The Staggered Six-Vertex Model: Conformal Invariance and Corrections to Scaling

We study the emergence of non-compact degrees of freedom in the low energy effective theory for a class of $\mathbb{Z}_2$-staggered six-vertex models. In the finite size spectrum of the vertex model this shows up through the appearance of a continuum of critical exponents. To analyze this part of the spectrum we derive a set of coupled nonlinear integral equations from the Bethe ansatz solution of the vertex model which allow to compute the energies of the system for a range of anisotropies and of the staggering parameter. The critical theory is found to be independent of the staggering. Its spectrum and density of states coincide with the $SL(2,\mathbb{R})/U(1)$ Euclidean black hole conformal field theory which has been identified previously in the continuum limit of the vertex model for a particular 'self-dual' choice of the staggering. We also study the asymptotic behaviour of subleading corrections to the finite size scaling and discuss our findings in the context of the conformal field theory.


I. INTRODUCTION
Studies of two-dimensional vertex models with local Boltzmann weights satisfying the Yang- Based on a particular solution of the Yang-Baxter equation vertex models on lattices of arbitrary sizes can be defined using the co-multiplication property of the Yang-Baxter algebra. Introducing inhomogeneities -either by shifts in the spectral parameter and/or by using different representations of the underlying algebra for the internal local degrees of freedom [4,31] -allows to generalize these models further while keeping their integrability or to uncover relations to different systems: the equivalence of the q-state Potts model on the square lattice and a staggered six-vertex model [6,40] allowed for the solution of the Potts model by means of the Bethe ansatz [5]. Other choices for the staggering parameter in the six-vertex model lead to integrable quantum chains with longer ranged interactions [18,38] and have been found to appear in the zero charge sector of a vertex model based on alternating four-dimensional representations of the quantum group deformation of the Lie superalgebra sl(2|1) [17].
Recently, the critical properties of the staggered six-vertex model have been investigated for a particular choice of the staggering parameter corresponding to one of the integrable manifolds of the antiferromagnetic q-state Potts model [23,27]. For this 'self-dual' case the model has an additional discrete Z 2 -invariance leading to a conserved charge which can be used to classify the spectrum. Remarkably, it has been found that although the lattice model is defined in terms of (compact) spin-1/2 degrees of freedom the low energy effective theory has a continuous spectrum of critical exponents. From a finite size scaling analysis and the computation of the density of states in the continuum the latter has been identified with the non-compact SL(2, R) k /U (1) sigma model, a conformal field theory (CFT) on the two-dimensional Euclidean black hole background [9,24]. The appearence of a non-compact continuum limit of a lattice model with finite number of states per site has also been observed in staggered vertex models with supergroup symmetries, see e.g. Refs. 8, 15-17. The spectrum for finite lattices is necessarily discrete. In the staggered six-vertex model the continuous spectrum emerges in the thermodynamic limit by closing the gaps between critical exponents as 1/(log L) 2 . As a consequence the investigation of this scenario requires to compute energies for very large system sizes. For the integrable models mentioned above this is only possible after formulation of the spectral problem in terms of nonlinear integral equations (NLIEs) in which the system size enters as a parameter only. For the self-dual staggered six-vertex model such NLIEs have been derived and solved numerically to identify the low energy effective theory [9].
In this paper we derive a different set of NLIEs for the finite size spectrum of the spin chain from the Bethe ansatz solution of the staggered six-vertex model with anisotropy 0 < γ < π/2. These equations hold for arbitrary values of the staggering parameter γ < α < π − γ, in particular away from the self-dual line α = π/2. After recalling what is known about the finite size spectrum of this model we propose a parametrization of its emerging continuous part in terms of the eigenvalues of the quasi-momentum operator. We solve the NLIEs numerically for systems with up to 10 6 lattice sites to verify this proposal. Both the finite size spectrum and the density of states are found to depend only on the anisotropy γ but not on the staggering parameter α. This suggests that the effective theory for the staggered model is the black hole sigma model CFT SL(2, R) k /U (1) at level k = π/γ > 2, independent of α. We also study the subleading terms appearing in the finite

II. THE STAGGERED SIX-VERTEX MODEL
The six-vertex model on the square lattice, see Fig. 1, is defined through the R-matrix The spin variables lying on the horizontal and vertical links of the lattice take values from V 1 and V 2 , respectively.
Similarly, we associate the spectral parameters λ and µ to horizontal and vertical lines while γ parametrizes the anisotropy of the model.
In the vertex model depicted in Fig. 1 we have introduced an additional staggering in the vertical direction. The corresponding double-row transfer matrix is the product of commuting operators On two lines in the space of coupling constants γ and α this staggered six-vertex model exhibits additional quantum group symmetries: the first occurs for α = γ. In this case the product of neighbouring R-matrices in (2.2) degenerates and the underlying quantum group symmetry is that present in the integrable spin-1 XXZ chain [47]. A second special line follows from the spectral equivalence of the models with staggering parameter α and π − α [17]. This leads to the presence of a discrete Z 2 -invariance on the 'self-dual' line α = π/2 where the model is equivalent to one of the integrable manifolds of the antiferromagnetic q-state Potts model [5,23]. On this line the model has a quantum group symmetry related to the twisted quantum algebra U q [D 2 ] with q = e 2iγ [17]. The commuting operators generated by the double-row transfer matrix (2.3) can be written as sums over local (i.e. finite-range on the lattice) interactions, independent of the system size: as a consequence of R(λ, λ) being the permutation operator on V 1 ⊗ V 2 , t (2) (ξ) acts as a two-site translation operator. Therefore we can define the momentum operator as P = −i log t (2) (ξ) = −i log(t(ξ) t(ξ + iα)). The next term in the expansion of log t (2) (ξ) is the spin chain Hamiltonian with nearest and next-nearest neighbour interaction. In terms of local Pauli matrices the Hamiltonian reads up to an overall factor (2.5) Note that this Hamiltonian differs from the one given by Ikhlef et al. [23] for α = π/2 by a unitary rotation of spins on one sublattice. Following Refs. 9, 24 we also define a quasi-shift operator τ ≡ t(ξ) [t(ξ + iα)] −1 and the corresponding quasi-momentum Similar as in the self-dual case τ acts as a diagonal-to-diagonal (light-cone) transfer matrix.

III. BETHE ANSATZ SOLUTION
In the six-vertex model the number of down arrows is conserved. This U (1) symmetry allows to diagonalize the transfer matrix starting from the reference state |0 ≡ 1 0 ⊗2L by means of the algebraic Bethe ansatz [30]. The resulting eigenvalues in the sector with M ≤ L down arrows are (3.2) In Eq. (3.1) the rapidities λ j are solutions to the Bethe equations Note that these equations ensure the analyticity of the transfer matrix eigenvalues Λ(λ) at the points λ = λ j , j = 1, . . . , M .
Since the expressions above depend only on the difference λ − ξ we are free to choose ξ ≡ i(γ − α)/2. This results in the symmetric Bethe equations , j = 1, . . . , M .
Similarly, the energy eigenvalues E of the Hamiltonian (2.4) and the eigenvalue K of the quasimomentum operator (2.6) are found to be: Using the expression (3.1) for Λ(λ) in terms of the Bethe roots {λ j } they are found to be sums over contributions from single rapidities, i.e. E = M j=1 ǫ 0 (λ j ) and K = M j=1 k 0 (λ j ), where ǫ 0 (λ) = 4 sin γ ch(2λ) cos α − cos γ ch(2λ) − cos(α − γ) ch(2λ) − cos(α + γ) , k 0 (λ) = log ch(2λ) − cos(α + γ) ch(2λ) − cos(α − γ) . In the following we shall analyze the ground state and lowest excitations of the model with Hamiltonian (2.4) for staggering γ < α < π − γ (denoted as phase B in Ref. 17). In this regime the low energy states have been found to be described by configurations involving two types of Bethe roots for all anisotropies 0 ≤ γ ≤ π/2, namely {λ j } with Im(λ j ) = 0 or π/2: Among these the ground state of the staggered spin chain is realized in the sector with M = n 1 + n 2 = L and roots ν j , µ j filling the entire real axis. Taking the thermodynamic limit L → ∞ with fixed n 1 /L, n 2 /L their distributions are described by densities ρ 1 (ν) and ρ 2 (µ) which are determined through coupled linear integral equations [14,44]. In the range of parameters considered here these equations have been solved to give [17] ρ 1 (ν) = sin π(α−γ) Integrating these expressions we find the total densities of the two types of roots in the ground state to be (3.10) Note that under the action of the duality transform α → π −α the two types of roots are exchanged.
This allows to restrict our analysis to staggering parameters γ < α ≤ π/2 in the following. (note that m = n 1 + n 2 − L ∈ Z by construction), and their momentum which can be parametrized by a single vorticity w ∈ Z. Also within the root density approach the energy of these excitations has been found to be [17] (see also Refs. 23, 27 for the self-dual case) Here ε ∞ is the bulk energy density of the state with root densities (3.9) is the Fermi velocity of the low energy modes, n ± are non-negative integers characterizing particle-hole type excitations and the corrections to scaling R(L) vanish as L → ∞.
Similarly, the eigenvalue K of the quasi-momentum operator can be computed within the root density approach. For the ground state K is proportional to the system size, the corresponding 'quasi-momentum density' of the ground state in the thermodynamic limit is (3.14) Note that k ∞ = 0 on the self-dual line α = π/2 as a consequence of k 0 (λ) = −k 0 (λ + iπ/2) and To analyze the finite size spectrum (3.12) further and to identify the low energy effective theory for the staggered six-vertex model one has to study the system size dependence of the coupling valid for anisotropies 0 ≤ γ < π/2. 1 Numerical data for general staggering suggested that the finite-size spectrum is in fact independent of α ∈ (γ, π/2] and consistent with this expression [17]. To go further, Candu and Ikhlef [9] have reformulated the spectral problem for the self-dual model in terms of nonlinear integral equations. In the next section we shall derive such integral equations which are valid for arbitrary staggering γ < α ≤ π/2 and which are then used to analyze the continuous part of the spectrum and the corrections to scaling R(L) in (3.12). , are outside the shaded strips | Im z| ≤ γ 2 and | Im z − π 2 | ≤ γ 2 of the complex plane.

IV. NONLINEAR INTEGRAL EQUATIONS
To be specific we consider solutions to the Bethe equations corresponding to low energy excitations (3.12) with (m, w) = (0, 0). These configurations contain a total of M = n 1 + n 2 ≡ L roots (3.8) distributed symmetrically around the imaginary axis Fig. 2 for an example. Following Refs. 28, 29 we introduce auxiliary functions and a 2 (λ) ≡ a 1 (λ + iπ/2), thereby encoding the Bethe roots (3.8) in the zeroes of (1 + a 1 )(λ) and (1 + a 2 )(λ), respectively. The additional zeroes of these expressions are called hole-type solutions, c.f. Fig. 2. This allows to rewrite the Bethe equations (3.4) in the sector of n 1 + n 2 = M = L roots and parameter ranges 0 < γ < α ≤ π/2 in terms of coupled nonlinear integral equations (NLIEs) Here we have chosen the branch cut of the logarithm along the negative real axis. Note that unlike in the NLIEs derived earlier for the self-dual model [9] the convolution integrals in (4.3) and (4.4) are computed along a fixed non-intersecting closed contour C (c.f. Fig. 3). As a consequence, the system size L enters only as a parameter in the driving terms. The kernels have no poles which need to be considered in the contour integration. Furthermore, their Fourier transformations on the contour are regular which is particularly useful for the numerical evaluation of the convolution integrals.
The Bethe numbers and thus the auxiliary functions a 1 and a 2 fix the eigenvalue of the transfer matrix. For λ inside the closed contour C this yields .
The proof is straightforward and only involves Cauchy's theorem. Considering the logarithmic derivative of a(λ), the summation part can be cast into an integral representation involving the auxiliary functions: for any analytic f (λ) the relations hold if the contour encloses all Bethe roots but none of the hole-type solutions and α-dependent (similarly {µ j } n2 j=1 for a 2 ) one has to introduce a distance parameter ε > 0 to avoid singularities at the border Im λ = ±η/2, e.g. from the kernels. Extending C to ±∞ covers all system sizes L. according to Fig. 3. As the expression d(λ) is analytic within C Eq.
the summation part can be cast into an integral representation according to (4.7) as long as λ and λ − iγ remain outside the closed contour. For λ inside the closed contour C the additional term ∂ log 1 + a 1 (λ) can be absorbed into the contour integral involving a 1 by Cauchy's theorem.

A. Mixed Eigenvalues
The energy E and quasi momentum K of the eigenstates are related to the logarithmically combined expressions (3.6) evaluated at the points ξ and ξ + iα being zeroes of a(λ). The bulk parts can be split off by considering the mixed eigenvalues (4.8) Due to iπ-periodicity of (3.1) the mixed eigenvalues are periodic, log Λ mix (λ + iπ/2) = log Λ mix (λ), and antiperiodic, log Ω mix (λ + iπ/2) = − log Ω mix (λ) with respect to iπ/2 satisfying the functional and respectively. To finally evaluate (3.6) for general staggering the equations (4.9) and (4.10) can be solved in Fourier space for Λ mix and Ω mix using the transformation pair As Λ mix (λ) and Ω mix (λ) are analytic 2 in λ ∈ C π+2γ 4 < Im λ < π−γ 2 and the region enclosed by the contour C a standard manipulation in Fourier space yields with the iπ/2-(anti)periodicity 2 valid for the range 0 < γ ≤ α 2 ; similar results can be obtained for α 2 < γ < α ≤ π 2 where we used the Fourier transform of log d(λ), (4.14) Note that this system (4.12) and (4.13) already describes the energy E and quasi momentum K according to (3.6) in the self dual case α = π/2. However, as Λ mix (λ) and Ω mix (λ) are composed from simple eigenvalues Λ(λ), c.f. (4.8), one can solve the system for Λ x− iπ 4 + iγ 2 and Λ x+ iπ 4 + iγ 2 . Recombining after suitably shifting the arguments (3.6) reads in Fourier representation for general staggering γ < α ≤ π/2. From the bulk parts of these expressions we can read off the energy density ε ∞ (3.13) and quasi momentum densities k ∞ (3.14) already obtained within the root density approach above.
Using the relation a −1 (−λ) = a(λ) provided by the Bethe root's symmetry (4.1) energy and momentum reduce for all 0 < γ < α ≤ π/2 to with p, q ∈ N prime to each other (p = q = 1 on the self-dual line) we have to choose L c being an integer multiple of (p + q). For this choice of parameters the ground state corresponding to (m, w) = (0, 0) and m = 0 is described by integer numbers n (0) 1 = Lp/(p + q) and n (0) 2 = Lq/(p + q) of Bethe roots on the real line and with Im(λ j ) = iπ/2, respectively. Excitations can be constructed by shifting t roots between these two sets, i.e. with n 1,2 = n (0) 1,2 ± t, resulting in m = 2t. Similarly, we can consider the spectrum in sectors where the commensurability condition is not satisfied: let L = ℓ 0 (p + q) + r with ℓ 0 ∈ N, r = 1, 2, . . . , (p + q − 1). The Bethe states are described by n 1 = ℓ 0 p + t + r and n 2 = ℓ 0 q − t roots of the two types. From (3.11) we obtain m = 2t + 2rq/(p + q) for these states. Together this allows to vary m in steps of 2q/(p + q) for the staggering parameter (5.1).
The observed presence of a continuous component of the spectrum implies that there should be a corresponding continuous quantum number in the thermodynamic limit L → ∞. For the self-dual model, i.e. α = π/2, it has been shown that this quantum number is in fact related to the conserved quasi-momentum K of the corresponding excitation [9,24]: for large L the number m characterizing the Bethe configuration (3.11) is related to the rescaled quasi-momentum s ≡ π−2γ 4πγ K as Here L 0 is a non-universal length scale which only depends on the anisotropy γ while the function B(s) determines the (finite part) of the density of states in the continuum part of the spectrum.
This line of arguments can be implemented in a straigthforward way for staggering away from the self-dual line: here the quasi-momentum has a non zero value (3.14) in the ground state ((m, w) = (0, 0) and m = 0). Since the expression (3.15) has been found to be consistent with numerical results [17] for the finite size spectrum for α ∈ (γ, π/2] we propose that the quasi- shown to agree with the known result for the SL(2, R)/U (1) sigma model [22,34] ρ BH (s) = 1 π log ǫ + ∂ s (s B BH (s)) , (the term log ǫ arises from the regularization needed to handle divergencies arising in the string theory).
From its effect on the finite size spectrum a staggering α = π/2 is an irrelevant perturbation of the low energy effective theory. This is consistent with the assumption that the critical theory for the entire phase 0 ≤ γ < α < π − γ is the Euclidean black hole sigma model CFT. Due to the presence of a continuous spectrum, however, it is not sufficient for this identification to rely on the finite size spectrum alone. In addition the density of states in the continuum has to be computed.
According to the considerations at the beginning of this section, the allowed values of m for given staggering α and system size L differ by multiples of 2. Therefore, the density of states in the continuum follows from (5.2) to be (see also Refs. 9,24) ρ(s) = 1 2 ∂ s m = 2 π log L L 0 + ∂ s (sB(s)) . H * of the CFT by terms involving irrelevant operators [11] and therefore should provide additional insights into the particular lattice regularization of the CFT considered. If the deviations are small these terms can be written as where Φ b are conformal fields with scaling dimension X b = h b +h b > 2 and conformal spin The coupling constants g b are in general unknown.
The effect of these terms on the finite size spectrum can be studied within perturbation theory [2,11]: to second order one finds Here conformal invariance has been used to compute the matrix elements of the perturbation (6.2) to be present in any theory [10,11,39]. Based on these insights the irrelevant operators present in lattice formulations of various unitary models have been identified: the 'analytic' corrections to scaling resulting from operators in the conformal block of the identity have been studied to even higher orders as in (6.3), c.f. [25,39,42]. For the six-vertex model including its higher spin variants subject to various boundary conditions some of the deviations from the respective fixed point have been identified based on numerical studies of the finite size spectrum [2,3,37]. Very recently, it has been shown that similar arguments as above can also be applied to non-unitary models where the spectrum may contain zero norm states and therefore the scalar product used in conformal perturbation theory has to be adapted to properly deal with Jordan cells, see e.g. Ref. 13. With such a modification the perturbative analytical corrections to scaling in a logarithmic minimal CFT with central charge c = −2 describing critical dense polymers have been found to coincide with the exactly known spectrum for a lattice model [26].
Corrections to scaling similar to (6.3) are expected to arise in systems with a continuous spectrum [46].  Fig. 10. Therefore, even larger system sizes are needed for a quantitative analysis of the asymptotic behaviour of R(L). Based on our data, however, we find an algebraic decay consistent with the conjecture (6.5).
To interpret these findings for the staggered six-vertex model with its low energy description in terms of the Euclidean black hole sigma model CFT the considerations leading to (6.3) need to be for the presence of a perturbation of the fixed point Hamiltonian by a continuum of conformal fields. Furthermore, since the ground state of the lattice model is not the (non-normalizable) vacuum of the CFT but rather the state corresponding to the lowest conformal weight (h 0 ,h 0 ), first order corrections cannot be excluded to contribute to R 0 (L). This is consistent with the fact that the asymptotic behaviour of all states considered is governed by the same exponent (6.5).
Taking into account these modifications to (6.3) would lead to the conclusion that the numerical data for R(L) for the lattice model are the first order effect of a perturbation of the fixed point Hamiltonian by a descendent of the identity operator with dimension X I = 4 and a continuum of operators starting with dimension X k = 2π/γ − 2 = 2(k − 1). The latter, however, is not in the spectrum (5.5) of the SL(2R)/U (1) coset model 3 : while the γ-dependence of (6.5) could be realized by a perturbation through fields with vorticity w = ±2 there is no sign of the divergence due to the contribution of the non-compact degree of freedom to the conformal weights as γ → π/2 (or k → 2).
To resolve this discrepancy one has to consider additional ways how the regularization of the CFT in terms of the staggered six-vertex model on a finite lattice can affect the asymptotic Ldependence of the corrections to scaling, Eq. (6.3). Here one has to take into account that the latter are given -apart from the coupling constants appearing in the perturbation (6.2) -in terms of universal quantities such as the scaling dimensions and OPE coefficients of the CFT.
Let us now assume that the perturbation of the fixed point Hamiltonian H * present in the staggered six-vertex model is given in terms of an operator from the conformal block of the identity with dimension X I = 4 and operators from the continuum of fields with quantum numbers (m, w) = (0, 2) and SL(2, R)-spin j = (−1/2+ is). On a finite lattice the latter is quantized as a consequence of (5.2) with ∆s ≃ π/(2 log L). Then, for sufficiently large L, the OPE coefficients appearing to the first order expression for the corrections to scaling in the ground state of the lattice model Wess-Zumino-Novikov-Witten model, the OPE coefficients have been found to be given in terms of double Gamma functions depending on combinations of the spins j a = (−1/2 + is a ) and k = π/γ [12,41,45]. With (6.6) this gives a rise to an additional L-dependence in the individual terms contributing to the corrections to scaling (6.3) which may account for the observed asymptotics with exponent (6.5).
Finally, we note that the exponent (6.5) vanishes as γ approaches π/2 indicating the appearance of a marginal operator in the perturbation of the fixed point Hamiltonian which leads to a different low energy effective theory. In the staggered six-vertex model some of the vertex weights vanish in this limit and the lattice model has an OSP (2|2) symmetry [23].

VII. DISCUSSION
We have investigated the finite size spectrum of the staggered six-vertex model for the range of parameters 0 ≤ γ < α < π − γ. As has been noted in previous works the continuous component of this spectrum leads to a strong logarithmic size dependence [17,23,27]. Therefore both a formulation of the spectral problem allowing for numerical studies of large system sizes and insights into the parametrization of the low energy degrees of freedom in terms of the parameters of the lattice model are needed. For the self-dual model, α = π/2, these points have been addressed before and provided evidence for the proposal that the critical theory of the model is the SL(2R)/U (1) sigma model at level k = π/γ > 2 describing a two-dimensional Euclidean black hole [9,24].
We have derived a set of coupled NLIEs (4.3) and (4.4) which generalize the ones obtained previously for the self-dual case α = π/2 [9] to the range of staggering given above. The kernel functions appearing in the NLIEs used here are regular in Fourier space. As a consequence this formulation is particularly suitable for their numerical solution: we can compute the energies of the ground state and in the continuum above it for chains with up to 10 6 lattice sites for arbitrary staggering γ < α ≤ π/2. Based on our numerical data we have extended the proposal [24] for the quantum number for the continuous part of the spectrum in terms of the conserved quasimomentum of the vertex model for staggering away from the self-dual line, Eq. (5.3). With this input we were able to compute the density of states of the model from the finite size spectrum obtained by numerical solution of the NLIEs. Together with the existing data for the finite size spectrum [17] this shows that the model is in the same universality class as the self-dual model independent of the staggering γ < α < π − γ. Both the finite size spectrum and the density of states agree with what is known about the Euclidean black hole sigma model. Finally, we have extended previous studies of the finite size spectrum [17,23] by considering the corrections to scaling due to irrelevant perturbations of the fixed point Hamiltonian appearing in the lattice model. Such perturbations are expected to lead to subleading power-laws in the finite size spectrum which can provide additional information on the operator content of the continuum model and insights into the emergence of the continuum of critical exponents in the thermodynamic limit of the lattice model. Again, our numerical data suggest that the variation of the staggering parameter does not change the critical theory: different values α only lead to small changes in the non-universal coupling constants g in (6.2). As for the interpretation of our numerical results summarized in the conjecture (6.5) for the asymptotic algebraic decay of the corrections to scaling, however, we find that the known predictions for theories with purely discrete spectrum have to be modified here. These modifications appear to be closely related to the way how the non-compact degree of freedom is dealt with in the regularization of the field theory leading to the staggered six-vertex model. To make progress in this direction additional work from the CFT side is called for, in particular with respect to the operator product expansion in theories with non-compact target space.
A natural extension to our work would be the finite size scaling analysis of the lattice model in sectors with non-zero magnetization, i.e. with m = n 1 + n 2 − L = 0, or non-zero vorticity w.
In the derivation of the corresponding NLIEs this amounts to consider hole-type solutions of the Bethe equations (3.4) appearing inside the integration contours which lead to additional logarithmic driving terms. Another direction for future work is to consider more general lattice models which develop a continuous spectrum of critical exponents in the thermodynamic limit. Known examples with such a behaviour are the supersymmetric vertex models based on alternating representations of U q [gl(2|1)]. These models are known to contain the staggered six-vertex model studied in the present work as a subsector [15][16][17]. Apart from general insights into the critical properties of quantum spin chains based on super Lie algebras and conformal field theories with non-compact target space this may also provide a basis for an improved understanding of some topical problems in condensed matter physics, e.g. the quantum phase transitions in two-dimensional disordered systems [19,21,35,48] or possibly the deconfinement of U (1) gauge fields coupled to the Fermi surface of a two-dimensional system [32,36], in the context of an exactly solvable model.