4), let us stress that typicality and ETH are two distinct concepts. On the one hand, while the ETH is assumed to hold for few-body operators and nonintegrable models [5], [48], [49], [50], [51], [52], [53], [54], [55], a rigorous proof for its validity is still absent. On the other hand, typicality is no assumption and essentially requires the largeness of the effective Hilbert-space dimension. This difference becomes particularly clear from the following point of view: since the distribution of the

Since typicality is independent of the validity of the ETH, it can be used in integrable or many-body localized models, where the ETH is expected to be violated [56], [57], [58], [59]. As a side remark, typicality can also be used to test the ETH [60].

Eventually, let us emphasize that the choice of the specific basis

## 3 DQT as a numerical tool

We now discuss the use of DQT as a numerical method. To begin with, we discuss in Sec. 3.1 the iterative forward propagation of pure states in large Hilbert spaces. Afterward, as a first application, we demonstrate in Sec. 3.2 how typicality can be used to study the (local) density of states. In Sec. 3.3, we then show how DQT can be used to evaluate equilibrium correlation functions within the framework of LRT. Sec. 3.4 is concerned with the out-of-equilibrium dynamics in certain quantum-quench scenarios. Eventually, in Sec. 3.5, we discuss how DQT can be combined with other approaches such as numerical linked cluster expansions or projection operator techniques.

### 3.1 Pure-state propagation

From a numerical point of view, a central advantage of the typicality approach comes from the fact that one can work with pure states instead of having to deal with full density matrices. This fact leads to a substantial reduction of the memory requirements, since it is possible to efficiently generate time and temperature dependencies of pure states. (Note that, while it is always possible to purify a density matrix, the DQT approach in contrast does not require to square the Hilbert-space dimension [62].)

Specifically, let us consider the pure state *t* can be subdivided into a product of consecutive steps,

where

The four auxiliary states

and the error of the approximation (8) scales as

Apart from RK4, other common and more sophisticated methods to propagate pure states without diagonalization are, e. g., Trotter decompositions [34], [63], Krylov subspace techniques [64], as well as Chebyshev polynomial expansions [65], [66], [67], [68], [69]. A unifying property of all these methods and RK4 is the necessity to calculate matrix-vector products, i. e., to evaluate the action of the Hamiltonian

### 3.2 Calculating the (local) density of states

As a first useful application, let us describe how pure states, in combination with a forward propagation in real time, can be used to evaluate the (local) density of states [33]. To begin with, we note that the density of states of some Hamiltonian

where we have used the definition of the *δ* function. In the spirit of Eq. (4), we can approximate the trace in Eq. (11) by a scalar product with a randomly drawn pure state

such that Eq. (11) can be approximated as

where

In fact, the relation (13) turns out to be useful for any arbitrary pure state

where

Relying on the forward propagation of pure states discussed in Sec. 3.1, it is thus possible to access

As an example, let us consider the spin-

where *L* is the number of lattice sites, *z*-direction. In Figure 1, the density of states

### 3.3 Time-dependent equilibrium correlation functions

Let us now turn to quantum many-body dynamics within the framework of LRT. Within LRT, central quantities of interest are time-dependent correlation functions *A* and *B* evaluated in equilibrium,

where

where we have introduced two auxiliary pure states,

and

In the context of transport, an interesting quantity is the current autocorrelation function *j* is the current operator. Note that the Fourier transform of

For concreteness, let us (again) consider the XXZ chain (16). In this case, the spin current operator *j* takes on the form [73],

In Refs. [29], [74], *β* and *L*, even in the semi-logarithmic plot used. (For further numerical data of

In addition to the XXZ chain, DQT has been used to study

Another interesting quantity in the context of transport are the spatio-temporal correlation functions

While a calculation of

where *c* is chosen such that

Thus, it is possible to calculate

As an example, the equal-site spin-spin correlation function

As another example, the full time-space profile

A very similar example is shown in Figure 5, where the spatio-temporal correlations for spin and energy densities are depicted at fixed times. Yet, the model is a spin-

where

In addition, DQT has been used to obtain spatio-temporal correlation functions

### 3.4 Applications to far-from-equilibrium dynamics

Nonequilibrium scenarios in isolated quantum systems can be induced via explicitly time-dependent Hamiltonians or, e. g., by means of quantum quenches [83]. For instance, the system can be initially in an eigenstate of some Hamiltonian

Here, we discuss an alternative type of quench, where the system starts in a Gibbs state with respect to (w.r.t.) some initial Hamiltonian

We then consider a quantum quench, where

The post-quench Hamiltonian can, for instance, be created by adding or removing a static (weak or strong) force of strength ϵ to the initial Hamiltonian, i. e., *A* is conjugated to the force [13], [58], [84], [85]. The resulting expectation value dynamics of, e. g., the operator *A* is given by

and its evaluation in principle requires complete diagonalization of both

which mimics the density matrix (26), and the reference state

It is worth pointing out that the (simple) quench protocol above can be modified by additional changes of the Hamiltonian in time. A static force switched on at time

In Figure 7, the nonequilibrium dynamics

### 3.5 DQT and its extensions

In addition to the direct applications discussed above, DQT also is a useful tool to “boost” other (numerical or analytical) techniques, which can profit from accurate data for large system sizes. Two examples of such techniques, which have recently been combined with DQT, are numerical linked-cluster expansions (NLCE) and projection operator techniques.

#### 3.5.1 NLCE

The key feature used in NLCE is the fact that the per-site value of an extensive quantity on an infinite lattice can be expanded in terms of its respective weights on all linked (sub-)clusters that can be embedded in the lattice. While NLCE is described in detail and generality in [86], [87], this section focuses on practical aspects of NLCE, particularly on its combination with DQT to calculate, e. g., current-current correlation functions of one-dimensional systems. The starting point of a corresponding NLCE is the expression

where *c* with multiplicity

where the weights of all embedded clusters *s* are subtracted from *c*.

Since the maximum treatable cluster size is naturally limited by the available computational resources, the sum in Eq. (31) has to be truncated to a maximum size

As demonstrated in Ref. [88], this rather simple formula can have a better convergence towards the thermodynamic limit than a standard finite-size scaling at effectively equal computational cost.

As shown in Figure 8, the current autocorrelation function for the spin-1/2 Heisenberg chain directly obtained by DQT for a large system with

When studying thermodynamic quantities, for which the NLCE was originally introduced, using larger cluster sizes similarly improves the convergence of the expansion down to lower temperatures [87], [88], [89], [90], [91]. Either way, it is thus desirable to access cluster sizes as large as possible and DQT can be used to evaluate the contributions of clusters beyond the range of ED. Since the difference in Eq. (33) could be sensitive to small statistical errors, it might be recommended to average the DQT results over multiple random pure states, in particular in higher dimensions, where the NLCE expression is not just a single difference.

#### 3.5.2 Projection operator techniques

The DQT approach can also be used in the context of projection operator techniques, e. g., the so-called time-convolutionless (TCL) projection operator method. These techniques can be applied to situations where a closed quantum system with Hamiltonian *λ*, such that the total Hamiltonian takes on the form

In this setting, one then chooses a suitable projection on the relevant degrees of freedom to obtain a systematic perturbation expansion for the reduced dynamics. We refer to [92], [93], [94], [95] for a detailed description of the TCL method and do not discuss it here in full generality.

Choosing a simple projection onto *A* only and considering the specific initial conditions

where the second-order damping rate

and the index

The calculation of Eq. (36) can be conveniently done using typical states and becomes especially simple in the case where the observable of interest is conserved under the unperturbed Hamiltonian, i. e.,

with the auxiliary states

In [92], the quality of the second-order prediction (35) was numerically studied for the example of the current autocorrelation functions

## 4 Conclusion

To summarize, we have discussed several applications of DQT and its usefulness as a numerical approach to the real-time dynamics of quantum many-body systems. The main idea of this typicality approach is to approximate ensemble expectation values via single pure states which are randomly drawn from a high-dimensional Hilbert space. In particular, time (temperature) dependencies of expectation values can be obtained by iteratively solving the Schrödinger equation in real (imaginary) time, e. g., by means of Runge-Kutta schemes or more sophisticated methods.

First, we have described that DQT can be used to study the (local) density of states as well as equilibrium correlation functions for long time scales and comparatively large system sizes beyond the range of standard ED. Especially in the context of transport, the calculation of current autocorrelations and density-density correlations by means of DQT is possible. Furthermore, we have outlined that DQT is suitable to investigate also the far-from-equilibrium dynamics resulting from certain quench protocols. For instance, an initial Gibbs state is properly imitated by a typical pure state and nonequilibrium conditions are induced by removing or adding an external force. Eventually, we have discussed that DQT can additionally be combined with other approaches. As one example, we have shown that the convergence of NLCE can be improved by evaluating the contributions of larger clusters by means of DQT. As another example, we have discussed that DQT allows to compute memory kernels which arise in projection operator methods such as the TCL technique.

While this paper has illustrated the usefulness of DQT for selected applications in the context of transport and thermalization, we should stress that there certainly are other applications of DQT which have not been mentioned by us. One such application, as done in, e. g., [96], is the spreading of quantum information measured by so-called out-of-time-ordered correlators (OTOCs) of the form [97],

where the operators *A* and *B* are, for instance, local magnetization densities

In conclusion, the concept of DQT offers a rather simple yet remarkably useful approach to study the real-time dynamics of quantum many-body systems. It is our hope that the examples discussed in this paper motivate its application in other areas as well.

**Funding source: **Deutsche Forschungsgemeinschaft

**Award Identifier / Grant number: **355031190397067869

## Acknowledgments

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) – Grants No. 397067869 (STE 2243/3-1), No. 355031190 – within the DFG Research Unit FOR 2692.

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**Received:**2020-01-09

**Accepted:**2020-03-11

**Published Online:**2020-04-27

**Published in Print:**2020-05-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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