UvA-DARE (Digital Academic Repository) Exact out-of-equilibrium central spin dynamics from integrability

We consider a Gaudin magnet (central spin model) with a time-dependent exchange couplings. We explicitly show that the Schrödinger equation is analytically solvable in terms of generalized hypergeometric functions for particular choices of the time dependence of the coupling constants. Our method establishes a new link between this system and the


Introduction
The problem of describing the coherent out-of-equilibrium evolution of driven many-body quantum systems has attracted a great deal of attention in recent years. This interest was spurred by the recent advances in cold atoms and semiconductor physics, which made experimental observations possible. The attention of the community has mostly been devoted to the investigation of two limiting cases: the quench regime [1], where the variation of the parameters of the Hamiltonian is very fast with respect to all the other time scales of the problem, and the adiabatic regime, where it is slow (see [2] for a recent overview). Outside of these two extreme situations, very little is known, either analytically or numerically. This is unfortunate, because this driven regime has the most potential for novel physics.
There are natural obstacles for direct studies of non equilibrium quantum many-body systems: only a few solutions of the single-particle Schrödinger equation with time-dependent parameters are known even in the single-particle case. The situation is even worse in the manybody case. While integrable many-body systems provide considerable insight into equilibrium physics in one dimension, their non equilibrium behavior is still difficult to analyze because of the complexity of their solution. Numerical treatments of time-dependent systems (like e.g. by the time-dependent density-matrix renormalization group) are limited by the quantum entanglement which grows while the system evolves in time starting from some initial state [3][4][5][6]. To our knowledge there is a only a single subclass of systems where the full time dependence of parameters can be kept to some extent and where the dynamics can be understood in its full complexity. The dynamics of these systems can be mapped to the dynamics of a different systems which have no explicit time-dependence of parameters by an appropriate transformation of the coordinates, time, and wave-functions [7][8][9][10][11][12]. While this class of models is limited, it provides a clue about certain interesting and fundamental dynamical effects, like e.g. dynamical fermionization, and moreover it does not rely on integrability of the time-independent model.
To go beyond this class of models some new ideas are needed. Here we make an effort in this direction by suggesting to use a fact that the wave functions of a broad class of many-body quantum-mechanical models can be represented in terms of the correlation functions of some field theories with known properties. These connections were discovered and used in the context of the quantum Hall effect, where the quantum wave functions can be related to conformal blocks of the two-dimensional conformal field theories (CFTs) [13]. Interestingly, the wave functions of some integrable spin models can also be related to the correlators of certain CFTʼs [14,15]. Here we extend and use these observations further to study non equilibrium dynamics of those spin models. Since these spin models belong to a broader class of a so-called Gaudin systems, our observations can be applied to that class as well. The central ingredient of our approach here is the fact that the conformal blocks of the 2D Wess-Zumino-Witten (WZW) model are solutions of the Knizhnik-Zamolodchikov (KZ) equation. For a broader class of CFTʼs (without internal symmetry) these equations should be replaced by the Belavin-Polyakov-Zamolodchikov system. We believe that the approach we explore here can be generalized further for systems with a more general matrix-product like structure of the wave functions.
Implementing the above ideas concretely, we investigate a model of N spin −1/2 degrees of freedom coupled by time-dependent exchange parameters J t ( ) Label 0 refers to the 'central spin' which is coupled to the − N 1 other (mutually uncoupled) spins. For time-independent couplings, this model is known as the central spin model, a Gaudin magnet [16][17][18]. Crucially, this Hamiltonian is directly relevant to experiments in quantum dots [19,20] and nitrogen vacancy centers in diamond [21], in which time-dependent couplings are intrinsic to the experimental protocols (respectively via time-dependent gate voltages and external electromagnetic fields). The model (1) is only one of the broad class of models where dynamics can be treated using our method here. Other Gaudin-type models can be directly studied in a similar way.
The aim of this article is therefore twofold. On the one hand, we identify a time-dependent protocol for which it is possible to obtain analytical information (i.e. the exact many-body wavefunction) for a class of Hamiltonians related to the (1). While the requirements of this protocol are restrictive, they nonetheless allow to go beyond the adiabatic or sudden approximation. On the other hand, our technique points to an intriguing link between the timedependent central spin Hamiltonian and the WZW model, a well-known CFT, opening the door to further applications of CFT techniques to driven nonequilibrium physics.
We note that here we will restrict our interests to the dynamics of the total spin-singlet subspace, = S 0 2 . This is a good starting point because (see [22,23] and the recent review [24]) this subspace plays a crucial role in quantum information theory: the decoherence-free dynamics naturally occurs in this subspace, while the qubits can be encoded into its basis states.

Main results
We begin from the fact that the conformal blocks of the WZW model [25] satisfy the KZ equations [26]. For a ( ) is the N-point holomorphic conformal block of primary field φ, while k is a number known as the level of the Kac-Moody algebra. If k is a positive integer, the WZW model is a rational CFT. Interestingly, there exist integral representations of solutions to the KZ equations that can be analytically continued to any nonzero complex k [27,28]. Choosing = k iv where ∈ v  and considering the ansatz ψ for a many-body wavefunction, we see that ψ t ( ) N can in fact be reinterpreted as a timedependent Schrödinger equation ψ is chosen to be the sole time-dependent parameter, It is shown in appendix A that this choice of time-dependent parameters z t ( ) 1) is uniquely dictated by the form of (1). Notice that the hermiticity of the Hamiltonian forces all the z i to be on the same line in the complex plane (for example, we can take them to be all real).
Let us emphasize the main features of our approach. First of all, the main ingredient for an explicit solution of the time-dependent Schrödinger equation is a solution of the KZ equations that can be analytically continued to imaginary k. For small systems, this can be done explicitly, using standard CFT techniques. For larger systems, we can rely on a class of integral representations. Quite interestingly, these representations rely crucially on the integrability of the time-independent Hamiltonian, i.e. the off-shell Bethe equations. Therefore, the solubility of the time-dependent Schrödinger equation seems to be a signature of the underlying integrability of the model that survives also when the couplings are time dependent. Indeed, this interpretation is confirmed by the fact that, as we will discuss later on, the solvability of the time-dependent Schrödinger equation is not a special feature of the central spin model: our arguments apply also to the broader class of XXZ Gaudin magnets. Moreover, our results establish a new connection between the ( ) SU 2 WZW model (admittedly, for the quite unusual imaginary k case) and the time-dependent central spin Hamiltonian. It is worth to note here that the ( ) SU 2 WZW model is known to be related to integrable [14] and nonintegrable [29,30] time-independent spin Hamiltonians. Finally, it is important to stress that-by constructionthis approach works only if the time dependence of the J t ( ) i is finely tuned: essentially, the time evolution is 'geometric', i.e. ∫ dtH t ( ) can be written as a curvilinear integral in the space of the z j .
Our paper is organized as follows. First of all, in section 3 we provide a detailed analysis of a simple system of four spins. Thanks to the connection between the central spin Hamiltonian and the WZW model, we are able to analyze the time of evolution of the subspace of zero total spin in terms of hypergeometric functions. In this way, we can see our approach explicitly in action and understand some mathematical property of our solution (i.e. completeness and nontriviality). Therefore, in section 4, we move to a more general setting: a N particle XXZ Gaudin magnet with time-dependent couplings (4). Here, we take advantage of an integral representation of the solution of the (generalized) KZ equations to provide an integral representation for the time-dependent many-body wavefunction. While this representation is not (yet) amenable to an quantitative evaluation, it allows us to consider two interesting situations: the adiabatic and the semiclassical limit, thus gaining insight on the completeness of our solutions (section 4.1) . Finally, we present our conclusions in section 5, while some of the more technical details are discussed in the appendices.

A simple example: a four spins system
The class of Hamiltonians under consideration has a quite specific time-dependent coupling constant (4). Moreover, as we will see, in the general case, while it is possible to write down an integral representation for the wavefunction, it is not easy to extract physical predictions from it. The reader could wonder if this class of Hamiltonians can be solved only because their physics is trivial or if, instead, we can expect some interesting phenomenology that might motivate a further investigation of these systems. In this section, we want to address this point by studying one quite simple representative of this class of Hamiltonians: a central spin Hamiltonian with four constituents, Since the total spin is conserved by the time evolution, we can restrict ourselves to the . In the following, we would like to show that, indeed, the WZW correlators provide solutions that describe the whole zero spin subspace.
The computation of the four point conformal blocks Ψ z z z z ( , , , 3 of the WZW model is a standard exercise of CFT (see [25]). The detailed calculation is reported in appendix B, where Ψ z z z z ( , , , ) on a basis of the = S 0 2 subspace given by the two states We thus can write , where c 1,2 are constants determined by the initial conditions, while i k k . Therefore, as discussed above, the wavefunction 3 (with = k iv) is a solution of the time-dependent Schrödinger equation for Hamiltonian (5).
As an example, let us consider the following protocol. At time t = 0, the spins S j , = j 1, 2, 3, are at a distance j from the central spin S 0 . Their couplings J j are taken to be As an application, an interesting quantity to look at is the modulus square of the overlaps of the wavefunction with the basis vectors , which are simply computed. We can expect that, if these overlaps are almost constant in time, then the time evolution is essentially trivial. As a signature of the nontriviality of the time evolution, we look the crossing of a t ( ) > ′ t t : this means that for < ′ t t the state i is more important than the state j, while the opposite is true for > ′ t t . Two interesting examples are shown in figure 1 for the dipolar interaction (left) and for the shell model (right). In both cases, the initial condition is chosen in such a way that it is possible to observe one (dipole interaction) or two (shell model) crossings of the overlaps a t ( ) i . Another interesting quantity to understand the dynamics of the system is the fidelity   (6)).

General solution
It this section, we would like to outline our strategy for getting an integral representation of the solution to the KZ equations (2) or, more precisely, a generalized version of these equations. Our arguments are a straightforward generalization to the ones of [15,27,32]. The XXZ Gaudin magnets are defined from the Gaudin algebra , while the λ i are complex numbers. Notice that X and Z are not arbitrarily functions, but they have to satisfy a set of quadratic equations that come from the Jacobi identities for the generators of the Gaudin algebra (8). The solutions to these equations are known, and the simplest one is the rational one λ λ (for a detailed discussion of Gaudin magnets, the reader is referred to [18]). For example, in order to describe a spin or fermionic system, the Here z i are a set of complex numbers (the disorder variables) that are directly linked to the coupling constants of the Hamiltonian, while S i are the familiar spin operators. Instead, a bosonic system is described by a ( ) where the K i satisfy a ( ) su 1, 1 algebra. Of course, mixed representations are also available, in order to describe a system where the spin degrees of freedom interact with a bosonic bath (i.e. the Dicke model and its generalizations). In the following, we will use the notation These H i are a family of commuting operators, and each of them can be considered as the Hamiltonian of a quantum system. Since H w z ( , ) can be diagonalized using the algebraic Bethe Ansatz, the Hamitonians H i are exactly solvable. In the rational case, the integrals of motion for a spin system reduce to satisfy the Schrödinger equation with Hamiltonian . Quite nicely, this integral representation of the solution is based on the integrability of the model. Let us introduce the Bethe state (λ λ λ The functions h i and α f can be derived from a Yang-Yang action: more precisely, Usually, one imposes the on-shell condition = α f 0 (Bethe equations), thus obtaining a basis of eigenstates of the Hamiltonian. Here instead, we take advantage of the existence of the action  and we define where the closed contour γ is chosen in such a way that the branch of the integrand at the end point of γ is the same as that at the initial point. It is quite easy to show that (20) is indeed a solution of (16). Notice that due to the multi-valuedness of the integrand, the path of integration is usually highly nontrivial, this being the major technical difficulty of our approach. In the rational case, these integrals represent multivariable hypergeometric functions [33,34], and our hope is that this connection could be exploited to evaluate explicitly (20).

The k →0 limit and the completeness of the integral representation
Unfortunately, a direct evaluation of (20) is beyond our present ability. However, in the → k 0 limit, the only contribution to the integral comes from the stationary points of the action (19), i.e. from the on-shell Bethe state [35]. Therefore, it is quite interesting to discuss the physical meaning of this limit for our time-dependent Schrödinger equation. The most natural interpretation of this limit is as an adiabatic one. As an example, let us consider the central spin limit with coupling constant (4). If we parametrize = k i v and Ω  is the corresponding dynamical phase. Moreover, by choosing properly the contour γ, we can select any eigenstate, and therefore our solution is complete in the adiabatic limit. This is at least a strong hint that our solution is complete for any time dependency of the coupling constants. Quite interestingly, the → k 0 limit can be interpreted also as a semiclassical limit, if we use a different parametrization. Indeed, if where  0 has the dimensions of an action, we have a central spin model with coupling constants

Conclusions
In this article, we have studied a class of time-dependent Hamiltonians which possess manybody wavefunctions given by solutions to the associated KZ equations. The underlying timeindependent integrability and link with CFT allow us to provide an explicit integral representation of the solution to the time-dependent Schrödinger equation. For a small system, these solutions reduce to the familiar hypergeometric functions, allowing us to easily study the dynamics of the system, as we did for a four spins model. This specific example shows that the exact solubility of these time-dependent systems is not due to their triviality. Instead, their physics appears to be quite rich, as you could expect for a full time-dependent problem. Of course, from a practical point of view, the most interesting thing would be to solve a Nparticle model. In order to do so, we have to deal with the complicated integral (20). Unfortunately, we are not able to do it at the present time. However, in the rational case, this integral reduces to generalized hypergeometric functions, that have been extensively studied in the mathematical literature. Another possible line of investigation could be to compute the corrections to the adiabatic limit, that could teach us something about this complicated integral representation.
While our construction works only if the time dependence of the Hamiltonian is finely tuned, it provides an intriguing starting point for understanding the consequences of quantum integrability in time-dependent physics.
Here, we would like to elaborate more on this point. In particular, one could wonder if there exists a more general wavefunction ψ

, satisfies a Schrödinger equation with Hamiltonian
It turns out that This identity can be proved by a brute force calculation (itʼs just a trivial algebraic calculation), but it is possible to obtain it in a clever way. First of all,we notice that the lhs has poles in = 3 , we get = A 1. Therefore, the KZ equation reduces to