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A new class of exactly solvable interacting fermion models in one dimension

We investigate a model containing two species of one-dimensional fermions interacting via a gauge field determined by the positions of all particles of the opposite species. The model can be solved exactly via a simple unitary transformation. Nevertheless, correlation functions exhibit nontrivial interaction-dependent exponents. A similar model defined on a lattice is introduced and solved. Various generalizations, e.g. to the case of internal symmetries of the fermions, are discussed. The present treatment also clarifies certain aspects of Luttinger's original solution of the ``Luttinger model''.

Exactly solvable models [1][2][3][4] have played an important role in the current understanding of one-dimensional interacting many-particle systems. These together with the idea of dominant low energy bosonic excitations of Fermi systems [5] gave rise to the emergence of the "Luttinger liquid" [6] as a unifying concept. Nevertheless, the technicalities of these exact solutions (bosonization, Bethe ansatz) are often rather complex. In the present paper we wish to introduce a class of interacting models which can be diagonalized by a simple (pseudo-)unitary transformation, yet exhibit nontrivial Luttinger-liquid behavior.
The models can be defined both in the continuum and on a lattice, and can have rather arbitrary single-particle bandstructure. Only the interactions are constrained to be of a particular "gauge form". The long-distance asymptotics of correlation functions can then be determined either by direct calculation or by a mapping on an effective Luttinger liquid description. Our investigation was inspired by Luttinger's original treatment of the "Luttinger model", and we will comment on this connection further below.
We start by considering the simplest model in our class, a one-dimensional fermion model with two species of particles, designated by a pseudospin index σ = ±, having coordinates x σi and momenta p σi = −i∂ x σi . The Hamiltonian of our model then is where we have introduced a "covariant momentum" Π σi = p σi + σA σ (x σi ), i.e. in this model, particles interact via a gauge potential, given for a particle at The potential V is an even function and vanishes at infinity. On a ring of length L, we will assume a potential V that is periodic in L. Clearly, the Hamiltonian is not time-reversal invariant, but it is invariant under simultaneous time reversal and charge (σ) conjugation.
The model can now be straightforwardly diagonalized by a (in general pseudo-)unitary transformation: noting that one can chose S so as to eliminate the interaction in eq.(1) by where E is the indefinite integral of the interaction potential: The transformed Hamiltonian then takes the form The eigenfunctions ofH clearly are Slater determinants of plane wave states |{k −,i }, {k +,j } , characterized by the sets of wavenumbers {k −,i } and {k +,j } for the − and + particles respectively. Consequently, the eigenfunctions and eigenvalues of the original Hamiltonian are obtained straightforwardly: At first sight it thus appears that the spectrum of the interacting Hamiltonian is independent of the interaction. Conformal field theory, or equivalently Luttinger liquid theory, then would imply that the asymptotic form of correlation functions (which is directly determined by the eigenvalue spectrum) is also interaction-independent. This conclusion is however incorrect: periodic boundary condition have to be treated carefully. In fact, keeping all other coordinates fixed, one easily finds S( where L is the length over which periodic boundary conditions are applied, N + is the total number of + particles, and the phase shift δ is given by An analogous result, with N + and δ replaced by N − and −δ, holds for the phase shift of the + particles. Consequently, the quantization condition on the wavenumbers is given by where the n ±,i are integer quantum numbers analogous to those used in the noninteracting case. Clearly, particles of one given "spin" orientation give rise to an effective Aharonov-Bohm flux acting on the other species, the value of the flux depending on the number of particles present. It should now also be clear why we refer to the transformation eq.(2) as pseudounitary: unless δ is "accidentally" an integer multiple of 2π, the plane wave states of the interacting and noninteracting problems obey different boundary conditions and therefore define different Hilbert spaces.
The ground state energy E 0 can be found in any sector with N ± particles, as follows. In where n 0 ±,i are the quantum numbers in the absence of the interaction, and we denote is not extensive, but on the scale expected from a magnetic field applied to the ring.
An effective Luttinger liquid description in terms of a bosonic field theory for the lowenergy properties can be obtained from the low-energy excited states. [6] To be precise, we start from a ground state with N ±0 = 2n 0 + 1 and assume that N ±0 δ is an integer multiple of 2π. We now add n ±R (n ±L ) particles at the right (left) Fermi points of the ± particles. Introducing particle number and current quantum numbers N ± = n ±R + n ±L and J ± = n ±R − n ±L the second order variation of the ground state energy is Up to quantum fluctuations, N ± and J ± are related to bosonic fields and their conjugate momentum density via N ± = −(L/π)∂ x φ ± and J ± = LΠ ± . The effective Hamiltonian including the low-energy quantum fluctuations then takes the form where n = 4n 0 /L is the particle density. Introducing new variables the Hamiltonian takes an apparently non-interacting form (eq.(10) with δ = 0). However, the expression of single-fermion operators [6] is changed and therefore the asymptotic decay law of the single particle Green function is obtained as with α = δ 2 /(2π 2 ). Thus, for any non-vanishing δ the decay is faster than 1/x, leading, amongst other things, to the well-known power-law singularity of the momentum distribution function at k F . The correctness of eq.(12) can be checked independently using the eigenfunctions of eq.(6): one obtains a Toeplitz determinant of the form previously considered by Luttinger [2], and which has the same asymptotic power law as obtained by the bosonization approach.
Similarly, correlations of two-particle operators decay as x −η , with interaction dependent exponent η. Specifically: The most slowly decaying correlations identify the dominant incipient instabilities. In the spin language, for positive δ then spiral spin-density wave correlations and opposite-spin We can now comment on Luttinger's original solution of his model. [2] In first-quantized form his Hamiltonian only contains first derivatives: We first remark that this Hamiltonian is a conserved quantity as far as the Hamiltonian (1) is conserved, and shares all (non degenerate) eigenfunctions. It is unbounded from below though, unlike H in Eq(1). Hence the issue of finding its groundstate is replete with difficulties familiar from relativistic field theories. The second-quantized version of the model H Lutt can be solved consistently and exactly by filling the Dirac sea and using bosonization. [3] This leads, amongst other things, to an asymptotic decay exponent of the single particle Green function α = 1/ 1 − (δ/π) 2 − 1. In a first-quantized framework, a consistent but different solution can be obtained if one is willing to consider quasi-groundstates where single-particle states below a certain very negative cutoff energy E cutoff are left empty (evidently, the model does not have a conventional groundstate). This "rapidity cutoff" in fact is frequently used in the Bethe ansatz solution of field theoretical models [7,8]. be found using Luttinger's original paper and lead to the same asymptotic decay exponent α of the Green function as in eq.(12). This result was in fact obtained in Luttinger's paper, [2] i.e. Luttinger's result in fact applies to the first-quantized solution of the model described here. [9] The same result for correlation exponents can also be obtained by considering variations of the energy with particle number, similar to what we described above. We note that the Mattis-Lieb and Luttinger results for α, though different in general, agree to the lowest nontrivial order in δ. The differences at higher order clearly have to be attributed to the different cutoff procedures used in the two calculations.
We now turn to similar models defined on a one-dimensional lattice. Specifically, we will consider the Hamiltonian where α is a periodic function, α m+L = α m , and the number operator n m = c † m c m . This corresponds to a lattice where the up electrons feel a "gauge potential" due to the down electrons, and vice versa. Here periodic boundary conditions are implied, i.e. c L+1,σ = c 1σ .
We notice that for the particular case where only α 0 is nonzero, this lattice model only involves two-particle interactions, contrary to our continuum model where some three-body interaction is unavoidable.
The interaction term V can be either zero, or one of two non trivial functions which retain the exact solvability. We will consider the XXZ model and the Hubbard model, i.e.
and will show below that these are exactly solvable. The XXZ model corresponds to two copies of the usual model, wherein the two species of particles (spin up and down) only talk to each other via the phase factors. The Hubbard model corresponds to the usual two body interaction.
We now perform a unitary transformation induced by U = exp(iS) where S = 1≤l,m≤L β l,m n l↑ n m↓ . It is easy to see that β i,j = −β j,i , i.e. an odd function is appropriate, and we will assume it to be so. Thus we find c mσ → e iS c mσ e −iS = c mσ exp −iσ l β m,l n l,−σ (20) The transformed Hamiltonian takes the form We now use the freedom in defining β to cancel the interior terms in the phase factor by choosing β m+1,l − β m,l = α m−l for 1 ≤ m ≤ L − 1 and 1 ≤ l ≤ L.
The hop across the L ↔ 1 bond has a total phase It is in fact not necessary to solve explicitly for β, although it is easy enough to do so for simple choices of α. We can add eqs. In the presence of a nonzero extra interaction V , previous work [10] can be used where the Bethe Ansatz has been adapted to the case of a "spin twist", which is precisely the case needed here. We write the solution immediately: for the XXZ model and with the usual phase shift and the usual integers I n . For the Hubbard model we have a pair of equations. With E = − cos k n with 1 ≤ n ≤ N, and N = N ↑ + N ↓ we find In these models, one has non Fermi liquid behavior even in the absence of α r , and adding this changes the exponents, and indeed even the symmetries of the model, e.g. for the Hubbard model we have less than SU(2) invariance. The detailed behavior of the lattice models and the resulting exponents will be reported elsewhere.
There is a number of further possible generalizations of the present model: first, the unitary transformation (2) will in fact put Hamiltonians containing arbitrary powers of the covariant momenta Π σi into diagonal form. Provided the highest nonvanishing power is even, these models have a well-defined groundstate. One could thus study models with complicated bandstructures, involving e.g. more than two Fermi points. Similarly, in the lattice model certain forms of hopping terms beyond nearest neighbors can be included.
One can further solve the case where the number of flavors for the + and − particles is different. In