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On the Thermodynamic Limit of the Lipkin Model

The thermodynamic limit of the Lipkin model is investigated. While the limit turns out to be rather elusive, the analysis gives strong indications that the limit yields two analytically dissociated operators, one for the normal and one for the deformed phase. While the Lipkin Hamiltonian is hermitian and has a second order phase transition in finite dimensions (finite particle number), both properties seem to be destroyed in the thermodynamic limit.


INTRODUCTION
The Lipkin model [1], even though it appeared more than forty years ago in the literature, has, up to present, given rise to a vast amount of research papers. This continuing interest is due to its basic properties in that it demonstrates the mechanism of a phase transition including symmetry breaking for a schematic many body system. Originally introduced as a model in nuclear physics [2] it has found applications in a broad range of other topics: statistical mechanics of quantum spin systems [3], Bose-Einstein condensates [4] as well as quantum entanglement [5], to name but a few. Recent interest is focused upon its large N behaviour with N being the particle number. Progress has been achieved using a variety of methods such as continuous unitary transformations [6], a semiclassical approach [7] or the Bethe ansatz [8].
We briefly list here the major results relevant to the present paper. In its original form the Lipkin model considers interacting Fermions occupying two Ω-fold degenerate levels.
One major appeal is due to its easy solubility. This is achieved by rewriting the kinetic energy term of the Fermion Hamiltonian, viz. k,σ σa † k,σ a k,σ and the two body interaction term in terms of the SU(2) operators J z and J 2 ± , respectively. In dimensionless form the Hamiltonian thus reduces to where the 2j + 1 = N + 1-dimensional representations of the SU(2) operators are used with N ≤ Ω being the particle number. Here the interaction is scaled by N to ensure that H is extensive, the operators J 2 + and J 2 − effectively scale as N 2 . The Hamiltonian allows reduction into two spaces: m integer and m half-integer, with m the eigenvalues of J z ; it corresponds to N even and odd, respectively and is denoted as parity. For λ > ∼ 0 the N-even and N-odd levels are obviously separated and remain so for all λ < 1 while the levels become degenerate for λ > 1. The phase at λ < 1 is called the normal phase while the symmetry (parity) breaking phase at λ > 1 is called the deformed phase. For λ > ∼ 1 this phase transition is confined to the lower part of the spectrum in that higher up in the spectrum the normal phase still prevails. In fact, for a specific λ > 1 there is an energy E kc (λ) such that the ordered spectrum E k is for k < k c associated with the deformed phase while for k > k c the energies relate to the normal phase [9]. In other words, the transition moves up in the spectrum from the ground state E 1 to E N/2 when λ increases from unity to infinity (we confine ourselves to the lower half of the spectrum as the symmetry of the model yields the same pattern in the upper half but mirror reflected). For finite N, E kc is clearly characterised by the minimum of the level distance; this minimum gap vanishes for N → ∞. In fact, at λ = 1 the large N scaling behaviour [7,8] is found whereas for λ > 1 the gap around E kc behaves as [7] We mention that a mean field approach yields the result [10] being obviously unable to reproduce the richer structure described above; however, the last two equations do describe the lower part of the spectrum remarkably well.
In the present note we aim at a more complete description of the thermodynamic limit of the Lipkin model.

THE SPECTRUM
A closer look at the essential features of the spectrum for large N may elucidate state- The range 0 > ǫ > −1 is the normal phase while ǫ < −1 corresponds to the deformed phase. For better illustration we do not display the lower end of Fig.1, it suffices to note that [7] for λ > 1 the ground state assumes the value The critical line is crossed by each level sequence at a certain x c (λ) associated with λ > 1; the larger λ the larger x c , i.e. the phase transition occurs higher up in the spectrum and higher above the ground state given in Eq. phase transition through the singularity, but it is also associated with a specific state vector being identical to the unperturbed ground state once the limit N → ∞ is attained. In fact, where the second term of the right hand side vanishes in the limit. The right hand side also becomes independent of λ when N → ∞. Recall that for finite N all levels and state vectors are analytic functions of λ and analytically connected in the λ-plane. As a special result we further note that this state is perfectly localised; all other states are more or less extended superpositions of vectors |j, m , −j ≤ m ≤ j; this localisation was noted also in [7].
The singularity discussed above for the level density is found as well when a particular are analytically connected, in fact, each E k (λ) represents the values on a particular Riemann sheet of one analytic function [12].
In Fig.3 we illustrate the exceptional points for a few small values of N. Two aspects are of importance: (i) for increasing N some EPs accumulate along the real axis λ ≥ 1 and, (ii) at the EPs of the sequence along the real axis, starting at λ = 1, the levels coalescing are E 1 and E 2 , E 2 and E 3 , E 3 and E 4 and so forth.
The second point clarifies immediately some observations made in the previous section.
Take, for instance, λ = 2. From Fig.3 we read off that there are N/8 EPs along the real axis between λ = 1 and λ = 2. This implies that the Riemann sheet (energy level) with k = N/8, that is for x = 0.25, has a singularity at λ = 2. In general terms, there is just one x c associated with a particular value of λ.
With regard to the first point we note two aspects that are, however, related. In the limit N → ∞ there is a dense set of EPs, say at λ = 2, for x = 0.25. The singularity, being an accumulation of square root branch points, cannot be expected to be of a square root nature. The more likely behaviour is that of a logarithmic singularity. This is supported by a similar, albeit simpler situation [13], where an accumulation of square root branch points has been demonstrated analytically to lead to a logarithmic branch point. There is further strong numerical evidence: a perfect fit is obtained of the curves in Figs.1 and 2 using the trial functions employing three terms or less. The trial function (8) is used to fit the spectra of Fig.1 (after a shift upward by one unit to make the critical line occurring at zero). In a subsequent acid test the derivative of (8) is compared with the respective curves for ∆ǫ in Fig.2 yielding an excellent agreement. Note that the modulus of x − x c is being used implying that the function is non-analytic at this point. Depending on the sign of x − x c a separate fit must be made. As it was mentioned in the previous section the same pattern is found when a level is considered as a function of λ. A good fit is obtained replacing x and x c by λ and λ c , respectively, in (8).
Not only do we have evidence for a very special singularity, the fact that the EPs accumulate on the real axis indicates that the thermodynamic limit of the Hamiltonian (1) cannot be a hermitian operator. Moreover, if the limit is taken either in the deformed or in the normal regime, the two regimes become disconnected for real values of λ.

CONCLUSION
We summarise our findings: the thermodynamic limit N → ∞ of the Hamiltonian (1) leads to a specific singular behaviour associated with the phase transition. It is immaterial as to whether a specific level is considered as a function of λ or whether the whole spectrum is considered for a fixed value of λ. In either case, a logarithmic singularity is found at a particular value of λ being uniquely related to a specific level. Note that the singularity is non-analytic in that, on the real λ-axis, the behaviour on the left hand side of the singularity is not the continuation of that on the right hand side. The singularity produces an infinity in the respective second derivative. As such, it is no longer a second order phase transition as it is found for finite N.
Furthermore, the fact that the EPs accumulate on the real λ-axis in the limit, renders the operator nonhermitian in the limit. In fact, depending on how the limit is taken, a different result emerges: one operator describing the normal phase and the other describing the deformed phase. In contrast to the case of finite N, the two are dissociated. This is in contrast to finite N, where all energies and state vectors are analytically connected.
In particular, when the limit N → ∞ is attained, the state vector |j, −j (see Eq. (7)) is common to all spectra, irrespective of λ. As a particular feature, this vector has, for λ > 0, a non-zero overlap with any other vector of either the normal or the deformed sector. This is in contrast to finite N, where (1) yields the usual complete orthonormal basis for any given λ.
It is worth to mention that the semiclassical approach [7] has explicitly given rise to ambiguities of a hermitian thermodynamic limit. An unambiguous limit could be obtained only to lowest order in ∼ 1/N.
We refrain from speculating about physical consequences of our findings. Rather we feel that the thermodynamic limit of the Lipkin model is unsuitable to describe a physical system despite its beauty for finite yet large values of N. We see the value of this analysis, apart from its own sake, in a specific contribution of the general understanding of models describing quantum phase transitions.

Acknowledgment
The author gratefully acknowledges helpful discussions with Hendrik Geyer and Frederic Scholtz.