Two-Dimensional Spin Liquids with Z2 Topological Order in an Array of Quantum Wires

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PHYSICAL REVIEW B 94, 195130 (2016)

Two-dimensional spin liquids with Z2 topological order in an array of quantum wires

Aavishkar A. Patel1,* and Debanjan Chowdhury2,† 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 15 August 2016; published 16 November 2016)

Insulating Z2 spin liquids are a phase of matter with bulk anyonic quasiparticle excitations and ground-state degeneracies on manifolds with nontrivial topology. We construct a time-reversal symmetric Z2 spin liquid in two spatial dimensions using an array of quantum wires. We identify the anyons as kinks in the appropriate Luttinger-liquid description, compute their mutual statistics, and construct local operators that transport these quasiparticles. We also present a construction of a fractionalized Fermi liquid (FL*) by coupling the spin sector

of the Z2 spin liquid to a Fermi liquid via a Kondo-like coupling.

DOI: 10.1103/PhysRevB.94.195130

I. INTRODUCTION for time-reversal symmetric QSLs as ground states of exactly solvable (but somewhat artificial) Hamiltonians [23,24], which Mott insulators without any broken symmetries, commonly provide a complementary and useful point of view on the above referred to as quantum spin liquids (QSLs), have been studied approaches. theoretically for more than four decades now. Starting with the In this paper, we take yet another route to arrive at the original theoretical proposal for the resonating valence bond description of a gapped Z QSL, which does not rely on liquid by Anderson [1], much of the interest in QSLs has been 2 either of the above two approaches. This approach involves driven by the study of high-temperature superconductivity constructing an interacting phase in (2+1) dimensions starting [2–6] and quantum magnetism in low dimensions [7]. Many from a set of decoupled Luttinger-liquid wires in (1+1) dimen- interesting insights have been gained by using the notion of sions and turning on nonperturbative interactions between the topological order [8] in order to draw parallels between gapped wires. It has been applied remarkably successfully to describe spin liquids [9,10] and other interesting phenomena such as and construct, e.g., electronic liquid crystalline phases in the fractional quantum Hall effect [11]. doped Mott insulators [25], the Laughlin state in the fractional On the experimental side, a number of quasi-two- quantum Hall (FQH) effect [26], and, more recently, even the dimensional materials have been proposed to host QSL ground non-Abelian and compressible FQH states [27,28]. By using states [7]. One of the most well-studied and promising such this route, we obtain a fully gapped time-reversal symmetric materials is herbertsmithite, consisting of spin-1/2 moments Z QSL and identify the local operators that correspond arranged in a kagome lattice. There are indications from 2 to and transport the bulk quasiparticles and compute their theoretical studies that the ground state of the nearest-neighbor mutual statistics.1 A similar approach has been used to Heisenberg model (supplemented by next-nearest-neighbor construct chiral spin liquids [29] (with broken time-reversal interactions) on the kagome lattice is a gapped Z spin 2 symmetry) [30,31], Abelian topological phases in higher liquid [12,13], even though the question is far from being than two spatial dimensions [32,33], and even non-Abelian settled definitively. At the same time, inelastic neutron scat- topological spin liquids [30,34]. tering [14,15] and nuclear magnetic resonance (NMR) [16] The rest of this paper is organized as follows: In Sec. II, measurements on herbertsmithite have detected the existence we summarize the key features of Z spin liquids using a of a spinon continuum over a broad energy window and a spin 2 Chern-Simons effective field theory description. In Sec. III,we gap, respectively. propose a purely bosonic coupled wire construction for the Z Theoretical descriptions of QSL ground states usually 2 spin liquid or, more specifically, the toric-code model, in (2+1) rely on a “parton” description. Within this prescription, the dimensions. Section IV summarizes our key results for the bulk canonical fermionic operator is fractionalized in terms of quasiparticles, their mutual statistics, and the edge physics in excitations that carry its spin and charge separately along the insulating Z spin liquid within the wire construction. In with the introduction of an emergent gauge field that encodes 2 Sec. V, we fermionize the above description in order to arrive the nontrivial entanglement in the system. The spin-liquid at a coupled wire construction of a Z fractionalized Fermi phase corresponds to the deconfined phase of an appropriately 2 liquid (FL*) via a “Kondo”-like construction. We conclude in defined gauge theory; examples of gauge groups that often Sec. VI with a summary of our results and an outlook for some arise in descriptions of various interacting models include Z , 2 future directions. U(1), and SU(2) coupled to matter fields that are either gapped, gapless at special points, or gapless along an entire contour in momentum space (see, e.g., Refs. [17–22] for a few repre- sentative examples). There also exist alternative descriptions 1We note in passing that it is, in principle, possible to construct a gapped spin-liquid phase starting from the decoupled (Jz = 0) and *[email protected] gapless limit of Kitaev’s honeycomb model [24] and then study the † [email protected] effect of a finite Jz perturbatively.

II. PRELIMINARIES III. BOSONIC COUPLED-WIRE CONSTRUCTION

In this section, we review key features of Z2 spin liquids In this section, we arrive at a description of an insulating in terms of the low-energy effective theory for its topological Z2 spin liquid using a purely bosonic construction; we defer states in terms of a Chern-Simons action in imaginary time a discussion of the excitations and edge physics to the next (τ)[35,36], section. We begin by considering an array of uncoupled identical one-dimensional (1D) quantum wires (labeled = i i S = 2 I J + I CS dτd x μνλaμKIJ∂ν aλ tI Aμμνλ∂ν aλ . 1,2,...), where each wire consists of two chains, a and 4π 2π b [see Fig. 1(a)]. Each of these chains is described by a (1) nonchiral Luttinger liquid (LL). We denote the bosonic fields associated with the LL on the th wire and on chains a or b In the above action, I,J are indices extending from 1,...,N a,b a,b as {θ (x),ϕ (x)}; they satisfy the following commutation and aI are N U(1) gauge fields, with A a fixed external μ μ relations: “probe” gauge field. The above action realizes an insulating = a(b) a(b) π Z2 spin liquid for N 2 with a K matrix given by θ (x),ϕ (y) = i sign(x − y)δ . (8) 2 02 K = , (2) We now carry out a series of transformations on the above 20 fields, introducing new degrees of freedom at each stage, as and where the ground-state degeneracy on a torus is given follows. by |detK|. The electromagnetic charge of the quasiparticles is We first define a new set of variables, determined by the vector tI , 1 = a + b 2 = b + a φ ϕ mθ ,φ ϕ m θ , (9) 1 1 a b 2 b a t = . (3) φ¯ = ϕ − m θ , φ¯ = ϕ − mθ . I 0 The commutation relations for these variables are It is possible to integrate out the internal gauge fields {aI }, μ 1 2 =− ¯1 ¯2 leading to φ (x),φ (y) φ (x),φ (y) π T −1 2 i = i (m + m )sign(x − y)δ , (10) SCS = (t K t) dτd x μνλAμ∂ν Aλ . (4) 2 4π which follow trivially from Eq. (8). We also note that the T −1 The Hall response is then given by σxy = (t K t) in units definitions in Eq. (9) are chosen such that of e2/2π, which is identically zero for the Z spin liquid as it 2 1 ¯2 = ¯1 2 = preserves time-reversal symmetry. φ (x),φ (y) φ (x),φ (y) 0, (11) The quasiparticle excitations of the theory are characterized 1 ¯1 = ¯2 2 = φ (x),φ (y) φ (x),φ (y) 0. by an integer vector l, such that they couple minimally to the I Now introduce a “new” array of wires, defined on the “dual”- combination: I lI aμ. The self-statistics of a quasiparticle is determined by lattice sites, j ≡ + eˆy /2, with the bosonic fields {θ,ˆ φˆ} and { } T −1 θ ,φ [see Fig. 1(b)]. They are defined as θself = lK l , (5) 1 1 2 2 φ − φ¯ + φ + φ¯ + with θ = 0(mod2π) for bosons and θ = π(mod2π)for θˆ = 1 , φˆ = 1 , self self j 2 j 2 fermions. The mutual statistics between two different quasi- (12) φ2 − φ¯2 φ1 + φ¯1 particles (“1” and “2”) is given by = +1 = +1 θj ,φj . = T −1 2 2 θmutual 2πl1 K l2, (6) Using the commutation relations in Eqs. (10) and (11), the = with θmutual π(mod2π) for mutual semions. In particular, commutation relations for the bosonic fields on the dual lattice the Z2 spin liquid has the following quasiparticle excitations: e, m, and ε, with 1 0 1 l = , l = , l = . (7) e 0 m 1 ε 1 It is straightforward to show that e,m are bosons, while ε is a fermion. All of the above quasiparticles are mutual semions. Before we present our construction, the reader might wonder how to obtain a nonchiral and fully gapped state (a)(b) starting from an array of coupled wires. As will become clear in the next section, one of the key ingredients is to be FIG. 1. (a) Representation of intra- and interwire scattering terms. able to find a set of modes with vanishing self- and mutual Vertical arrows represent the tunneling of bosons between wires (e.g., commutators. It then allows us to add independent sine-Gordon ∼ a − a ϕ ϕ +1). The circular arrows represent backscattering within b b terms for each of these modes in the action, pinning the fields a wire (e.g., mθ ,m θ +1). The dark (dashed) arrows represent a O O to certain classical values simultaneously and gapping out the combination of all the processes involved in 1(x)[ 2(x)] in Eq. (18). edge excitations [27,37]. (b) Lattice of “dual” wires labeled by j.

195130-2 TWO-DIMENSIONAL SPIN LIQUIDS WITH Z2 . . . PHYSICAL REVIEW B 94, 195130 (2016) sites are given by combination of interwire boson hoppings and scatterings off boson density fluctuations, ˆ ˆ = [θj (x),φj (y)] [θj (x),φj (y)] a/b a/b + a/b − a/b ∼ 2iθ (x) 2iπρ¯ x π ρ (x) ρ¯ e , (19) = i (m + m )sign(x − y)δjj , (13) 4 in the wires [27,37], where we have taken the average densities and all other fields commute, i.e., ρ¯a/b to be independent of . Imagining the wires to be one-dimensional lattices with lattice constant a0,wesetthe ˆ = ˆ [θj (x),θj (y)] [φj (x),φj (y)] average densities of bosons,ρ ¯a/b, at commensurate values so iπ(m±m)¯ρa/bx = ˆ = ˆ = that the oscillatory factors e are equal to 1. Then, [θj (x),φj (y)] [θj (x),φj (y)] 0. (14) oscillatory factors do not appear in the combinations of hop- Thus far we have kept the description in terms of m,m (∈ pings and scatterings used to achieve Eq. (18) and they are thus integers) completely general. As will become clear later in not trivially rendered irrelevant in the long-wavelength limit. Sec. IV, we require from Eq. (13) that m + m = 4 in order for We note that the scattering terms have been cleverly chosen the bulk anyonic quasiparticles to be mutual semions with a such that they gap out all possible single-site modes; this fol- relative phase of π. Moreover, in order to make the definitions lows from the observation that (a ϕa + b ϕb + c θ a + d θ b) = = symmetric, it is natural to choose m m 2. can never commute with all the terms in HIC simultaneously The remainder of our discussion will be based on the wires for any nontrivial choice of a ,b ,c ,d [37]. Hence the bulk labeled j. In particular, the usual LL Hamiltonian for these of the system will be gapped—one of the criteria for realizing decoupled wires is given by a Z spin liquid. 2 vˆ 1 HIC can therefore be most simply expressed as H = j dx (∂ θˆ )2 + gˆ (∂ φˆ )2 0 2π gˆ x j j x j = ˆ + j j HIC [Cj,1 cos(2θj ) Cj,2 cos(2θj )], (20) j v j 1 2 2 + dx (∂x θ ) + g (∂x φ ) , (15) and the entire system is described in terms of the following 2π g j j j j j Hamiltonian: wherev,v ˆ are the effective velocities and g,gˆ represent the HSL[θ,ˆ φ,θˆ ,φ ] = H0 + HF + HIC. (21) Luttinger parameters for each individual wire. gˆ ,g In addition, we also allow for forward-scattering terms By appropriately tuning the values of j j , both of the coefficients Cj,α can be made relevant. A simple choice is between different wires, = = = to set HF 0 and to set gˆj gˆ and gj g . This choice ∂ φˆ produces independent sine-Gordon models for each of the = ˆ ˆ ˆ x j HF dx(∂x φj ∂x φk)Mjk j ∂ φˆ wires. Then we have the following renormalization-group j=k x k (RG) flow equations for these coefficients [27,38]: ∂ φ + + x j dCj,1 m m dx(∂x φj ∂x φk)M jk , (16) = 2 − gˆ Cj,1, = ∂x φk dl 2 j k (22) + ˆ × dCj,2 m m where the matrices Mjk,M jk represent 2 2 matrices that = 2 − g Cj,2. describe interactions between wires labeled j and k. The theory dl 2 described in H0 + HF is quadratic in the fields {θ,ˆ φˆ}, {θ ,φ } Therefore, at low energies, the system flows to a gapped phase and describes a “sliding” LL phase. in which both θˆ and θ are localized in the respective wells of Let us now add to the above Hamiltonian further inter- the cosine potential if gˆ(l = 0),g(l = 0) < 4/(m + m); this is channel scattering terms; it is useful to go back briefly to the made possible by the additional fact that these fields commute. description of our system in terms of the original wires labeled We will henceforth make Cj,1 and Cj,2 independent of j as [Fig. 1(a)]. Then it is possible to write a term of the form well, and drop the j label on them. Moreover, in the remainder of this paper, we shall set m = m = 2, unless stated otherwise. = O HIC dxC ,α α(x), (17) ,α=1,2 IV. BULK AND EDGE EXCITATIONS O with two specific choices of α(x): Let us now investigate the nature of the excitations that arise in the system described by H [θ,ˆ φ,θˆ ,φ ]. In particular, our O (x) ∼ cos φ1 − φ¯1 = cos ϕa − ϕa + mθ b + m θ b SL 1 +1 +1 +1 aim is to identify the bulk anyonic quasiparticles along with = cos(2θˆj ), the operators that transport them and study the fate of the edge excitations. O ∼ 2 − ¯2 = b − b + a − a 2(x) cos φ φ +1 cos ϕ ϕ +1 m θ mθ +1 = cos(2θj ). (18) A. Bulk quasiparticles and Wilson loops The solid and dashed arrows in Fig. 1(a) depict the scattering It is clear from the form of the term in Eq. (20) that processes involved above. For our bosonic wires, we need quasiparticles (in the bulk) correspond to kinks in θˆj and ≡ ˆ m,m 0 (mod2) so that the above terms can be written as a θj , where they jump by π; the states described by θj and

195130-3 AAVISHKAR A. PATEL AND DEBANJAN CHOWDHURY PHYSICAL REVIEW B 94, 195130 (2016)

Let us now construct the local operators that hop quasiparticles from wire j to wire j + 1,

a i(φˆj −φˆj+1−θ −θ + )/2 −2iθ + ˆ j,j+1 = e j j 1 = e l 1 , (24) − − ˆ − ˆ − b = i(φj φj+1 θj θj+1)/2 = 2iθl+1 j,j+1 e e ,

which are again proportional to the previously discussed (a) (b) scatterings off density ﬂuctuations on the original a,b wires [Eq. (19)]. The operators that transfer quasiparticles from x1 FIG. 2. (a) Adiabatic transport of quasiparticles gives statistics. to x2 along wire j are given by Here we take a θˆ around a loop, picking up a Berry phase proportional to the number of θ s enclosed. (b) The presence of a “vison” (θ ) x2 −i dx(∂x φˆj )/2 ˆ = x1 induces a branch cut in the hopping of a “spinon” (θˆ). ζj (x1,x2) e , (25) x2 −i dx(∂x φ )/2 ζ (x ,x ) = e x1 j , ˆ → ˆ + j 1 2 θj θj π are energetically equivalent (similarly for θj ). The quasiparticle density operators are then given by which can again be expressed in terms of the a,b boson currents ∂ θˆ ∂x θ and densities. ρˆ = x j ,ρ = j . (23) j π j π The mutual statistics of the bulk quasiparticles is easily generated by computing the phase generated by taking a ∓ ˆ The operators e iφj /2 create and annihilate θˆ quasiparticles, quasiparticle around a loop adiabatically [27]. Such a process ∓iφ /2 while the operators e j create and annihilate θ quasiparti- is illustrated in Fig. 2(a). The Berry phases generated by such cles, respectively. processes are

⎛ ⎞ ⎛ ⎞† iˆ = ⎝ ˆ ⎠ˆ ⎝ ˆ ⎠ ˆ e j,j+1(x2) ζj2 (x2,x1) j,j+1(x1) ζj1 (x1,x2),

j1j

ˆ ˆ corresponding to only half a revolution. Thus, this additional where Nj , Nj are the number of θ and θ quasiparticles inside the loop on wire j. composite quasiparticle is a fermion and can be identiﬁed as A phase of −π is picked up for each quasiparticle of the ε introduced earlier in Sec. II. the other kind inside the loop (and −π/2 for those on the We now place the array of n wires (i.e., j = 1,...,n) boundaries of the loop along the wires). Moreover, the phase on a torus of dimensions (Lx ,n) (Fig. 3). The Wilson loop accumulated is 0, in the absence of any quasiparticles inside the loop, thereby establishing mutual semionic statistics. We can identify the above quasiparticles as the e and the m introduced in Sec. II above. At this point, there is nothing in our construction that distinguishes between the two quasiparticles. We note that the above ﬁelds commute mutually and hence there can be an additional composite quasiparticle, associated ˆ with a simultaneous kink in θj ,θj . The density operator for this ˆ + quasiparticle is given by ∂x (θj θj )/(2π) and it is created and ∓i(φˆ +φ )/2 annihilated by e j j , respectively. Repeating the above procedure, we see that this quasiparticle has semionic statistics with each of the θˆ and θ quasiparticles. Similarly, we also note that each such quasiparticle within the loop contributes a phase of −2π to the one being taken around. Since this involves a FIG. 3. Wire array of Fig. 1(b) on a torus. The Wilson loop − full revolution, it implies an exchange statistical angle of π, operators Wˆ y and Wˆ x are shown in red and blue, respectively.

195130-4 TWO-DIMENSIONAL SPIN LIQUIDS WITH Z2 . . . PHYSICAL REVIEW B 94, 195130 (2016) operators [35,39,40] are then given by parton constructions of Z2 spin liquids [43]. A different model n of coupled spin chains without Z2 topological order but with spinons capable of hopping between chains was previously Wˆ y (x) = ˆ j,j+1(x), Wˆ x = ζˆ1(0,Lx ), proposed in Ref. [44]. j=1 (27) n = = B. Bulk gap Wy (x) j,j+1(x),Wx ζ1(0,Lx ). j=1 The bulk quasiparticle excitations are gapped, with a finite energy required to create them. The gaps are nonuniversal and + ≡ We use periodic boundary conditions to identify n 1 1 are, in general, different for θˆ and θ , which would correspond ≡ and Lx 0. They obey the algebra to different gaps for the spinons and visons. As we show below, ˆ =− ˆ they depend on the details of the renormalization-group flows Wx Wy (x) Wy (x)Wx , (28) of the sine-Gordon models on the j wires. The flow equations ˆ =−ˆ Wx Wy (x) Wy (x)Wx , for C1,C2 are given by Eq. (22); the equations for g,gˆ are [38] with all other combinations commuting. This operator algebra is easily realized by two independent sets of Pauli matrices, dgˆ =−A C2gˆ3, signaling the fourfold degeneracy of the ground state on the dl 1 1 (32) torus. dg =− 2 3 One possible choice for the action of the time-reversal A2C2 g , operator T on the bosonic fields is [37] dl T a → a b →− b a →− a b → b + where A1 and A2 are nonuniversal numerical√ constants. Defin- : θ θ ,θ θ ,ϕ ϕ ,ϕ ϕ π. (29) = − = − ⊥ = ⊥ = ing √z1 2gˆ 2, z2 2g 2, z1 C1/ 8A1, and z2 The wires labeled b can then be thought of as being derived C2/ 8A2, we have the Kosterlitz-Thouless RG equations for | | from the bosonization of XX spin-1/2 chains [38,41], while small z1,2 , the wires labeled a are simply neutral spinless bosons.

This is consistent with the above choice of time reversal dz1,2 ⊥ 2 4iπρ¯bx ≈−(z ) , and the requirement that e not oscillate. Then, the dl 1,2 interwire terms in Eq. (18) are only invariant under the SO(2) (33) dz⊥ rotations of the spin components in the XY plane, given by 1,2 =− ⊥ b → b + b z1,2z1,2. ϕ ϕ f , and not under the SO(3) rotations that mix ϕ dl b and θ [42]. When z < 0 and z⊥ |z |, the system flows to strong This choice of time reversal sends θˆj →−θˆj , while leaving 1,2 1,2 1,2 θ invariant; since we wish to identify the time-reversal odd coupling and the bulk is gapped. Additionally, when j − excitations of the toric code with physical spin densities, z1,2 > 1, the low-energy excitations in the bulk are the ∼ we call θˆ “spinons” and θ “visons”. With this choice, the kinks we discussed previously, and the bulk gaps 1,2 kink/antikink creation operators transform as ⊥ + z1,2/(z1,2 2) [38]. Note that the RG equations do not have ±iφˆ / ∓iφˆ / ±iφ /2 ±iφ /2 T : e j 2 →∓ie j 2,e j → e j , a stable fixed point, and hence the flows will be stopped by (30) nonuniversal scales. The kinks are of the form 2 ±iφˆ /2 ±iφˆ /2 ±iφ /2 ±iφ /2 T : e j →−e j ,e j → e j , −1 θˆ(x) ∼tan (x/w1), keeping in mind that T involves complex conjugation. Thus (34) ∼ −1 spinon kinks get switched to antikinks, and vice versa, since θ (x) tan (x/w2), the physical spin density is odd under T . It also follows that ∼ ⊥ + T 2 =−1fortheε (fermion) quasiparticle. where w1,2 1/ z1,2(z1,2 2) [38]. However, we will as- Since the bulk is gapped, we can perturbatively add sume that the widths w1,2 of the kinks are much smaller than interwire hoppings for the spinons to the Hamiltonian, the other length scales in our model, and treat the kinks as sharp step functions. hop ˆ − ˆ =− ij,j+1(x) i(φj φj+1)/2 + Hj,j+1(x) te e H.c., (31) =− + C. Physics at the edges j,j+1(x) [ θj (x) θj+1(x) ]/2. As long as t is much smaller than the bulk gaps, this In a wire array where runs from 1 to n, the fields ¯1 ¯2 1 2 should not destabilize the coupled-wire fixed point. Let us φ1 ,φ1 and φn,φn living on the edges do not appear in the O consider the hopping of a spinon at x from wire j − 1to sine-Gordon terms α. Thus, these nonchiral modes are j + 1, in the presence of a vison located at x = 0onwire gapless and also commute with the Hamiltonian. In the absence j [Fig. 2(b)]. The hopping amplitude for this process is of additional symmetries, we are free to add sine-Gordon terms 2 i[j−1,j (x)+j,j+1(x)] to the edges to gap these modes out; for example, we can hj−1,j+1(x) ∝ t e . Since the presence of the = add vison causes θj to jump by π at x 0, we can see that h − + x> =−h − + x< j 1,j 1( 0) j 1,j 1( 0). Thus the vison induces = ¯1 + 1 a branch cut for the spinon hopping, as we know already from Hedge dx D1 cos 2φ1 Dn cos 2φn , (35)

195130-5 AAVISHKAR A. PATEL AND DEBANJAN CHOWDHURY PHYSICAL REVIEW B 94, 195130 (2016) and tune the kinetic terms on the edges, as we did before written as (σ =↑ , ↓) ¯1 for the bulk, to make these relevant. This localizes φ1 and 1 Fj 0 + + φ R = √ i[kF x φjσ(x) θjσ(x)] n. Due to the nonvanishing commutators [Eq. (10)] between ψjσ(x) e , φ¯1,φ¯2 and φ1,φ2, fluctuations of φ¯2 and φ2 are maximized 2πxc 1 1 n n 1 n (40) due to the uncertainty principle, and these modes are conse- † 1 2 Fj − 0 + − ¯ ¯ L = √ i[ kF x φjσ(x) θjσ(x)] quently gapped [37]. Note that φ1,φ2 (or φn,φn) cannot be ψjσ(x) e . simultaneously localized owing to their nonvanishing mutual 2πxc commutators [Eq. (10)]. Thus, our results are consistent The Fj represent the Klein factors that ensure anticommutation with the usual expectations for the edge of the toric code, on different wires and xc is a short-distance cutoff; one possible which can either be of the m or e type, but not both choice for the Klein factors is [27] [45,46]. N R +N L ¯1 →−¯1 1 →− 1 σ,lelectrons form a Fermi − † † S = 1 ψR ψR + ψL ψL , liquid with a “small” Fermi surface. In the limit of a weak j 2 j↓ j↑ j↓ j↑ Kondo exchange between the local and itinerant electrons, the which can be reexpressed in terms of the bosonic fields as resulting FL* phase violates Luttinger’s theorem [48], which + 1 − ˆ + ˆ − ˆ − ˆ can be understood as arising from the presence of background = i(φj θj ) + i(φj θj ) Sj [e e ], topological order [49]. 4πxc In order to fermionize the spinons, we first add a new set (44) − 1 ˆ + ˆ ˆ − ˆ of bosonic “chargon” fields θ c,φc to the wires labeled by j S = [ei(φj θj ) + ei(φj θj )]. j j j 4πx which satisfy c On the other hand, the z component is given by c c = − θj (x),φj (y) iπsign(x y)δjj . (36) 1 † † † † ∂ θˆ Their Hamiltonian is given by z = R R − R R + L L − L L = x j Sj ψj↑ ψj↑ ψj↓ ψj↓ ψj↑ ψj↑ ψj↓ ψj↓ . c 2 (2π) v 1 c 2 c c 2 c Hc = ∂x θ + g ∂x φ + Cc cos 2θ , (45) 2π gc j j j j Thus we have S →−S under T . (37) j j S± switch antikinks (≡↓) to kinks (≡↑), and vice versa. with gc chosen so that Hc is gapped, and HSL → HSL + Hc. Thus, they create and annihilate two spinons at a time We then consider the θ,ˆ φˆ and θ c,φc to respectively describe the respectively. The spin-sector sine-Gordon term maps to the long-wavelength spin and charge sectors of spinful fermionic backscattering term, Luttinger-liquid wires, ˆ → ˜ L† R R† L + C1 cos(2θj ) C1ψj↑ ψj↑ψj↓ ψj↓ H.c., (46) θˆj = θj↑ − θj↓, φˆj = φj↑ − φj↓, (38) 2ik0 x c c which does not have any oscillatory e F factors and hence is θ = θ ↑ + θ ↓,φ= φ ↑ + φ ↓. j j j j j j not trivially rendered irrelevant in the long-wavelength limit. Given the commutation relations in Eqs. (13), (14), For the charge sector, we have and (36), we demand that the fields introduced above satisfy 0 L† L† c → 4ikF x ˜ R R + the following commutation relations: Cc cos 2θj e Ccψj↑ ψj↑ψj↓ ψj↓ H.c. (47) π [θjσ(x),φj σ (y)] = i sign(x − y)δjj δσσ . (39) Imagining the fermions to live on a lattice with lattice constant 2 0 = a0 as before, we tune to half filling kF π/(2a0) to eliminate These are the canonical Luttinger-liquid commutators. Thus, the oscillatory factor in the above so that we can have both the fermion creation and annihilation operators may then be charge and spin gaps.

195130-6 TWO-DIMENSIONAL SPIN LIQUIDS WITH Z2 . . . PHYSICAL REVIEW B 94, 195130 (2016)

Thus we have

HSL → Hf + Hv, † ∂ † ∂ † † † † H = dx v ψR −i ψR − ψL −i ψL + C˜ ψL ψR ψR ψL + H.c. + C˜ ψL ψR ψL ψR + H.c. , f f jσ ∂x jσ jσ ∂x jσ 1 j↑ j↑ j↓ j↓ c j↑ j↑ j↓ j↓ j,σ v 1 H = j dx (∂ θ )2 + g(∂ φ )2 + C cos(2θ ), (48) v 2π g x j x j 2 j j

where Hv corresponds to the vison piece unaffected by the used in the original description of the FL* [47]. We analyze fermionization. two different cases below.2 The spinons (together with chargons) may be hopped between wires by adding perturbative nonchiral hoppings of A. Kondo Hamiltonian the fermions, We begin by using a local Kondo coupling, HK = hop,f L† JK [S · s] , that preserves the SU(2) spin rotation sym- H (x) =−teij,j+1(x) ψ (x)ψL (x) j j j,j+1 j+1,σ jσ metry. Then, + R† R + ψ + (x)ψ (x) H.c., (49) ∗ j 1,σ jσ HFL = Hel + HSL + HK , JK with the phase of the hopping amplitude given by Eq. (31)as HK = dxj · τj , 4 the chargons have trivial mutual statistics with the visons. j,σσ (52) To realize the FL*, we add another set of wires to the sites R† L† = τ R + τ L labeled by j, carrying the conduction electrons labeled by j ψjσ σσ ψjσ ψjσ σσ ψjσ , R/L R† L† c ; we also add nonchiral hoppings between these wires so = τ R + τ L jσ τj c σσ cjσ c σσ cjσ , that the electrons form a quasi-1D Fermi surface (Fig. 4). The jσ jσ conduction electrons are described by where HSL is as described in Eq. (21) earlier. Even though the Hamiltonian looks like a standard Kondo-type Hamiltonian, there is a subtlety associated here with the speciﬁc construction † ∂ † ∂ H = dx v cR −i cR − cL −i cL used to arrive at the description of the Z spin liquid. The el F jσ ∂x jσ jσ ∂x jσ 2 j,σ spin-spin coupling in HK does not commute with HIC [in HSL; x,y ˆ see Eq. (21)] as Sj depend on φj after bosonization. However, − R† R + L† L + t1 cj+1,σ cjσ cj+1,σ cjσ H.c . (50) we appeal to our physical intuition here; since the spin-liquid background is gapped, the phase obtained by coupling it to a Fermi liquid will be perturbatively stable as long as the The spin density corresponding to the conduction electrons is Kondo coupling is small compared to the typical gaps (i.e., denoted JK min{1,2}). Thus in the small JK limit, we realize the Z2 FL* phase without any broken symmetries. However, it 1 R† R L† L = τ + τ remains an interesting open problem to study the fate of this sj 2 cjσ σσ cjσ cjσ σσ cjσ . (51) phase when the above condition is not satisﬁed. We now couple the spin sector of the electrons to the spinons via a local spin-spin coupling, similar to the Kondo coupling B. Ising limit There is a special limit in which the complications described z x,y above can be circumvented. Suppose JK JK as a result of easy-axis anisotropy. The Kondo coupling then essentially z = z z involves only a local HK JK j [S s ]j coupling. We then have

∗ = + + z HFL Hel HSL HK , (53) J z H z = K dxzτ z. - K 4 j j - j (a) (b)

FIG. 4. (a) The additional set of wires (dashed blue line) carrying the conduction electrons. Nonchiral tunneling between these wires 2Recently, a construction for an FL* was proposed starting from a (red arrows) allows the electrons to form a two-dimensional Fermi set of decoupled wires in Ref. [50]. However, the phase obtained in liquid, coupled to the spin-liquid background (FL*). (b) Schematic the above paper is not a Z2 FL* and does not discuss the topological Fermi surface of such a Fermi liquid. structure or nature of its anyonic excitations.

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z ˆ It would be interesting to explore the possibility of Since Sj depends only on θj in the bosonized language, it commutes with HIC [in HSL;seeEq.(21)]. Thus, the spin- realizing other time-reversal symmetric spin-liquid phases in two spatial dimensions with gapless excitations in the bulk. liquid background is stable for any reasonable value of JK , as long as it is not strong enough to drive Kondo screening. A particular example is the U(1) spin liquid with a spinon Moreover, we do not expect the gapped spinons to induce Fermi surface [22], which can potentially be constructed any non-Fermi-liquid behavior for the electrons. Therefore, in a manner similar to the one proposed for the half-ﬁlled Landau level [28]. The fate, or even the existence, of the for small values of JK , we realize once again a Z2 FL* [that explicitly breaks the SU(2) spin rotation symmetry] with a strong-coupling ﬁxed point for the above spin-liquid problem Fermi surface of the type shown in Fig. 4(b). remains unanswered [51] and it would be interesting to see if a complementary approach, such as the one proposed here, can address some of these unresolved questions. VI. DISCUSSION ACKNOWLEDGMENTS In this work, we have tried to extend the general program of constructing two-dimensional correlated phases of matter We thank S. Sachdev for helpful suggestions and discus- by coupling together an array of one-dimensional wires. In sions that inspired this work and M. Barkeshli for useful particular, we have demonstrated that it is possible to explicitly discussions. A.A.P. is supported by the NSF under Grant No. construct a time-reversal symmetric phase of matter that has DMR-1360789. D.C. is supported by a postdoctoral fellowship the following characteristics: (i) Energy gap in the bulk and from the Gordon and Betty Moore Foundation, under the at the edge, (ii) three bulk anyonic quasiparticles which are EPiQS initiative, Grant No. GBMF-4303, at MIT. We also mutual semions, and (iii) nontrivial ground-state degeneracy acknowledge support by the “2016 Boulder Summer School on on a torus and is the Z2 spin liquid. In the limit of a weak Condensed Matter Physics—Topological phases of quantum Kondo-type coupling to an itinerant Fermi sea, we have also matter” through NSF Grant No. DMR-13001648 during which constructed a Z2 FL*. part of this work was completed.

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