Translation symmetry-enriched toric code insulator
Pok Man Tam,1 J¨ornW. F. Venderbos,2, 3 and Charles L. Kane1 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 2Department of Physics, Drexel University, Philadelphia, PA 19104, USA 3Department of Materials Science & Engineering, Drexel University, Philadelphia, PA 19104, USA We introduce a two-dimensional electronic insulator that possesses a toric code topological order enriched by translation symmetry. This state can be realized from disordering a weak topological superconductor by double-vortex condensation. It is termed the toric code insulator, whose anyonic excitations consist of a charge-e chargon, a neutral fermion and two types of visons. There are two types of visons because they have constrained motion as a consequence of the fractional Josephson effect of one-dimensional topological superconductor. Importantly, these two types of visons are related by a discrete translation symmetry and have a mutual semionic braiding statistics, leading to a symmetry-enrichment akin to the type in Wen’s plaquette model and Kitaev’s honeycomb model. We construct this state using a three-fluid coupled-wire model, and analyze the anyon spectrum and braiding statistics in detail to unveil the nature of symmetry-enrichment due to translation. We also discuss potential material realizations and present a band-theoretic understanding of the state, fitting it into a general framework for studying fractionalizaton in strongly-interacting weak topological phases.
I. INTRODUCTION as topological orders [10, 11]. Well-known examples of topological order include fractional quantum Hall states Over the past decades, symmetry and topology have and quantum spin liquid states [12, 13]. In the case of the emerged as two central and interwoven organizing prin- former, recent experimental evidence for the fractional ciples in the study of condensed matter physics. In the statistics of Laughlin quasiparticles has been reported case of weakly interacting systems, symmetry-protected [14, 15]. Perhaps the simplest example of topological or- topological (SPT) phases—a class which includes topo- der, however, is of the Z2 type, which was first studied in logical insulators (TIs) and superconductors (TSCs)— the context of frustrated quantum antiferromagnets [16– have been predicted theoretically and in a number of 19] and later reconsidered in the form of Kitaev’s toric cases discovered experimentally [1,2]. Internal symme- code toy model, as well as Wen’s plaquette model [20, 21]. tries, such as time-reversal and particle-hole symmetries, Given the compelling appeal of the toric code model, in give rise to so-called strong SPT phases with protected this paper we refer to the Z2 topological order as toric gapless boundary modes [3,4]. In addition to strong code topological order. It features four types of anyons: 1, e, m and f = e m, where e and m are self-bosons SPT phases, there exist weak SPT phases, which can be × viewed and constructed as stacks of strong SPT phases which obey a mutual π-braiding (semionic) statistics as from a lower dimension. Importantly, the distinction of well as the Z2 fusion rule. Remarkably, as pointed out weak SPTs from a trivial phase requires an additional by Hansson et al. [22], toric code topological order not discrete translation symmetry along the stacking direc- only arises in spin systems, but also in conventional su- tion, which prevents hybridization of pairs of stacked lay- perconductors, where f is interpreted as the Bogoliubov ers. Prototypical examples of weak SPTs include the fermion and m as the superconducting vortex. It is then three-dimensional (3D) weak TI [5–7], which harbors natural to wonder to what extent and how the structure an even number of surface Dirac modes, and the two- of the topological order is modified in unconventional dimensional (2D) weak TSC [8,9], which harbors a pair superconductors. More specifically, it is natural to ask of counter-propagating Majorana edge modes. While the whether a 2D weak TSC can provide a platform for a basic properties of weak topological phases are well un- toric code topological order enriched by translation sym- derstood in the weakly-interacting regime, less is known metry. Here we answer this question in the affirmative arXiv:2107.04030v1 [cond-mat.str-el] 8 Jul 2021 about the effect of strong interactions, which may lead by constructing an insulator from a strongly interacting to exotic correlated phases and phenomena either on the weak TSC and showing that the resulting anyon spec- boundary or in the bulk. The purpose of this paper is trum exhibits symmetry-enrichment. to study the effect of strong interactions on weak topo- In general, a symmetry-enriched topological (SET) or- logical phases, and in particular to address the interplay der can exhibit many interesting properties in addition to between translation symmetry and topology. To this end, the fusion and braiding properties of anyons [23–28]. For we focus on the paradigmatic example of a strongly in- instance, it can feature fractionalized symmetry quantum teracting weak TSC in two dimensions. numbers, such as the electric charge of Laughlin quasi- Strong interactions can give rise to correlated quan- particles. In this paper, we focus on another aspect: the tum phases with emergent fractionalized quasiparticles permutation of anyon types by a symmetry transforma- known as anyons. Such quantum phases are referred to tion. This has been famously demonstrated in spin sys- 2
m-particles (and vice versa) under a discrete translation.
This phenomenon is sometimes known as “weak symme- (c)
Hall state, which possesses a charge sector of a Laugh- m
ϕσ = ϕ1 ϕ2, θσ = θ1. (2.2b) B. Outline of the paper − Note that the commutation relations are preserved and that the charge-sector variable ϕ is the superconducting The rest of the paper is organized as follows. In Sec.II ρ phase. The Hamiltonian of a single wire is chosen to be we present a microscopic model for the toric code insula- = + , with tor using the coupled wire construction. In Sec.IIIA, the Hwire Hρ Hσ anyonic spectrum and braiding statistics are analyzed. In vρ 2 1 2 ρ = [gρ(∂xϕρ) + (∂xθρ) ], (2.3a) particular, by constructing local operators that transport H 4π gρ anyons, we discover a constrained motion for the visons vσ 2 1 2 that lead to symmetry-enrichment. An interesting con- σ = [gσ(∂xϕσ) + (∂xθσ) ] + u cos ϕσ + v cos 2θσ. sequence is a size-dependent ground state degeneracy on H 4π gσ (2.3b) torus, which we elucidate in Sec.IIIB. In Sec.IV, we provide a survey of possible material platforms for realiz- Here, gρ,σ are the Luttinger parameters of the charge ing the toric code insulator, and discuss a band-theoretic and neutral Luttinger liquids. The term with coupling perspective that connects to recent proposals of attaining constant u describes the pairing interaction between the fractionalization in semimetals. We conclude in Sec.V. two fluids, which turns two electrons into a Cooper pair and vice versa. The interaction with coupling constant v describes the single electron back-scattering. II. WIRE MODEL In the limit where v u, the unpaired electrons are depleted by back-scattering and the wire becomes Our approach to constructing the toric code insula- a trivial 1D superconductor which is gapless only to two- tor relies on a coupled wire model for an array of spin- electron excitations. Instead, in the opposite limit u v less single-channel quantum wires. Since the 2D weak the electrons are weakly-paired in the sense that charge-e TSC can be viewed as a stack of 1D TSCs, the coupled and charge-2e fluid coexist, and the wire is a topological wire model provides a natural description for the weak superconductor (with a fluctuating phase). As pointed TSC, and as we will see, it also provides a description for out in Ref. 41, the weakly paired phase is adiabatically the double-vortex condensation that leads to the toric connected to a single-channel Luttinger liquid with at- code insulator. In this section, we begin by introducing tractive interactions, and is gapless to both one-electron a two-fluid model for a single wire. Then, by considering and two-electron excitations. The gapless charge-e exci- the competition between the inter-wire Josephson cou- tation corresponds to a composite operator given by
1 pling and the intra-wire charge-density wave ordering, a i[ (ϕρ+ϕσ )±θρ] ψ± = e 2 , (2.4) Z2 gauge structure emerges. This leads us to a three- fluid wire model for a disordered weak TSC, which is the which is a composite of adding a bare electron and tun- symmetry-enriched toric code insulator. neling a vortex across the wire (because e±iθρ introduces a 2π phase-slip in ϕρ). The gaplessness of this compos- ite± one-electron excitation simply reflects the fractional A. Two-fluid model for a single wire Josephson effect in a 1D TSC: a 2π phase-slip switches the fermion-parity of the ground± state. Let us first review a bosonized theory for a 1D TSC, As a next step, we couple an array of 1D TSCs which is developed in Ref. 41. A single 1D TSC can by Josephson-tunneling between neighboring wires, re- be described by a two-fluid model, where a Luttinger sulting in a 2D weak TSC, and then consider phase- liquid of charge-e fermions (i.e. the spinless electrons) disordering the superconductor. We label the wires by j 4
and links between the wires by `, which is related to wire which the single vortices are represented as the magnetic label as ` = j + 1/2. The Josephson coupling (coupling flux of a Z2 gauge field, is equivalent to the original for- constant J) between the wires can then be expressed as mulation. It has the benefit that the charge and neutral variables are treated as independent, but are coupled via X = J cos(∆ ϕ ) , (2.5) the gauge field. In this sense, our approach is similar HSC y ρ ` ` in spirit to previous approaches, such as slave-particle representations, which introduce a gauge symmetry to where we have defined the discrete “derivative” ∆y of liberate new degrees of freedom [38]. wire variables as ∆yϕρ ϕρ,j+1 ϕρ,j. The ∆y deriva- To further develop the gauged two-fluid model, it is in- tive is associated with link≡ ` = j +− 1/2 and we thus write structive to consider the emergence of the Z2 gauge field (∆yϕρ)`. A vortex with winding number n, which binds at the level of the partition function. In particular, con- nh/2e-flux, corresponds to a 2nπ-kink in ∆yϕρ (and thus sider the Josephson tunneling term (2.5) in the partition lives on the links). Hence, a pinned charge-density wave function, which may be rewritten using Villain’s method (CDW) on wire j of the form as [44]
−J cos ∆ ϕ X J (∆ ϕ −2nπ)2 CDW,j = w cos 2θρ,j (2.6) e y ρ e 2 y ρ H ∼ n∈Z competes with SC (since ϕρ and θρ are conjugate vari- X (2.7) ables) by tunnelingH double vortices across wire j. In the e−4J cos(∆y ϕρ/2−Ay ). ∼ large-w limit, double vortices are condensed and super- Ay =0,π conductivity is destroyed by the rapid phase fluctuations. However, since double vortices correspond to 4π-kinks, In the first line the cosine is approximated using Villain’s there still remains a binary degree of freedom± that dis- prescription, which is reversed in the second line after re- tinguishes configurations with ∆ ϕ = 0 from those with arranging the sum over integers into separate sums over y ρ even and odd integers. By re-expressing the partition ∆yϕρ = 2π. The latter are remnants of the uncondensed single vortices. To properly treat these as dynamical function in this way, we have introduced a Z2 gauge field quantum excitations and to study the properties of the Ay 0, π , which is defined on the links, as is ∆yϕρ, and∈ enters { } as a minimal coupling to the superconducting insulating ground state, we now introduce a Z2 gauge theory, which will be treated as the third fluid in the phase. As is evident from (2.7), a 2π phase slip in ∆yϕρ wire model. must be accompanied by a shift of Ay by π, which reflects the aforementioned identification of magnetic gauge flux with the superconducting vortex. It is worth noting that Eq. (2.7) is similar in spirit to the gauge theory devel- B. Z2 gauge theory Z2 oped by Senthil and Fisher [38], which they introduced by splitting the Cooper pair in half (i.e., ϕ ϕ/2), at To see how a gauge sector emerges from the Lut- Z2 the expense of a gauge degree of freedom. → tinger liquids, consider the compactification of the fields Gauging the Z2 symmetry on individual wires leads to ϕ1 and ϕ2. In the above formulation, both fields are de- the minimal coupling of the gauge field components Ax fined on a circle such that ϕ1 ϕ1 +4π and ϕ2 ϕ2 +2π. ≡ ≡ and At to the phase variables ϕ2: ∂t,xϕ2 2At,x. Anal- As mentioned, in this case the minimal density operators − are e2iθ1 and eiθ2 , respectively. Given the compactifi-
j+1
j
are coupled, and not independent. f
Anyon Symbol Operator Hopping operator Self-statistics Mutual π-braiding with 1 1 i( ϕσ +θσ )j i( ∆y ϕσ +∆y θσ +Θ)` Neutral fermion f e 2 e 2 Fermion me , mo i 1 ϕρ,j i( ∆y ϕρ−Θ)` Chargon e e 2 e 2 Boson me , mo i Φ` Vison (even-link `) me e 2 see Eq. (3.9) Boson mo , f , e i Φ` Vison (odd-link `) mo e 2 see Eq. (3.9) Boson me , f , e
TABLE I. Summary of the spectrum and anyon statistics in the toric code insulator, which exhibits charge-statistics separation and symmetry-enriched topological order. Neutral fermions and chargons live on wires (labeled by j), while visons live on links (labeled by ` = j + 1/2), with even/odd-link corresponding to even/odd integer j. The local hopping operator moves an anyon across one link, along the direction perpendicular to the wires. In particular, without creating additional excitations, a vison has to hop across two wires at a time, thus differentiating between me and mo. are referred to as the hopping operators, and their ex- resembles the fractionalization noticed by Senthil and plicit form dictates the braiding statistics of the anyons. Fisher in the Z2 gauge theory of cuprate superconduc- The hopping operators for motion alongx ˆ are easily con- tors [38], hence a similar nomenclature has been adopted structed. For instance, a chargon can move along the here. More explicitly, the bare electron operator in Eq. wire by (2.1) can be written as
Z x1 i 1 † i ϕρ i( ϕσ +θσ ) e (x ) e (x ) = exp[ dx ∂ ϕ ], (3.2) ψe = e 2 e 2 = e f. (3.6) j 1 j 2 x ρ,j × × 2 x2 There is a local Z2 symmetry that transforms (ϕρ, ϕσ) which is indeed a local operator as ∂xϕρ represents the (ϕ + 2π, ϕ + 2π), or equivalently 7→ current operator in the Abelian bosonization. Similarly, ρ σ one can argue that f (x )f †(x ) is a local operator. Vi- j 1 j 2 Z2 :(e, f) ( e, f), (3.7) i Φ 7→ − − sons, which are created by e 2 ` , can move alongx ˆ by applying the y-component of the electric field operator, which leaves the electron operator invariant. This is the namely origin of the Z2 gauge field that we introduce in Sec. IIB, which couples to both the chargon and the neutral † i 2 [Φ`(x1)−Φ`(x2)] m`(x1) m`(x2) = e fermion. Its gauge flux is known as the “vison”. From Z x1 (3.3) Eqs. (3.4) and (3.5), one should notice a phase factor i ±iΘ = exp[ Ey,` dx ]. e ` being picked up by transporting e/f. Since a vison 2 x2 on link-` corresponds to a π-kink in Θ`, this implies a π- The more interesting hopping operators are for the mo- braiding between e/f and m, as illustrated in Fig.3(a). tion perpendicular to the wires, which explicitly encode Despite the above similarities with the Senthil-Fisher information about the braiding and exchange statistics. model, the toric code insulator just constructed has an From the partition function in Eq. (2.7), along with the additional feature of symmetry-enrichment. To see that, let us analyze how visons move alongy ˆ, and derive a identification of Ay,` = Θ`, we can identify the local op- erator that transports a chargon across link-`: mobility constraint. A local operator that hops a π-kink iEx,j /2 in Θ` can be obtained from the twist operator, e , 1 i( ∆y ϕρ−Θ)` introduced in Sec.IIB. From the Gauss’s law constraint T`(e) = e 2 . (3.4) in Eq. (2.9b), we have By composing this with the electron hopping operator i 1 1 1 2 Ex,j i(θρ−θσ − 2 ∆y Φ)j exp i∆y( 2 ϕρ + 2 ϕσ + θσ), we obtain the local operator e = e , (3.8) that transports an f-particle across link-`: hence the right-hand side provides a physical operator in 1 i( ∆y ϕσ +∆y θσ +Θ)` T`(f) = e 2 . (3.5) the our chosen gauge: Ax = 0. However, this operator − i ∆ Φ not only hops a vison (by e 2 y ), but also creates an 1 i( 2 ϕσ +θσ ) −iθσ We first notice that the above combination e f-particle, which is a 2π-kink in ϕσ as created by e . for the f-particle suggests that it, as a 2π-kink in ϕσ, is This is the reincarnation of the fractional Josephson ef- i ϕ a self-fermion, since e 2 σ is acting as a Jordan-Wigner fect: an m-particle simply cannot hop across a single wire string. In contrast, the charge-e e-particle is a self-boson, without changing the fermion-parity of the wire. Never- since θρ does not show up in Eq. (3.4). The above con- theless, by applying the above twist operator twice, a siderations suggest that an electron can fractionalize into vison can hop across two neighboring wires (j and j +1), a bosonic chargon (which carries charge-e of the electron) or equivalently across a link (` = j + 1/2), without cre- and a neutral fermion (which carries the fermionic self- ating excitations as long as the hopping operator in Eq. statistics of the electron) in the toric code insulator. This (3.5) is also applied to annihilate additional f-particles. 7
j
B. Ground state degeneracy on torus: Namely, we consider the local operator an even-odd effect
i (Ex,j +Ex,j+1) 2iθσ,j T`(m) = e 2 T`(f) e Related to the fusion and braiding properties of × × (3.9) i( 1 ∆ ϕ +Σ θ +Θ) i (Φ −Φ ) = e 2 y σ y ρ ` e 2 `−1 `+1 , anyons, another prominent feature of a topological order × is the ground state degeneracy (GSD) on a high genus where (Σyθρ)` θρ,j+1 + θρ,j. In the last line, the first Riemann surface. Below, we focus on the GSD on torus part is a phase≡ factor, and the second part has the action (T 2), by imposing periodic boundary conditions on both of tunneling a vison across two wires. We have made use directions of the 2D system. For Kitaev’s toric code, the of f 2 = 1 by attaching a factor of e2iθσ,j , so as to com- GSD on torus is well-known to be 4 [20]. However, with a pletely eliminate θσ in the hopping operator, and hence non-trivial interplay between translation symmetry and no additional excitation is created during the tunneling the topological order, there can be a size-dependent GSD process. as in the case of Wen’s plaquette model [21, 47, 48]. The upshots of the above analysis are three-fold: (1) For the toric code insulator just constructed, the GSD there are two types of visons, one lives on the even-links depends on the parity of the number of wires, L, as fol- (me) and one lives on the odd-links (mo). While they lows: are related by a translation symmetry, there is no local ( 4, for even L; operator that turns one into the other without creating GSD on T 2 = (3.11) additional excitations. This leads to a pattern of weak 2, for odd L. symmetry breaking akin to the one in Wen’s plaquette The above result can be understood using a Wilson-loop model and Kitaev’s honeycomb model [21, 29, 30]; (2) argument. Let us consider the action of creating a pair By the virtue of Eq. (3.8), a local operator can create a of anyons from a ground state, then bringing one of the composite of three anyons: me, mo and f. This implies anyons all the way around a non-trivial cycle i (i = x, y) the following fusion rule: of T 2 and back to re-annihilate with its partner,C finally returning the system back to a ground state. We denote me mo = f ; (3.10) × the corresponding operator for the a-anyon as a. Since Wi Together with Eq. (3.6), this implies that a physical the fundamental anyons are me and mo, while other exci- electron can fractionalize into one chargon e and two tations can be treated as composites of these (or together symmetry-related visons me and mo. (3) The braid- with the trivial electron), we shall focus just on the al- ing statistics related to the m-particle is encoded in the gebra generated by the operators associated to these two 8
that the minimal GSD is indeed realized in each case. In future numerical studies, this feature could be useful for identifying the symmetry-enriched toric code insulator.
IV. POSSIBLE REALIZATIONS
We now discuss possible routes towards the experimen- tal realization of a symmetry-enriched toric code insula- FIG. 4. Toric code insulator with an odd number of wires. tor. Since the 2D weak TSC has provided the conceptual Due to the periodic boundary condition, an even-link vison starting point for our analysis, we focus on three promis- (me) turns into an odd-link vison (mo) after going around Cy ing experimental platforms for realizing the weak TSC in once. two dimensions. A separate question concerns the pre- cise mechanism by which vortex-condensation might oc- anyons. In the case of an even number of wires, from the cur, thus phase-disordering the superconductor and po- mutual braiding statistics discussed above, we have: tentially giving rise to the toric code insulator. This ques- tion is intimately related to the nature of superconduc- me mo mo me tivity, i.e., whether it is intrinsic or proximity-induced. x y = y x , (3.12a) W W −W W We leave this question for future study, and instead fo- mo me = me mo , (3.12b) Wx Wy −Wy Wx cus more broadly on realizations of weak TSCs. while other combinations of operators commute. The In addition, we present a tight-binding model realiza- above algebra demands at the minimal a four- tion for the weak TSC on a square lattice. This serves as dimensional ground-state Hilbert space: n, m a toy model for certain available experimental platforms, {| i | and furthermore exposes an intriguing connection with n, m Z2 , where the Wilson loop operators can ∈ } me strongly-interacting gapped semimetals, both in two and be shown to act as: x n, m = n + 1, m , mo W me | i | m i three dimensions. x n, m = n, m + 1 , y n, m = ( 1) n, m W | mo i | ni W | i − | i and y n, m = ( 1) n, m . InW case| of ani odd− number| i of wires, the situation is quite different. Notice that by going around once, A. Relevant experimental platforms Cy me and mo are exchanged, as illustrated in Fig.4. In me mo other words, y and y are ill-defined, as the m- As a first example of a possible experimental platform, particle cannotW annihilateW with its partner by just going consider the side surface of a weak topological insula- around y once. Instead, we are forced to consider a new tor (TI). The weak TI can be viewed as a stack of two- C m dimensional quantum spin Hall insulators protected by operator fy , which pair-creates two me/o-particles and W time-reversal symmetry (TRS) and translation symme- transports one of them around y twice, and finally re- annihilates them. The Wilson operatorsC that act on the try in the stacking direction. This implies that a side ground-state subspace then obey the following relations surface, whose normal is perpendicular to the stacking (again derived from the braiding statistics): direction, realizes a stack of counter-propagating helical (i.e., spin-momentum locked) edge modes, with one pair me m m me of counter-propagating modes per layer. Viewing the he- x fy = fy x , (3.13a) W W −W W lical edge modes as quantum wires establishes a connec- mo fm = fm mo , (3.13b) Wx Wy −Wy Wx tion with the wire model introduced in the previous sec- tion, albeit with the important difference that the helical while other combinations of operators commute. This quantum wires can only exist as boundary modes of a two time the minimal ground-state subspace is - topologically nontrivial bulk system protected by TRS. dimensional: n n Z2 , where the operators act as: Breaking translation symmetry allows for the hybridiza- me {|moi | ∈ } m n x n = x n = n + 1 and fy n = ( 1) n . tion of two helical wires and leads to a trivial gapped InW fact,| i oneW can show| i that| mi e andW m|o iare related− | byi Wx Wx surface. Opening up a pairing gap in a single helical wire a large gauge transformation in this case, so they are ef- gives rise to a topological superconducting phase simi- fectively the same operator which we shall denote as fm. lar to Kitaev’s model for a spinless superconductor in Wx The only non-trivial algebraic relation for the Wilson op- one dimension. Notably, evidence for such a 1D phase m m erators is that fx anti-commutes with fy , demanding has been recently reported in a WTe2-NbSe2 proximity- a ground stateW subspace with minimal dimensionW 2. coupled heterostructure [49]. This similarity with the The Wilson-loop argument thus provides support Kitaev wire suggests that the physics of the 2D weak to Eq. (3.11), as a consequence of the translation TSC can be emulated on the surface of a 3D weak TI. symmetry-enrichment. In AppendixB, we present a A promising experimental candidate is the recently re- counting argument using the three-fluid wire model to ported weak TI bismuth iodine (β-Bi4I4), which has a explicitly demonstrate this even-odd effect, confirming quasi-1D structure and side surface states with weak dis- 9 persion in one of the two momentum directions [50]. As ture of non-s-shell orbitals can give rise to directionally a result, our wire model may provide a useful starting anisotropic and strongly quasi-1D hopping. When elec- point for describing such surfaces. trons on such quasi-1D Fermi surfaces form unconven- The fate of the weak TI surface in the presence of tional pairing states of p-wave type with a full pairing strong interactions was explored in Ref. 51, which showed gap, the resulting phase can realize a weak TSC. This sce- that a symmetric gapped surface is necessarily topolog- nario has indeed been proposed for Sr2RuO4 [66], which ically ordered. The minimal Abelian order studied in has quasi-1D Fermi surfaces coming from dxz,yz orbitals, Ref. 51 is of the Z4 type, whose fundamental anyons in addition to a 2D Fermi surface sheet derived from a a and d are self-bosons and have mutual π/2-braiding dxy orbital. Although the nature of the superconducting statistics. In particular, in this Z4 topological order order parameter of Sr2RuO4 remains an unsettled ques- the d-particle transforms non-trivially under translation, tion, at least partially, one compelling proposal for the very much like the m-particle in the toric code insula- pairing state assumes dominant pairing of the quasi-1D tor. However, the Z4 topological order is anomalous, in Fermi surfaces [67], which would imply nontrivial weak the sense that it can only be realized on the surface of a indices [66]. 3D system but not in a strictly 2D system provided that When taking the spin degree of freedom of electrons both TRS and translation symmetry are preserved. This into account and assuming absence of spin-orbit coupling is a consequence of the bulk topology. Breaking either in the normal state, such pairing leads to a state which symmetry allows for a 2D topological order. In fact, by is equivalent to two copies of a weak TSC, one for a breaking TRS, one can condense the d2-anyon (which re- spin sector [66]. As argued before, each copy can be quires back-scattering of helical modes), and reduce the disordered into a Z2 toric code insulator. Hence, in the Z4 state to the Z2 toric code. As long as the translation limit of vanishing spin-orbit coupling, a possible fate of symmetry is still preserved, the surface topological order such systems is a Z2 Z2 fractionalized insulator with × is then identical to our symmetry-enriched toric code in- translation symmetry-enrichment. Being a fully gapped sulator. This again suggests that 2D side surfaces of 3D topological order, this exotic state is expected to survive weak TIs are possible venues for realizing the toric code when SOC is switched-on adiabatically. We thus believe insulator. that Sr2RuO4, and systems alike, could provide a route A second route towards a weak TSC makes use of to exploring generalizations of the toric code insulator an array of nanowires, with each wire realizing a 1D introduced in this work. TSC (i.e. Kitaev chain). Two broad experimental plat- forms for assembling such 1D quantum wires have at- tracted much attention: proximitized quantum wires B. Model realization with strong spin-orbit coupling (e.g. InAs and InSb) [52–54], and magnetic atomic chains on superconductors To further aid the identification of experimental plat- [55–61]. Both classes of systems have provided promis- forms, we now introduce a square lattice tight-binding ing experimental evidence for the existence of Majorana model for the weak TSC. Without pairing, this square end states [62, 63]. Platforms based on the combination lattice model describes the transition between a trivial of helical magnetism and superconductivity may provide and a (inversion symmetric) weak TI in 2D, which nec- the most fruitful route to exploring 2D architectures and essarily occurs via a 2D Dirac semi-metallic phase. In- weak TSC phases. Aiming to generalize the setup based cluding pairing terms of the Kitaev type gives rise to on 1D magnetic atomic chains, initial theoretical pro- a weak TSC. Notably, as we will argue below, the 2D posals have explored the possibility of exploiting modu- model presented here enables an interesting connection lated magnetic phases in 2D systems to generate effective with recent proposals for achieving fractionalized phases spin-orbit coupling and local Zeeman splittings. In prin- in 3D weak TSCs, particularly for those realized by gap- ciple, this allows for the realization of effectively spinless ping 3D Weyl semimetals [68, 69]. This suggests that the gapped p + ip superconductors [64] as well as nodal 2D toric code insulator fits into a broader and more general superconductors [65]. The latter may be viewed as an framework for studying and realizing fractionalization in intermediate nodal phase separating a trivial SC from a strongly interacting weak topological phases. weak TSC. It is important to note, however, that all these Here we consider a tight-binding model of spinless elec- proposals and platforms or realizing engineered 1D or 2D trons on the square lattice with alternating hoppings t1,2 TSCs rely on proximity coupled (conventional s-wave) in the x direction, as depicted in Fig.5. The alternating superconductivity. This presents a important challenge, hoppings give rise to a two-site unit cell and we label the since our construction of the toric code insulator relies two sublattices as A and B. The two-component electron on intrinsic superconductivity with a fluctuating phase. † operator ck and the Hamiltonian H0 are given by Hence, even if realized, disordering the resulting weak TSC may be challenging. † † † X † ck = (ckA, ckB),H0 = ckhkck, (4.1) A third class of systems relevant to 2D weak TSCs k is characterized by quasi-1D Fermi surfaces associated with orbital degrees of freedom. The anisotropic na- The Hamiltonian matrix can be expressed as hk = 10
ty
hk = εkσ + 2t sin k σ , (4.3) 1 X † † x 0 x y H = (∆k) c c + H.c.. (4.4) ∆ 2 αβ kα −kβ k where εk = 2(tx cos kx + ty cos ky) describes an anisotropic square lattice dispersion. The Hamiltonian Here, α, β label the sublattice degree of freedom and ∆k T has two important symmetries: time-reversal ( ) and is the pairing potential, which satisfies ∆−k = ∆k. inversion ( ) symmetry. -symmetry is definedT by We focus on pairing along the wires in the x direction− ∗ P T h = h−k, and -symmetry by σ hkσ = h−k. and consider a nearest neighbor pairing potential of the k P x x To understand the electronic phases described by this form ∆k = ∆0 cos kxσy. Setting the chemical potential model it is instructive to consider two special cases, which µ to zero, we obtain a quasiparticle spectrum with four ± 2 2 correspond to setting either tx or ty to zero. First, con- branches given by the equation (Ek ) = (2t0 sin kx) + 2 sider the case t = 0. We may then view the system (εk ∆ cos k ) . To understand the quasiparticle spec- y ±| 0| x as a collection of decoupled 1D wires in the x direction, trum, it is useful to consider the case tx = 0 and examine each realizing a Su-Schrieffer-Heeger (SSH) chain with the existence of nodes. Given the solution for the quasi- Hamiltonian h(kx) = 2tx cos kxσx + 2t0 sin kxσy. In each particle spectrum, the condition for vanishing pairing chain two distinct insulating phases are possible: a triv- gap is 2t cos k ∆ = 0, which implies that nonzero y y ± | 0| ial phase and a topological phase. The topological phase (and small) pairing ∆0 splits each Dirac node into two is characterized by a fractional charge polarization and Bogoliubov-Dirac nodes.| | The Bogoliubov-Dirac nodes is protected by inversion symmetry [70, 71]. This moti- move in opposite directions along the k axis as ∆ is y | 0| vates an interpretation of the square lattice model as a changed, as is indicated in Fig.5(b). When ∆0 = 2 ty collection of 1D wires of spinless electrons stacked and the four Bogoliubov-Dirac nodes merge and annihilate| | | in| coupled in the y direction. pairs at both Γ and M, and the resulting fully gapped A second special case is realized when tx = 0 and superconductor realizes a weak TSC in 2D. This may be corresponds to an ordinary square lattice model with understood by taking ty to zero, which does not close anisotropic nearest neighbor hopping in the x and y di- the quasiparticle gap, but yields decoupled 1D supercon- rections and a flux Φ = π piercing through each square ducting wires, each realizing a Kitaev chain [35]. This plaquette. As a result of the π-flux, the spectrum exhibits shows that for sufficiently strong BCS pairing of the Ki- 11 taev type (i.e., nearest neighbor pairing along the wires), V. DISCUSSION AND CONCLUSION the square lattice model describes a weak TSC composed of coupled 1D Kitaev chains coupled in the y direction. In this paper we have introduced and analyzed a Whereas zero momentum BCS pairing does not im- coupled wire construction for a translation symmetry- mediately lead to a full gap (but instead gives rise to enriched Z2 topological order, which is termed the toric nodal points), finite momentum FFLO pairing directly code insulator. Our construction is based on the vortex- gaps out the quasiparticle spectrum, and similarly leads condensation approach, for which the weak TSC in 2D to a weak TSC phase. To describe this, we consider finite (i.e., a gapped superconductor equivalent to an array of momentum pairing of the form 1D TSCs) serves as the conceptual starting point. The 1 X † † nature of the 2D weak TSC naturally suggests a cou- H = (∆k) c c + H.c., (4.5) ∆ 2 αβ kα −k+Qβ pled wire model description. A key feature of the weak k TSC is the fractional Josephson effect, which gives rise to where Q = (0, π) is the wave vector connecting Γ to a distinction between two types of vortices: vortices on M, see Fig.5(b). The wave vector Q also connects the even and odd links. In this way, the fractional Josephson Dirac points located at K and K when t = 0, and effect is inextricably linked to the translational proper- − x we therefore set tx to zero at first. Including the FFLO ties of weak TSC, as well as to the implementation of pairing term, we obtain the Hamiltonian matrix translation symmetry in the toric code insulator phase. Starting from the weak TSC, the latter is the result of hk ∆k proliferating double-vortices. k = , (4.6) H ∆† hT To describe double-vortex condensation, we have in- k − −k+Q troduced a three-fluid model for the coupled wires. In T addition to a Luttinger liquid of charge-e fermions and a where ∆k satisfies ∆−k+Q = ∆k due to Fermi statis- tics. As a result, the admissible− nearest neighbor pair Luttinger liquid of charge-2e Cooper pairs in each wire, potential takes the same form as in the case of BCS this model consists of a third fluid, which arises from a pairing. As before, we focus on pairing of the form Z2 gauge field defined on the link. The Z2 gauge field is a ∆k = ∆0 cos kxσy and determine the quasiparticle spec- consequence of gauging the superconducting phase-shift trum; we find two degenerate branches given by symmetry and achieves a decoupling of the charge and neutral sectors of the two Luttinger liquids. In this sense, E2 = (2t sin k )2 + (2t cos k )2 + ∆ 2 cos2 k . (4.7) k 0 x y y | 0| x the three-fluid model–particularly the introduction of the This describes a fully gapped superconductor for arbi- Z2 gauge field—bears resemblance to previously consid- trary strength of pairing ∆0 and the resulting gapped ered slave-particle approaches and parton constructions phase is a weak TSC. Note that due to the commensu- [38]. Importantly, the magnetic flux of the Z2 gauge field rability of Q, this FFLO pairing state does not break corresponds to a single vortex in the phase of the Cooper- translation symmetry. Indeed, density modulations of fi- pair fluid. nite momentum Q pairing have wave vector 2Q, which Our central result is the analysis of the three-fluid is equal to a reciprocal lattice vector. model and the demonstration that it leads to a 2D It is interesting and enlightening to connect this anal- gapped insulator with topological order—a new phase of ysis to recent work on Weyl semimetals and weak topo- matter which we refer to as the toric code insulator. The logical superconductors in 3D. Conceptually, T -breaking topological order is of the Z2 type akin to the toric code, Weyl semimetals can be viewed as gapless phases describ- with the following anyons: charge-e chargon (e), neutral ing the topological transition between a trivial insulator fermion (f), and two types of visons (m). An electron ψe and a 3D quantum Hall insulator, i.e., a weak topolog- can fractionalize into a chargon and a neutral fermion, ical phase equivalent to stacks of 2D Chern insulators. thus exhibiting a charge-statistics separation. Moreover, Recent work explored different ways of introducing a full there is a translation symmetry-enrichment manifest in pairing gap in this type of Weyl semimetals, with the aim the toric code insulator: translation symmetry relates of realizing a 3D weak TSC equivalent to a stack of 2D two types of visons, me (on even-links) and mo (on odd- chiral p + ip superconductors [68, 69]. In particular, it links), which have a semionic mutual braiding statistics. was shown that commensurate finite momentum FFLO The toric code insulator is thus a tensor product of a pairing leads to a direct pairing gap [68, 72, 73], whereas physical electron and a toric code topological order en- zero momentum BCS pairing requires sufficiently strong riched by translation symmetry: ψe me, mo, f . The ⊗ { } pairing in order to induce a merging and pairwise an- complete topological data is summarized in Table.I. nihilation of Bogoliubov-Weyl nodes at high symmetry Our work suggests that strongly-interacting weak SPT points [69, 74]. In both cases, the resulting gapped su- phases are promising platforms for realizing SET phases perconductor realizes a 3D weak chiral TSC with one chi- enriched by translation symmetry. In particular, neither ral Majorana mode per stacked layer on any side surface. the Hamiltonian nor the ground state breaks translation The case of the T -breaking Weyl semimetal is the direct symmetry, yet translation has a non-trivial effect as per- 3D analog of the square-lattice model we have introduced muting anyons (here me mo). This phenomenon has and analyzed in this section. been termed “weak symmetry↔ breaking” [29], and is an 12 intriguing consequence of symmetry-enrichment in topo- ACKNOWLEDGMENTS logical order. There are celebrated spin lattice models, such as Wen’s plaquette model and Kitaev’s honeycomb This work is in part supported by the Croucher Schol- model [21, 29], which feature this effect. Recently, Rao arship for Doctoral Study from the Croucher Foundation and Sodemann have proposed to understand the weak (P.M.T.) and a Simons Investigator grant from the Si- breaking of translation symmetry in these models as a mons Foundation (C.L.K.). consequence of the emergent spinons forming a weak TSC, which leads to a mobility constraint for m-particles in the toric code [30]. This is essentially the same pat- Appendix A: The Gauss’s law constraint tern of symmetry-enrichment as realized in the toric code insulator introduced here. There is, however, a crucial In the main text, we have treated the y-component of difference: our system is built out of itinerant electrons, gauge field Ay,` as the density variable Θ` of a Luttinger instead of localized magnetic moments on a lattice. This liquid, and the conjugate electric field is then Ey,` = perspective has enabled us to consider several material ∂xΦ`. To facilitate this Luttinger liquid representation, realizations, which include the surface of 3D weak TI, we have to adopt a gauge-fixing condition: Ax = 0, but 2D array of nanowires or magnetic adatom chains, and then the operator eiEx/2 which tunnels a π-flux (of B = correlated materials with quasi-1D Fermi surfaces. ∂xAy ∆yAx) is incompatible with this gauge. We thus look for− a re-writing of this operator using the Gauss’s law constraint, which we now derive. Furthermore, our work has an interesting connection Due to the discreteness of the gauge fields, one should with recent proposals for strongly correlated fractional- not attempt to perform an infinitesimal variation of At in ized phases in 3D Weyl semimetals [68, 69]. In particular, hope of obtaining an equation of motion that represents Ref. 68 explored the possibility of realizing a 3D frac- the Gauss’s law. Instead, the Gauss’s law is obtained by tional quantum Hall effect in Weyl semimetals by disor- integrating out At in a discretized fashion. This proce- dering a 3D weak TSC. The 3D weak TSC can be realized dure can be carried out straightforwardly by viewing the by pairing fermions within each Weyl node and is equiv- Z2 gauge theory (coupled to a compact ϕ2) as a Z gauge alent to a stack of 2D chiral p+ip Read-Green supercon- theory (coupled to a non-compact ϕ2). In this case, the ductors. This is analogous to the 2D weak TSC, which “basic” configuration of At takes the following form: can be understood as a stack of 1D Kitaev TSCs. As demonstrated in the previous section, the 2D weak TSC At(j, x, t) = πnjH(x x0)δ(t t0), (A.1) is intimately related to topological semimetals in 2D. Our − − work therefore shows that gapping and disordering topo- with nj Z and H(x) being the Heaviside step function. ∈ logical semimetals by proliferating defects should be con- Here x0 and t0 label a reference space-time position where sidered a general route towards realizing novel types of the gauge field changes its discrete value. A more general topological order, both in 2D and 3D. The two canon- form of At results from the superposition of these “basic” R ical examples of topological semimetals in 2D and 3D configurations. Now the functional integral At can P D suggest possible generalizations to other types of topo- be replaced by a discrete sum nj . Let us only consider logical semimetals, for which spatial symmetries play a the part of the action associated to At, which arises from prominent role. minimal coupling, and integrate it out:
Z n X Z i o A exp ∂ θ (∂ ϕ 2A ) E ∂ A E ∆ A Z ∼ D t 2π x 2,j t 2,j − t,j − x,j x t,j − y,` y t,j j x,t (A.2) Y X n 1 1 o Y X exp in θ E (∆ Φ) = 4πδ 2θ E (∆ Φ) 4πm ∼ j 2,j − 2 x,j − 2 y j 2,j − x,j − y j − j j nj ∈Z j mj ∈Z
The second to last expression is obtained upon substi- should counts all possible configurations, the Gauss’s law tuting Eq. (A.1). The last equality is obtained from the constraint holds in general. The π-flux hopping operator Poisson summation, and the delta function imposes the can thus be written as in Eq. (3.8): Gauss’s law constraint as quoted in Eq. (2.9b). In the i E i[(θ −θ ) − 1 (∆ Φ) ] above derivation, we tentatively choose a field configura- e 2 x,j = e ρ σ j 2 y j . (A.3) tion as in Eq. (A.1), so the obtained constraint holds at Notice the right-hand side is consistent with the A = 0 (x0, t0) for every wire j. Since we do not have to spec- x gauge, so this serves as a physical operator in our Lut- ify (x0, t0), and indeed the complete functional integral tinger liquid formalism. This is used in the Sec.IIIA 13 to study the motion of visons in the toric code insu- ground states are ϕσ, θρ, Θ , hence we will only focus lator, which reveals a pattern of translation symmetry- on these variables.{ Next, there} is an intertwined identi- enrichment. fication of fields on neighboring wires, as revealed by the action of T`(f) introduced in Eq. (3.5). We have
Appendix B: Counting GSD in the wire model I :(ϕ , ϕ ) (ϕ + 2π, ϕ + 2π). (B.4) B,` σ,j σ,j+1 ≡ σ,j σ,j+1 The Wilson-loop argument in Sec.IIIB suggests there Finally, by virtue of the gauge redundancy, is a topologically-protected lower bound to the ground Z2 state degeneracy (GSD) of the symmetry-enriched toric 2 code insulator on torus (T ). Due to the interplay be- IC,j :(ϕσ,j, Θ`, Θ`−1) (ϕσ,j + 2π, Θ` + π, Θ`−1 + π), tween topology and translation symmetry, the torus GSD ≡ (B.5) depends on the parity of the number of wires, which is which is also revealed by the action of the twist operator referred to as the even-odd effect as summarized in Eq. eiEx,j /2 in Eq. (3.8). The above three types of identifi- (3.11). Here we count the GSD on T 2 explicitly using cation fully capture the redundancies in the definition of the three-fluid wire model developed in Sec.IIC. fields. For each wire j there are identifications IA,j and Ground states are determined by the interaction po- IC,j that can be performed to relate equivalent ground- tentials in Eq. (2.10): state configurations. Similarly, for each link ` there is an identification IB,` that can be used to relate equivalent L 3L X configurations. Starting from the 2 counting, each in- = (u cos ϕ + w cos 2θ + h cos 2Θ ). (B.1) Hint σ,j ρ,j ` dependent identification reduces the number of distinct j=1 ground states by a factor of 2. with ` j + 1/2. We consider the system with L wires Crucially, not all redundancies introduced above are on a torus≡ by imposing periodic boundary conditions in independent. Due to the torus geometry, both the x-(along the wire) and the y-(along the stacking of wires) directions, so wire j = L + 1 is identified with L Y wire j = 1. In the strong-coupling limit, ground states I 1 = 1. (B.6) B,j+ 2 are gapped and characterized by the condensed values j=1 of ϕσ,j, θρ,j and Θ`, which are respectively pinned to the bottom of the cosine potentials. Hence, ϕσ,j 2πZ, ∈ This is because applying I on every link leads to a θρ,j πZ and Θ` πZ. Notice that the bosonic fields B,` are compact:∈ ∈ 4π-shift of ϕσ for every wire, which have been consid- ered already in Eq. (B.2). Hence there are only (L 1) − ϕ ϕ + 4π; (B.2a) independent redundancies associated to Eq. (B.4). σ,j ≡ σ,j θ θ + 2π; (B.2b) Now it comes the even-odd effect. If L is even, there ρ,j ≡ ρ,j Θ Θ + 2π, (B.2c) is one more relation (thus one less redundancy): ` ≡ ` 3L which imply that there are at most 2 distinct ground L L/2 states. Y Y IC,j = I 1 , (B.7) B,2i+ 2 On top of the above compactifications, there are other j=1 i=1 redundancies in the definition of the bosonic fields, which lead to identification in the 23L states we just naively counted. The redundancies in the bosonic fields are re- as the net effect of the left-hand side (after modding out vealed by local operators that create kinks in them. Con- the compactification of Θ`) is to induce a 2π-shift of ϕσ sidering the local electron operator ψe, in Eq. (2.1), we on every wire, which is equivalent to applying Eq. (B.4) see that the charge and neutral sectors have an inter- on every alternating link. Everything considered, the twined identification: GSD on T 2 in the three-fluid wire model is thus
I :(ϕ , θ ) (ϕ + 2π, θ + π). (B.3) ( A,j σ,j ρ,j ≡ σ,j ρ,j 23L 2−(3L−2) = 4, for even L; GSD = × (B.8) To be precise, the above should also be accompanied by 23L 2−(3L−1) = 2, for odd L. × a π-shift in θσ,j, but for simplicity we would leave it im- plicit. After all, what matter to the determination of
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