Quantum quenches and Generalized Gibbs Ensemble in a
Ansatz solvable lattice model of interacting
We consider quantum quenches in the so-called -boson lattice model. We argue that the Generalized Eigenstate Thermalization Hypothesis holds in this model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description of the stationary states in the long time limit. For a special class of initial states (which are the pure Fock states in the local basis) we are able to provide the GGE predictions for the resulting root densities. We also give predictions for the long-time limit of certain local operators. In the limit the calculations simplify considerably, the wave functions are given by Schur polynomials and the overlaps with the initial states can be written as simple determinants. In two cases we prove rigorously that the GGE prediction for the root density is correct. Moreover, we calculate the exact time dependence of a physical observable (the one-site Emptiness Formation Probability) for the quench starting from the state with exactly one particle per site. In the long-time limit the GGE prediction is recovered.
The problems of equilibration and thermalization of closed quantum systems have attracted considerable interest recently [1, 2]. One of the central questions is whether the principles of statistical physics can be derived from the unitary time evolution of the quantum system. Research in this field has been motivated partly by new experimental techniques (for example with cold atoms ) where an almost perfect isolation from the environment can be achieved, and therefore equilibration induced by the system itself can be studied.
Equilibration in a quantum mechanical system means that the expectation values of physical observables approach stationary values in the long time limit. Thermalization happens when these coincide with predictions obtained from a thermal ensemble. One dimensional integrable models comprise a special class of systems which possess a family of higher conserved charges in addition to the usual ones. These extra conservation laws prevent thermalization in the usual sense. Instead, it was proposed in  that the stationary values of local observables should be described by the Generalized Gibbs Ensemble (GGE). This ensemble includes all the charges with Lagrange-multipliers fixed by the mean values of the charges in the initial state.
Since its inception the idea of the GGE has attracted considerable interest and sparked many discussions. A large body of numerical evidence for its validity was found in the lattice model of hard-core bosons [4, 5, 6, 7] (see also ) and it was proven to be true in free theories or models equivalent to free fermions [9, 10, 11, 12, 13, 14, 15, 16]. However, it was found in  that in the interacting spin-1/2 XXZ chain the GGE (built on the strictly local charges) gives different predictions than the Quench Action (QA) method , which (as opposed to the GGE) is built on first principles and does not involve any assumptions or approximations. Furthermore conclusive evidence was found in a case of a specific quench problem in  that while the predictions of the QA method coincide with results of real-time simulations, the GGE predictions [20, 21, 22] are not correct.
It was argued in  that equilibration to the GGE predictions can be explained by the Generalized Eigenstate Thermalization Hypothesis (GETH), which roughly states that if there are two states which have almost the same mean values of the conserved charges, then local correlations in the two states should be also close to each other. If the GETH holds than the GGE is valid for quenches from any initial state satisfying the cluster decomposition principle. It was recently shown in  that the failure of the GGE in the XXZ spin chain can be attributed to the failure of the GETH. It was argued in [23, 24] that this is a generic property of integrable models with multiple particle species.
The question remains whether the GETH and the GGE can be correct in any genuinely interacting integrable model. To find an example one should certainly look for models with one particle type. An obvious choice would be the Lieb-Liniger (LL) model, which is a continuum theory of 1D interacting bosons . Quenches in the LL model have been investigated in a number of papers recently [26, 27, 28, 29, 30, 31, 32, 33, 34]. However, it was found in  that for an interaction quench from zero to finite coupling the expectation values of the higher charges are divergent, and therefore the GGE can not be defined in that case. The problem was circumvented by applying a lattice regularization using the so-called -boson model [35, 36, 37]. The QA solution of this interaction quench was later given in , where it was argued that the GGE can not be correct in the LL model due to the aforementioned divergences and the observed logarithmic singularities in the Bethe root densities, which can not be captured by the GGE.
In this paper we investigate quantum quenches in the -boson model, without the goal of taking the continuum limit towards the Bose gas. In this lattice model there is only one particle type in the spectrum, and the infinities encountered in the LL model do not appear here. Therefore it is an ideal testing ground for the GETH and the GGE.
The paper is organized as follows. In Section 2 we review the Bethe Ansatz solution of the model and the construction of the higher conserved charges. In 3 we construct the GGE density matrix for this model and argue that the GETH holds, therefore the GGE is valid. In 4 we consider specific quench problems and provide the GGE predictions for a class of initial states. The limit of the model is investigated in 5, where the equilibrium properties of the model are established. Quenches in the limit are investigated in 6, where we confirm the GGE predictions in two simple cases by analyzing the exact overlaps. Also, we derive an analytic formula for the time dependence of a simple physical observable, the one-site Emptiness Formation Probability. The long-time limit of this quantity is found to agree with the GGE prediction. The limit of the model (the case of free bosons) is considered in 7. Finally, Section 8 includes our conclusions, a number of remarks about our results, and a list of open problems.
2 The model and its Bethe Ansatz solution
Consider a lattice consisting of sites such that the configuration space of each site is a single bosonic space. Let us define the canonical Bose operators , , acting on site by the usual commutation relations
The action of these operators on the local states , is given by
We also define the operators , by their action
The parameter is an arbitrary real number. In the present work we will consider the cases and we will use the parametrization , . It is easy to check that the following commutation relations hold:
These equations are the defining relations of the so-called -boson algebra . The canonical Bose operators are recovered in the limit:
The -boson Hamiltonian is defined as
where periodic boundary conditions are assumed. Even though the Hamiltonian (2.1) has the form of a free hopping model, there are interactions between the particles due to the fact that the and are not the canonical Bose operators and the hopping amplitudes depend on the local occupation numbers. The model can serve as a lattice regularization of the Lieb-Liniger model [35, 36, 37, 39, 29]; however, in the present work we focus on the lattice model only.
The -boson Hamiltonian was solved in  by the Algebraic Bethe Ansatz (ABA). The coordinate Bethe Ansatz wave functions were later calculated in ; they are given by Hall-Littlewood functions. Here we review the (ABA) solution, our exposition follows that of .
Let us consider an auxiliary space and define the so-called Lax operator, which is a matrix in auxiliary space with matrix elements being operators in the bosonic Fock spaces:
Here is the rapidity parameter and . The Lax operator satisfies the Yang-Baxter equation
with the -matrix
The central object of the ABA is the monodromy matrix, which is given by
It follows from (2.2) that the monodromy matrix satisfies the RTT-relation
A direct consequence of (2.4) is that the transfer matrix defined as
It is easy to see that
where . It follows from the definition of the transfer matrix and the form of the Lax operator that is a sum of local operators which span at most sites. The first two examples are
A formula for is also given in .
These operators are not Hermitian. It is useful to define the charges with negative indices as their adjoint:
They can be obtained by expanding the transfer matrix around .
The particle number operator
commutes with all of the charges, which follows from the fact that the transfer matrix only includes terms with an equal number of and operators. We define . The Hamiltonian can then be written as
Eigenstates of the system are constructed using the -operators of the monodromy matrix:
where is the Fock vacuum. The parameters are the rapidities of the interacting bosons. A state of the form (2.9) is an eigenstate of the transfer matrix if the rapidities satisfy the Bethe equations:
The eigenvalues of the transfer matrix on the Bethe states are
Eigenvalues of the local charges are easily obtained using the definition (2.7). It is easy to see that they can be expressed as sums of single particle eigenfunctions:
It is useful to parametrize the rapidities as . This way the Bethe equations take the form
In the case of considered in the present work all solutions to (2.13) are real numbers and they can be chosen to lie in the interval .
In terms of the -variables the single particle eigenvalues of the charges take the form
The single particle energy is
The -variables are the physical pseudo-momenta on the lattice, because the single particle eigenvalue for the translation by one-site is .
For the sake of completeness we note that the local -boson operators at sites and can be reconstructed from the off-diagonal elements of the monodromy matrix as
2.1 Thermodynamic limit
We will be interested in physical situations where there is a large number of particles in a large volume such that the particle density is finite. As usually we introduce the densities of Bethe roots and holes such that in a large volume the total particle density is given by
It follows from the Bethe equations that
Expectation values of the charges in the thermodynamic limit are then calculated as
It is a special property of this model that the local charges measure the Fourier components of the root distribution.
3 Quantum quenches and the Generalized Gibbs Ensemble in the -boson model
We are interested in non-equilibrium situations in the -boson model. We assume that at the state is prepared in the initial state
which is not an eigenstate of the Hamiltonian. It can be the ground state of a local Hamiltonian, or any other state prepared according to certain rules. Examples will be given in Section 4.
The time evolution of physical observables is given by
We are interested in the large time behaviour of the observables in the thermodynamic limit. Neglecting degeneracies in the spectrum the long-time average in a finite volume case can be written as
The sum over eigenstates on the r.h.s. above can be interpreted as a statistical physical ensemble, and it is called the Diagonal Ensemble (DE). The weights of the DE are given by the squared overlaps with the initial state, and it is an important question whether the predictions of the DE coincide with those of a statistical physical ensemble, at least in the thermodynamic limit. The system thermalizes, if the DE gives the same mean values as a canonical or grand-canonical Gibbs Ensemble (GE). This is the expected behaviour for a generic non-integrable model.
The situation is different in integrable models, where the existence of the higher conserved charges prevents thermalization. It was suggested in  that in these models the DE predictions should agree with those of a Generalized Gibbs Ensemble (GGE) which includes all higher charges.
In the case of the -boson model the GGE density matrix can be defined as
where the Lagrange-multipliers are fixed by the requirement
We also require , such that is Hermitian. This condition is consistent with (3.3).
The GGE hypothesis states that for any local operator
Time averaging is only required in finite volume, and it can be omitted in the infinite volume limit.
The GGE was proven to be correct for free theories or models equivalent to free fermions [4, 5, 6, 7, 8, 13, 14, 43, 44, 16], but counterexamples were found in [17, 19] in the case of the XXZ spin chain. It is open question whether the GGE holds in other interacting models, and what the precise conditions are for its validity.
In  it was proposed that the GGE is valid whenever the Generalized Eigenstate Thermalization Hypothesis (GETH) holds. This hypothesis states that the mean values local operators in the excited states only depend on the mean values of the charges. In other words, if two excited states have mean values of the charges that are close to each other, then all local correlations in the two states will be close as well. Given that the initial state satisfies the cluster decomposition principle, the Diagonal Ensemble (3.1) will be populated by states which have the same conserved charges as the initial state, such that the mean deviation for the densities of the charges becomes zero in the thermodynamic limit [6, 23]. On the other hand, the GGE density matrix (3.2) produces states with the prescribed charges by definition. Therefore, if the GETH holds then the two ensembles give the same results for the local operators, because the dominating states of both ensembles will have the same local correlations. This is the reason why the validity of the GGE follows from the GETH . We stress that the weights of the DE need not be directly related to the generalized Boltzmann-weights of the GGE.
We now argue that the GETH is valid in the repulsive -boson model. In this model there is only one particle type in the spectrum, and the density of Bethe roots is described by a single function . According to (2.17) the conserved charges measure the Fourier components of and the root density can be reconstructed from the charges as
The charges thus uniquely determine the root density, and in order to prove the GETH we need to show that in the thermodynamic limit the local correlators only depend on . We are not able to prove this statement in full generality, but experience with other Bethe Ansatz solvable models suggests that it is in fact true. In subsection 3.1 we prove it for a class of non-trivial local operators.
With this we have established that the GETH holds in the -boson model. As a consequence, the GGE should also hold for any initial state satisfying the cluster decomposition principle.
Equation (3.4) shows that the charges uniquely determine the root density and there is no need to obtain explicit expressions for the Lagrange-multipliers entering the GGE density matrix. However, for the sake completeness we show how to compute them.
where is the pseudoenergy defined as
the Lagrange-multipliers are obtained simply by Fourier transformation. Note that all Lagrange-multipliers have a finite well-defined value.
We wish to remark that on the lattice all charges have finite mean values, therefore the problem of infinities encountered for interaction quenches in the continuum Bose gas [29, 31] does not occur in the -boson model. Also, equation (3.4) holds even if there are logarithmic singularities in the root density, because such functions are still members of and therefore their Fourier series is well defined and converges almost everywhere. Logarithmic singularities were encountered earlier in other models [31, 17], but in the -boson model they do not obstruct the validity of the GGE.
In the following subsection we show how to compute the GGE predictions for a set of simple local observables. Specific quench problems are considered in Section 4.
3.1 Local correlators in the GGE
For the -boson model there are no results in the literature for the excited state mean values of short range correlation functions. Here we apply the Hellmann-Feynman theorem  to compute mean values of certain local operators in states with arbitrary root density . The GGE predictions are then calculated by substituting the root density obtained from (3.4) into the results presented below. The method we apply was previously developed independently in  and .
As a first example consider the Hermitian operator
which is the operator density for . The finite volume mean values in an arbitrary Bethe state are
We use the Hellmann-Feynman theorem to obtain the mean values of the operator
where and are defined by their action on the Fock states:
Taking the derivative of (3.8) with respect to leads to
The derivatives can be obtained from the logarithmic form of the Bethe equations:
The quantum numbers specify the state and can not change as we vary , therefore
where is given by (2.16) and
In a large volume , where is the so-called shift function. It follows from (3.10) that it satisfies
Using (2.15) we obtain
This linear equation uniquely determines .
Finally, the mean value of is expressed as
Completely analogous results hold if we apply the Hellmann-Feynman theorem to the remaining higher charges. Defining to be the operator density of the Hermitian combination we obtain
where and the shift function is given by the solution of (3.11). We refrain from writing down the in terms of local operators, as they are easily obtained from the expressions of the charges 111At present there are no closed form results known for with arbitrary . The cases and were computed in , the case is given in eq. (2.8). Therefore, at present explicit expressions can be written down only for , and . Higher charges and higher could be computed from the definition (2.7). . We just stress that all are non-trivial local operators which span at most sites.
4 GGE for a class of initial states
In this section we derive the GGE solution for a special class of initial states, which are given as tensor products of one-site particle number eigenstates. Translationally invariant cases are considered in 4.1, whereas 4.2 deals with states that break the translational invariance.
4.1 Translationally invariant cases
We consider the states in which there is exactly particle at each site:
They are not eigenstates of the -boson Hamiltonian. They can be considered as ferromagnetic states pointing in a certain direction in the infinite dimensional Fock space. Also, they are the ground states of the infinitely repulsing Bose-Hubbard model at a given integer filling. The physically most relevant case is which is a state of uniform particle density 1. Quantum quenches in the Bose-Hubbard model with initial state were studied in .
In the following we evaluate the -boson GGE predictions for quenches starting from the . First we compute the expectation values of the charges, then we reconstruct from (3.4), and finally we give predictions for the local operators introduced in the previous section.
It is easy to see that all charges , are built from local operators which have the form
such that , and the dots stand for operators acting on sites with . In other words, the leftmost and rightmost operators are “unpaired”: a combination of the form can only occur in the middle, but never on the two ends of the operator product. This follows simply from the definition of the transfer matrix (2.3): regarded as a power series in the transfer matrix has terms of the form
and after formally taking the logarithm only terms of the form (4.1) can arise.
As a consequence we obtain the remarkably simple result
which follows from for arbitrary , and site .
Applying (3.4) we find that in the quantum quench starting from the state the resulting root density is constant and given simply by the total particle density:
The hole density can be calculated from (2.15):
where we used . The filling fraction for these states is therefore
A physical interpretation can be given as follows. In the initial states all particles have well-defined positions. The initial states are not eigenstates, so the real-time dynamics of the system is non-trivial. Due to the initial sharp localization in real space we can expect that the eigenstates contributing to the dynamics will be spread out maximally in momentum space. Finding a completely constant root density in an interacting system is remarkable nevertheless, and it is a special property of the system and the initial states chosen.
The operators are special in the sense that they also have zero mean value in the initial states:
This follows from the fact that their structure is essentially the same as those of the charges. However, they are not conserved in time. For example it can be checked easily that
Therefore the prediction that all approach zero in the long-time limit is a highly non-trivial statement, and it can be used as a test of the GGE.
For the sake of completeness we compute the Lagrange-multipliers for this GGE. All relevant functions are constant and from (3.7) we obtain
To conclude this subsection we remark that any small departure from the sharp localization of the one-site particle numbers changes the resulting root densities. For example if the initial state is given by
then the mean values of the two simplest charges are
According to (3.4) the root density will have a non-vanishing first Fourier component for any .
4.2 Breaking translational invariance
Let us define initial states which are not translationally invariant, but still have fixed one-site particle numbers. As examples we consider the states
These states are invariant under translation by two sites. It is an important question whether the full translational invariance gets restored in the long-time limit. In Section 7 we consider the free boson theory and demonstrate on a few simple examples that mean values of local operators become translationally invariant indeed. In the cases most of the hopping amplitudes in the Hamiltonian are smaller than in the free case, but the particles can still hop from any site to its neighbours irrespective of the occupation number. Also, there is no one-site potential or any other term in the Hamiltonian which could “freeze” the artificial order of the initial state. Therefore we conjecture that translational invariance gets restored for any .
The GGE predictions for the steady state are derived easily. Mean values of the charges , are all zero due to the same reasons as in the previous subsection. Therefore the resulting root densities are constant and only depend on the average particle number. For example and for and , respectively.
It follows from our considerations that the predictions of the GGE are completely the same for any two initial states, if both are products of one-site particle number eigenstates and the overall particle density is the same. For example the initial states and should lead to the same long-time limit for any local quantity. This is a surprising statement, and it can be used as a check of the GGE.
5 Large limit: Equilibrium properties
In this section we treat the limit of the model. We review the special properties of this limiting case, and also establish new results for a specific local operator: the Emptiness Formation Probability. These results serve as a basis to study quantum quenches in the large limit, which is considered in the next section.
In the limit the local -boson operators behave as
where the operators , are defined by their action
The Lax-operator is written as
and the Hamiltonian is
This model attracted considerable attention, partly because it is closely related to the combinatorial problem of plane partitions [49, 50, 51, 52]. In the literature it is often called the phase model.
The Algebraic Bethe Ansatz solution was first given in , where equilibrium correlation functions were computed as well. The coordinate space wave functions were first computed in , where it was shown that they are given by Schur polynomials. In our notations the coordinate Bethe Ansatz wave function can be written as
where the coefficients are
The Bethe equations take the following simple form:
If the rapidities satisfy the Bethe equations, then the norm of the Bethe state (5.2) is
whereas the eigenvalues for the higher charges are
We also note that in the thermodynamic limit the relation between the root and hole densities is simply
Results for correlation functions were also computed in  and [49, 50, 51]. Here we consider the -site Emptiness Formation Probability (EFP), which is the probability to have zero occupancy number on neighbouring sites. We derive new formulas for the one-site and two-site EFP in arbitrary excited states. These results are used in the next section to give the GGE predictions in the quench problems.
Let us define the operators which project to the zero-particle state on site . Their action is simply
The m-site EFP operator is given by
In  it was shown that the normalized -site EFP in a Bethe state is
and . The thermodynamic limit of this expression is a Fredholm determinant.
Here we consider the two simplest cases and and show that the determinant can be expressed using single sums over the particles. In the thermodynamic limit we thus obtain the EFP’s as sums of products of simple integrals. This is a huge simplification as opposed to the original result of a full Fredholm determinant. For simplicity we only consider states with zero total momentum, but this does not change the conclusions.
In the case we have
Multiplying the th row with and the th column with leads to
The matrix has rank 2, therefore in the expansion of the determinant we only have terms where at most 2 elements are chosen from . This leads to
where we used . For the finite volume EFP we thus obtain
Let us define renormalized higher charges as
where is given by (5.6). In the thermodynamic limit we have
Using this definition the one-site EFP is expressed simply as
We now calculate the two-site EFP. As a first step we write the corresponding determinant as
The matrix has at most rank 3, because it is a sum of three matrices with rank 1. Therefore in the expansion of the determinant it is enough keep terms where at most 3 elements are chosen from . This leads to
After some tedious but elementary calculations we obtain the EFP as
Using the definition (5.9) this can be expressed as
This result remains valid in the thermodynamic limit if the definition (5.10) is used.
It is a special property of this system that the 1-site (or 2-site) EFP could be expressed using the mean values of the first (or first two) charges, respectively, and that the EFP does not depend on the other details of the states. We now argue that this is a general pattern: the -site EFP only depends on the first charges and the overall particle density. Also we show how to obtain the results (5.11) and (5.12) directly in the thermodynamic limit.