Semi-classical theory for quantum quenches in the O(3) non-linear sigma model

Semi-classical theory for quantum quenches in the O(3) non-linear sigma model

Abstract

We use the semi-classical approach to study the non-equilibrium dynamics of the O(3) non-linear sigma model. For a class of quenches defined in the text, we obtain the order-parameter dynamical correlator in the thermodynamic limit. In particular we predict quench-dependent relaxation times and correlation lengths. The approach developed for the O(3) non-linear sigma model can also be applied to the transverse field Ising chain, where the semi-classical results can be directly compared to both the exact and the numerical ones, revealing the limits of the method.

1 Introduction

The out-of-equilibrium physics of many-body systems has attracted a lot of interest in recent years, not least because of the experimental realizations of ultracold atomic gases in optical lattices [1, 2, 3, 4, 5, 6]. These experiments have observed the dynamics of many-body systems on a long time scale after a quantum quench, finding essentially unitary time-evolution. In three-dimensional systems fast relaxation towards a thermal steady state has been observed. On the contrary, in quasi-one-dimensional systems the relaxation process is normally much slower and leads to a peculiar non-thermal stationary state [2]. These results have led to a huge theoretical push [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] to address fundamental questions such as whether there is an asymptotic stationary state, and, if it exists, which ensemble characterizes it. The belief is that observables of non-integrable systems effectively thermalize, which implies that their stationary state is characterized by a thermal Gibbs ensemble. Numerical works on non-integrable systems confirm this expectation, even if some contradictory results point out that some issues have not been completely understood yet [19, 20, 21, 26]. On the other hand, in integrable systems, because of the existence of local integrals of motion, the stationary state is expected to be described by a generalized Gibbs ensemble (GGE), where each mode associated with a conseved quantity is characterized by its own temperature. So far results on integrable systems have been focussed on free fermion models, such as the transverse field Ising chain and the quantum XY chain [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].
For the transverse field Ising chain in thermal equilibrium Sachdev and Young first introduced a semi-classical description of the physical properties of the model in terms of ballistically moving quasi-particles [45]. This approach turned out to be incredibly accurate in predicting the temperature dependence of correlation length, relaxation time and in general to compute the order-parameter two point function in the ferromagnetic and paramagnetic phases. For global quenches this technique has been used to describe the dynamics of the transverse field Ising chain and the quantum XY model, with great accuracy [30, 37, 39, 44]. In these works quantitative features of the relaxation process have been explained with a quasi-particle picture, which had been introduced before in particular to study the evolution of the entanglement entropy [32, 46, 47, 48]. In practice the quench injects an extensive amount of energy into the system, which creates quasi-particles homogeneously in space, that then move ballistically with constant velocity. Because of momentum conservation, these quasi-particles are created in pairs with opposite momenta and are quantum entangled (within each pair). Their dynamics can be treated classically as long as they do not collide. But since collisions are unavoidable in one dimension, every scattering process has to be treated quantum mechanically. We shall use this semi-classical approach to calculate the dynamical correlation functions analytically for the O(3) non-linear sigma model. The dynamics of this model have already been studied at finite temperature in equilibrium [49, 50], where the order-parameter correlation function shows a universal form. Here we shall study its behaviour after a quantum quench, preparing the system in a state which is not an eigenstate of the Hamiltonian. In addition a section will be dedicated to the case of the transverse field Ising chain, whose order-parameter two-point correlation function can be derived straightforwardly from the general one obtained for the O(3) non-linear sigma model.
The paper is organized as follows: first we introduce the model and the general form of the initial states we shall consider throughout the whole article, before introducing and commenting the main result of the paper, concerning the O(3) non-linear sigma model. Then we describe the basic ideas behind the semi-classical technique and see in detail how to apply them in the present case.

2 The Model

Let us start by considering the one-dimensional quantum rotor chain, which is described by the following Hamiltonian:

(1)

where is the position operator of the rotor on site with the constraint , and is the angular momentum operator. is an overall energy scale and is a positive coupling constant. The operators which appear in the Hamiltonian satisfy the usual commutation relations and , where represent the three spatial directions. The continuum limit of this lattice model is the O(3) non-linear sigma model (), whose Lagrangian density reads:

(2)

where are three scalar fields and is a (bare) coupling constant. Here we have already set the maximal propagation velocity . This model is O(3)-symmetric, renormalizable and asymptotically free, and it has three massive particles in the O(3)-multiplet. The exact -matrix of this model is known [51], with any scattering event involving no particle production and the general -particle -matrix factorizes into a product of two-particle amplitudes. It is worth noticing that the O(3) non-linear sigma model also provides a description of the low-energy excitations of the one-dimensional antiferromagnetic Heisenberg chain [52], whose dynamical correlation functions’ lineshape can be measured experimentally. In the limit, in the ground state of Hamiltonian (1) all rotors must be in the state to minimize the kinetic energy. The low-energy excitations above the ground state form a triplet with quantum numbers . It is worth remarking that the structure of the low-energy spectrum is the same for any and the system has a gap . A finite gap is a necessary condition to apply a semiclassical approximation, as we shall see in the next section (for a deeper introduction to the semi-classical method and its range of applicability see [50] and references therein).
In this article we study the dynamical correlator of the order parameter after having prepared the system in a squeezed coherent state, namely1:

(3)

where is the ground state of the model, is the amplitude relative to the creation of a pair of particles with equal and opposite momenta, are creation operators of an excitation with quantum number (the -component of the angular momentum) and momentum . The main reason for choosing such an initial state comes from its relation with boundary integrable states: as shown by [10, 47] some dynamical problems can be mapped into an equilibrium boundary problem defined in a strip geometry, where the initial state acts as a boundary condition. In integrable field theories the most natural boundary states preserve the integrability of the bulk model, or in other words, do not spoil the integrals of motion. These boundary states were originally studied by Ghoshal and Zamolodchikov [53], and are supposed to capture the universal behaviour of all quantum quenches in integrable models. Furthermore in [18] it was shown that any quantum quench of an integrable field theory with this kind of initial states leads to a steady state which is described by a GGE ensemble. Precisely they show that long time limit of the one point function of a local operator can be described by a generalized Gibbs ensemble2. In short, their result is strong evidence that Rigol et al.’s conjecture does hold for integrable field theories.
The amplitude in expression (3) is a regular function which must satisfy a set of constraints that depend on the -matrix, such as crossing equations, boundary unitarity and boundary Yang Baxter equation. Different solutions of these equations form the set of integrable boundary conditions of the theory.3
The integrable boundary states belong to the class (3), but the semi-classical approach applies to the larger class of quenches whose initial states are expressed by a coherent superposition of particle pairs. In this case the only requirements on the amplitudes are that they must be peaked around and fast enough decreasing functions as , so that we can properly define the probabilities in (13).
The order parameter in (2) may be written as [50]:

(4)

where is the (time-dependent) Fourier transform of and the ellipses represent multiparticle creation or annihilation terms, which will be considered negligible because they are subdominat in the long-time limit. Relation (4) comes from the observation that the operator either creates a quasiparticle () with at position with some velocity or destroys one already present in the system, as will be discussed in the section (4).
The quench protocol is the following: At time we prepare the system in the initial state (3) and for this state evolves according to the Hamiltonian (1):

(5)

On the other side the time-evolution of an operator is written as:

(6)

We are going to analyze the two-point correlator:

(7)

The autocorrelation function is obtained for , whereas the equal-time correlation function is defined by imposing , even though in this paper the emphasis will be given to the non-equal time case.

3 Summary and discussion of the results

In this section we summarize the main result of the paper. The semi-classical method is based upon the existence of a small parameter, namely the average density of excitation pairs with quantum numbers , in the initial state :

(8)

which is valid in the limit , . This limit also defines what we call small quench, and it is also the first one we take in our semi-classical approach. The explicit computation of (8) will be carried out in appendix A. In the limit of small quenches we obtain the following expressions for the (non-equal time) two-point function of the order-parameter of the O(3) non-linear sigma model (7) in the limit :

(9)

where corresponds to the coherent propagation of a quasipaticle, while is the relaxation function, which describes the scattering with the excited quasiparticles. By definition and (without any loss of generality we assume both to be positive). Whilst is valid for all , the classical relaxation which yields the factor makes sense only within the light-cone (). Strickly speaking the validity of formula (9) requires , where multi-particle contributions are subdominant and can be neglected.4 In practice we shall see for the subcase of the Ising model that formula (9) works for almost everywhere, except for very small. We expect this to be the case for the O(3) non-linear sigma-model as well. The pre-factor in (9) reads:

(10)

where is a form factor and . For the general lattice model of equation (1) is not known. For our continuum theory, neglecting the multi-particle terms in the spectral density [56], we have:

(11)

where is a non-universal quasiparticle residue. This gives , with the modified Bessel function. The relaxation function is instead given by:

(12)

is defined by , , is the velocity of the particle with momentum and , where

(13)

Formula (12) is the main result of this article, and it describes the long-time behaviour of any initial state of the form (3). Nonetheless if we want the state to respect the O(3)-symmetry we must impose (), and all other probabilities equal to zero.
The most general expression for the relaxation function for generic values of the times will be given in the appendix B. Formula (12) defines the quench-specific time and length scales:

(14)

We also note that the correlator of equation (12) is never thermal unless one prepares the system at in a thermal inital state. This result confirms the belief that one-dimensional integrable systems relax towards a peculiar non-thermal distribution, namely the generalized Gibbs ensemble (GGE).

4 The Semi-classical Theory

4.1 From quantum correlators to classical probabilities

The main idea behind the semi-classical approach to the computation of out-of-the-equilibrium correlators is basically encoded in the following representation of the correlator (7):

(15)

where the function , is the probability density of having quasiparticles at position at time with momentum and quantum number , being the total number of quasi-particle pairs in a given initial configuration. The matrix element in the equation represents the value of the correlator once we specify a particular initial configuration for the excitations, which is specified by the set . The summation over averages over all possible quantum numbers of the particles, which are the three possible values of . We will start by assuming the system to be finite with free boundary conditions, and only later will we take the thermodynamic limit. In spite of this we will assume the system to be translationally invariant, corrispondingly corrections due to the presence of boundaries will be considered negligible (for large enough systems). Therefore the correlator of equation (7) becomes a function of , whilst the same cannot be said for the time dependence, contrary to the equilibrium case.
In equation (15) we have substituted a complicated time-dependent matrix element with a sum over all possibile initial states, which can be represented by pairs of ballistically-moving quasiparticles. This technique has already been used extensively to compute finite-temperature correlators (see for example [49, 50, 57]), and dynamical correlation functions [37, 39]. The general idea is the following: each quasiparticle carries a momentum and a quantum number , and it is created at time together with a partner with equal and opposite momentum. If the square modulus of Fourier transform of the amplitude is a fast enough decreasing function as , the probability of creating these quasiparticles far apart from each other becomes negligible. For this reason in the follow we will always assume quasiparticles to be created in pairs in the same position , where the index labels the quasiparticle pairs.
We start by considering this gas of quasiparticles to be very dilute by tuning the amplitudes , therefore the quasiparticle are entangled only within each single Cooper pair. This means that the probability of a particular initial state can be factorized in a product of single-pair probabilities:

(16)

where in the last equality we made use that the system is homogeneous in space (corrections will be present close to the boundaries, but they are negligible in the thermodynamic limit). The quantity is the probability for a single pair to be created at a certain position with momenta and quantum numbers . When dealing with Cooper pairs it is convenient to introduce the notation , whose meaning is:

(17)

where are the quantum numbers associated to the two particles in the Cooper pair. This notation is graphically represented in figure (1).
Without any loss of generality we can also suppose , thanks to the (quasi)-translational invariance of the system.

Figure 1: Graphical representation of the probability function .

Calculating the matrix elements for the operator (which, as a classical vector would be proportional to the cosine of the azimuthal angle ) with the first few sherical harmonics, one finds that this operator either creates a quasiparticle with () at and with some velocity or destroys one already present in the initial state . We assume the latter to be negligible, because the excitations are very dilute in space and therefore their density is very small (in other words we will use as an expansion parameter). This adjoint particle can be created by either inbetween two different quasiparticle pairs or within two quasiparticle that belong to the same Cooper pair (i.e., two quasiparticles that originated in the same point at time ). A possible dynamical scenario is pictured in figure (2). Due to the collisions with the other excited particles there are only certain configurations where the quantum mechanical overlap in will be non-zero. Similar to other models the non-linear sigma model in the long-wavelength limit has a purely reflective scattering matrix [56]:

(18)

This means that in this limit the particles are impenetrable and the sequence of does not change in time (see figure (1)). Here we are implicitly assuming that the momentum distribution is a peaked function around zero. The operator creates a quasiparticle with a probability amplitude , where the subscript refers to the quantum number of the created particle and is its momentum. This particle, together with the other quasiparticles, propagates under the action of and collides with them (in one-dimensional systems collisions amongst particles can never be ignored). The -matrix takes on an exchange form (18), and therefore particles only exchange their velocities while conserving their internal quantum numbers (see figure (2)).
As a consequence, at any time precisely one of the particles will have the velocity of the particle which was created by , and will be at position . This very particle must be annihilated at time by , otherwise the final state after the action of will be orthogonal to the initial one. The probability amplitude that this particle is annihilated is proportional to , where is the quantum number of the particle that is removed by . As we shall see below, has to be equal to zero. This request is automatically guaranteed by another requirement, namely that the internal quantum numbers in the final state be exactly the same as those of the initial one. If this does not happen the final state is orthogonal to the initial one, and that particular configuration of quasiparticles does not contribute to the average (15).

Figure 2: Propagation and scattering of quasiparticles in the semi-classical picture of the non-linear sigma model. Lines drawn with different colours represent trajectories of different particles. The two states before and after the forward and backward time evolution must be identical, and this imposes constraints on the quantum numbers of the particles. Notice that in this example the particle with label is removed at time .

Let us now consider the case in which at time the point is located inbetween particle with quantum numbers and (see figure (2)). Consider now a line connecting the points and (this line does not correspond to any real particle trajectory). Define as the number of intersections between this line and the other lines which correspond to particle trajectories, where are intersections from the right and those from the left. By definition is given by:

(19)

where the labels correspond to the trajectories of two quasiparticle originated at the same point and ’s are Heaviside step functions. Then at time the operator creates a quasiparticle () with at with some momentum . The adjoint particle will be created inbetween particles and . Suppose that . The generalization of the following results to the case of negative is straightforward. The action of the operator on the initial sequence of quantum numbers may be written as:

(20)

At time the operator will remove the particle with label from the set of excitations, where is the difference between the number of intersections from the right and those from the left of the line connecting points and . This quantity is given by:

(21)

This number tells us which particle is moving along this line at time , particle that will be removed by the action of :

(22)

Comparing the very last sequence of quantum numbers with the initial one (that at time ), we end up with the following constraint on the set of quantum numbers:

(23)

In practice we have a sequence of quantum numbers that must all be equal to , which starts at position . To identify the contribution of a particular configuration of quasiparticles we must consider the phase factors. Quasiparticles in the initial state generate a phase factor under the action of ( being the particle energy). This phase factor, however, completely cancels under the action of , except for the quasiparticle added by the operator . This gives a factor . Moreover every collision results in a sign change of the many-body wave function, but all these signs cancel under the forward and backward propagations, except those that are associated with collisions with the extra particle. These give an extra sign , which can be conveniently re-expressed as:

(24)

Collecting all these pieces together we obtain the following expression for the general correlation function (7):

(25)

where is the relaxation function, which is given by:

(26)

where the sums with indices and are given by equations (19) and (21) respectively, while the index is configuration-dependent, signaling where the adjoint particle is created in each set 5. The appearing in the previous equation is exactly that introduced in (24). The average reads:

(27)

From equation (4.1) we can immediately see that only those configurations which have a sequence of quantum numbers all equal to contribute to the correlator (7). We first tackle the problem of explicitly computing the average over the quantum numbers in equation (4.1), keeping the total number of quasi-particle pairs fixed. In the next section we shall see how to release this constraint. Let us consider separately the four cases in which are even and/or odd numbers, which are labelled from to .

  • Case ( even and even): when is an even integer the adjoint particle is created inbetween two different quasiparticle pairs, thus we have a sequence of quasiparticle pairs which are all forced to have their quantum numbers equal to . The average over the set of quantum numbers can be written as:

    (28)

    It is worth noticing that the labels in the Kronecker deltas in the previous equation refer to the quasiparticles, while the labels in the summations and in the integrals refer to quasiparticle pairs. In order not to mix up these two different notations we recognize that in this case is an even number, therefore particle with label belongs to the pair, and so on. From this pair to the right we have a sequence of quasiparticle pairs all with quantum numbers . In addition the probability distribution factorizes into single-pair probabilities. Let us remember that the average is taken over all possible initial conditions, therefore specifies the value of the momentum of the particle at time , where is a small positive quantity. Roughly speaking is the value of the momentum of a particle right after time and before the first scattering process. Equation (28) can be easily evaluated to give:

    (29)

    where .

    Figure 3: Computation of : an example with . Red trajectories represent those particles whose quantum number must be equal to .
  • Case ( even and odd): also in this case the adjoint particle is created inbetween two different pairs, but when is odd the result of equation (28) becomes:

    (30)

    In practice the last particle of the sequence of belongs to a pair whose right-side partner can carry all possible quantum numbers.

  • Case ( odd and even): let us consider the case in which is an odd integer. In this case the adjoint particle is created within a quasiparticle pair, and the average over quantum numbers gives ():

    (31)

    The case simply gives a sequence of different ’s. When is odd and even in the sequence of quantum numbers with the first and the last one belong to pairs whose partner has no constraint on its quantum number . This is the reason for the factors and in the previous expression.

  • Case ( odd and odd): in this case the average over the set of quantum numbers gives:

    (32)

We are now ready to write down the global expression for the quantum number average:

(33)

This expression is a function of and the set of quantum momenta of a particular quasiparticle configuration. It is worth noticing that and enter the expression for , in different ways. While tells us how long the sequence is, gives us information about the starting point of the sequence and can easily be absorbed into the definition of . As we shall see, for the actual computation of the relaxation function the starting position of the sequence is not important, what matters is only the length of the sequence itself.

Figure 4: Computation of : an example with .

4.2 The semi-classical average in action

We must now compute the average of the previous quantity over all possible initial positions and momenta. Namely we must take the following average:

(34)

where the quantity in the square brackets is given by equation (33). We can split this average into four pieces, let us call them and , one for each respectively, where . We shall do the computation of the term which contains in some detail; the other cases are straightforward modifications.
Let us compute explicity , the first addend of the right side of equation (33), where the first summation over gets immediately canceled out by the corresponding Kronecker delta because itself enters the function trivially, therefore we have:

(35)
Figure 5: Computation of : an example with odd and .

where is specified by (19). can itself be splitted into two pieces, say and , with which correspond to the two contributions to the integration of the terms into the bracket, namely:

(36)

and

(37)

Let us start by computing the contribution of . By making use of the following integral representation of the Kronecker delta