PHYSICAL REVIEW B, VOLUME 64, 045120
Sliding Luttinger liquid phases
Ranjan Mukhopadhyay, C. L. Kane, and T. C. Lubensky Department of Physics, University of Pennsylvania, Philadephia, Pennsylvania 19104 ͑Received 8 February 2001; published 9 July 2001͒ We study systems of coupled spin-gapped and gapless Luttinger liquids. First, we establish the existence of a sliding Luttinger liquid phase for a system of weakly coupled parallel quantum wires, with and without disorder. It is shown that the coupling can stabilize a Luttinger liquid phase in the presence of disorder. We then extend our analysis to a system of crossed Luttinger liquids and establish the stability of a non-Fermi- liquid state: the crossed sliding Luttinger liquid phase. In this phase the system exhibits a finite-temperature, long-wavelength, isotropic electric conductivity that diverges as a power law in temperature T as T→0. This two-dimensional system has many properties of a true isotropic Luttinger liquid, though at zero temperature it becomes anisotropic. An extension of this model to a three-dimensional stack exhibits a much higher in-plane conductivity than the conductivity in a perpendicular direction.
DOI: 10.1103/PhysRevB.64.045120 PACS number͑s͒: 71.10.Hf, 71.10.Pm, 74.20.Mn
I. INTRODUCTION wires.9 Coupled Luttinger liquids have been studied exten- sively for the past thirty years, mostly in the context of quasi For over two decades a central theme in the study of one-dimensional conductors. Single-particle hoppings as correlated electronic systems has been the drive to under- well as pair-hopping correlations were shown to destabilize stand and classify electronic states that do not conform to the Luttinger liquid phase. Following Anderson’s sugges- Landau’s Fermi-liquid theory. A clear example of such tions regarding high-temperature superconductors, there ‘‘non-Fermi liquid’’ physics occurs in one dimension ͑1D͒,1 have been a series of studies on coupled quantum chains, where arbitrarily weak interactions destroy the Fermi surface which confirm this view10 though the issue remains and invalidate the notion of independent quasiparticles at controversial.11 It has recently been proposed12,13 that for a low energy. Away from charge- and spin-density-wave insta- range of interwire charge and current interactions, there is a bilities, the 1D interacting electron gas forms a Luttinger smectic-metal ͑SM͒ phase in which Josephson, charge- and liquid. The discontinuity in occupation at the Fermi energy is spin-density-wave, and single-particle couplings are irrel- replaced by a power-law singularity, and the low-lying exci- evant. This phase is a two or three-dimensional anisotropic tations are bosonic collective modes in which spin and sliding Luttinger liquid whose transport properties exhibit charge decouple. power-law singularities like those of a 1D Luttinger liquid. It In this paper, we investigate how such non-Fermi liquid is the quantum analog of the sliding phases of coupled clas- behavior could arise in two- and three-dimensional systems sical XY models first identified in the context of phases of consisting of arrays of quantum wires or chains. Our analysis 14,15 is largely motivated by the unusual normal-state properties DNA-lipid complexes. The study of coupled Luttinger liquids becomes particularly relevant in the context of striped of high Tc materials. Prominent among these are ͑ ͒ phases, which have recently been found in quantum Hall 1 Resistivity perpendicular to the CuO2 plane much 16 17 larger than the in-plane resistivity.2 systems and in the cuprates. These phases are character- ͑2͒ Linear temperature ͑T͒ dependence of resistivity along ized by inhomogeneous distributions of charge and spin, the conducting planes, and 1/T2 divergence of the Hall where, for example, the charge carriers might be confined to angle.3 separate linear regions, thus resembling stripes. Sliding Lut- ͑3͒ Angle-resolved photoemission data showing a tinger liquid phases might also be relevant for a variety of pseudogap and absence of dispersion in the c direction.4 other systems such as quasi-one-dimensional organic con- ͑4͒ Linear temperature dependence of the nuclear mag- ductors and ropes of nanotubes. 5 netic resonance relaxation 1/T1. The work on sliding Luttinger liquid phases has also been Anderson has suggested that these unusual normal-state extended to a square network of 1D wires formed by cou- properties of the cuprates are the result of non-Fermi-liquid pling two perpendicular smectic metals,18 which has been physics in two dimensions.6 The study of non-Fermi liquids shown to exhibit a crossed sliding Luttinger liquid ͑CSLL͒ in dimensions greater than one has, however, proven to be phase. At finite temperature T, the CSLL phase is a 2D Lut- quite difficult. Since the Fermi liquid is stable for weak in- tinger liquid with an isotropic long-wavelength conductivity teractions, perturbative methods about this state fail to locate that diverges as a power law in T as T→0. At Tϭ0, it is non-Fermi liquid states.7 Moreover, generalizations of the essentially two independent smectic metals. This model bosonization technique to isotropic systems in higher dimen- could be realized in man-made structures constructed from sions have indicated that Fermi-liquid theory survives pro- quantum wires such as carbon nanotubes. The extension of vided the interactions are not pathologically long ranged.8 the model to a three-dimensional stack may be relevant to An alternative approach has been to study anisotropic sys- the stripe phases of the cuprates. Based on neutron and x-ray tems consisting of arrays of parallel weakly coupled 1D scattering measurements, it has been suggested that spin-
0163-1829/2001/64͑4͒/045120͑18͒/$20.0064 045120-1 ©2001 The American Physical Society RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120 charge stripes in the adjacent CuO2 plane are orthogonal to interactions that we neglect are expected to be much smaller each other.17 than those we consider here, their relevance becomes impor- In order to understand the nature of the sliding phase, it is tant only at very small temperatures. We delay a more com- easiest to think in terms of the classical analog. Imagine a plete study of their effects to a future publication. Through- stack of 2D XY models coupled by the Josephson coupling out the paper we will use the term ‘‘stable’’ somewhat Ϫ loosely to refer to stability with respect to a restricted set of cos( n nϩ1) where n is the XY-angle variable in layer n. Such a system always goes directly from a 3D ordered phase interactions that include all two-wire interactions. In Sec. III to a completely disordered phase as a function of the tem- we extend the above analysis to two identical coupled Lut- perature T. However in the presence of interlayer gradient tinger liquid arrays, arranged such that the wires of one array run perpendicular to those of the other. In this case we es- coupling terms of the form “ m•“ n , the system may have an intermediate sliding phase,15 that exhibits an in-plane 2D tablish the stability of a CSLL phase to a large class of op- order with power-law decay of correlations along the planes. erators. The interarray density-density couplings effectively In the absence of Josephson couplings, in the sliding phase, renormalize the intraarray couplings; however, the stability in neighboring layers freely slide over each other with no of the CSLL phase turns out to be identical to the stability of energy cost, and two-point correlation functions are identical the smectic-metal phase for each array, but with the intraar- in form to those of a stack of decoupled 2D layers. When ray couplings replaced by the renormalized couplings. We interlayer Josephson couplings are present, though irrelevant, then generalize our analysis to a three-dimensional stack of two-point correlation functions decay exponentially perpen- arrays, with wires on each array running perpendicular to the dicular to the layers. Thus the sliding phase has in-plane 2D wires of the consecutive array, and obtain once more a slid- order but is disordered in the third direction. For sliding Lut- ing phase that appears stable. In Sec IV we explore the trans- tinger liquid models, the 2D layers are replaced by 1D quan- port properties of sliding phases. In particular, we focus on tum wires, and interwire, current-current, and density-density the power-law singularities of the conductivity as a function → couplings play the role of the gradient couplings. Thus, in of T as T 0. Finally Sec V sums up our principal results. the sliding Luttinger liquid phases, the correlation functions Appendix A sketches out the steps used to obtain intrawire along a given wire exhibit the same power-law functional correlation functions, Appendices B and C present details of form as in a 1D Luttinger liquid. some integrals used in Secs. II and III, and Appendix D In this paper, we review and discuss in detail, results on presents details of conductivity calculations using the Kubo the sliding Luttinger liquid phase. Our discussion of the par- formula. allel Luttinger liquids follows Ref. 13, but it establishes the stability of the sliding phase to a more complete set of inter- II. COUPLED PARALLEL LUTTINGER LIQUIDS wire operators and also to disorder. For the CSLL phase, we provide detailed calculations that were presented only briefly Our goal is to construct non-Fermi-liquid electron sys- in Ref. 18. The stability and transport properties of the CSLL tems in two and three dimensions. Our basic building blocks phase form the central results of the paper. We also extend are one-dimensional quantum wires that exhibit Luttinger- the analysis to a three-dimensional stack of crossed Luttinger liquid phases with non-Fermi-liquid power-law decay of cor- liquids, and show that we can still obtain a sliding Luttinger relation functions. We couple these wires together in arrays liquid phase, with hopping between planes being irrelevant. in such a way that they retain their one-dimensional non- In this phase, the in-plane and perpendicular conductivity Fermi-liquid character yet allow nonvanishing interwire can be quite different and exhibit different temperature de- electron transport at nonzero frequency or temperature. We pendencies. This could be of relevance to the normal state of will consider several types of arrays, and it is useful to in- the cuprates. troduce a nomenclature for them. The long axis of parallel This paper is organized as follows. In Sec. II we explore wires defines one direction in space. The wires can be ar- the stability of the sliding phase for an array of parallel ranged in either a one- or a two-dimensional Bravais lattice coupled Luttinger liquids with respect to a large ͑but not of points in the plane perpendicular to that direction. We, complete͒ set of interwire operators. For a one-dimensional therefore introduce the notation dЌ :1 to denote an array of system of interacting spin-1/2 fermions, the spin excitations parallel wires centered on lattice points in a dЌ-dimensional could either be gapped or gapless. In the spin-gapped Luther- space. The resulting structure occupies a dЌϩ1 dimensional Emery regime, the system can by described by a single Lut- space. Thus a 1:1 array is a planar array of equally spaced tinger liquid for charge. In the gapless case, both spin and parallel wires, and a 2:1 array is a three-dimensional colum- charge are dynamical degrees of freedom, and there are two nar array of wires. As stated, we will restrict our attention to Luttinger parameters ( , ), and two velocities (v , v). arrays in which the wires lie on a dЌ-dimensional periodic We study coupled Luttinger liquids, both for the spin-gapped lattice. For 2:1 arrays, we will only consider, in detail, those and the gapless case, and in each case demonstrate the sta- based on a two-dimensional rectangular lattice. Both 1:1 and bility of the sliding phase. In addition, we study the effect of 2:1 arrays can exhibit anisotropic sliding phases for an ap- disorder. We find that density-density and current-current in- propriate choice of interwire potentials. teractions that stabilize the sliding phase, also make disorder In this section we will consider anisotropic sliding metal more strongly irrelevant. Note, however, that in this paper, phases in 1:1 and 2:1 arrays. In the succeeding sections, we we do not establish stability with respect to all multiwire will discuss how crossed 1:1 and 2:1 arrays can produce two operators. Since the overall strength of these higher-order and three-dimensional sliding phases with C4v symmetry and
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isotropic long-wavelength finite-temperature conductivity. 1 ͒2ͬ ץ͑ ͒2ϩ ץ͑ ϭ ͵ ͫ We begin our discussion with a review of 1D Luttinger liq- S, dx d v x , 2 v uids. 1 1 ͒ ͑ ͒2ͬ ץ͑ ͒2ϩ ץ͑ ϭ ͵ ͫ A. Review of one-dimensional Luttinger liquids S, dx d v x . 2.6 2 v One-dimensional Luttinger liquids are most easily de- scribed in terms of collective bosonic charge and spin- In the spin sector, we might either have a spin gap corre- density modes. The bosonized form of the fermion operators sponding to the Luther-Emery regime, or have a gapless near the left and right Fermi points are19 phase where the spin excitations are described by a Hamil- tonian of the same form as Eq. ͑2.4͒ with the parameters and v . In the gapless phase, SU(2) symmetry imposes the 1 ⌽ constraint ϭ1. ͒ϭ ikFx i R,s, j(x) Rs, j͑x R,s, je e , ͱ4⑀ B. The spin-gapped 1:1 array In this subsection we consider the simplest array of quan- 1 Ϫ ⌽ L ͑x͒ϭ e ikFxei L,s, j(x), ͑2.1͒ tum wires, the two-dimensional 1:1 array of Luttinger liquids s, j ͱ ⑀ L,s, j 4 in the spin-gapped phase. It has been suggested that this case might describe the striped phases of high-temperature where R and L stand for the right and left moving electrons, superconductors.21 In this subsection, all bosonic variables ⑀ respectively, s is the spin index, j denotes the wire number, refer to the charge sector, and we do not write explicitly the is some intrachain cutoff, and R/L,s, j are the Klein factors. subscript . In general, we expect density-density and We can write down an effective theory for the low-energy current-current interactions between the wires, which can be ⌽ excitations in terms of the boson operators . It is conve- represented by an action of the form nient to define 1 ϭ ͵ ͑ ͒ ˜ ͑ Ϫ Ј͒ ͑ ͒ ϭ͑⌽ ϩ⌽ ͒/ͱ4, Sint ͚ dxd j,n x, W n n j,nЈ x, , s, j R,s, j L,s, j 2 n,nЈ, ͑2.7͒ ϭ ⌽ Ϫ⌽ ͒ ͱ ͑ ͒ ץs, j ͑ R,s, j L,s, j / 4 , 2.2 ϭ ϭ where j,n ͓ n(x, ),Jn(x, )͔ with n x ,n(x, ) the (x,) the current on the nth wire. The ץdensity and J ϭ n x n where s, j is a phase variable, and s, j is the conjugate den- 20 density-density interaction is an effective interaction gener- sity variable. Since the bosonic excitations can be charac- ated by both the screened Coulomb and the electron-phonon terized in terms of charge and spin excitations, we further interaction. In the striped phases, stripe fluctuations lead to define current-current interactions.21 These current-current and density-density interactions are marginal and should be in- ϭ ϩ ͒ ͱ ϭ Ϫ ͒ ͱ ͑ ͒ , j ͑ ↑, j ↓, j / 2, , j ͑ ↑, j ↓, j / 2, 2.3 cluded in the fixed-point action. They are invariant under the → ϩ␣ → ϩ␣Ј ‘‘sliding’’ transformations n n n and n n n . and similarly for the variables. Here characterizes the Equations ͑2.4͒ and ͑2.7͒ define the fixed-point action of the charge excitations, and the spin excitations. For a single smectic-metal phase,12 which can be written as Luttinger liquid, we can write down an effective Hamiltonian for the charge sector ͒ ץ͑ ϩ ͒ϩ͚ V ץ͑͒ ץ͑ Sϭ͚ ͵ dx d͚ͫ V j x n x n j j x n n j j 2 ͒ ץ͑ v ͒ ͑ ͒2ͬ ץ͑ ϭ ͵ ͫ x ϩ H dx x , 2.4 ͒ͬ. ͑2.8͒ץ͑͒ ץ ϩ ͒ϩ2i͑ ץ ϫ͑ 2 x n j x n n where v is the velocity, and is a Luttinger liquid param- Upon integration over or , respectively, the effective ͑ n n eter this is the inverse of the usual Luttinger liquid param- action for and become eter g). Alternately, we could write down the action, in a n n path-integral formulation, as d3Q 1 1 ϭ ͵ ͑ ͒ͭ 2ϩ ͑ ជ ͒ 2ͮ ͉͑ ͉͒2 S 3 qЌ v qЌ qʈ Q , ͑2͒ 2 v͑qЌ͒ 2 ͒ ץ͑ v ͒ ץ͑ ͒2ͬϪ ץ͑ ϭ ͵ ͭ ͫ x ϩ S dx d x 2i x 2 d3Q 1 1 ϭ ͵ ͭ 2ϩ ͑ ជ ͒ 2ͮ ͉͑ ͉͒2 S 3 v qЌ qʈ Q , ͑2͒ 2͑qЌ͒ v͑qЌ͒ ͒ ͑ ͒ ͑ ͮ͒ ץϫ͑ . 2.5 2.9
where Qϭ(,qʈ ,qЌ), with qʈ the momentum along the Integrating out either the or the variables, we obtain chain and qЌ perpendicular to the chains. Here
045120-3 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
͑qЌ͒ϭͱV ͑qЌ͒/V ͑qЌ͒,
v͑qЌ͒ϭͱV ͑qЌ͒V ͑qЌ͒, ͑2.10͒ where V (qЌ) and V (qЌ) are the Fourier transforms of V j ͑ ͒ and V j with respect to the wire index j. Equations 2.8 and ͑2.9͒ define the action of the ideal 2D sliding Luttinger liquid or SM phase. Even though they include interwire interactions described by Eq. ͑2.7͒, they yield power-law correlations characteristic of a 1D Luttinger liquid. We could consider, for example, correlation functions in- volving the density variable . We note that
ץϵ†ϭ † ϩ † ϩ † ϩ Ӎ R R L L ͓L R c.c.͔ x ϩ ϩ ͱ ͒ ϩ ͑ ͒ ͕exp͓i2kFx i 2 ͑x, ͔ c.c.͖. 2.11 Thus the density has two pieces: the first piece is the course- grained density and the second is modulated at 2kF , where FIG. 1. A schematic depiction of a two-dimensional array of kF is the Fermi wave vector. Correspondingly, the density- density correlation has two pieces. We consider G(x,); quantum wires. the component of the density-density correlation function with 2kF modulation. Thus H ϭ ͵ ͓͑ † † ϩ † † ͒͑ SC,n ͚ dx R↑, jL↓, j R↓, jL↑, j R↑, jϩnL↓, jϩn j ͒ϵ ͱ ͒Ϫ ͒ ϩ ϩ G͑x, ͗exp͕i 2 ͓ j͑x, j͑0,0 ͔ i2kFx͖͘ c.c. ϩR↓ ϩ L↑ ϩ ͔͒ϩc.c., ͑2.17͒ It easy to show that for large x ͑see Appendix A͒ , j n , j n and ͒ A1cos͑2kFx G͑x,0͒Ϸ ⌬ , ͑2.12͒ x CDW,ϱ H ϭ ͵ ͓͑ † ϩ † ͒͑ † CDW,n ͚ dx R↑, jL↑, j R↓, jL↓, j L↑, jϩnR↑, jϩn j where † ϩL↓ ϩ R↑ ϩ ͔͒ϩc.c. ͑2.18͒ , j n , j n dqЌ 1 ⌬ ϭ ͵ ͑ ͒ CDW,ϱ , 2.13 At low temperatures, the spin variable is effectively fro- Ϫ 2 ͑qЌ͒ zen, and these interactions depend only on i . Their associ- and A1 is a constant. Alternately, for large , ated actions can be expressed as Ϫ⌬ ͑ ͒Ϸ CDW,ϱ ͑ ͒ G 0, A2 , 2.14 ϭJ ͵ ͓ͱ ͑ Ϫ ͔͒ SSC,n n͚ dx d cos 2 i iϩn , i where A2 is a different constant. In general, G(x, ) can be written in the scaling form ϭV ͵ ͓ͱ ͑ Ϫ ͔͒ SCDW,n n͚ dx d cos 2 i iϩn . Ϫ⌬ x i ͑ ͒Ϸ CDW,ϱ ͩ ͪ ͑ ͒ G x, x F , 2.15 ͑2.19͒
where The relevance of these terms are determined by the scaling ͱ dimensions of the corresponding operators, cos͓ 2 ( i ͒→ → Ϫ ͔ ͓ͱ Ϫ ͔ F͑y A1 as y 0 iϩn) and cos 2 ( i iϩn) , which are, respectively, ⌬ → CDW,ϱ →ϱ ͑ ͒ A2y as y . 2.16 dqЌ ⌬ ϭ ͵ ͑ Ϫ ͒͑ ͒ ͑ ͒ SC,n 1 cos nqЌ qЌ , A variety of interactions, other than those of Eq. 2.7 , Ϫ 2 couple neighboring wires ͑see Fig. 1͒. The sliding phase is stable only if the interactions are irrelevant, that is, only if dqЌ ͑1Ϫcos nqЌ͒ they scale to zero with system size. We will now investigate ⌬ ϭ ͵ ͑ ͒ CDW,n . 2.20 perturbatively the relevance of these interchain interactions Ϫ 2 ͑qЌ͒ to determine under what conditions the sliding phase is The exponent ⌬ follows from stable. Due to the spin gap, single-particle hopping between SC,n chains is irrelevant, and the interchain interactions that could 2 ͗cosͱ2͑ Ϫ ϩ ͒͘ϭexp͓Ϫ͗͑ Ϫ ϩ ͒ ͔͘, become relevant are the Josephson ͑SC͒ and charge-density i i n i i n ͑2.21͒ wave ͑CDW͒ couplings, which are represented by operators of the form and
045120-4 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120
where the charge-density varies from wire to wire. For small but positive min , the system is close to the transverse CDW instability. As pointed out in ͑Ref. 13͒, a transverse CDW would frustrate the crystallization of fermions, since kF is now a function of the chain index j. Strong fluctuations of this kind prevent the locking in of density fluctuations along the wires. Thus, in order to stabilize the sliding Luttinger liquid phase, we need to tune min to be very small compared ⌬ ⌬ to average (qЌ), so that SC,n and CDW,n can both be made large. Note that in addition, we should also consider † † interchain operators of the form R↑, jR↓, jR↑, jϩnR↓, jϩn . These interactions, however, turn out to be automatically ir- relevant if the superconducting and CDW interactions are irrelevant, and hence merit no further consideration. In addition to the lowest-order interactions between pairs of chains described by SSC,n and SCDW,n , there can, in prin- ciple, be higher-order multichain interactions with actions of the form
FIG. 2. (q) as a function of q/ , with a minimum, min at q S ϭ͚ ͵ dx dJ cosͫͱ2ͩ ͚ s ϩ ͪͬ, ϭq . SC,s s p i p 0 p i p p
1 d3Q ͗͑ Ϫ ͒2͘ϭ ͵ ͉͗˜͑ ͉͒2͓͘ Ϫ ͑ ͔͒ i iϩn 3 q, 1 cos qЌn 2 ͑2͒ S ϭ͚ ͵ dx dV cosͫͱ2ͩ ͚ s ϩ ͪͬ, CDW,s s p i p p i p p dqЌ ͑2.26͒ ϭ ͵ ͭ ͑qЌ͒͑1Ϫcos qЌn͒ Ϫ 2 where J are the interchain Josephson couplings, V the sp sp ͑ ͒ dqʈd 1 interchain particle-hole CDW interactions, and s p is an ϫͫ ͵ ͬͮ , ͑2.22͒ ͑2͒2 vq2ϩ2/v integer-valued function of the chain number p satisfying ʈ ͚ ϭ ͱ ͚ ps p 0. The scaling dimensions of cos͓ 2 ( ps p iϩp)͔ ͗͘ ͑ ͒ ͱ ͚ where denotes averaging with respect to S in Eq. 2.8 . and cos͓ 2 ( ps p iϩp)͔ are, respectively, The integral in the square brackets diverges logarithmically 22 ϳ with system size L ( C ln L). Using this, we find dqЌ ⌬ ϭ ͵ ͑ ͒ ͕͑ Ϫ Ј͒ ͖ SC,s qЌ ͫ ͚ s ps pЈcos p p qЌ ͬ, Ϫ⌬ p ͱ Ϫ ͒ ϳ SC,n ͑ ͒ Ϫ 2 ͗cos 2 ͑ j jϩn ͘ L , 2.23 p,pЈ where ⌬ is given by Eq. ͑2.20͒. From the above equation, SC,n dqЌ 1 it follows that ⌬ ϭ ͵ ͕͑ Ϫ Ј͒ ͖ CDW,s ͫ ͚ s ps pЈcos p p qЌ ͬ. p Ϫ ͑ ͒ Ϫ⌬ 2 qЌ p,pЈ ͗S ͘ϳL2 SC,n. ͑2.24͒ SC,n ͑2.27͒ ⌬ Similar calculations produce CDW,ϱ . For a stable smectic- metal phase, these terms have to be irrelevant, implying These perturbations are irrelevant if ⌬ Ͼ ⌬ Ͼ ͑ ͒ ⌬ Ͼ ⌬ Ͼ ͑ ͒ CDW,n 2, SC,nЈ 2 2.25 CDW,s 2, SC,sЈ 2 2.28 p p for all n and nЈ. ⌬ Ͻ for all sets of s and sЈ . The relevance of higher-order terms If any SC,n 2, the SM phase is unstable to the forma- p p ⌬ of this form is a subtle issue, and we will elaborate on this in tion of an anisotropic 2D superconductor. If any CDW,n Ͻ2, the SM phase will flow to a 2D longitudinal CDW- a future publication. However the strength of these terms, as measured, for example, by V , is small and they become crystalline phase with 2kF density modulations along the sp wires and the phase locked from wire to wire. Notice that if important only at very small temperatures even if they are ⌬ (qЌ) is uniformly small, the SC,n’s are small and the relevant. In this paper we will restrict ourselves to pairwise ⌬ ⌬ ͑ ͒ CDW,n’s are large, and for large , the SC,n’s are large and CDW couplings of the form given by Eq. 2.17 ; however, ⌬ the CDW,n’s small. For a stable sliding phase we need all the we comment on the higher-order superconducting terms be- ⌬ ⌬ SC,n’s and CDW,n’s to be greater than two. Our strategy to low. create a stable sliding phase is to choose (qЌ) of the form To explore the regions of stability of the SM phase, we shown in Fig. 2, with having a minimum, min ,atqЌ follow Refs. 13 and 15 and take ϭ q0. When min becomes zero, the system undergoes a tran- ͑ ͒ϭ ͓ ϩ ͑ ͒ϩ ͑ ͔͒ ͑ ͒ sition to a transverse CDW modulation with wave vector q0, qЌ K 1 1cos qЌ 2cos 2qЌ . 2.29
045120-5 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
The parameters 1 and 2 can be tuned to set the value q0 of qЌ , at which (qЌ) reaches a minimum and the minimum ϭ ⌬ ⌬ value of (qЌ) and (q0) K . For a specific value of and k,
Ϫ⌬͒ 4͑1 cos q0 ϭϪ , 1 ϩ 2 ͒ ͓1 2cos ͑q0 ͔
͑1Ϫ⌬͒ ϭ , ͑2.30͒ 2 ϩ 2 ͒ ͑1 2cos q0 ϭ ϩ ⌬ unless q0 0or , in which case only 1 2 is fixed by . ͑ Þ We note that, in the above equation i.e., for q0 0, ), while ⌬Ͻ 2 is always positive for 1, 1 can be either positive or negative. Typically positive ’s correspond to repulsive in- teractions. In this paper we treat K as the control parameter. For small K the system becomes superconducting, while for large K the system goes into a CDW crystalline phase. We will ϵ  show that through judicious tuning of and it is possible FIG. 3. Plot of KCDW /KSC as a function of q0 / . For 1 2 Ͼ to have an intermediate window of K where the smectic- 1, there exists a region of K over which the non-Fermi-liquid metal phase is stable. For this purpose, we define a phase is stable. sn ϭ⌬ /K and b ϭ⌬ K, where a and b depend Ϫ5 SC,sn n CDW,n sn n where C has been defined in Eq. ͑2.34͒.Weset⌬ϭ10 . only on 1 and 2, and We consider the range of q0 / lying between 0.25 and 0.75. This range can be broken into three sections: 1 2 ͑1͒ 0.25Ͻq /Ͻ0.41957: In this range Ͻ0 and ͉ ͉ a ϭ͚ ͓s ϩs ͑s ϩ ϩs Ϫ ͒ /2 0 1 1 sp p p p 1 p 1 1 Ͼ p 2 2. Here the most relevant superconducting term corre- ͓ͱ ϩ Ϫ ϩ ͑ ϩ ͒ ͔ ͑ ͒ sponds to the multichain operator cos 2 ( i iϩ1 iϩ3 s p s pϩ2 s pϪ2 2/2 . 2.31 Ϫ 23 iϩ4)͔. The dimension of this operator sets the minimum of a . Thus, in this range, mina ϭ(2ϩ Ϫ /2) and The SM phase becomes unstable to interchain Josephson sp sp 1 2 ϭ ϭ ϩ Ϫ couplings for K less than KSC maxs (2/as ) and unstable to KSC 1/(2 1 2/2). p p ͑ ͒ Ͻ Ͻ ͉ ͉Ͻ interchain CDW interactions for K greater than KCDW 2 0.41957 q0 / 0.5804: In this region 1 2.We ϭ find that a is smallest for the set s ϭ␦ Ϫ␦ . Thus, in minn(bn/2). Thus the smectic-metal phase is stable over a sp n n,0 n,2 Ͻ Ͻ ϭ Ϫ window of K and KSC K KCDW , provided this range KSC 2/(1 2/2). ͑ ͒ Ͻ Ͻ Ͼ 3 0.5804 q0 / 0.75: Here 1 2, and as is the a b p KCDW sp m smallest for the set s ϭ␦ Ϫ␦ . Thus K ϭ2/(1Ϫ /2). ϵ ϭ ͉ Ͼ1. ͑2.32͒ n n,0 n,1 SC 1 min. wrt.m&sp  KSC 4 In Fig. 3 we plot as a function of q0. The minima of the ϭ curve corresponds to q0 2 l/m, where l and m are integers. We note, once more, that in this paper we consider stability Ϫ Also, note that since 1 has the same sign as ( cos q0), with respect to all superconducting terms, but only pairwise there are regions of stable smectic phase for positive as well CDW terms of the form given by Eq. ͑2.17͒.IfϽ1, the as negative values of 1. system goes directly from a 2D SC phase to a CDW crystal Having established a stable smectic phase for the pure as K is increased, without passing through the SM phase. For system, we now study the relevance of quenched disorder in a stable sliding phase, we need to make ⌬ very small. The ⌬ ⌬ this phase. Disorder gives rise to a random electron potential value of CDW,n for small is determined by values of qЌ D(x), with associated action near q0. We can therefore set (qЌ) ͑ ͒Ϸ ͓⌬ϩ ͑ Ϫ ͒2͔ ͑ ͒ ϭ ͵ ͑ ͒ ͓ͱ ͔ ͑ ͒ qЌ K C qЌ q0 , 2.33 Sdis ͚ dx d D j x cos 2 j . 2.36 j where D(x) can be treated as a Gaussian random variable, with ͒ zero mean and local fluctuations such that Љ͑q0 Cϵ ϭ2 sin2q . ͑2.34͒ 2K 2 0 D͑x͒ϭ0, This gives us ͒ ͒ϭ⌬ ␦ Ϫ ͒ ͑ ͒ D͑x D͑xЈ D ͑x xЈ , 2.37 ͱ K͓1Ϫcos͑nq ͒eϪn ⌬/C͔ ⌬ Ӎ 0 ͑ ͒ where the over line signifies averages over the randomness. CDW,n , 2.35 ͱC⌬ By a generalization of the Harris criterion,24 it can be shown
045120-6 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120
quite easily that ͑also, see Giamarchi and Schulz25͒ disorder ϭJ ͵ ͓ͱ ͑ Ϫ ͔͒ is irrelevant if ⌬ ϱϾ3, where SSC,n n dx d ͚ cos 2 , j , jϩn CDW, j
ϫcos͑ͱ2 ͒cos͑ͱ2 ϩ ͒, dq 1 1 1 , j , j n ⌬ ϵ ͵ Ӎ ͑ ͒ CDW,ϱ . 2.38 Ϫ2 ͑q͒ K ͱ ⌬ C ϭV ͵ ͓ͱ ͑ Ϫ ͔͒ SCDW,n n dx d ͚ cos 2 , j , jϩn j For the range of parameters we are considering where the ϫ ͑ͱ ͒ ͑ͱ ͒ ͑ ͒ ⌬ cos 2 , j cos 2 , jϩn . 2.40 SM phase is stable, CDW,ϱ is large and thus disorder is strongly irrelevant. This is an important point. For a single The variables now contribute to the dimensions of these Ͼ Luttinger liquid in the repulsive region, 1 and disorder is terms. Because is constrained to be one, the contribution always relevant. However, interwire interactions can drive of the variables is trivial, and the dimensions of these disorder irrelevant for (qЌϭ0)Ͼ1, even in regions of terms are given by phase space where all interactions are repulsive. ⌬ ϭ⌬(gap)ϩ Thus, there is a small but finite region of phase space SC,n SC,n 1, where the smectic-metal phase appears stable. We should ⌬ ϭ⌬(gap) ϩ ͑ ͒ note that over a larger region of phase space, the only rel- CDW,n CDW,n 1, 2.41 evant operators involve nonlocal interactions of the form where ⌬(gap) and ⌬(gap) are given by Eqs. ͑2.13͒ and ͑2.14͒ V cos͓ͱ2( Ϫ ϩ )͔, where V is expected to be expo- CDW,n SC,n n i i n n nentially small, for large n, in the bare Hamiltonian. Though with replaced by . Since the variables are no longer gapped, single- relevant, these operators would only play a role for kBT V electron tunneling is no longer irrelevant. Single-particle smaller than some energy scale set by n . So, for example, † there will be a range of temperatures, where we will only hopping is described by operators such as R j,↑R jϩn,↑ , which V J can be represented by terms of the form need to consider the relevance of 1 and 1. These can be made irrelevant over a reasonably large region of phase space ͑see Ref. 13͒. Thus, even though the region of phase ϭ ͵ T ͫ Ϫ ͱ ͑ Ϫ ͒ͬ Sel,n dx d ͚ nexp i , j , jϩn space where the smectic phase is strictly stable is highly j 2 restricted, at finite temperature and for weak coupling, we expect a much larger region of phase space whose behavior ϫ ͫ Ϫ ͱ ͑ Ϫ ͒ͬ exp i ϩ is governed by the sliding Luttinger liquid ground state. 2 , j , j n ϫͭ ͫ Ϫ ͱ ͑ Ϫ ϩ Ϫ ͒ͬͮ exp i ϩ ϩ . C. The gapless 1:1 array 2 , j , j n , j , j n We now consider 1:1 arrays of wires in which both charge ͑2.42͒ and spin excitations are gapless. In this case, there are two Luttinger liquid parameters ( ,) for the charge and spin The expectation value of the term in the curly bracket goes Ϫ1/2 modes, respectively, and two velocities (v ,v) on each as L as system size L goes to infinity. Thus ͗Sel͘ Ϫ⌬ wire. To maintain gapless Luttinger liquids and SU(2) spin ϳL2 el,n, where symmetry, we do not include any marginal spin-spin cou- ⌬ ϭ 1 ͓⌬(gap) ϩ⌬(gap)͔ϩ 1 ͑ ͒ pling terms in the Hamiltonian. Thus the spin degrees of el,n 4 CDW,n SC,n 2 . 2.43 freedom are represented by the fixed-point action Regions of phase space, where SSC,n is relevant, corresponds to the superconducting phase, whereas regions of relevance 2 ͒ of SCDW,n correspond to the CDW crystal phase. Regionsץ͑ ͬ ͒2ϩ , j ץ͑ ϭ ͵ ͫ S, ͚ dx d v x , j where both of these are irrelevant, but single-particle hop- j v ping is relevant, correspond to the Fermi-metal phase. For a ͑ ͒ 2.39 stable smectic-metal phase, we require that all these opera- tors be irrelevant. The superconducting and CDW coupling with ϭ1. In a more general treatment, one could include terms are irrelevant if spin-spin coupling terms and consider their relevance, main- ⌬(gap)Ͼ ⌬(gap) Ͼ ͑ ͒ taining, however, the SU(2) symmetry of the spin sector. SC,n 1, CDW,nЈ 1, 2.44 We leave that for a future consideration. The charge modes for all n and nЈ The condition for single-particle hopping to are still represented by Eq. ͑2.9͒, with (qЌ) and v(qЌ) be irrelevant is that replaced by (qЌ) and v(qЌ). The form of is still given ͑ ͒ by Eq. 2.29 . ⌬(gap)ϩ⌬(gap) Ͼ6 ͑2.45͒ We again consider the relevance of single-particle, CDW SC,n CDW,n and SC tunneling. The SC and CDW tunneling were already for all n. considered in the previous subsection. When the spin vari- We now proceed exactly as for the gapped case, assuming ables are included, Eqs. ͑2.11͒ and ͑2.12͒ become to have a form as given by Eq. ͑2.29͒. As before, we may
045120-7 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
ϭ (n1 ,n2). We will focus on the spin-gapped case, though extensions to the gapless case proceed exactly as in the pre- vious section. Taking into account interwire density-density and current-current interactions, we could write down an ac- tion of the form ͑2.7͒, but where we now have sums over → columns in a 2D lattice, with j,n(x, ) j,n(x, ) and n ϭ (nx ,ny). When transformed to Fourier space, this action becomes
1 d4Q ជ 2 2 ជ 2 2 Sϭ ͵ ͓V ͑qЌ͒q ͉͉ ϩV ͑qЌ͒q ͉͉ 2 ͑2͒4 ʈ ʈ Ϫ ϩ ͑ ͒ i qʈ͕ * x c.c.͖͔, 2.48 ជ ជ where Qϭ(,qʈ ,qЌ) with qЌ a vector in the first Brillouin zone of the 2D lattice of columns. We choose the x axis to lie ជ ϭ along wires, so that qЌ (qy ,qz). The or the variables may be integrated out, giving us the effective actions
⌬ 4 FIG. 4. A plot of the phase diagram in q0 , K space with d Q 1 1 Ϫ ជ 2 ជ 2 2 ϭ10 5. SC stands for superconducting, FL for Fermi-liquid, SM Sϭ ͵ ͑qЌ͒ͭ ϩv͑qЌ͒qʈ ͮ ͉͑Q͉͒ , ͑2͒4 2 ͑ ជ ͒ for smectic metal, and CDW for charge-density wave crystal. v qЌ
4 (gap) (gap) d Q 1 1 write ⌬ ϭa /K, ⌬ ϭb K, K ϭmax (1/a ), and 2 ជ 2 2 CDW,n n SC,n n SC n n Sϭ ͵ ͭ ϩv͑qЌ͒qʈ ͮ ͉͑Q͉͒ , ϭ Ͼ ͑2͒2 ͑ ជ ͒ ͑ ជ ͒ KCDW minn(bn). Provided KCDW /KSC 1, there is a win- 2 qЌ v qЌ Ͻ Ͻ ͑ ͒ dow of K, KSC K KCDW , where the system is stable with 2.49 respect to both the CDW and superconducting couplings. For where the smectic-metal phase to be stable, the single-particle hop- ping has to be irrelevant as well, which indicates that ជ ͱ ជ ជ ͑qЌ͒ϭ V ͑qЌ͒/V ͑qЌ͒,
an ϩb KϾ6. ͑2.46͒ ជ ϭͱ ជ ជ ͑ ͒ K n v͑qЌ͒ V ͑qЌ͒V ͑qЌ͒. 2.50
This condition is violated for K lying between KϪ and Kϩ , In three dimensions it turns out that the stability of the slid- ϭ ϭ ing phase with respect to the complete set of operators, re- where KϪ minnKϪ,n and Kϩ maxnKϩ,n , with quires an even further fine-tuning of the generalized current- Ϯͱ Ϫ 3 9 anbn current coupling terms. In particular, (qy ,qz) should have a KϮ ϭ . ͑2.47͒ ⌬ ϭ ϭ ⌬ ,n b minimum K at some qy q0,y , qz q0,z , with both and n the second derivative of /K being much smaller than unity The single electron hopping is relevant in a large region of at the minimum. Let us consider two examples of the form phase space, indicating an instability towards a Fermi-liquid that (qy ,qz) could assume in order to obtain a stable slid- ⌬ ͓ FL phase. We write 1 and 2 as functions of and q0 see ing phase ͑see Fig. 5͒. Eq. ͑2.30͔͒ and set ⌬ϭ10Ϫ5. Higher-order terms involving The first example is one that is symmetric with respect to and , in general, are less relevant in this case, and we qy and qz . We assume that the wires are arranged in a square do not need to consider the whole set of operators.26 Depend- or rectangular pattern, and align the y- and z- axes along the ing on q0, we have the following possibilities for phases as K edges of the rectangle. We consider the form is increased: ͑1͒ SC →FL→ CDW crystal, ͑2͒ SC →SM ͑ ͒ϭ ͓ ϩ ͑ ͒ϩ ͑ ͒ → CDW crystal, ͑3͒ SC →FL→SM→ CDW crystal, and qy ,qz K 1 1cos qy 1cos qz ͑4͒ SC →SM→FL→SM→ CDW crystal. ϩ cos͑q ͒cos͑q ͔͒2. ͑2.51͒ The phase diagram is complicated, and we plot a region 2 y z ⌬2 of K, q0 space in Fig. 4, in the absence of disorder. Back- 1 and 2 are adjusted such that has a minimum K at ⌬(gap) Ͼ ϭ ϭ scattering due to disorder is irrelevant for CDW,ϱ 2, which qy qz q0. This gives is automatically satisfied in the SM phase. ͑1Ϫ⌬͒ ϭϪ 1 ͒ , D. The three-dimensional anisotropic sliding phase cos͑q0 We now turn to three-dimensional 2:1 arrays with wires ͑1Ϫ⌬͒ ជ ϭ ͑ ͒ on a periodic 2D lattice with primitive translation vectors a1 2 2͑ ͒ . 2.52 ជ ជ ϩ ជ cos q0 and a2. Each wire occupies a position n1a1 n2a2 ona2D ϭ lattice and is labeled by the integer valued vector n Close to (qy ,qz) (q0 ,q0), we can expand as
045120-8 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120
FIG. 6. Schematic depiction of a two-dimensional crossed array.
plane conductivities at finite temperature are isotropic at long wavelengths. In this section we demonstrate the existence of a sliding phase in the crossed arrays that is stable if the sliding phase in the constituent arrays is stable. The correla- tion functions in this phase exhibit power-law decay along the planes, and the electric conductivity diverges as a power FIG. 5. A three-dimensional array of quantum wires. law in temperature T as T→0. Ӎ ⌬ϩ Ϫ ͒ Ϫ ͒ 2 ͑ ͒ K͓ 2͑qy q0 ͑qz q0 ͔ . 2.53 ⌬ ϭ ⌬ ϭ As before SC,m Kam and CDW,n Kbn , where m and n A. Crossed two-dimensional sliding phase are now vectors. The sliding phase is stable provided We consider now a square grid of wires, starting again with the spin-gapped case. The system consists of two arrays KCDW ambn ϵ ϭ ͉ Ͼ1. ͑2.54͒ of quantum wires, the X- and Y- arrays running, respectively, K 4 min. wrt.m&n SC parallel to the x and y directions. Each wire sees a periodic ⌬ϭ Ϫ2 ͑ ͒ For 10 or smaller there is a large range of q0, where one-electron potential from the array of wires crossing it. For the sliding phase is stable. simplicity we assume that this periodicity is commensurate One could also consider the highly anisotropic form with bands in the wire. This leads to a new band structure with new band gaps. It is assumed that the Fermi surface is ͑q ,q ͒ϭK͕͓1ϩ cos͑q ͒ϩ cos͑2q ͔͒2 y z 1 y 2 y between gaps so that the wires are conductors in the absence ϩ ϩ ͒ ͑ ͒ 3͓1 cos͑qz ͔͖. 2.55 of further interactions. By removing degrees of freedom with wavelengths smaller than the inverse wire separation, we ob- Again, for any , one can adjust and to produce a 3 1 2 tain a new effective theory whose form is identical to the stable sliding phase. We conclude by noting that in three theory before the periodic potential was introduced. Thus, in dimensions, obtaining a sliding phase, requires an even finer the absence of two-particle interactions between crossed ar- adjustment of parameters than in 2D. At finite temperature, as before, the region of phase space controlled by the rays, the system could be in a phase consisting of two smectic-metal fixed point is expected to expand consider- crossed, noninteracting smectic-metal states. ably. We will now demonstrate the existence of a stable sliding phase in the crossed arrays. In addition to the interwire cou- plings within each array, we need to consider Coulomb in- III. CROSSED SLIDING LUTTINGER LIQUID PHASE teractions between wires on the X array and wires on the Y Having established regions of stability of the sliding metal array. These interarray couplings are marginal and should be phases formed from arrays of quantum wires, we now turn to included in the fixed point. They do not, however, change the the investigation of sliding phases formed from crossed ar- dimensions of the operators, except by renormalizing (qЌ). rays of wires. We consider two basic configurations: one a For a stable sliding phase, additional interactions between two-dimensional system formed from two coupled 2D slid- the two arrays, such as the Josephson and CDW couplings, ing phases ͑1:1 arrays͒ oriented at right angles to each other have to be irrelevant. We will show that it is possible to tune and the other a 3D system formed by stacking the crossed (qЌ) such that this is indeed the case ͑see Fig. 6͒. two-dimensional system. The latter three-dimensional sys- The Coulomb interactions between electrons on intersect- tem can be constructed from two interpenetrating 3D aniso- ing wires give rise to a term in the Hamiltonian of the form c ͓ ͔ tropic sliding phases ͑2:1 arrays͒, of the type discussed in Vm,n(x,y) x,m(x) y,n(y), where x,m(x) y,m(y) is the Sec 2 D, oriented at right angles to each other. Both the 2D electron density on the mth wire in the X(Y)-array at posi- c c and 3D systems have C4v symmetry. As a result, their in- tion x (y). We expect Vm,n(x,y) to have the form V (x
045120-9 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
Ϫna,yϪma), where a is the distance between parallel wires. 1 2 2 Dϭ ͩ ϩ 2 ͪͩ ϩ 2 ͪ Ϫ͑ c ͒24 ͑ ͒ vxqx vyqy VR 3.6 Thus, we represent the interaction between the X and Y array x y vx vy as is the determinant of GϪ1. In order to determine the dimensions of operators, we ͔ ץ͒ c͑ Ϫ Ϫ ץ͓ ͵ dx dy x x,mV x na,y mb y y,n . calculate the leading dependence of correlation functions ͗2 ͘ such x (r,t) on system size L. Thus we can consider the If all parameters for the X and Y arrays are the same, the function crossed-grid action as a functional of the and variables, can be written as 2 ͩ ϩ 2 ͪ vyqy dqx dqy d vy 1 d dq dq ͗2͑r,t͒͘ϭ ͵ . ͑3.7͒ x y 2 2 2 2 x ͑ ͒3 D Sϭ ͵ ͓V ͑q ͒q ͉ ͉ ϩV ͑q ͒q ͉ ͉ 2 y 2 ͑2͒3 y x x x y y The leading L dependence is related to the infrared diver- ϩ ͑ ͒ 2͉ ͉2ϩ ͑ ͒ 2͉ ͉2 V qy qx x V qx qy y gence of the integral just introduced. This infrared diver- ϩ c͑ ͒ ͕ ϩ ͖Ϫ ͕ ϩ ͖ gence comes purely from the integration over qx , .Wecan V qx ,qy qxqy x *y c.c. i qx x* x c.c. 2 ϩ 2 write the integrand as x /( /vx vxqx ) plus a remaining Ϫ ͕ ϩ ͖͔ ͑ ͒ part. The integral of the remaining part is free of infrared i qy *y y c.c. 3.1 singularities ͑see Appendix C for details͒. Thus, it is easy to ϭ with obvious definitions for x x( ,qx ,qy), y , x , and see that the leading L dependence goes as y . It should be noted that this is an effective theory with Ϫ/aϽq ,q Ͻ/a. Integrating out the variables, we are dqy x y ͗2͘ϳ ln͑L͒ ͵ ͑q ͒, ͑3.8͒ left with an effective action, which is conveniently expressed x 2 y in matrix form as ϭ where (qy) x(0,qy). Notice that this is precisely what we 1 had for a single array of parallel wires. A similar analysis 2 Ϫ1 Sϭ ͵ d kd ͑G ͒ * , ͑3.2͒ ͗2͘ϭ͗2͘ 2 a ab b yields y x . Also note that cross correlations of the form x- y are finite as L goes to infinity. where aϭx,y and bϭx,y. Here We also need to consider correlation functions in the variables. To do so, we start with the action of Eq. ͑3.1͒, and 1 2 integrate out the variables. The effective action as a func- ͩ ϩv q2 ͪ ϪVc 2 v x x R tional of the variables is Ϫ1ϭ x x ͑ ͒ G ͩ 2 ͪ , 3.3 c 2 1 ϪV ͩ ϩ 2 ͪ 1 d dqx dqy Ϫ R vyqy ϭ ͵ ͑ 1͒ ͑ ͒ v S a G ab b , 3.9 y y 2 23 where with aϭx,y and bϭx,y. Here
␥͑q͒ 2 ͑ ͒ϭͱ x q ͑ ͒ ͑ ͒, ¯ ͩ ϩ¯v q2 ͪ Vcq q V qx V qy x ¯ x x x y Ϫ vx G 1ϭ , ͑3.10͒ ͩ 2 ͪ V ͑q ͒␥͑q͒ ͑ ͒ϭͱ y Vcq q ¯ ͩ ϩ¯v q2 ͪ vx q ͒ , x y y ¯ y y V ͑qx vy where Vc͑q͒ Vc ͑q͒ϭ , R ␥͑q͒ ¯ ͒ϭ ͒ ͒ 1/2 ͑qy ͓V ͑qy /V ͑qy ͔ ,
␥ ͒ϭ ͒ ͒Ϫ c ͒ 2 ͑ ͒ ͑q V ͑qx V ͑qy ͓V ͑q ͔ , 3.4 ¯ ͒ϭ ͒ ͒ 1/2 ͑ ͒ v͑qy ͓V ͑qy V ͑qy ͔ . 3.11 ϭ ϭ ϭ and y(q) x(Pq), vy(q) vx(Pq) where Pq P(qx ,qy) Note that ¯ is different from defined for the correlation ϭ ͑ ͒ Ϫ1 (qy ,qx). From Eq. 3.3 for G , we can calculate Ϫ1 functions. From G , we calculate 2 1 2 ͩ ϩ 2 ͪ Vc 2 vyqy R ¯ ͩ ϩ¯v q2 ͪ ϪVcq q 1 v y y y x y ϭ y y ͑ ͒ 1 ¯v G ͩ 2 ͪ , 3.5 y D 1 Gϭ , c 2 ϩ 2 D¯ ͩ 2 ͪ V ͩ v q ͪ R v x x ϪVcq q ¯ ͩ ϩ¯v q2 ͪ x x x y x ¯ x x vx where ͑3.12͒
045120-10 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120
where XYϭ ͵ J XY ͕ͱ ͓ ͑ ͒Ϫ ͑ ͔͖͒ SSC ͚ d cos 2 x,m na y,n ma , m,n 2 2 D¯ ϭ¯ ¯ ͩ ϩ¯v q2 ͪͩ ϩ¯v q2 ͪ Ϫ͑Vc͒2q2q2 x y ¯ x x ¯ y y x y vx vy XY ϭ ͵ V XY ͕ͱ ͓ ͑ ͒Ϫ ͑ ͔͒ ͑ ͒ SCDW ͚ d cos 2 x,m na y,n ma 3.13 m,n
Ϫ1 ϩ ͑ Ϫ ͖͒ ͑ ͒ is the determinant of G . G can be used to calculate di- 2kF ma na . 3.17 mensions of operators involving . For example, the expec- tation value The dimensions of the cosine operators in the integrands are, respectively,
2 2 dq ¯ ͩ ϩ¯v q ͪ ⌬ ϵ ͑ ͒ϭ x ¯ y y SC,ϱ ͵ q K, dqx dqy d vy Ϫ 2 ͗2͑r,t͒͘ϭ ͵ , ͑3.14͒ x ͑2͒3 D¯ dq 1 1 1 ⌬ ϵ Ӎ ͑ ͒ where again we take /LϽ͉q ͉,͉q ͉,͉͉. In the integral, as CDW,ϱ ͵ , 3.18 x y Ϫ 2 ͑q͒ K ͱC⌬ before, the infrared divergence comes purely from the inte- gral over qx , . Once more, the infrared divergent part goes where we assume that (q) has the form given by Eq. ͑3.17͒, as ⌬ ϵ is defined as before, and C Љ(k0)/2K.If is chosen ͑ ͒ XY such that Eq. 3.16 is satisfied for each array, then SSC and dq 1 XY ͗2͘ϳ ͑ ͒ ͵ y ͑ ͒ SCDW are automatically irrelevant. Thus, we do not need any x ln L ͒ , 3.15 2 ͑qy further fine tuning of to get a stable CSLL phase. Having established a region of stability of the CSLL ϭ phase, we now investigate the nature of the correlation func- where (qy) x(0,qy) is the same function appearing in ͗2͘ ͑ ͒ tions. Consider once more the correlation function x , Eq. 3.8 . Thus, correlation functions for and can be calcu- x y X ͑ ͒ G͑x,yϭma͒ϵ͗exp͓i ͑x,͒Ϫi ͑0,͒ϩ2k x͔͘ lated directly from Eq. 3.2 . x- y cross correlations are x,m x,0 F nonsingular, whereas, - and - correlations have sin- x x y y ϩc.c., ͑3.19͒ gular parts with exactly the same functional forms as they have in the absence of coupling between layers, but with the which corresponds to the component of the density-density (q) function in expressions for the scaling exponents re- correlation function modulated at 2kF . In the absence of placed by XY terms such as SCDW , this correlation function vanishes for Þ XY y 0. Thus, though irrelevant, the presence of SCDW changes ͒ϭ ͒ϭ ͒ ͑ ͒ ͑qЌ x͑0,qЌ y͑qЌ,0 . 3.16 the nature of the correlation functions. In its presence, to lowest order in V XY, we obtain The same holds for - correlation functions. Thus correla- tion functions within a given array have the same functional ͑V XY͒2 X i2k (xϩy) cϭ G͑x,y͒Ϸ e F form as for V 0 but with different definitions of . Other 4 c than renormalizing (q), the coupling Vm,n between the two arrays leaves the dimensions of all operators unchanged. ϫ ͵ dx ͓͗exp͕iͱ2͓ ͑x,yϭma͒ This means that it is possible to choose interchain interac- 1 x tions within the X and Y grids so that these grids form 2D Ϫ ͑ ϭ ͔͖͒͘ ͗ ͕ ͱ ͓ ͑ ͒ anisotropic sliding phases even in the presence of the inter- x x1 ,y ma 0 exp i 2 y x1 ,y c ͑ ͒ ͑ ͒ grid coupling Vm,n . Equations 3.2 and 3.16 define a 2D Ϫ ͑x ,0͔͖͒͘ ͗exp͕iͱ2͓ ͑x ,0͒ non-Fermi-liquid with scaling properties to be discussed in y 1 0 x 1 Ϫ ͒ ϩ ͑ ͒ the next section. x͑0,0 ͔͖͘0͔ c.c., 3.20 First, however, we must verify that it is possible to choose potentials so that this 2D non-Fermi liquid is stable with where ͗͘0 is the expectation value with respect to the CSLL respect to perturbations. All pairwise couplings within a fixed point. We need the asymptotic form of the correlation X X Y Y function for large x, y. Following Eq. ͑2.12͒, we obtain, for given array, i.e., SSC,n , SCDW,n , SSC,n and SCDW,n defined as obvious generalizations of Eqs. ͑2.19͒, can be rendered irrel- example, evant by choosing (qЌ), as in the case of an individual array. We must also consider Josephson and CDW couplings A ͗exp͕iͱ2͓ ͑x,0͒Ϫ ͑0,0͔͖͒͘ Ϸ ⌬ . ͑3.21͒ between the two arrays, which operate at the points of cross- x x 0 x CDW,ϱ ing (x,y)ϭ(na,ma) of wire m in the X array and wire n of the Y array, respectively. These take the form Similarly
045120-11 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
1 1 d dq dq 1 2 ͗ ͕ ͱ ͓ ͑ ͒Ϫ ͑ ͔͖͒͘ ϳ ϭ ͵ x y ͫ ͩ ͑ ͒ 2ϩ ͪ ͉ ͉2 exp i 2 x,y x,0 ⌬ . S v q q y x 0 CDW,ϱ ͒3 ͒ x x ͒ x y 2 ͑2 x͑q vx͑q ͑3.22͒ 1 2 ϩ ͩ ͑ ͒ 2ϩ ͪ ͉ ͉2 ͑ ͒ vy q qy ͑ ͒ y ⌬ y q vy q Note that in the sliding phase, CDW,n is greater than unity and thus in Eq. ͑3.20͒, the largest contribution to the integral Ϫ͕ XY͑ ͒2 *ϩ ͖ͬ ͑ ͒ comes from x1 close to 0 and to x. It thus follows that for VR q x y c.c. , 3.25 large x and y, G goes as where
C cos͓2k ͑xϩy͔͒ ␥͑q͒ X F ϭͱ Ϸ ͑ ͒ ͑q͒ , G ⌬ ϱ , 3.23 x ͒ ͒ ͑xy͒ CDW, V ͑qy ,qz V ͑qx ,qz
͒␥ ͒ V ͑qy ,qz ͑q where C is a constant that is proportional to (V XY)2. Thus v ͑q͒ϭͱ , x V͑q ,q ͒ the correlation function G decays as a power law in all x z directions. Notice that G is not isotropic but exhibits a VXY͑q͒ square C4 symmetry. XY͑ ͒ϭ v VR q , The stability of the CSLL phase for the gapless case fol- ␥͑q͒ lows along the same lines. If there are no marginal interarray ␥͑ ͒ϭ ͑ ͒ ͑ ͒Ϫ͉ XY͉2 ͑ ͒ spin-dependent coupling terms, then ϭ1, and we define a q V qx V qy V 3.26 renormalized (qЌ). The stability of the CSLL phase is ϭ ϭ and y(q) x(Pq), vy(q) vx(Pq), where Pq identical to the stability of the smectic phase on a single ϭ ϭ P(qx ,qy ,qz) (qy ,qx ,qz). Proceeding exactly as in the array, with a fixed-point action described by the renormal- previous case, we find that ized function (qЌ). Also, proceeding as in Eqs. ͑3.20͒ to ͑ ͒ 3.23 , we now expect the single-electron correlation func- dq dq ͗2͘ϳ ͑ ͒ ͵ y z ͑ ͒ ͑ ͒ tions to exhibit power-law decay in all directions. x ln L x 0,qy ,qz 3.27 ͑2͒2 and B. Crossed 2:1 array
The above analysis can also be extended quite easily to a dqy dqz 1 ͗2͘ϳ ln͑L͒ ͵ . ͑3.28͒ three-dimensional stack of alternate 2D X and Y arrays. We x 2 ͑ ͒ ͑2͒ x 0,qy ,qz could also think of such a stack as a three-dimension array of wires running along the X axis, intermeshed with a 3D array The stability of the three-dimensional crossed stack is pre- of wires running in the Y direction. Thus the fixed point cisely the same as the stability of a three-dimensional stack ϩ ϩ action would be of the form SX SY SXY where SX and SY of parallel quantum wires with the Luttinger liquid parameter are the actions for the 3D arrays formed by wires running (qy ,qz) set equal to x(0,qy ,qz) of the crossed stack. As along the X axis and Y axis, respectively, where SXY repre- before, there are no additional singularities due to the cou- sents the interarray Coulomb interactions. Thus the fixed- pling between the crossed arrays. point action is IV. TRANSPORT PROPERTIES 1 d dqx dqy dqz We now investigate the transport properties of the sliding Sϭ ͵ ͓V͑q ,q ͒q2͉ ͉2 2 ͑2͒4 y z x x Luttinger liquid phases. The conductivities of an array of parallel wires has been considered by Emery et al.12 In a ϩ ͑ ͒ 2͉ ͉2ϩ ͑ ͒ 2͉ ͉2 V qx ,qz qy y V qy ,qz qx x pure system, the conductivity along a wire is infinite. In the presence of impurities, the resistivity along the wires van- ϩ ͑ ͒ 2͉ ͉2ϩ͕ XY͑ ͒ V qx ,qz ky y V qx ,qy ,qz qxqy x *y ishes as:25
ϩc.c.͖Ϫiq ͕* ϩc.c.͖Ϫiq ͕* ϩc.c.͖͔, ␣ʈ x x x y y y ʈϳT , ͑4.1͒ ͑ ͒ 3.24 with ␣ ϭ⌬ Ϫ ͑ ͒ ʈ CDW,ϱ 2. 4.2 where x,y and x,y are functions of , qx , qy , and qz , and XY V (qx ,qy ,qz) represents the interactions between the X and The conductivity perpendicular to the wires for an array of Y arrays. For interactions only between nearest-neighbor lay- parallel wire can be calculated12,27 using the Kubo formula, XY ϭ c ϩ iqz ers, we obtain V (qx ,qy ,qz) V (qx ,qy)(1 e ). Inte- giving us grating out the variables, we are left with an effective ␣Ќ action ЌϳT ͑4.3͒
045120-12 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120
where T X and T Y are current densities ͑current/area͒ in- jected, respectively, into the X and Y grids. If no currents are Xϭ YϭϪ injected, then this equation is solved by V V E•x to produce a total in-planar current density ϵ Xϩ Yϭ͑X ϩY ͒ ϭ͑ ϩ ͒ ͑ ͒ Ji Ji Ji ij ij E j ʈ Ќ Ei . 4.8 Thus under a uniform electric field, the double layer behaves like an isotropic 2D material with in-plane conductivity ϭʈϩЌӍʈ , or equivalently with an isotropic resistivity ␣ʈ that vanishes as ʈϳT . We could also consider currents that are spatially nonuni- form, as they are, for example, when current is inserted at one point and extracted from another. In that case, there is a FIG. 7. A schematic depiction of the 2D non-Fermi liquid as a crossover from isotropic to anisotropic behavior at length resistor network, with two parallel arrays of wire running along the scale x and y axes, with nodes in the z direction. ʈ Ϫ ␣ ϩ␣ ͑ ͒ ␣ ϭ ⌬ Ϫ ⌬ lϭaͱ ϳT ( ʈ c)/2 ͑4.9͒ with Ref. 28 Ќ 2 SC 3, where SC is the minimum of ⌬ ⌬ ͑ ͒ c SC,1 and SC,2 . For details see Appendix D. The conduc- → tance c , arising from the Josephson coupling at the contact that diverges as T 0. To illustrate this crossover, we calcu- between the crossed wires, can be calculated similarly using late explicitly the case where a current I is inserted at a point the Kubo formula, and satisfies r1 on the X array and extracted at another point r2 on the X array. Then ␣ ϳT c, ͑4.4͒ c T Xϭ ␦ Ϫ ͒Ϫ␦ Ϫ ͒ I͓ ͑r r1 ͑r r2 ͔, ␣ ϭ ⌬ Ϫ where c 2 SC,ϱ 3. In this section we focus on the T Yϭ ͑ ͒ gapped case. In the gapless case, ʈ , Ќ , and c still exhibit 0. 4.10 power-law behavior even though the major contribution to Using Eqs. ͑4.7͒ and ͑4.10͒, one can solve for the resistance perpendicular conductivities may come from single-particle between these two points: hopping. Thus we can model our 2D non-Fermi liquid as the resis- ˙ Ϫ VX͑r ͒ϪVX͑r ͒ d2q 1Ϫeiq(r2 r2) tor network depicted in Fig. 7 with nodes at the vertical ϭ 1 2 ϭ ͵ R 2 2 , Josephson junctions between the arrays at (x,y)ϭ(na,ma). I ͑2͒ g͑q͒ ͑ ͒ The nodes of the X(Y) array are connected by nearest- 4.11 Ϫ1 neighbor resistors with conductances ʈϭʈ , if they are where parallel to the x(y) axis and Ќ , if they are perpendicular to the x axis (y axis͒. Nearest-neighbor nodes of the X and Y c ͓͑ ϩ ͒ 2ϩ͑ ϩ ͒ 2͔ arrays are connected by resistors of conductance . In the ʈ Ќ qx ʈ Ќ qy c a2 continuum limit, the 2D current densities in the plane of the ͑ ͒ϭ ␣ ␣ ␣ g q ␣ grids (␣ϭX,Y)isJ ϭ E , where i ij j c Ϫ 2Ϫ 2 Ќqx ʈqy a2 ʈ 0 Xϭͩ ͪ ͑ ͒ 2 2 2 2 , 4.5 ͑Ќq ϩʈq ͒͑ʈq ϩЌq ͒ 0 Ќ x y x y Ϫ . ͑4.12͒ c 2 2 ϪЌq Ϫʈq Ќ 0 2 x y Yϭͩ ͪ , ͑4.6͒ a 0 ʈ For and E␣ is the in-plane electric field in plane ␣. The current 2 2 ϭ 2 /a ӷʈq , ͑4.13͒ per unit area passing between the planes is Jn ( c /a ) c X Y ϫ(V ϪV ), where V is the local voltage. In this limit, the ϩ 2 ͉ Ϫ ͉ӷ g(q) takes the simple form ( ʈ Ќ)q .If r1 r2 l, with local voltages satisfy l defined in Eq. ͑4.9͒, then the integral over q in Eq. ͑4.11͒ is ͑ ͒ ͉ dominated by small q satisfying Eq. 4.13 . Thus for r1 Ϫ ͉ӷ Xϩ c ͑ XϪ Y ͒ϭT X r2 l, the system has approximately the same resistance ץ ץ ϪX ij i jV V V , a2 as an isotropic conductor with conductivity ʈ .Ifwein- serted current at a point on the X array and extracted it from ͑ ͒ the Y array, we would have the form Eq. 4.11 for the re- c VYϪ ͑VXϪVY ͒ϭT Y, ͑4.7͒ sistance with a different function qЈ(q) whose small q limit ץ ץ ϪY ij i j 2 ϩ 2 ͉ Ϫ ͉ӷ a is still given by ( ʈ Ќ)q . Thus for r1 r2 l, the resis-
045120-13 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
tance is approximately independent of whether the current is knowledge useful discussions with S. Kivelson, E. Fradkin, inserted into ͑or extracted from͒ the X or the Y array. More and A. Vishwanath. generally, in a region where Tϭ0, by inspection of Eq. ͑4.7͒ it can be seen that inhomogeneities in the voltage ͑or differ- APPENDIX A ence between VX and VY) would heal over the lengthscale l as defined in Eq. ͑4.9͒, and at longer lengthscales the system In this appendix we sketch out the steps leading to the ϭ would behave isotropically. This length diverges as T→0 asymptotic form for the correlation function G(x, 0) for ͓ ͑ ͒ ͑ ͔͒ and at Tϭ0, current can only be carried along the wires; the large x see Eqs. 2.12 and 2.13 . resistance between wires in a grid or between grids is infi- ͑ ͒ϭ͗ ͕ ͱ ͓ ͑ ͒Ϫ͑ ͔͒ϩ ͖͘ϩ nite. G x,0 exp i 2 j x,0 0,0 i2kFx c.c. We could, in addition, investigate the frequency- ϭ Ϫ ͒Ϫ ͒ 2 ͒ ͑ ͒ exp͕ 2 ͓͗ j͑x j͑0 ͔ ͖͘cos͑2kFx . A1 dependent zero-temperature conductivity. By arguments similar to those used with Eqs. ͑4.1͒ to ͑4.4͒, we obtain It can be easily checked that
␣ʈ ͓͗ ͑ ͒Ϫ ͑ ͔͒2͘ ʈ͑͒ϳ , j x j 0 Ϫ ͑ ͒ ␣Ќ dqʈ dqЌ d 1 cos xqʈ Ќ͑͒ϳ , ϭ2 ͵ . ͑A2͒ ͑2͒3 2 ͑ ͒ͫ ͑ ͒ 2ϩ ͬ ␣ qЌ v qЌ qʈ ͒ϳ c ͑ ͒ ͑ ͒ c͑ . 4.14 v qЌ ␣ ␣ ␣ ʈ , Ќ , and c are the same as before, though the coeffi- Next we carry out the integration over qʈ and obtaining, cients are now different ͑and complex, in general͒. At finite for large x, , the long-wavelength resistivity is isotropic as before, and ␣ Ϫ ϳ ʈ 1 cos͑xqʈ͒ log͑x͒ vanishes as ʈ( ) . ͵ Ӎ ϩ ͑ ͒ dqʈ d 2 F qЌ , We could also consider extensions of these calculations to ͑qЌ͒ ͑ ͒ͫ ͑ ͒ 2ϩ ͬ qЌ v qЌ qʈ three-dimensional stacks of crossed arrays. As we saw in the v͑qЌ͒ previous section, it is possible to get a stable sliding phase in ͑A3͒ such a system. The conductivity now has a three-dimensional where F(qЌ) is some function of qЌ , which depends on , character, with conductivity along the planes given by ʈ /d, ͑ ͒ ͑ ͒ d being the separation between adjacent X arrays, but with v(qЌ), and the momentum cutoff. From Eqs. A1 and A3 it follows that conductivity in the third direction given by c /a. Thus the conductivity along the planes is much larger than the perpen- dqЌ 1 dicular conductivity. ͓͗ ͑ ͒Ϫ ͑ ͔͒2͘Ӎͫ ͵ ͬ ͑ ͒ϩ j x j 0 log x const. 2 ͑qЌ͒ ͑A4͒ V. CONCLUSION Using this, we obtain In conclusion, we have demonstrated the existence of non-Fermi metallic phases in two and three dimensions, that ͒ A1cos͑2kFx are stable with respect to a wide class of perturbations. We G͑x,0͒Ϸ ⌬ , ͑A5͒ x CDW,ϱ consider both spin-gapped systems and gapless systems that exhibit spin-charge separation. Our central results pertain to where the stability and properties of the CSLL phase. This is a remarkable phase, which could be identified as a two- dqЌ 1 ⌬ ϭ ͵ ͑ ͒ CDW,ϱ . A6 dimensional Luttinger liquid. The correlation functions in Ϫ 2 ͑qЌ͒ this phase exhibit power-law decay along the planes, and the finite-temperature long-wavelength electric conductivity, Equations ͑2.14͒ and ͑2.15͒ follow along similar lines. which is isotropic along the planes, diverges as a power law in temperature T as T→0. The importance of this paper is APPENDIX B that it provides a perturbative access to non-Fermi-liquid fixed points in two- and three-dimensional systems, some- Here we outline how the integral thing that has proven to be quite difficult in the past.7 This dq ͑1Ϫcos nq͒ paper could be of significant relevance for higher- ϭ ͵ ͑ ͒ In , B1 dimensional strongly correlated electron systems in general, Ϫ 2 f ͑q͒ and to the normal conducting phases of the cuprates in par- needed to calculate ⌬ , can be solved exactly. Here ticular. CDW,n ͒ϭ ϩ ͒ϩ ͒ ͑ ͒ f ͑q 1 1cos͑q 2cos͑2q , B2 ACKNOWLEDGMENTS ͑ ͒ using expression 2.30 for 1 and 2. We can rewrite R.M. and T.C.L. acknowledge support from the National ͒ϭ ϩ ϩ͒ ϩ Ϫ͒ ͑ ͒ Science Foundation under Grant No. DMR97-30405. We ac- f ͑q 2 2͑cos q u ͑cos q u , B3
045120-14 SLIDING LUTTINGER LIQUID PHASES PHYSICAL REVIEW B 64 045120 where APPENDIX C
Here we consider the infrared divergence of the integral 1 1 uϮϭ ͫ ϮiDͬ, ͑B4͒ 2 2 2 2 2 dq dq d ͑ /v ϩv q ͒ ϭ ͵ x y y y y ͑ ͒ I ͑ ͒3 D , C1 and 2 y where 2 ⌬ 1 2 2 D2ϭϪ ϩ Ϫ2ϭ . ͑B5͒ 42 1 2 2 2 2 Dϭ ͩ ϩ 2 ͪ ͑2 ϩ 2͒Ϫ͑ c ͒24 vxqx /vy vyqy VR . x y vx It is easy to check that ͑C2͒ 1 At first sight, it may be appear that the integral is divergence ϭϪ ϩ ͑ ͒ In Im Jn B6 free, since by power counting, there are two powers of Q ͱ2⌬ ͓ ϭ 3 2 where Q ( ,qx ,qy)] in the numerator multiplying d Q, and four powers of Q in the denominator. This seems to with indicate that the integral is finite as L→ϱ. Notice, however, that if Vc is set equal to zero in D, the integral can be written 2 dq 1Ϫcos͑nq͒ 1 R ϩϭ ͵ ϭ ͓ ϩ ͔ as Jn ϩ J1,n J2,n , 0 2 u ϩcos q i ͒ dqx dqy d x͑qx ,qy Iϭ ͵ , ͑C3͒ dz 23 2/v ϩv q2 J ϭ Ͷ , x x x 1,n 1ϩ2uϩzϩz2 which is clearly infrared divergent. This divergence comes purely from the integration over q ,. It turns out that even ͓ ϩ 2n͔ x 1 dz 1 z in the presence on Vc , the divergence comes purely from the J ϭϪ Ͷ ϩ , ͑B7͒ R 2,n n ϩ ϩ 2 2 z 1 2u z z integration over qx , . To obtain the infrared divergent part we write where zϭeiq, and the z integral is over the unit circle cen- tered about the origin. The integrands have poles at z ͑2 ϩ 2͒ ͒ /vx vxqx x͑0,qy ϭ ϩ Ϫ ϭ ϩR, ͑C4͒ (0,z0 ,z0 ): D 2 ϩ 2 x /vx vxqx ϮϭϪ ϩϮ ͱ Ϫ͑ ϩ͒2 ͑ ͒ z0 u i 1 u . B8 where R is the remaining piece, and our task is to show that ϭ its integral has no infrared divergence. Let us write R R1 ͑ ͒ ϩϫ Ϫϭ ϩ Using Eq. B8 , it is easy to check that z0 z0 1. Thus, R2, where ϩ Ϫ ͑ either z0 or z0 lies inside the contour of integration the unit ͒ Ϫ ͑q ,q ͒Ϫ ͑0,q ͒ circle . Using the method of residues, it is now straightfor- ϭ x x y x y R1 2 ϩ 2 , ward to calculate the integrals. We simply need to sum over /vy vyqy the residues of the poles enclosed within the contour of in- tegration. In order to express our results, we distinguish two ͑2 ϩ 2͒ /vy vyqy 1 x y cases: R ϭ ͫ Ϫ ͬ. ϩ 2 D ͑2 ϩ 2͒͑2 ϩ 2͒ ͑1͒ ͉z ͉Ͻ1. Then y /vx vxqx /vy vyqy 0 ͑C5͒ ϩ nϪ1 2 ͑1ϩ͑z ͒2n͒ 1 The integral of R has no infrared divergence. To check that ϩϭ ͫ Ϫ 0 ͬϪ 1 Jn ϩ Ϫ 1 ϩ ͚ ϩ ϩ Ϫ Ϫ ͑ Ϫ ͒ 2͑z ͒n ϭ ͑z ͒m 1͑z ͒n m this is true for R2 as well, we note that in the expression for z0 z0 0 m 0 0 0 R2 the term in the square brackets can be written as 2 ϭ ͓ Ϫ͑ Ϫ͒p͔ ͑ ͒ c 4 ϩ Ϫ 1 z0 . B9 x yV ͑z Ϫz ͒ R ͑ ͒ 0 0 D͑2 ϩ 2͒͑2 ϩ 2͒ . C6 /vx vxqx /vy vyqy Ϫ ͑2͒ ͉z ͉Ͻ1. In this case 0 Now, by noticing the powers of ,qy , it is easy to see that the integration of R has no divergence. Thus the infrared Ϫ Ϫ 2 2 ͑1ϩ͑z ͒2n͒ n 1 1 divergent part of I can be written as ϩϭ ͫ Ϫ 0 ͬϪ Jn Ϫ ϩ 1 Ϫ n ͚ ϩ mϩ1 Ϫ nϪm ͑z Ϫz ͒ 2͑z ͒ mϭ0 ͑z ͒ ͑z ͒ 0 0 0 0 0 dq dq d ͑0,q ͒ ͵ x y x y ͑ ͒ ͑ ͒3 2 ϩ 2 . C7 2 ϩ 2 /vx vxqx ϭ ͓1Ϫ͑z ͒p͔. ͑B10͒ ͑ ϪϪ ϩ͒ 0 z0 z0 Equation ͑3.8͒ now follows easily.
045120-15 RANJAN MUKHOPADHYAY, C. L. KANE, AND T. C. LUBENSKY PHYSICAL REVIEW B 64 045120
APPENDIX D The next step is to calculate the Matsubara correlator ⌸ (x,). To begin with, we only consider nearest-neighbor We demonstrate explicitly the leading dependence of the M hoppings between wires. To lowest order in J and T: perpendicular conductivity on temperature. According to the Kubo formula, the transverse conductivity is given by
⌸ ͑x,͒ϭ͚ ͗JЌ͑x, j,͒JЌ͑0,0,0͒͘, ͑D7͒ i n e2 M ͑͒ϭ ͫ⌸ ͑͒ϩ 0 ͬ ͑ ͒ j Ќ Ќ , D1 m where where the first term represents the paramagnetic contribution with ͒ϭ J ͒Ϫ ͒ J͑x, j, a 1͕sin͓ j͑x, jϪ1͑x, ͔ ϱ it ϩsin͓ ϩ ͑x,͒Ϫ ͑x,͔͖͒. ͑D8͒ ⌸Ќ͑͒ϭϪi͚ ͵ dx͵ dt⌰͑t͒e j 1 j j Ϫϱ Here a is the distance between adjacent wires, and the ex- ϫ͓͗JЌ͑x, j,t͒,JЌ͑0,0,0͔͒͘ ͑D2͒ pectation value is taken with respect to the sliding fixed- being the retarded current-current correlator, and the second point Lagrangian. The correlator can be written as term represents the diamagnetic contribution. The step func- ͕ Ϫ ϩ Ϫ ͖ ⌰ ͓ ϩ ͔ ⌸ ͒ϭJ 2 2 i [ 1(x, ) 1(0,0)] [ 0(x, ) 0(0,0)] tion (t) may be written as 1 sign(t) /2, where sign(t)is M͑x, a ͗e ͘ ϩ1 for positive t, and Ϫ1 for negative t. In the spin-gapped ϭJ 2 2 Ϫ ͒ ͑ ͒ case, the contribution to the paramagnetic part comes from a exp͓ f ͑x, ͔, D9 superconducting pair hopping. The paramagnetic and dia- magnetic terms can be combined to give where
i ͑ ͒ϭ͓͗ ͑ ͒Ϫ ͑ ͔͒2Ϫ͓ ͑ ͒Ϫ ͑ ͔͒ ͑͒ϭ ͫ ͵ ͵ it͓ ϩ ͑ ͔͓͒⌸Ͼ͑ ͒ f x, 1 x, 1 0,0 1 x, 1 0,0 Ќ dx dte 1 sign t x,t ϫ ͒Ϫ ͒ ͓ 0͑x, 0͑0,0 ͔͘ Ϫ⌸Ͻ͑x,t͔͒Ϫ͑ϭ0͒ͬ, ͑D3͒ 1 ϭ ͵ ͕͗͑ ͒ ͑ ͒͘ ͚ dqЌ dqx qx ,qЌ , * qx ,qЌ ,  where ϫ Ϫ ϩ͒ ϫ Ϫ ͒ ͑ ͒ ͓1 cos͑qxx ͔ ͑1 cos qЌ ͖. D10 Let us first consider a simpler case, where the velocity vs ⌸ϾϭϪi͚ ͗J͑x, j,t͒J͑0,0,0͒͘, j has no dependence with qЌ . Then
⌸ϽϭϪi͚ ͗J͑0,0,0͒J͑x, j,t͒͘. ͑D4͒ j
Since ⌸ϾϪ⌸Ͻ is odd in t, the real part of the conductivity is given by
i Ј ͑͒ϭ ͫ ͵ ͵ it͓⌸Ͼ͑ ͒Ϫ⌸Ͻ͑ ͔͒ͬ Ќ dx dte x,t x,t . ͑D5͒ Note that the dc transverse conductivity is purely real, and Ͼ can be obtained from ЌЈ () by taking the limit →0. ⌸ is related by analytic continuation to the Matsubara cor- ⌸ relator M(x, ) in the upper-half-plane of complex t space, ⌸Ͻ ⌸ and is related to M(x, ) in the lower-half-plane. Thus we could view the integral in Eq. ͑D5͒ as an integral over the Keldyish contour shown in Fig. 8͑a͒. This contour can be distorted to the contour shown in Fig. 8͑b͒. Note that ⌸Ͼ(t ϩ  ϭ⌸Ͻ Ϫ  ϭ i /2) (t i /2), where 1/kBT. Thus, we obtain
1 ϱ  i Ј ͑͒ϭ ͵ ͵ ͫ it ͩ ͪ ⌸Ͼͩ ϩ ͪͬ FIG. 8. ͑a͒ The Keldyish contour in complex t-plane, with real Ќ dx dt e sinh x,t . Ϫϱ 2 2 time along the X axis. In ͑b͒ we depict how the contour is deformed ͑D6͒ in order to evaluate the integral.
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⌬ Ϫ ⌸ ͒ (2 SC,1 3) M͑x, ЌЈ ͑ϭ0͒ϳT . ͑D14͒ ͒2 ͑ Tax /v When the velocity v is a function of qЌ , the integral can ϭa2J 2 , 1 ͕sinh͓T͑x/vϩi͔͒sinh͓T͑x/vϪi͔͖͒ no longer be solved exactly. However, the leading T depen- dence of the conductivity remains unchanged. To check this, ͑D11͒ we follow the previous set of steps and arrive at the expres- ϭ␦ sion where SC,1 , and ax is the spatial cutoff along x. Thus we may write sinh͑/2T͒ Ј ϭ 2J 2 ͑ ͒2͑ ͒Ϫ2 Ќ ay 1 Tax T sinh͑/2͒ Ј ϭ 2J 2 ͑ ͒2͑ ͒Ϫ2 ͵ ˜ ˜ Ќ ay 1 Tax /v T v dx dt ˜ ϫ ͵ dx˜ dt˜ͫ eit(/T) ˜ eit(/T) ϫ , ͑D12͒ ͓cosh͑˜xϩ˜t ͒cosh͑˜xϪ˜t ͔͒ 1 ϫ͟ ͬ, ⌬ ͓ ͑ ͒ ͑˜ ϩ˜͒ ͑˜ Ϫ˜͔͒ qЌ where ˜xϭxT/v and ˜t ϭtT ͑Fig. 8͒. By introducing new qЌ v qЌ cosh x/v t cosh x/v t variables ˜xϩ˜t and ˜xϪ˜t , we carry out the above integrals, ͑D15͒ giving ˜ϭ ˜ϭ ͚ ⌬ ϭϭ⌬ where x xT, t tT, and qЌ qЌ SC,1 . The result sinh͑/2T͒ can be expressed in the scaling form Ј ϭJ 2a2͑va ͒2 v͑T͒2Ϫ3 Ќ 1 y x 8 /T ЌЈ ͑,T͒ϭT F͑/T͒. ͑D16͒ ⌫2͑/2ϩi/4T͒⌫2͑/2Ϫi/4T͒ In the limit →0, the integral is finite and T independent, ϫ , ⌫2͑͒ implying F(0) is finite. Thus Ќ(ϭ0)ϳT , where ϭ⌬ for nearest-neighbor hopping. In exactly the same ͑D13͒ SC,1 manner, we can calculate the contribution to ЌЈ from next- ϭ⌬ → where SC,1 . The 0 limit of the above expression nearest neighbor pair hopping. It has the same scaling form ⌬ ⌬ yields as before with SC,1 replaced by SC,2 .
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Ref. 12, and define as the phase variable, and as the conju- 26 We should note, however, that the set of possible interchain op- gate density variable. erators is huge and we have not explicitly checked for all of 21 V.J. Emery et al., Phys. Rev. B 56, 6120 ͑1997͒. them. 22 C.S. O’Hern, T.C. Lubensky, and J. Toner ͑unpublished͒. 27 A. Georges, T. Giamarchi, and N. Sandler, Phys. Rev. B 61, 23 Ϫ In these expressions, i(x) corresponds to i(x) i,0 , where i,0 16 393 ͑2000͒; A. Lopatin, A. Georges, and T. Giamarchi, Phys. ϭ refers to the uniform part of . Hence, we exclude the qʈ 0 Rev. B 63, 075109 ͑2001͒. modes from the integrals. 28 In this calculation we have ignored the effect of diorder. At low 24 A.B. Harris, J. Phys. C 7, 1671 ͑1974͒; A.B. Harris and T.C. temperatures, we expect the temperature dependence of Ќ to be ͑ ͒ Lubensky, Phys. Rev. Lett. 33, 1540 1974 . modified due to forward scattering from impurities. In particular, 25 A. Luther and I. Peschel, Phys. Rev. Lett. 32, 922 ͑1974͒. See for very low temperatures, ␣Ќϭ2⌬ Ϫ2. also, T. Giamarchi and H. Schulz, Phys. Rev. B 37, 325 ͑1988͒. SC
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