Relaxation timescales and prethermalisation in d-dimensional long-range quantum spin models

Relaxation timescales and prethermalisation in -dimensional long-range quantum spin models


We report analytic results for the correlation functions of long-range quantum Ising models in arbitrary dimension. In particular, we focus on the long-time evolution and the relevant timescales on which correlations relax to their equilibrium values. By deriving upper bounds on the correlation functions in the large-system limit, we prove that a wide separation of timescales, accompanied by a pronounced prethermalisation plateau, occurs for sufficiently long-ranged interactions.

Empirically it is well established that, after a sufficiently long time, most physical many-body systems, whether isolated or coupled to an environment, will equilibrate. In many cases the equilibrium is well described by a Gibbs state, and this observation is at the basis of equilibrium statistical mechanics. An understanding of the microscopic process leading to thermalisation is, however, still incomplete. Recent experiments with cold atoms, ions, and molecules Kinoshita et al. (2006); Hofferberth et al. (2007); Struck et al. (2011); Islam et al. (2011); Trotzky et al. (2012); Britton et al. (2012); Islam et al. (2013); Yan et al. (2013) have sparked a revival of interest in questions related to the foundations of equilibrium statistical mechanics. Substantial progress, often based on typicality techniques, has been made on the theoretical side in the past few years. Results in various physical settings have been reported, proving that equilibration and/or thermalisation takes place for typical quantum systems of sufficient size Reimann (2008); Linden et al. (2009); Goldstein et al. (2010); Reimann (2010); Reimann and Kastner (2012); Ududec et al. (2013).

While these results establish that equilibration and/or thermalisation will happen eventually, the time scale of such a relaxation process remains unspecified. Only very recently has it become apparent that typicality techniques can also be applied for analysing the time scales of equilibration. The results of these efforts, while pioneering a promising approach, are not yet fully satisfactory, as they either over- Short and Farrelly (2012) or—in a different setting—underestimate Goldstein et al. (2015) the timescales by many orders of magnitude. These results indicate that physically realistic models or observables are not “typical” in the mathematical sense.

In this paper we approach the problem of equilibration time scales from a different angle, reporting the results of a model study. This work is a continuation and extension of a previous paper van den Worm et al. (2013) where exact, analytical expressions for equal-time two-point correlation functions have been computed for long-range interacting Ising models in a longitudinal magnetic field. While these results are very general and exact, the long-time asymptotics relevant for the relaxation to equilibrium is not at all obvious from the analytical expressions. Motivated by recent ion-trap experiments where the Ising model with long-range interactions can be realised, we have derived in van den Worm et al. (2013) upper bounds on the time-evolution of the various spin–spin correlation functions of the two-dimensional long-range Ising model on a triangular lattice. For the long-range Ising model on this specific lattice, the asymptotic long-time behaviour can be read off easily from the upper bounds. In the present paper we report generalisations of these upper bounds to arbitrary regular lattices and dimensionality.

We consider long-ranged coupling constants decaying like a power law with the distance between lattice sites and . The exponent in some sense quantifies the range of the interactions, from infinite range in the case to nearest-neighbour interactions in the limit . The upper bounds derived in this paper are stretched or compressed exponentials in time for all values of . On a -dimensional lattice and for , we find that some of the spin–spin correlation functions relax to their equilibrium values in a two-step process, governed by two widely separate time scales, while single-spin expectation values relax already on the faster of the two time scales. This kind of behaviour, characterised by a long-lived quasi-stationary state in which only some of the expectation values have already relaxed to their equilibrium value goes under the name of “prethermalisation” and has been discussed extensively in the past few years Berges et al. (2004); Moeckel and Kehrein (2008); Eckstein et al. (2009); Gring et al. (2012); Marino and Silva (2012); Smith et al. (2013); Gong and Duan (2013); Marcuzzi et al. (2013).

The stretched or compressed exponentials that upper-bound the spin–spin correlation functions do not only depend on the exponent , but they do so in a nonanalytic, transition-like manner: the long-time asymptotic behaviour of the spin–spin correlation functions switches from one kind of behaviour to a different functional form at the values and . The first of these threshold values was already discussed in van den Worm et al. (2013) for the triangular lattice in two dimensions, and it is related to the occurrence of widely separated time scales and prethermalisation. The second threshold value, at , has not been described before, as it becomes relevant (i.e., occurs at a positive -value) only for lattice dimensions .

I Long-range Ising model

Consider a lattice consisting of sites, to each of which is assigned a spin- degree of freedom. Each spin is modelled by a Hilbert space , and the composite system is described by the Ising-type Hamiltonian


on the tensor product Hilbert space . The parameter denotes the magnitude of a homogeneous external magnetic field in the direction, and is the -Pauli operator acting on the th component of the tensor product space . At this point the coupling strengths between lattice sites are arbitrary, but we will specialise to power law-decaying long-range interactions at a later stage.

As initial states we choose density operators that are diagonal matrices in the tensor-product eigenbasis,


where denotes the identity operator on . The indices , , , in (2) are summed over the lattice sites. This choice of has been exploited previously Emch (1966); Kastner (2011, 2012); van den Worm et al. (2013), as it leads to particularly simple calculations and results, although generalisations are possible. Starting from , exact analytic expressions have been reported in van den Worm et al. (2013) for the time-evolution of the various spin–spin correlation functions, e.g.




where we have set for simplicity. Other correlation functions either behave similarly (like ), or simpler (like ), or are constant in time (like ); see van den Worm et al. (2013) for details. In the following we restrict the presentation to the -correlation function given in (3), but similar techniques can be applied to other correlation functions. Upper bounds on one-point functions like have been reported in the Supplemental Material accompanying Ref. Bachelard and Kastner (2013).

Ii Upper bound on the correlations

The expressions in (4) are quasi-periodic in , and it is therefore precluded that converges in the long-time limit for any finite lattice. Only in the thermodynamic limit of an infinite number of lattice sites do we have a chance of observing convergence towards an equilibrium value. To derive such a result, we consider regular -dimensional lattices. Without loss of generality we consider the lattice constants normalised such that there is on average one lattice site per unit (hyper)volume in the limit of large lattice size. We choose coupling constants decaying like a power-law with the Euclidean distance between lattice sites and ,


Without loss of generality we set . Under these conditions and in the limit of large lattice size, we obtain the bounds


with positive constants and as defined in (23). The proof of the inequalities (6a) and (6b) is postponed to Sec. III.

Figure 1: Plots of the bound on the normalised correlator as a function of time. The examples are for lattice dimension , and the -values are chosen such that they furnish examples for the three different regimes of relaxation behaviour, as discussed in the text.

Depending on the value of , the bounds (6a) and (6b) decay like stretched or compressed exponentials. By numerically evaluating the exact expressions (4), we find that the functional form of the bound agrees well, although the numerical constants in the bound overestimate, as expected, the exact values. From Eqs. (6a) and (6b) one can read off that there are three different regimes of -values, each with a different relaxation or scaling behaviour:

: and both decay like a Gaussian in time, and both do so on time scales that are -dependent. The two time scales of relaxation are widely separated, with decaying much slower than . The form of the resulting upper bound on is shown in Fig. 1a. This regime occurs for positive only in lattice dimensions .

: Again, relaxation takes place in a two-step process with widely separated time scales. The fast process described by still decays like a Gaussian in time, on a time scale that is -dependent. The slow time scale corresponding to is independent of the system size, with a decay in the form of a compressed exponential. The form of the resulting upper bound on is shown in Fig. 1b.

Figure 2: For fixed instances of time, the bound is shown as a function of the exponent . The example is for dimension and a lattice of sites. The different shaded regions correspond to the three ranges of -values discussed in the text.
Figure 3: Contour plot of the bound as a function of and for a three-dimensional lattice of sites. The dashed lines separate the three regions of -values discussed in the text.

: Both terms and decay to zero like stretched or compressed exponentials. Relaxation takes place in a single step, as both relevant time scales are very similar and independent of . The form of the resulting upper bound on is shown in Fig. 1c.

Figs. 2 and 3 show further graphical representations of the bound, highlighting in particular the qualitative changes that occur upon variation of the exponent .

On the basis of these results, physical properties of the model—including dephasing dynamics, prethermalisation, and others—can be discussed similarly to the two-dimensional case reported in van den Worm et al. (2013), and the reader is referred to this reference for details.

Iii Proof of Eqs. (6a) and (6b)

The starting point for the derivation is the exact expression (4), where is given as a product (over all lattice sites) of cosine terms. Since , we can upper bound the absolute value of this quantity by a product over a subset of lattice sites,


where we have defined


This subset is chosen such that, for all , we can make use of the inequality


valid for all , to write

Figure 4: Sketch of the regions and and of the chosen coordinate system as used for the proof in Sec. III.

We restrict the -summation even further by excluding the hyperslab sketched in Fig. 4 1. The occurrence of Euclidean distances in the couplings and then suggests to parametrise the lattice sites by hyperspherical coordinates,


with the origin of the coordinate system placed at lattice site and the -axis chosen along the line connecting and . The couplings can then be written as


where denotes the distance between sites and . It is then convenient to further restrict the -summation in (10) by excluding the hyperslab sketched in Fig. 4. Exploiting also the reflection symmetry of the problem, we arrive at the bound


with the -summation restricted to the lattice sites in the half plane


For large lattices, we can bound the sum in (13) by an integral,


where the prefactor


originates from the integrations over .

The lower limit of the -integration still needs to be determined such that the region is excluded. I.e., we need to determine such that


for all . We are interested in the long-time asymptotic behaviour, and in this limit large values of are required to satisfy the above inequality. Hence we can assume that is much larger than and expand


to leading order in the small parameter , yielding


Inserting these asymptotic expressions into (18), we obtain


valid for sufficiently large .

For similar reasons we can insert the expansions (19a) and (19b) into the integrand of (15). The integrations become elementary in this case, yielding


in the limit of large and . Interestingly, the bound in (21a) is independent of the distance between the lattice sites.

Depending on the sign of the exponents and in the -terms, either the first or the second term in the square brackets of (21a) and (21b) will give the dominant contribution in the limit of large system size . As a result, the asymptotic behaviour of the bounds is different for different ranges of ,


Inserting these expressions into the inequality (10) and defining the positive constants


the main results (6a) and (6b) of the paper follow.


  1. This exclusion is not necessary, but it simplifies the calculation.


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