Energy-level crossings and number-parity effects in a bosonic tunneling model

An exactly solved bosonic tunneling model is studied along a line of the coupling parameter space, which includes a quantum phase boundary line. The entire energy spectrum is computed analytically, and found to exhibit multiple energy level crossings in a region of the coupling parameter space. Several key properties of the model are discussed, which exhibit a clear dependence on whether the particle number is even or odd.


Introduction
The symmetric two-site Bose-Hubbard model has been studied widely for some time [1][2][3][4][5][6]. The Hamiltonian reads where = 0, for i, j = 1, 2. Above I denotes the identity operator, and N j = b † j b j . Setting N = N 1 + N 2 , it can be verified that [H, N] = 0. The model has a simple interpretation through two terms describing particle interactions with coupling k, and a tunneling process between two wells with interaction strength J. Without loss of generality we take J ≥ 0. Though it is simple, the Hamiltonian has been successfully used as a model for experimentally realised tunneling phenomena [7].
Several studies have identified a quantum phase transition in the attractive regime k < 0, using a variety of approaches including semiclassical methods [8,9], mean-field approximation [10], entanglement [11][12][13], fidelity [12,13], fragmentation [14,15], NMR simulations [16], and exact results using Bethe Ansatz methods [12,17]. One way to characterise the two phases is through the energy gap between the ground state and the first excited state. Setting k = 0 in (1) it is not difficult to check that the ground-state energy is −JN/2, and the gap to the first excited state is J. At the other extreme when J = 0 and k < 0, the ground state is two-fold degenerate, so the gap is zero. The transition between these extremes is abrupt. Setting λ = (kN )/(2J) the transition takes place at λ = −1 [12].
In recent times a generalised version of (1) has been studied which includes a second-order tunneling process [18][19][20]. The extended Hamiltonian is where the coefficient Ω is the coupling for second-order tunneling. The inclusion of such a term can be justified on physical grounds, but it often neglected because the coupling Ω is much weaker than k and J [21,22]. Nonetheless, the model has been employed [18] to account for the observation of second-order tunneling in the low-particle number limit [23]. From the mathematical perspective, (2) offers a richer structure than (1). Analyses of bifurcations of fixed points in the classical limit show there are three expected phases, which will be referred to as Josephson, self-trapping, and phaselocking [19,20]. Multiple energy-level crossings were found in the phase-locking phase through the studies of [20]. Such crossings are a new feature not found in the studies of the Hamiltonian (1). The main objective of this work is to investigate the boundary between the phaselocking and self-trapping phases. Energy-level crossings are also found to occur on this boundary, and they can be precisely identified. The energy levels can be computed analytically. The character of the set of energy levels is dependent on whether the particle number is even or odd. We will study some of the consequences of this finding, which may have implications for few-body bosonic systems. The results complement those for few-body fermions systems, that have attracted recent attention [24,25].
In Sect. 2 we begin by establishing that the phase-locking and self-trapping phases exhibit a duality. The boundary between them is a self-dual line with an enhanced symmetry. In Sect. 3 we recall a Bethe Ansatz equations for the model, which are easily solved on the self-dual line. This solution is used in Sect. 4 to examine the nature of the ground-state energy gap, and in Sect. 5 a supersymmetric structure within the model is unveiled. Number-parity effects in the computation of dynamical expectation values are investigated in Sect. 6, and concluding remarks are given in Sect. 7.
To reveal the duality between the phase-locking and self-trapping phases, introduce the su(2) realisation satisfying the relations for which the Casimir invariant C = 2(S z ) 2 + S + S − + S − S + has eigenvalue Λ = N(N + 2)/2. In terms of this realisation, the Hamiltonian is expressed as This Hamiltonian will now be transformed by a composition of three unitary operators: and U 2 (H) = H. It is easily checked that, up to the inclusion of an N-dependent term, U maps Hamiltonians between the phase-locking and self-trapping phases, while Hamiltonians in the Josephson phase are mapped back to the Josephson phase under the action of U. This shows that there is a 1-1 correspondence between the energy spectra in phase-locking and self-trapping phases. Hamiltonians on the line γ = λ, or equivalently Ω = k/2, are invariant under the action of U. Along this line, which includes the boundary between the phase-locking and self-trapping phases, analytic expressions for the entire energy spectrum can be obtained, as we describe below.

Exact solution
The Bethe Ansatz solution derived in [20] gives the energy eigenvalues and eigenvectors as Here, the parameters {u j : j = 1, ..., N} satisfy the Bethe Ansatz Equations (BAE) Note that the form (6) is different to the BAE presented in [20], which reads Eqs. (6) and (8) are equivalent whenever there are no root multiplicities in (7) . The more general form (6), which accommodates root multiplicities, will be required for the analysis below.
Hereafter set Ω = k/2, which is the self-dual line identified in the previous section. For this constraint the BAE (6) are solved with the choice u 2 j = 1 for all j = 1, . . . , N. There are N +1 solutions where q of the roots are chosen to taken the value −1, while the remaining N −q are chosen to take the value 1. This gives a complete set of (normalised) eigenstates with the corresponding energies While the structure of the states and spectrum through (9,10) is very simple, we can see that the system is non-trivial from the following analysis. Fixing J in (10), by setting E(N, q) = E(N, q ′ ), we see that all energy levels corresponding to labels q, q ′ ∈ 0, 1, . . . , N 2 will cross at values Moreover, energy levels corresponding to q and q ′ = q + 1 cross when which decreases as q increases. This then implies that for all k > J/(1 − N), the label q = 0 corresponds to the ground state, noting that for these values of k there are no further energy level crossings for this state. By a similar argument, for all corresponds to the ground state. For labels q = 1, 2, . . . , N 2 − 1, the ground state occurs when This is easily seen using standard calculus techniques. In other words, all the ground state energy level crossings occur from the lowest value The level crossings predicted by our analysis can be seen in Fig. 1. From the diagram we can see that when k = 0 the energies are equally spaced, e.g. see Fig. 1(a). For negative k, as |k| increases, a sequence of level crossings occurs, e.g. see Fig. 1(b). Also for negative k, for sufficiently large |k| the energies form a system of bands. When N is odd the number of energy levels is even, and the energy level bands occur in pairs |N, q and |N, N − q , each with separation J(N − 2q). When N is even, however, the number of levels is odd, and there is a single unpaired state, e.g. see Fig. 1(c). This points towards a prospect for number-parity effects, which will be explored below. We also remark that from (9), it is straightforward to calculate certain correlation functions. For example, for each q = 0, 1, ..., N

Continuum approximation
One straightforward approach to analyse the system is to introduce the variable l = q/N, 0 ≤ l ≤ 1, and treat this as varying continuously. This approximation is expected to be a valid in the limit of large N. To leading order in N (10) becomes For λ ≥ −1/2 the minimum value of energy occurs at l = 0, while for λ ≤ −1/2, the minimum occurs at We note that (14) is consistent with (12) in the large N limit. The following expression are then found for the ground-state energy and correlations: in this subsection The fact that the value of l as given by (14) is a function of λ is a reflection of the level crossings. It is also has the effective a treating the system a being gapless when λ ≤ −1/2. While this is correct in some sense, the above treatment does not capture the full physical properties of the model.

Ground-state energy gap
Define the ground-state energy gap ∆ to be the difference between the first excited-state energy and the ground-state energy. From previous discussion, we know that for fixed J and q = 1, 2, . . . , N 2 − 1, the energy level |N, q is the ground state for values of k given by (13). In the following, we only consider this range of k and q values, so that |N, q is the ground state. In this case, it is straightforward to show that E(N, q+1) = E(N, q−1) occurs when The difference in the energy corresponding to k P and the ground state is found to be J/ (N − 2q). Also, the difference in energy between the ground state |N, q and the state |N, N − q is J(N − 2q) which is greater than J/(N − 2q) for the given values of q. It follows that peaks in the gap must occur at the values k P given in (15), corresponding to the crossing of |N, q − 1 and |N, q + 1 . Fig. 2 plots ∆ as a function of k for N = 4, 10, 100, and by contrast, Fig. 3 gives the same plot for odd values N = 5, 11, 101. For the odd case, the maximum value of the gap is J for negative values of k, and for the even case the gap is unbounded as   k → −∞. It is given by ∆ = −J − k whenever k ≤ −J. Furthermore, these figures illustrate that the gap converges to a "sawtooth" function, however the cases of even N and odd N do not converge to the same function. Indeed, the relationship is one in which the locations of the zeros and peaks of the sawtooth functions are interchanged. In both instances the convergence is pointwise, which can be proved rigorously. In neither case, however, is the convergence uniform with respect to the ||.|| ∞ norm. This is an explicit example of a number parity effect. Fig. 4 plots ∆ as a function of λ for N = 5, 11, 101. It indicates that the gap vanishes in the limit N → ∞, consistent with the analysis of Subsection 3.1. The gapless regime, which occurs for k < −1/2, arises independently of N being even or odd, but the convergence is not uniform. In other words, in the continuum approximation, the number parity effect is lost.

Continuum approximation
For other aspects of the system, however, the number parity effect still has a significant influence. We investigate some further consequences of numberr parity in the remaining sections.

Supersymmetry
When N is even, k = J, and recalling we have fixed Ω = k/2, the Hamiltonian possesses supersymmetry [26]. This is expected since at these values we observe multiple two-fold degeneracies and a single state, the ground state, which is non-degenerate, that can be observed in Fig. 1(b). To formalise the result, note that the crossing of energy levels associated with states |N, q and |N, q ′ occurs when (11) holds. Set q ′ = N + 1 − q and define Q = N/2 q=1 qq ′ |N, q N, q ′ |, It is easily verified that Define the Hamiltonian where C is the su(2) Casimir element. Recall that the eigenvalue of C is Λ = N(N +2)/2. It is easy to check using only (16) that if |Φ is an eigenstate of (17) with eigenvalue E then Q|Φ and Q † |Φ are either eigenvectors with the same eigenvalue, or null vectors.
Explicitly from (17) Setting k = J in (10) gives the spectrum of (17), as confirmed by (18). However, there is no supersymmetry point in the coupling parameter space when N is odd. One example where this particular parity property has a striking manifestation is in the study of quantum dynamics.

Quantum dynamics
which represents an initial state such that all particles are in the same site. Define the expectation value of the fractional atomic imbalance to be where t denotes time. It can be shown using that a simple expression for I is obtained: At the supersymmetric point k = J when N is even, it is apparent that 0 ≤ I ≤ 1. For odd N at the same value of coupling parameters it is apparent that −1 ≤ I ≤ 1. Thus the even N case exhibits a type of self-trapping behaviour, while the odd N does not. While the phenomenon of self-trapping is well-known [1,3], such a parity influence on self-trapping does not appear to have been previously identified.
Since the expression (19) is an even function of k, exactly the same dynamical behaviour occurs for k = −J. For even N this corresponds to the smallest value of k such that the ground-state energy gap is zero. In contrast, for odd N the smallest value of k for which the ground-state energy gap is zero is k = −J/2. Illustrative examples of the expectation values for the fractional atomic imbalance at these parameter values are provided in Figs. 5 and 6 for N = 10 and N = 11. It is clear that the number-parity significantly influences the character of the dynamical behaviour. This remains true for different parameter vales, although the effects are not so pronounced. An example is given in Fig. 7, with parameter values in the vicinity of k = −1/N for J = 1.

Conclusion
We have studied an extension of the familiar two-site Bose-Hubbard model that includes a second-order tunneling term. This model is known to exhibit three phases determined by fixed-point bifurcations, and in the present work a detailed analysis has been undertaken along a line of the coupling parameter space that includes the boundary between phase-locking and self-trapping phases. All energy levels on this line can be computed analytically, and from this result it was identified that significant number- parity effects are present. In particular, the influence of number-parity on the groundstate energy gap, and the dynamics of the fractional atomic imbalance, were investigated. Mathematically, the model considered here is equivalent to the Lipkin-Meshkov-Glick (LMG) model of nuclear physics, which can be seen through the spin representation (5). The LMG model been studied through an exact Bethe Ansatz solution [27], although the exact solution has a different form to that of [20]. In [27] the analysis was conducted using a choice of coupling parameters such that the region of level crossing is treated as a gapless region in the limit of large particle number. Our results indicate that there may be new insights to be gained for the LMG model by choosing a different form of coupling parameters. Such an approach has recently been applied to the attractive one-dimensional Bose gas [28], whereby a distinction is made between the zero density thermodynamic limit and the weakly interacting thermodynamic limit which are obtained by different scaling of parameters as the system size increases.