Fractionalization patterns in strongly correlated electron systems: Spin- charge separation and beyond
The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters
Citation Demler, Eugene, Chetan Nayak, Hae-Young Kee, Yong Baek Kim, and T. Senthil. 2002. “Fractionalization Patterns in Strongly Correlated Electron Systems: Spin-Charge Separation and Beyond.” Physical Review B 65 (15) (March 27). doi:10.1103/ physrevb.65.155103.
Published Version doi:10.1103/PhysRevB.65.155103
Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:27989551
Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA PHYSICAL REVIEW B, VOLUME 65, 155103
Fractionalization patterns in strongly correlated electron systems: Spin-charge separation and beyond
Eugene Demler,1 Chetan Nayak,2 Hae-Young Kee,2 Yong Baek Kim,3 and T. Senthil4,5 1Physics Department, Harvard University, Cambridge, Massachusetts 02138 2Physics Department, University of California Los Angeles, Los Angeles, California 90095–1547 3Department of Physics, The Ohio State University, Columbus, Ohio 43210 4Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030 5Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139 ͑Received 22 May 2001; published 27 March 2002͒ We discuss possible patterns of electron fractionalization in strongly interacting electron systems. A popular possibility is one in which the charge of the electron has been liberated from its Fermi statistics. Such a fractionalized phase contains in it the seed of superconductivity. Another possibility occurs when the spin of the electron, rather than its charge, is liberated from its Fermi statistics. Such a phase contains in it the seed of magnetism, rather than superconductivity. We consider models in which both of these phases occur and study possible phase transitions between them. We describe other fractionalized phases, distinct from these, in which fractions of the electron themselves fractionalize, and discuss the topological characterization of such phases. These ideas are illustrated with specific models of p-wave superconductors, Kondo lattices, and coexistence between d-wave superconductivity and antiferromagnetism.
DOI: 10.1103/PhysRevB.65.155103 PACS number͑s͒: 71.10.Hf, 74.20.Mn, 71.27.ϩa, 74.72.Ϫh
I. INTRODUCTION effect is the enormously rich number of exotic phases which display different patterns of fractionalization of the electron Electron fractionalization in strongly interacting electron and associated topological orders. In view of the similarity systems in dimensions larger than 1 has been an important between the theoretical characterization of quantum Hall subject of study since spin-charge separation was suggested states and fractionalized states in zero magnetic field, it is 1,2 as a mechanism of high-Tc superconductivity in the cu- tempting to investigate a similar possibility of a variety of prates. In particular, it was suggested that the electron is fractionalization patterns in other strongly correlated sys- splintered into a spin-carrying neutral excitation ͑‘‘spinon’’͒ tems. We explore this possibility in this paper. We describe and a charge-carrying spinless excitation ͑‘‘holons’’ or theoretically a few of the several different possible fraction- ‘‘chargons’’͒. There have been different proposals in regard alized phases that may exist in various different models of to this possibility, but the existence of such phases in the strongly interacting electron systems. cuprates is still controversial. Following the introduction of the Schwinger boson de- On the other hand, there exist clear experimental ex- scription of the Heisenberg model of quantum amples of phases in the quantum Hall regime of two- antiferromagnets,5 slave fermion6 formulations of doped an- dimensional electron systems where quantum number frac- tiferromagnets were introduced. In these formulations, it is tionalization has been well established. The low-energy assumed that the electron decays into a Bosonic, spin-1/2 excitations ͑quasiparticles͒ in these two-dimensional strongly spinon and a fermionic, charge-e holon. We will call this interacting electron systems carry fractions of the quantum phase CFSB ͑charged fermion, spinful boson͒. numbers of the original electrons. Different quantum Hall On the other hand, a phase with Bosonic holons and fer- liquid states can be characterized by different varieties of mionic spinons – which we will call CBSF ͑charged boson, topological order. The transitions between different quantum spinful fermion͒—naturally leads to superconductivity Hall states can be understood as topological-order-changing through the Bose condensation of Bosonic holons in the transitions which occur even in the absence of conventional presence of Fermionic spinon pairing. Consequently, much broken symmetries. The Hall conductance is but one of the attention has been focused on the description of such a frac- topological quantum numbers which characterize a given tionalized phase, especially in the context of the slave boson phase. Another important property of a topologically ordered description of the t-J model. The pairing symmetry of the state is the ground-state degeneracy of the system on higher resulting superconductor is dictated by the underlying sym- genus manifolds such as tori. For each topologically ordered metry of the spinon pairing. state, there are corresponding sets of characteristic excita- A Z2 gauge theory of Fermionic spinons and Bosonic ho- tions with different quantum numbers. lons was developed in the context of superconductivity in the It has become clear3,4 that the notion of topological order cuprates7 ͑see also Refs. 8–10͒. Spinons and holons are also provides a precise characterization of spin-charge sepa- coupled by an Ising gauge field. The deconfined phase of this rated and other fractionalized phases in spatial dimensions theory corresponds to the CBSF phase. Most importantly, the higher than one even in situations of zero or weak magnetic deconfinement-confinement transition of spinons and holons fields. One of the remarkable features of the quantum Hall occurs through the condensation of vortices in the Z2 gauge
0163-1829/2002/65͑15͒/155103͑26͒/$20.0065 155103-1 ©2002 The American Physical Society DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
field, or visons. In the deconfined phase, the visons exist as different models: p-wave superconductors, Kondo lattices, gapped excitations; when visons condense, the spinons and and XY magnets coupled to d-wave superconductors. holons are confined within electrons. The existence of The main results can be summarized as follows. gapped visons is crucial for the robustness of the topological ͑i͒ Both CBSF and CFSB phases can arise in a variety of order of the deconfined fractionalized phase.11,4 Although different models. this formalism was introduced in the context of cuprate su- ͑ii͒ Upon accepting the possibility of electron fractional- perconductivity, it is sufficiently flexible to permit a descrip- ization, one is led to consider a wide variety of fractionalized tion of other types of fractionalized phases including CFSB. phases. In the higher-level fractionalized phases, electrons These ideas have a physical manifestation in the context can be fractionalized in many different ways. For example, of quantum disordered magnets and superconductors. In this spinons and holons can be further fractionalized. Apart from picture, one visualizes fractionalized states in terms of the CBSF and CFSB phases, we discuss two others. One is nearby ordered states. In a broken ͑continuous͒ symmetry the CSBNF ͑charge- and spin-carrying boson, neutral fer- state, Goldstone modes can screen the associated quantum mion͒ phase, in which the electron breaks up into a boson number͑s͒.12 Thus it is possible for quasiparticles to be which carries both the spin and charge quantum numbers and stripped of some of their quantum numbers. One might a neutral fermion. This phase is at the first level of fraction- imagine that the destruction of order by quantum fluctuations alization along with the CBSF and CFSB phases. The other can preserve this screening of quasiparticle quantum num- is the CBSBNF phase, in which there exist spin-carrying bers. This occurs when those topological defects of the or- neutral bosons, charge-carrying spinless bosons, and dered state which braid nontrivially with the quasiparticles ‘‘statistics-carrying’’ neutral spinless fermions. The CBSBNF persist as gapped excitations even after the demise of the phase is at the second level of fractionalization. In principle, order.13 Indeed, this is precisely what happens when the state higher-level fractionalized phases exist. is topologically ordered. The neutral, spin-1/2 fermion of the ͑iii͒ We demonstrate the existence of some of these exotic CBSF state is viewed as the descendent of the phases in the context of the three different systems men- Bogoliubov–de Gennes quasiparticle; the vison, of the hc/2e tioned above—Kondo lattices, p-wave superconductors, and vortex. When considered in the context of spin-triplet super- models with both strong spin and d-wave pairing fluctua- conductors and their rich order-parameter structure, this im- tions. For the p-wave superconductor, the four fractionalized mediately suggests exotic phases such as CBSF, CFSB, and phases discussed here arise naturally and the order parameter even a third phase CBSBNF ͑charged boson, spinful boson, has a rich spectrum of topological defects which can con- neutral fermion͒, in which the charge- and spin-carrying ex- dense in a variety of ways, thereby giving rise to an array of citations are bosons and there is a neutral, spinless Fermionic fractionalized nonsuperconducting phases. excitation. Since these superconductors can break both ͑iv͒ The question of whether CBSF and CFSB are charge and spin symmetries—as do states in which singlet smoothly connected to one another or whether they are nec- superconductivity and magnetism coexist – one can envision essarily separated by a phase transition is a subtle and deli- the screening of both quantum numbers of a quasiparticle. If cate issue for reasons that will be discussed at length later. ͑the minimal͒ topological defects in the charge sector survive While we do not provide a definitive conclusion, we outline into a disordered state, then this disordered state has neutral, a possible scenario in which the distinction between CBSF spin-1/2 Fermionic excitations ͑CBSF͒; if topological defects and CFSB is similar to that between liquid and gas phases. in the spin sector survive into a disordered state, then this These phases are separated by a first-order transition line disordered state has charge-e spinless Fermionic excitations which terminates at a critical point. In principle, one can go ͑CFSB͒; if topological defects in the both sectors survive around the critical point from one phase to the other without into a disordered state, then this disordered state has neutral, encountering a phase transition. This scenario is supported spinless Fermionic excitations ͑CBSBNF͒. by a number of suggestive ͑though certainly not conclusive͒ The analysis of quantum dimer models2 and resonating arguments. valence-bond1,14,15 ground states led to conflicting claims On the other hand, the transition between the two phases that the CBSF ͑Ref. 15͒ or CFSB ͑Refs. 16 and 17͒ scenario can occur through another fractionalized phase with a higher- is realized in these models. These models have a Z2 vortex level fractionalization pattern. In this case, each transition in excitation17,18—which are precisely the visons described the process could be a continuous transition. We demonstrate above—which are relative semions with spinons and holons. that the transition between CBSF phase and CFSB phase can Thus a spinon or holon can change between Bosonic and occur through the CBSBNF phase. Fermionic statistics by forming a bound state with a vison. ͑v͒ In order to examine whether one can go from CBSF to This begs the question whether the CBSF phase discussed in CFSB through further fractionalized phases like CBSBNF, the context of superconductivity is the same as the CFSB one can design a gedanken flux trapping experiment similar phase considered in relation to magnetism. We reconsider to the one proposed in Ref. 19. This gedanken experiment this question in the context of recent progress in the under- clearly demonstrates the existence of a phase boundary be- standing of fractionalized phases described above. One tween CBSF and CFSB when these phases are close to might worry that the apparent differences between these CBSBNF. phases is an artifact of the formalisms employed. One might Topological order is robust against local perturbations also wonder if there are any further fractionalized phases. In such as impurities. Thus we will concentrate on general uni- this paper, we discuss the questions raised above using three versal properties of the fractionalized phases. One of our
155103-2 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103 goals will be to give a precise characterization of these † phases which is independent of the underlying microscopic HtϭϪ ͚ trrЈ͑cr␣crЈ␣ϩH.c.͒, ͑2͒ models where they may occur. We believe that these exotic ͗rrЈ͘ phases could play a role in the physics of 3He ͑Ref. 20͒ and the ruthenates21 as well as the cuprates,22 organic ϩ † HKϭJK͚ ͑Sr cr cr ϩH.c.͒, ͑3͒ superconductors,23 heavy fermion superconductors,24 spinor r ↓ ↑ Bose-Einstein condensates,25 and the crusts of neutron stars.26 J H ϭ Ϫ SϩSϪ ϩH.c.͒ϩJ SzSz . ͑4͒ The rest of the paper is organized as follows. In Sec. II, ex ͚ ͑ r rЈ z r rЈ rrЈ 2 we consider a Kondo lattice model and how the CFSB frac- tionalized phase can occur in this model using the language Here the ci␣ represent ‘‘conduction’’ electrons with spin ␣ at of a Z2 gauge theory. Some details are given in Appendix A. site i. The operators Sជ i are spin operators representing mag- In Sec. III, we suggest how this analysis can be generalized netic moments localized at the lattice sites. The first term is and discuss a hierarchy of fractionalized phases. Here we the usual conduction electron hopping, described in a tight- provide an overview of our results. In Sec. IV, we discuss binding approximation. The second term is a ‘‘Kondo’’ cou- how this hierarchy can be realized in p-wave superconduct- pling between the conduction electrons and the local mo- ing systems when the superconducting and spin order are ments. The third term is an explicit exchange interaction quantum disordered. This is done using the vortex conden- between the local moments. For simplicity, we have assumed sation formalism. In Appendix B, the same ideas are shown that system only has a U͑1͒ spin symmetry for rotations to apply to an XY magnet which is coupled to a d-wave about the z axis of spin ͓we will comment on situations with superconductor. In Sec. V, the fractionalized phases of Sec. full SU͑2͒ spin symmetry later͔. We are interested not so IV are further discussed in the framework of a Z2ϫZ2 gauge much in establishing the exact phase diagram of this particu- theory. In Sec. VI, we consider the question of the distinction lar model; rather our main interest here is in establishing the in principle between the putatively different fractionalized possible existence and stability in models of this kind of phases constructed in this paper. In Appendix C, we give quantum phases where the electron is fractionalized. To that some technical details of an argument using Z2ϫZ2 gauge end, we will think more generally about a class of models theory which supports our picture of the phase diagram. In which may be obtained from the model above by adding Sec. VII, we show how flux-trapping experiments ͑of the other local interactions which share its symmetries. If the variety suggested by Senthil and Fisher19͒ can be used to system is in a quantum phase in which both the symmetry of shed further light on the phase boundaries between these rotations about the z direction of spin and the charge conser- phases and could be used to detect them. We conclude in vation symmetry is unbroken, the excitations may be labeled Sec. VII. Appendix D contains an aside in which we discuss by their Sz and charge ͑Q͒ quantum numbers. Clearly, we can various interesting properties of unfractionalized phases oc- visualize two qualitatively different possibilities. First, the curring in the models considered in this paper. system may be in a phase in which the excitations are elec- 1 For other perspectives on fractionalization, see Refs. trons (Qϭ1,Szϭ 2 ) or composite objects made from elec- 1–4,7,8,12,13,15,17–19, and 27–33. trons ͑such as, for instance, a magnon which has Qϭ0,Sz ϭ1). This is a conventional phase of the kind familiar from II. FRACTIONALIZATION IN SPIN MODELS: textbooks ͑for instance, a Fermi liquid or a band insulator͒. SPIN-STATISTICS SEPARATION On the other hand, one could also imagine phases in which there are excitations which carry quantum numbers which In principle, there are several possible ways in which the are fractions of those of an electron. The simplest possibility electron can fractionalize in a strongly correlated system. In ͑the one we will focus on͒ is that there are excitations which the context of the cuprates, attention has been focused on the ϭ ϭ ͑ ͒ situation in which the electron splinters into two separate carry Sz 1/2,Q 0 spinons and others which carry Sz ϭ ϭ excitations—a charged spinless boson, and a neutral spinful 0,Q 1 ͑holons͒. In such a phase, the electron has been fermion. In this case, the charge of the electron is liberated fractionalized. In what follows, we will discuss several ways from its Fermi statistics. of thinking about such phases. Our focus will be on general In this section, we will briefly discuss another possible universal properties of such phases. In particular, we will be fractionalization pattern in which the spin, rather than the interested in obtaining robust precise characterizations of charge, of the electron is liberated from its Fermi statistics. fractionalized phases that are independent of the particular The electron splinters into a charged spinless fermion, and a microscopic models in which they possibly occur. spinful boson. As we will see, this phenomenon also requires It is extremely instructive to begin by just considering the physics of the local moments alone as described by the ex- the presence of a gapped topological Z2 vortex excitation. The issue of whether such a fractionalized phase is distinct change part of the Hamiltonian Hex . This Hamiltonian is from one in which the charge is liberated from the Fermi clearly invariant under a global spin rotation about the z axis statistics is a delicate one, and shall be discussed in Sec. VI. of spin. For technical simplicity, we will assume J,Jzу0. The physics of this particular Hamiltonian is well under- To motivate the discussion, consider a ‘‘Kondo lattice’’ ϩ model with the Hamiltonian stood: when Jz /J is small, there is long-range order in S . When Jz /J is large, the system breaks translational symme- z HϭHtϩHKϩHex , ͑1͒ try with ͗S ͘ being larger in one sublattice of the square
155103-3 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103 lattice than the other, but the U͑1͒ spin rotation symmetry is of states where Nr is even. Therefore we need to impose the Nr unbroken. The point JzϭJ can be mapped to the nearest- operator constraint (Ϫ1) ϭ1 at each site of the lattice. For- neighbor antiferromagnetic Heisenberg model with full mally this may be implemented through the projection op- SU͑2͒ spin symmetry on a bipartite lattice by rotating the erator spins on one sublattice by about the z axis. In the specific case of a square lattice ͑which we assume through out our ϭ͟ r , ͑10͒ discussion͒, this is known to develop Ne´el long-range order P r P in two spatial dimensions. Our interest here is not so much in the properties of this particular Hamiltonian as in the prop- 1 erties of an entire class of systems with the same symmetry, ϭ ͓1ϩ͑Ϫ1͒Nr͔. ͑11͒ r 2 and with short-ranged interactions between the spins. In par- P ticular, we will be interested in fractionalized phases in Note that ͓ ,H͔ϭ0. It is now convenient to pass to a func- P which the excitations are spinons with quantum number Sz tional integral formulation. We follow Refs. 7 and 27 closely ϭ1/2. To that end, we will reformulate the Hamiltonian di- to obtain for the partition function rectly in terms of spinon fields which carry spin Szϭ1/2. This naturally introduces a Z gauge symmetry. The result is 2 ϪS a theory of Bosonic spinon fields coupled to a Z gauge field Zϭ͚ e , ͑12͒ 2 ͵ D which can then be used to analyze the possibility of fraction- r alized phases and their universal properties. SϭS ϩS ϩS , ͑13͒ We may think of Sϩ,SϪ as the creation and destruction r B operators, respectively, of a hard-core boson on the sites of the lattice. Specifically, write Sϩϵb† , SϪ ϵb , and Sz r sr rЈ sr r Sϭ Jrcos͑r,ϩ⑀Ϫr͒, ͑14͒ † ͚,r ϭ1/2Ϫbsrbsr . Note that there is half a boson for each site on average. Now imagine relaxing the hard-core constraint on the bosons, and instead add a term Srϭ⑀ ͚ J cos͑2rϪ2rЈ͒, ͑15͒ ͗rrЈ͘ U 2 ͑2nrϪ1͒ ͑5͒ where ϭϮ1 may be interpreted as the time component of 2 ͚r r a Z2 gauge field that imposes the constraint on the Hilbert at each lattice site. Here nr is the boson number at each site. space, and ⑀ is the lattice spacing along the time direction. In the limit U ϱ, we recover the spin model exactly. For The constant J is determined by the original interaction → large but finite U, however, relaxing the hard-core constraint strength U. The term in the action Sr involving the spatial is expected to be innocuous. It is now convenient to go to a coupling may be decoupled by a Hubbard-Stratanovich number-phase representation for the bosons: we write bsr transformation: ir ϳe with ͓r ,nrЈ͔ϭi␦rrЈ . For simplicity, we also special- ize to the limit where Jzϭ0. The Hamiltonian then becomes Ϫ Ϫ⑀ ͚ 2ϩ ⑀ † ϩ e Srϭ ͵ De J ͗rrЈ͘, rrЈ( ) 2 J rrЈ( )[zr ( )zrЈ( ) c.c.]. U 2 ͑16͒ Hϭ ͚ ϪJ cos͑rϪrЈ͒ϩ ͚ ͑1Ϫ2nr͒ . ͑6͒ ͗rrЈ͘ 2 r Here rrЈ() is a real-valued field. We have omitted an un- This is clearly closely related to the original spin Hamil- important overall constant. tonian in Eq. ͑4͒. Now consider a formal change of variables We now proceed exactly as in Refs. 7 and 27, and replace which involves splitting the boson operator bsr into two the integral over the continuous variable by a sum over a pieces: discrete field rrЈ()ϭϮ1. As discussed in Refs. 7 and 27, this approximation respects all the symmetries of the action, ir 2 bsrϭe ϭzr , ͑7͒ and is expected to be innocuous. The resulting partition func- tion becomes ir i(r/2) zrϵe ϭsre ͑srϭϮ1͒. ͑8͒ ϭ ϪS We will refer to zr as the spinon destruction operator. Note Z ͚ ͵ e , ͑17͒ ij D that with these definitions, both r and r are defined in the interval ͓0,2). It is also convenient to define a number SϭSsϩSB , ͑18͒ operator for the spinons Nrϭ2nr which is conjugate to r . In terms of the spinon operator, the Hamiltonian becomes ϭϪ Ϫ U Ss ͚ Jijijcos͑i j͒. ͑19͒ 2 ͗ij͘ Hϭ ϪJ cos͑2rϪ2rЈ͒ϩ ͑NrϪ1͒ . ͑9͒ ͚ 2 ͚r ͗rrЈ͘ Here the i, j label the sites of a space-time lattice in three The change of variables above must be supplemented with a dimensions. The constants JijϭJ for temporal links, and constraint—clearly the physical Hilbert space consists only equals ⑀J for spatial links. SB is the Berry phase action
155103-4 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
i net is therefore expected to develop spin-Peierls order. We SBϭ ͑1Ϫij͒. ͑20͒ will not discuss such conventional phases very much in this 2 ͚ ˆ i, jϭiϩ paper. Note that the action ͑18͒–͑20͒ respects all the symmetries of Much further insight into the physics of the fractionalized the original model. The discrete field ijϭϮ1 may be phases may be obtained by the following considerations. We viewed as a Z2 gauge field. At this stage, this field does not begin by first considering ordered phases in which the sym- have any dynamics. However, it is natural to expect that metry of rotations about the z direction of spin has been upon coarse graining, some dynamics will be generated. The broken spontaneously. For simplicity, we consider a phase in simplest such term allowed by symmetry is which the spins have all lined up along some direction in the xy plane. The general properties of such a phase are well known. There are two distinct kinds of excitations. First, SKϭϪK͚ ͟ ij . ͑21͒ ᮀ ᮀ there is a gapless spin-wave mode with linear dispersion. We will therefore consider the full action Apart from these, there are also topological vortex excita- tions. On moving along any circuit that encloses a vortex, the
SϭSsϩSKϩSB . ͑22͒ direction of the spin in the xy plane winds by an integer multiple of 2. This integer winding number—the What we have achieved so far is an approximate reformula- vorticity—is conserved, and may be used to label the spec- tion of spin models with XXZ symmetry. This reformulation trum of excited states. States with different total vorticity is extremely useful to explore the various possible allowed belong to different topological sectors and are not mixed by phases in such models. However, the approximations made the dynamics generated by the Hamiltonian. Note that in this in obtaining this reformulation are severe enough that it is z not easy to see which one of these allowed phases will be ordered phase we can no longer label states by their S quan- obtained in any particular microscopic model. tum number. Consider the possible phases when the parameter K is These familiar properties of the XY ordered phase must be contrasted with those of the quantum paramagnet. First very large. When Kϭϱ, the Z2 flux through each plaquette is constrained to be 1. We may then choose a gauge in which consider a conventional paramagnet ͑i.e., one with no frac- tionalization͒. Clearly in this phase Sz is conserved, and is a ijϭ1 on every link. In this limit therefore, the action re- duces to good quantum number to label the excitation spectrum. On the other hand, the vorticity loses its meaning in the para- magnetic phase, and is no longer a good quantum number. SϭϪ͚ Jijcos͑iϪ j͒. ͑23͒ ͗ij͘ This suggests that one may view the paramagnet as a phase in which the vortex excitations have themselves condensed. This simply describes a quantum XY model in two spatial Condensation of the vortices implies that the vorticity is no dimensions. Note that the Berry phase term simply vanishes longer a good quantum number ͑just like condensation of z when all the ijϭ1. There clearly are two possible spin implies that S is no longer a good quantum number͒. ii phases—an XY ordered phase in which ziϭe has con- Indeed, these observations may be formalized precisely by densed, and a paramagnetic phase in which the excitations means of a duality transformation which reformulates the created by zi are gapped. Note that these excitations in the system in terms of the vortex fields rather than the spins. In paramagnetic phase carry spin Szϭ1/2. Thus the spin has this dual formulation, the paramagnet is described as a vor- been fractionalized in this phase. tex condensate, and the XY-ordered phase as a vortex insu- Now consider moving away from the limit Kϭϱ by mak- lator ͑in which the vortices are gapped͒. The physical exci- ing K large but finite. For finite K, as can be seen from the tations of the paramagnet which carry the Sz quantum arguments advanced in Ref. 7, the XY ordered phase where number appear as dual flux tubes of the vortex condensate in the spinon field has condensed is indistinguishable from a this language. conventional XY ordered XXZ magnet. The paramagnetic How are we to view the fractionalized quantum paramag- phase in which the spinons are uncondensed and deconfined net in this dual language? As the phase in question is a para- survives for large but finite K. When K is finite, it becomes magnet, it is clear that the vorticity has no meaning, imply- clear that this phase has another distinct excitation which ing that the vortices must have condensed. As pointed out in carries the flux of the Z2 gauge field. This Z2 vortex— Ref. 13, we may view the fractionalized phase as a conden- dubbed the vison—does not carry any physical spin, and has sate of paired vortices. This has the immediate consequence an energy gap of order K for large K. It has the important of halving the dual flux tube, i.e., of fractionalizing Sz as property that when a spinon is taken around it, the wave required. Furthermore, note that the unpaired ͑and uncon- function of the system acquires a phase of . densed͒ single vortex is still an excitation in the system. Its Upon decreasing K, at some critical value, the vison gap vorticity is screened by the ͑double strength͒ vortex conden- goes to zero. For smaller K, the visons condense leading to sate as is required in the paramagnet. However, its parity is confinement of the spinons. The resulting phase is a conven- still a good quantum number. Thus the unpaired vortex, tional quantum paramagnet with gapped Szϭ1 excitations. though a legitimate excitation of the fractionalized paramag- In this phase, the Berry phase term becomes important and net, carries only a Z2 quantum number—it is clear that it is leads to a breaking of translational symmetry—the paramag- the vison excitation discussed previously.
155103-5 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
The discussion above provides a description of a fraction- The other terms of the action are as given before. alized quantum paramagnet in the context of spin models Following the discussion above, for large K, we expect to with XXZ symmetry. We now return to the full model which have a phase in which the holons and spinons are liberated includes coupling to the ‘‘conduction’’ electron degrees of from each other. In such a phase, the electron is fractional- Ϫ freedom. As above, we first replace the operator Sr in the ized. However, in contrast to the fractionalized phase that is Kondo coupling at each site by the boson operators bsr most popular in the context of the cuprates, here the spin of ir ϩ the electron has been liberated from its Fermi statistics. Are ϳe ͑and similarly for Sr ). The Kondo coupling term then becomes these two phases actually the same? We will address this issue in subsequent sections.
† † Though we have based our discussion on models with HKϭJK ͑bsrcr cr ϩH.c.͒ ͑24͒ ͚r ↓ ↑ XXZ symmetry, we expect the fractionalized quantum para- magnetic phase to exist even in systems with full SU͑2͒ spin symmetry. Indeed, in the context of frustrated quantum 2† † 8 ϭJK͚ ͑zr cr cr ϩH.c.͒. ͑25͒ Sp(n) spin models in the large-n limit, Read and Sachdev r ↓ ↑ have argued for the stability of fractionalized paramagnetic In going to the second equation, we have introduced the phases with properties similar to that discussed above. spinon operators zr defined in Eq. ͑7͒. The Kondo coupling can be further simplified by another change of variables, III. A HIERARCHY OF FRACTIONALIZED PHASES
r ϵzrcr , ͑26͒ In Sec. II, we primarily discussed fractionalized phases in ↑ ↑ which the electron splinters into a spin-1/2 neutral boson and † r ϵzr cr . ͑27͒ a charged spinless fermion. For future convenience, we will ↓ ↓ refer to this as the CFSB ͑charged fermion, spinful boson͒ We will call the operators the holon operators. In terms of phase. This is to be contrasted with the fractionalized phases the holons, the Kondo coupling becomes which are popular in the context of cuprate physics in which the electron splinters into a spin-1/2 neutral fermion and a † charged spinless boson ͑see also Secs. IV and V͒. We will HKϭJK͚ ͑r r ϩH.c.͒. ͑28͒ r ↑ ↓ refer to this as the CBSF phase. In both cases, there is, in Note that the holons are actually spinless charge e fields addition, a Z2 vortex excitation ͑the vison͒ such that taking despite the presence of the label , . This is obvious from either the holon or spinon around it produces a phase change their definition in terms of the spinon↑ ↓ and electron operators of . above: the holon operators do not transform under spin rota- Having accepted the possibility of quantum number frac- tions about the z axis. Explicitly, the Kondo term mixes up and down holons so that their label , is changed by the dynamics. Therefore their spin label↑ has↓ no great signifi- cance, and they are correctly viewed as spinless fermions. We may use the following physical picture: the Kondo spins screen the spin of the conduction electrons. Under these changes of variables, the electron hopping term becomes
H ϭϪ t z†z † ϩ† ͒ϩH.c. . t ͚ rrЈ͓ r rЈ͑ rЈ rЈ r rЈ ͔ ͗rrЈ͘ ↑ ↑ ↓ ↓ ͑29͒ We now make approximations very similar to those used above for the exchange part of the Hamiltonian. They allow us to reformulate the system in terms of the spinons, holons, and a Z2 gauge field. Some of the details are outlined in the Appendix. The resulting action can essentially be guessed on symmetry grounds, and takes the form
SϭScϩSsϩSBϩSK , ͑30͒
c † † ScϭϪ͚ ijtij͑i j ϩi j ϩc.c.͒ ͗ij͘ ↑ ↑ ↓ ↓
† ϩJK͚ ͑i j ϩc.c.͒. ͑31͒ i ↑ ↓ FIG. 1. Hierarchy of fractionalized phases.
155103-6 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103 tionalization, one can imagine a wide variety of possible IV. FRACTIONALIZATION OF ELECTRON QUANTUM phases apart from the two mentioned above. In particular, NUMBERS WITH p-WAVE PAIRING one may consider exotic possibilities where the fractions of A. Order parameters and symmetries the electron in any given fractionalized phase themselves fractionalize. Such phases may be considered to have a Spin-triplet superconductors and their rich order- higher level of fractionalization. To see how these may be parameter structure offer the prospect of various exotic described in the same kind of formulation as discussed in phases. Since they break both charge and spin symmetries, this section, consider the following action: triplet superconductors exhibit features of both singlet super- conductors and of spin models. In particular, we can envision the restoration of the the UC(1) charge symmetry by quan- SϭS f ϩScϩSsϩS , tum fluctuations, thereby resulting in a spin-triplet insulating state. Alternatively, the spin symmetry ͑we will make the simplifying assumption that the system has only an easy- n ¯ ˜⌬ S f ϭϪ͚ ijij͕tijf i␣ f j␣ϩ t aij͓ f i f j Ϫ͑ ͔͒ϩc.c.͖ plane US(1) spin symmetry͒ can be restored, resulting in a ij,␣ ↑ ↓ ↑→↓ spin-singlet superconducting state. Finally, both symmetries can be restored, leading to a singlet insulating state. We be- ¯ Ϫ͚ f i␣ f i␣ , lieve that the gapped, symmetry-restored states will not be i␣ very sensitive to the precise symmetry of the spin sector, so we believe that our results apply to systems with full SU͑2͒ spin symmetry as well. In particular, when the symmetry is c increased ͑while keeping the size of the representation fixed͒, ScϭϪ͚ tijij͑bci*bcjϩc.c.͒, ij fluctuations are enhanced, and a system is more likely to be in a disordered state. In order for these symmetries to be restored separately, it will be necessary, as we discuss below, s for a type of topological ordering to occur. This topological SsϭϪ͚ tijij͑zi*z jϩc.c.͒, ij ordering is essentially spin-charge separation of the charge 2e, spin-triplet Cooper pairs. Depending on the way in which the symmetries are restored, it is possible for further topological ordering to take place, in which case the quan- SϭϪK͟ ijϪK͟ ijϪK͟ ijij . ͑32͒ ᮀ ᮀ ᮀ tum disordered states may support excitations with exotic quantum numbers. In such states, the spin and/or charge of the quasiparticles is screened by the Goldstone modes Here bc is a charge e spinless boson and z is a spin-1/2 chargeless boson. The f field represents a spinless, neutral ͑which are themselves separated from each other by the fermion ͑the spin index is just a label with no special signifi- higher-level topological ordering͒. As we describe in this pa- per, there are no fewer than nine phase which can result in cance͒. The ij and ij are two independent Z2 gauge fields. this way. The physical electron ci␣ϭbcizi f i␣ . Clearly if the field is To be concrete, let us consider the following p-wave su- confining, the f ␣ and bc get confined to form a Fermionic holon—we then recover the action discussed earlier in this perconducting state of electrons on a square lattice: section. On the other hand, if the field is confining, the Fermi statistics gets glued to the spinon (z )—the resulting z i ⌬ ϭ⌬ ei͑cos ϩi sin ␦ ͒sin k a. ͑33͒ theory is essentially that introduced in Ref. 7 in the context ␣ 0 ␣ ␣ y of cuprate physics and involves Bosonic holons and Fermi- onic spinons coupled to a Z2 gauge field. If both gauge fields This is the most general unitary triplet state in two and are deconfining, however, we have an exotic phase dimensions20 if we assume that there is only the U͑1͒ spin in which the fields bc ,z, f are all liberated. This phase will symmetry of rotations about the z axis, rather than the full also have two distinct vison excitations corresponding to the SU͑2͒. In Eq. ͑33͒, only ⌬ and ⌬ are nonzero. The lower ↑↑ ↓↓ fluxes of the two Z2 gauge fields. We may view this phase as symmetry could be the result of spin-orbit coupling. The a higher-level fractionalized phase as compared to the one symmetry-breaking pattern associated with this order param- discussed in Ref. 7 or that discussed earlier in this section. eter is: U (1)ϫU (1)ϫD Z ϫZ ϫD . The U (1) C S 4→ 2 2 2 C The connection between various fractionalized phases is charge symmetry is broken to Z2 by the condensation of a shown in Fig. 1. We use symbols ba and f a to label bosons charge 2e order parameter. The US(1) spin-rotational sym- and fermions that carry quantum numbers aϭn,c,s,cs ͑neu- metry is completely broken. The square lattice point group, tral, charge, spin, charge, and spin͒ and show the existence of D4, is broken to D2 by the orbital symmetry of ⌬. Finally, appropriate Z2 vortices in each phase ͑for more details see there is an additional Z2 since the order parameter is left Sec. V͒. invariant by ϩ, ϩ. As we discuss later, this → → i i In the sections which follow, we will show how an effec- can be understood as a Z2 gauge symmetry. From e and e tive action such as that of Eq. ͑32͒ can arise in the context of we can construct the following Z2-invariant order parameters p-wave superconducting systems and systems which feature whose presence or absence characterizes the phases which interplay between magnetism and superconductivity. we consider. In the absence of the triplet p-wave supercon-
155103-7 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
FIG. 2. -disclination–hc/4e vortex composite. ducting order parameter ͑33͒, we can characterize states by the charge-4e order parameter,
⌬4eϭ͑ei͒2, ͑34͒ FIG. 3. Order parameter for a sin ky p-wave superconductor. and the spin nematic order parameter, Gapless excitations exist at kជ Fϭ(ϮkF,0). Qϭcos 2. ͑35͒ with constant 0. Alternatively, we can form merons, which are trivial in the charge sector but not the spin sector, These order parameters define the following quantum- disordered states of triplet p-wave superconductors. i0 z ⌬␣͑r,͒ϭ⌬͑r͒e ͑cos ␣ϩi sin ␦␣͒sin kya • Charge-4e singlet superconductor: ⌬4e<0, Qϭ0. ͑39͒ 4e • Charge-4e nematic superconductor: ⌬ <0, Q<0. with constant . Finally, there are various composites 4e 0 • Spin-nematic insulator: ⌬ ϭ0, Q<0. formed from the above. A composite formed by nhc/2e • Spin-singlet insulator: ⌬4eϭ0, Qϭ0. vortices and m merons takes the form
⌬ ͑r ϱ,͒ϭ⌬ ein͑cos mz ϩi sin m␦ ͒sin k a. B. Topological defects ␣ → 0 ␣ ␣ y The quantum-disordered and topologically ordered states ͑40͒ which we will consider can be understood in terms of the condensation or suppression of various topological excita- If flux hc/4e vortex- disclination composites condense, tions. The most basic and fundamental topological excitation then UC(1) and US(1) are restored. The system will be in a is a composite formed of a flux hc/4e vortex together with a singlet insulating state and all excitations will have conven- disclination.34–36 tional quantum numbers. If, on the other hand, hc/4e vortex- Along a circuit about such an excitation, both and disclination composites are gapped and only complexes wind by so that any Z2-invariant combination is single- consisting of multiples of hc/4e vortex- disclinations ͑e.g., valued. If such an excitation is at the origin, and r, are nhc/2e-m meron composites͒ are condensed, then quantum polar coordinates in the plane, then the order parameter is of number separation is possible. If complexes consisting of a the form multiple of four hc/4e vortex- disclinations condense, then we will have the various versions of quantum number sepa- ration summarized in Figs. 3 and 4. ⌬ ͑r,͒ϭ⌬͑r͒ei⑀1/2 cos z ϩi⑀ sin ␦ sin k a, ␣ ͩ 2 ␣ 2 2 ␣ͪ y ͑36͒ where ⌬(0)ϭ0 and ⌬(ϱ)ϭ⌬0. The flux is into or out of the plane, respectively, for ⑀1ϭϮ1; the spins wind clockwise or counterclockwise, respectively, for ⑀2ϭϮ1. It is instructive to write this as
i⑀ϩ ⌬ ͑r,͒ϭ⌬͑r͒e sin kya, ↑↑ ͑37͒ i⑀Ϫ ⌬ ͑r,͒ϭϪ⌬͑r͒e sin kya, ↓↓ where ⑀Ϯϭ(⑀1Ϯ⑀2)/2. Hence, -disclination–hc/4e vortex composites are vortices in ⌬ or ⌬ alone ͑see Fig. 2͒. These excitations can be combined↑↑ ↓↓ to form an hc/2e vor- tex which is nontrivial in the charge sector but trivial in the spin sector,
i z ⌬␣͑r,͒ϭ⌬͑r͒e ͑cos 0␣ϩi sin 0␦␣͒sin kya ͑38͒ FIG. 4. Phases of quantum-disordered p-wave superconductor.
155103-8 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
c In action ͑42͒ we have included the electromagnetic field A and spin vector potential A which couple to the conserved electric and Sz currents. When hc/4e vortex- disclination composites are gapped, the Z2 symmetry ϩ, ϩ plays no role, and the other terms in Eq.→ ͑41͒ may→ be written in the form
1 ϪAc ͒2 ͑44͒ ץS ϭ d2xd͑ c 2 c ͵ and
1 ϪA ͒2. ͑45͒ ץS ϭ d2xd͑ 2 ͵
The conserved electric and Sz currents are given by
␦Stot c,ϭ j c, . ͑46͒ ␦A FIG. 5. Phases of quantum-disordered p-wave superconductor. Conservation of charge and the z component of spin require Note that the phase in which hc/2e vortices and merons are con- c, j ϭ0. ͑47͒ץ densed may be described as having hc/4e-vortex- composites condensed. The interactions between the Goldstone fields and the C. Quantum number separation quasiparticles are highly nonlinear in Eq. ͑42͒. This interac- tion can be made more tractable, following Ref. 28, if we The effective action of a p-wave superconductor may be define new fermion fields : written in the form z z ϭei /2ei /2. ͑48͒ StotϭS f ϩScϩS , ͑41͒ With this change of variables, we have defined a neutral, where S f is the action for the Fermionic quasiparticles and spinless fermion , which is governed by the action their interactions with the Goldstone modes, and Sc and S are the actions for the charge and spin Goldstone modes. ͔͒ ץϪv xx͑i ץϪv zi ץ͓†S ϭ d2xd Depending on the topology of the Fermi surface, the low- f ͵ ͩ F x ⌬ y energy spectrum of a p-wave superconductor may include 1 ϩ2v Ac͔ ץ Ϫ2AczϪv ץgapless Fermionic quasiparticles. Let us assume that the to- ϩ †͓z pology is such that the gap has nodes on the Fermi surface. 2 F x F x Focusing on the nodes, as shown in Fig. 5. We linearize the 1 . ϩ2v zzA͔ ץϪ2AzϪv zz ץaction ϩ †͓z 2 F x F x ͪ ͑49͒ ϭ 2 † Ϫ c zϪ z ϩ cϪ x vFAx A zץ A vF iץ͓ S f ͵ d xd The couplings between the Goldstone modes and the qua- s is x y ϩvFAx zzϪv⌬ e ͑cos ␣ϩi sin ␣͒ siparticles are now either trilinear or biquadratic, y͔͒. ͑42͒ 1ץϫ͑i Ϫ2Ac͒ ץϪ2Ac͒ϩJc͑ ץS ϭS0ϩ d2xd͓Jc͑ f f 2 ͵ 0 x x x sϭϮ and has a particle-hole index, acted on by Pauli xϪ2Ax ͔͒ ͑50͒ץ͑ Ϫ2A ͒ϩJxץ͑ matrices ជ; and a spin index, acted on by Pauli matrices ជ , ϩJ0 with c ជ ជ kFϩk 11 ↑ c † z c † † J0ϭ , JxϭϪvF cϪkជ Ϫkជ 21 F ↓ ͑51͒ ͑kជ ͒ϭ ϭ . ͑43͒ † z † z z a␣ J ϭ , J ϭϪvF . 12 ckជ ϩkជ 0 x F ↓ † ͫ 22ͬ The price that must be paid is that the change of variables ϪcϪជ Ϫជ ͫ kF k ͬ ͑ ͒ ↑ 48 is not single valued about a topological defect. In par-
155103-9 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
z † ticular, the charge part, exp(i /2), is double valued under where ⌽m is a meron creation operator. Analogous topologi- transport about a flux hc/2e vortex since winds by 2, cal objects in the spin sector have been discussed in the while the spin part, exp(iz/2), is double valued under context of quantum Hall systems38–40 and quantum transport about a meron since winds by 2. antiferromagnets.17,41–43 As we will see below, s are weakly coupled quasiparti- Actions ͑56͒ and ͑58͒ need to be supplemented by Chern- cles in those quantum disordered phases in which flux hc/2e Simons gauge fields which enforce the minus sign which is vortices and merons are gapped. acquired when a encircles a flux hc/2e vortex or a meron.28,44 With these additions, we obtain the following D. Defect condensation dual action: c s1 s2 Defect condensation is now implemented with dual repre- SDualϭSGL͑⌽v ,aϪa ͒ϩSGL͑⌽m ,aϪa ͒ sentations for the order parameters.28,13,29,37 In the dual de- 37 1 1 Ac Ϫ Jc ץ ⑀ scription of the XY model, the ordering field is replaced ϩ d d2x ͑ f c ͒2ϩac by a gauge field which parametrizes the total current, to- ͵ ͫ2 ͩ ͪ c c gether with a vortex field which accounts for the singularities 1 as1ϩ␣1 Jc ϩ d d2x ͑ f ͒2 ץ ⑀ of . ϩ2␣1 ͬ ͵ ͫ2 We use the conservation of charge to define the dual gauge field, 1 , as2ϩ␣2 J ץ ⑀ AϪ J ϩ2␣2 ץ ⑀ ϩa ϪAc ͒ϩJc , ͑52͒ ͩ ͪ ͬ ץ͑ ac ϭJtot cϭ ץ ⑀ c c with J from Eq. ͑51͒, and introduce the vortex current, ͑59͒ where ␣1,2 and as1,2 are the gauge fields that perform the flux 1 v attachement and enforce the minus sign. , ͑53͒ץץjϭ ⑀ 2 With this action in hand, we can now address the quantum which is not vanishing for a multivalued . With the last two disordered phases and quantum number separation. In es- equations we can relate the vortex current to the dual gauge sence, there are three quantum numbers: charge, spin, and field ac and quasiparticle current Jc , electron number modulo 2. These can separate in a variety of patterns. v Ϫ1 c c Ϫ1 c aϩAϪc J͔. ͑54͒␣ץ␣͓c ⑀ץjϭ⑀ • If ͗⌽v͘<0, flux hc/2e vortices condense. The Meissner Now a dual action for the charged degrees of freedom is effect associated with this condensate imposes c s1 easily constructed by requiring that its equations of motion aϩa ϭ0. ͑60͒ reproduce Eq. ͑54͒, c totc aϭJ0 (␣, ϭx,y) is the charge␣ץ␣⑀ Recalling that s1ϭ c a J0 is the quasiparticle density, we␣ץ␣⑀ density and 1 Sc ϭS ͑⌽ ,ac ͒ϩ d d2x ͑ f c ͒2 conclude that in this phase charge is attached to the s. Dual GL v ͵ ͫ2 c • If ͗⌽m͘<0, merons condense, and the Meissner effect as- sociated with this condensate imposes c c 1 c A Ϫ J , ͑55͒ s2ץϩa ⑀ ͩ ͪͬ aϩa ϭ0. ͑61͒ c tot aϭJ0 is the local spin density and␣ץ␣⑀ As where s2 a ϭJ0 we find that spin is attached to the ’s. All␣ץ␣⑀ the fermions carry spin. d Ϫia ͒⌽͉2ϩV͑⌽͒ ϭ ϭ ץ͉͑ S ͓⌽,a ͔ϭ d d2x GL ͵ ͩ 2 ͪ • If ͗⌽m͘ 0, ͗⌽v͘ 0, but ͗⌽v⌽m͘<0 hc/2e vortex, ͑56͒ meron composites condense. The Meissner effect associ- ated with this condensate imposes c c c † a . The field ⌽ may be thought of as a ץa Ϫ ץand f ϭ v ac ϩa ϭ0. ͑62͒ vortex creation field. The vortex current is given by In other words, spin and charge are confined, but the fer-
d 1 mion carries neither since does not acquire any phase Ϫac ⌽ ϩH.c. . ͑57͒ ץ †j v ϭ ⌽ 2 ͫ vͩ i ͪ v ͬ upon encircling this composite object, as evinced by the fact that ⌽v⌽m is not coupled to statistical gauge fields. 2 An identical construction is now used for with c re- • The condensation of other composites, such as ⌽m ͑i.e., c 2 placed by and J by J : skyrmions͒, ⌽v , etc., does not cause the confinement of any quantum numbers. 1 S ϭS ͑⌽ ,a ͒ϩ d d2x ͑ f ͒2 Dual GL m ͵ ͫ2 E. Exotic phases The order-parameter classification discussed after Eq. ͑35͒ 1 is incomplete; those states can occur in several varieties, A Ϫ J , ͑58͒ץϩa⑀ ͩ c ͪͬ classified by the allowed quantum numbers.45
155103-10 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
Charge-4e singlet superconductors, ⌬4e<0, Qϭ0: state, we find the exotic phases discussed in the previous ͑1A͒ If ͗⌽m͘<0, then the Fermionic excitations carry section. These have a simple description as the various de- spin 1/2. confining phases of the Z2ϫZ2 gauge theory. Readers who 2 ͑1B͒ However, if ͗⌽m͘ϭ0 but ͗(⌽m) ͘<0 then the ’s are uninterested in the technical details of our derivation may are spinless. Note that the charge quantum number of the skip directly to Eqs. ͑105͒, ͑106͒, and the subsequent discus- Fermionic excitation is not really well defined since U͑1͒ is sion. broken in the superconducting state; stated differently, the fermion can always exchange charge with the condensate. A. General formalism Spin-triplet insulator, ⌬4eϭ0: Q<0: We consider the following Hamiltonian that describes the ͑2A͒ If ͗⌽v͘<0, the ’s carry charge e. 2 equal spin pairing state of a p-wave superconductor: ͑2B͒ If ͗⌽v͘ϭ0 but ͗(⌽v) ͘<0, then the ’s are neutral. As in the previous case, the spin quantum number of the ’s HϭHtϩHuϩHvϩH⌬ ͑63͒ is not well defined. Spin-singlet insulator, ⌬4eϭ0: Qϭ0: with ͑3A͒ If ͗⌽v͘<0, ͗⌽m͘<0, then the ’s carry spin 1/2 and charge e: CSF phase. H ϭϪt c† c ϩH.c., 2 t ͚ r␣ rЈ␣ ͑3B͒ If ͗⌽m͘ϭ0 but ͗(⌽m) ͘<0 while ͗⌽v͘<0, then rrЈ,␣ the ’s are charge e, spinless Fermionic excitations: CFSB phase. 2 2 Huϭu͚ ͑NrϪN0͒ , ͑3C͒ If ͗⌽v͘ϭ0 but ͗(⌽v) ͘<0 while ͗⌽m͘<0, then the r ’s are neutral, spin-1/2 Fermionic excitations: CBSF phase. ͑3D͒ If ͗⌽ ⌽ ͘<0, then the ’s are neutral, spinless v m 2 Fermionic excitations, but spin and charge are confined into Hvϭv ͑M r͒ , ͚r a bosonic spin-1/2, charge e excitation: CSBNF phase. 2 ͑3E͒ Finally, if ͗⌽v͘ϭ0 but ͗(⌽v) ͘<0 and ͗⌽m͘ϭ0 2 but ͗(⌽ ) ͘<0, then the ’s are neutral, spinless Fermionic H ϭ ⌬↑↑ c c ϩ⌬↓↓ c c ϩH.c., ͑64͒ m ⌬ ͚ ͓ rrЈ r rЈ rrЈ r rЈ ͔ excitations. Bosonic charge e excitations, ei/2, and Bosonic rrЈ ↑ ↑ ↓ ↓ spin-1/2 excitations, ei/2 are also liberated: CBSBNF phase. where ⌬↑↑ and ⌬↓↓ represent the order-parameter fields for To summarize, we have the following phases with exotic rrЈ rrЈ the Cooper pairs with spin up-up and down-down pairs, re- quantum numbers: spectively. Here ␣ϭ , is the spin index. The term propor- •A charge-4e singlet superconductor with spinless Fermi- tional to u represents↑ the↓ on-site Coulomb repulsion. N is onic excitations. r the total number operator of electrons at the site r, N is the •A spin-triplet insulator with neutral Fermionic excita- 0 average electron number per site. M is the z component of tions. r the total spin operator. At equilibrium, ⌬ ϭ ⌬ . Note •Spin-singlet insulators with ͑i͒ charge e spinless fermions ͉ ↑↑͉ ͉ ↓↓͉ that there are two independent phases associated with ⌬ and spin-1/2 neutral bosons; ͑ii͒ spin-1/2 neutral fermions ↑↑ and ⌬ . We can rewrite H as and spinless charge e bosons; ͑iii͒ neutral spinless fermions, ↓↓ ⌬ Bosonic charge e spinless excitations, and Bosonic spin-1/2 ir ir neutral excitations; or ͑iv͒ neutral spinless Fermionic excita- H⌬ϭ⌬͚ arrЈ͓e ↑cr crЈ ϩe ↓cr crЈ ͔ϩH.c., tions and Bosonic charge e spin-1/2 excitations. rrЈ ↑ ↑ ↓ ↓ These result are summarized in the following diagrams ͑65͒ that describe various phases that can result from quantum where ⌬ϭ͉⌬↑↑͉ϭ͉⌬↓↓͉ and arrЈ is the form factor that gives disordering a p-wave superconductor. rise to the particular p-wave symmetry. The scenario proposed in this section for quantum number The fields r and r are canonically conjugate to the separation in p-wave superconductors may apply to other Cooper pair number↑ operators↓ of up-up and down-down systems, provided that they acquire nontrivial topological or- Cooper pairs, nr and nr : der in the spin and charge sectors, or in the language of this ↑ ↓ section when they have sufficiently strong quantum fluctua- ͓r ,nrЈ ͔ϭi␦rrЈ , ͓r ,nrЈ ͔ϭi␦rrЈ . ͑66͒ tions of spin and charge degrees of freedom simultaneously. ↑ ↑ ↓ ↓ The conserved charge densities for the electrons with spin In Appendix B we show that quantum disordered d-wave and are given by ↑ superconductor with easy-plane antiferromagnetic fluctua- ↓ tions may be treated in the same way as we treated p-wave Nr ϭ2nr ϩr , superconductors in this section. ↑ ↑ ↑
Nr ϭ2nr ϩr , ͑67͒ ↓ ↓ ↓ V. Z2ÃZ2 LATTICE GAUGE THEORY † where r␣ϭcr␣cr␣ is the quasiparticle number, which is not In this section, we derive a Z2ϫZ2 gauge theory repre- equal to the electron number. It is useful to remind the read- sentation of a model which gives rise to local p-wave super- ers that the Hamiltonian ͑64͒ does not conserve the quasipar- conducting fluctuations. In addition to the superconducting ticle number, since it contains terms that annihilate a pair of
155103-11 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103 them and create a Cooper pair. Only the total number of † † † † † † electrons of a given spin, given by Eq. ͑67͒, is conserved. HtϭϪt ͑br br zr zr f r f r ϩbr br zrzr f r f r ͒ϩH.c., ͚ Ј Ј ↑ Ј↑ Ј Ј ↓ Ј↓ This may be formulated as a conservation of the total charge rrЈ and z component of the total spin, H ϭ⌬ a b†b z†z f f ϩb†b z z† f f ͒ϩH.c. ⌬ ͚ rrЈ͑ r rЈ r rЈ r rЈ r rЈ r rЈ r rЈ NrϭNr ϩNr , M rϭNr ϪNr . ͑68͒ rrЈ ↑ ↑ ↓ ↓ ↑ ↓ ↑ ↓ ͑79͒ Let us define boson operators br␣ which carry charge e and spin ␣ϭ , : The Hamiltonian is invariant under the following local trans- ↑ ↓ formations: † ␣ ir␣ /2 ir␣ br␣ϭtr e ϭe , ͑69͒ ͑i͒ Z : b Ϫb ; f Ϫ f ; 2 r→ r r␣→ r␣ ␣ ͑ii͒ Z2 : zr Ϫzr ; f r␣ Ϫ f r␣ ; where tr ϭϮ1 are Ising variables and r␣ are defined in the → → † † ͑iii͒ Z2˜ : zr Ϫzr ; br␣ Ϫbr␣ . interval zero to 2. Note that the squares of br and br → → ↑ ↓ create the spin up-up and down-down Cooper pairs via the Only two of these transformations are independent, any one following relation: of them can be represented as a product of the other two.
† 2 ir␣ Together they form Z2ϫZ2 gauge symmetry, that has three ͑br␣͒ ϭe . ͑70͒ Z2 subgroups as reflected in three possible transformations One can also see that the canonical conjugates of r␣ are the above. These subgroups are distinct, but not independent. total densities of electrons with spin and . They satisfy Z2ϫZ2 local gauge symmetry is a consequence of the redun- ↑ ↓ the following commutation relations: dancy in the enlarged Hilbert space of f r␣ , br , and zr . There is a further redundancy in our description in terms of ͓r␣ ,NrЈ␣͔ϭi␦rrЈ . ͑71͒ br and zr because br ibr , zr Ϫizr also leaves all physi- cal quantities invariant.→ This identi→ fication allows for the ex- Similarly, the following commutation relations are also sat- istence of flux -disclination-hc/4e vortex composites isfied: which we discussed in Sec. IV B. As before, we assume that these topological defects are gapped so that we can safely ͓rc ,NrЈ͔ϭi␦rrЈ , ͓rs ,M rЈ͔ϭi␦rrЈ , ͑72͒ ignore this identification and take rc and rs as defined where rcϭ(r ϩr )/2 and rsϭ(r Ϫr )/2. At this from ͓0,2). ↑ ↓ ↑ ↓ † In order to get the correct Hilbert space of the electrons, stage, it is useful to define the fermion operators, f r␣ , as follows: we have to impose two constraints at each site.
† † † Nrϩr ϩr ϭeven number, cr␣ϭbr␣ f r␣ . ͑73͒ ↑ ↓ ͑80͒ † Note that f r␣ creates spinless neutral fermions due to the fact M rϩr Ϫr ϭeven number. ↑ ↓ that b† carries both the charge and spin of the electrons. r␣ These can be written as It is also useful to define rc and rs as follows: ͑Ϫ1͒Nrϩr ϩr ϭ1, ͑Ϫ1͒M rϩr Ϫr ϭ1. ͑81͒ ir irc irs ir irc Ϫirs ↑ ↓ ↑ ↓ e ↑ϭe e , e ↓ϭe e . ͑74͒ The constraints can be implemented in the path integral rep- Note that there is a Z2 symmetry associated with these defi- resentation of the partition function using the following pro- nitions of phase variables; rc rcϩ and rs rsϩ jection operators: ir ir → → do not change e ↑ and e ↓. Now we define boson opera- † † tors br and zr as cϭ͟ rc , sϭ͟ rs , ͑82͒ P r P P r P † irc/2 irc † irs/2 irs br ϭtre ϭe , zr ϭsre ϭe . ͑75͒ with Here trϭϮ1 and srϭϮ1 are Ising variables. Note that these operators satisfy the following identities: 1 Nrϩr ϩr rcϭ ͓1ϩ͑Ϫ1͒ ↑ ↓͔ † 2 irc † 2 irs P 2 ͑br ͒ ϭe , ͑zr ͒ ϭe . ͑76͒ 1 † † i(/2)(1Ϫ )(N ϩ ϩ ) Note also that br and br can be rewritten as ϭ e r r r r , ↑ ↓ ͚ ↑ ↓ 2 rϭϮ1 † † † † † br ϭbr zr , br ϭbr zr . ͑77͒ ↑ ↓ 1 ϭ ͓1ϩ͑Ϫ1͒M rϩr Ϫr ͔ Now the total Hamiltonian can be written as Prs 2 ↑ ↓
HϭHtϩHuϩHvϩH⌬ ͑78͒ 1 i(/2)(1Ϫr)(M rϩr Ϫr ) ϭ ͚ e ↑ ↓ . ͑83͒ with 2 rϭϮ1
155103-12 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
Using the projection operators, the partition function can 1 2 ¯ be written as St,⌬ϭ ⑀ ͚ ͕͓2͉rrЈ͉ ϪrrЈ͑br*␣brЈ␣ϩtfr␣ f rЈ␣ 4 rrЈ,␣ ϪH ZϭTr͓e c s͔. ͑84͒ 2 * P P ϩarrЈ⌬ f r␣ f rЈ␣͔͒ϩ͓2͉rrЈ͉ ϪrrЈ͑br␣brЈ␣ A Euclidean path-integral representation can be obtained by Ϫtf¯ f Ϫa ⌬ f f ͔͒ϩc.c.͖. ͑91͒ splitting the exponential into M number of time slices, r␣ rЈ␣ rrЈ r␣ rЈ␣ Rearranging terms, we get Ϫ⑀H M ZϭTr͓͑e c s͒ ͔, ͑85͒ P P 1 2 2 where ⑀ϭ/M. Now the partition function can be written as St,⌬ϭ ⑀ ͚ ͓2͉rrЈ͉ ϩ2͉rrЈ͉ Ϫ͑rrЈϩrrЈ͒br*␣brЈ␣ 4 rrЈ,␣ ¯ ¯ Ϫt͑rr Ϫrr ͒f r␣ f r ␣Ϫarr ⌬͑rr Ϫrr ͒f r␣ f r ␣ Zϭ dfi␣dfi␣dicdis Ј Ј Ј Ј Ј Ј Ј ͵ ͟i␣ ϩc.c.͔. ͑92͒ ϱ ϱ Rewriting this in terms of b and z , we get ϫ eϪS. ͑86͒ r r N ͚ϭϪϱ M ͚ϭϪϱ ͚ϭϮ1 ͚ϭϮ1 i i i i 1 2 2 Here iϭ(r,) runs over the 2ϩ1-dimensional space-time lat- St,⌬ϭ ⑀͚ ͚ ͓2͉rrЈ͉ ϩ2͉rrЈ͉ 4 rrЈ ͭ ␣ tice with ϭ1,2, . . . ,M time slices. The action S has the following form: ¯ Ϫt͑rrЈϪrrЈ͒f r␣ f rЈ␣ϪarrЈ⌬͑rrЈϪrrЈ͒f r␣ f rЈ␣͔ M † † † † f Ϫ ϩ ͒ b b z z ϩb b z z ͒ϩc.c. . ϭ ϩ cϩ sϩ ¯ ͑ rrЈ rrЈ ͑ r rЈ r rЈ r rЈ r rЈ S S S S ⑀ ͚ H͑N ,M ,c ,s , f ␣ , f ␣͒ ͮ ϭ1 ͑87͒ ͑93͒ with In order to decouple br from zr , another Hubbard- Stratanovich transformation is necessary. Using similar pro- M cedures, the term f ¯ Sϭ ͚ ͚ ͓ f ␣͑ϩ1ϩ1 f ϩ1,␣Ϫ f ␣͔͒, r,ϭ1 ␣ 1 Ϫ ⑀ ͓͑ ϩ ͒͑b†b z†z ϩb†b z z† ͒ϩc.c.͔ 4 ͚ rrЈ rrЈ r rЈ r rЈ r rЈ r rЈ M rrЈ ͑94͒ c S ϭ ͚ N cϪϪ1,cϩ ͑1Ϫ͒ , ͑88͒ r,ϭ1 ͩ 2 ͪ can be decoupled as
M 1 2 2 2 s Ϫ ⑀͚ ͑rrЈϩrrЈ͓͒2͉rrЈ͉ ϩ2͉rrЈ͉ ϩ2͉prrЈ͉ S ϭ ͚ M sϪϪ1,sϩ ͑1Ϫ͒ . 16 r,ϭ1 ͩ 2 ͪ rrЈ ϩ2͉q ͉2Ϫ͑ ϩ ͒z†z Ϫ͑p ϩq ͒z z† Here the spatial index r is suppressed for clarity. The Ising rrЈ rrЈ rrЈ r rЈ rrЈ rrЈ r rЈ variables and are defined on the links connecting ad- † Ϫ͑prr Ϫqrr ϩrrϪrr ͒b br ϩc.c.͔, ͑95͒ jacent time slices and can be regarded as the time component Ј Ј Ј r Ј of the Ising gauge fields. We now make a saddle-point approximation and keep the The sum of Ht and H⌬ can be decoupled using the Ising fluctuations around this saddle point. The natural choices are Hubbard-Stratanovich fields rrЈ and rrЈ ,
rrЈϪrrЈϭrrЈrrЈ f , Ϫ⑀(HtϩH⌬) * * ϪSt,⌬ e ϭ drrЈdrr drrЈdrr e . ͵ ͟ ͟ Ј Ј rrЈ ͑rrЈϩrrЈ͒͑rrЈϪrrЈϩprrЈϪqrrЈ͒ϭrrЈc , ͑96͒ ͑89͒ * * ͑rrЈϩrrЈ͒͑rrЈϩrrЈϩprr ϩqrr ͒ϭrrЈs , Using the expressions for Ht and H⌬ , Ј Ј where rrЈϭϮ1 and rrЈϭϮ1 are Ising fluctuations. We drop all of the constant terms and define the following vari- H ϭϪt ͑b† b f † f ϩH.c.͒, t ͚ r␣ rЈ␣ r␣ rЈ␣ ables: rrЈ,␣ ͑90͒ 1 1 1 1 † t f ϭ t f , tcϭ c , tsϭ s , t⌬ϭ ⌬ f H⌬ϭ⌬ ͚ arrЈ͑br␣brЈ␣ f r␣ f rЈ␣ϩH.c.͒, 4 16 16 4 rrЈ,␣ ͑97͒ we have to obtain
155103-13 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
eff ¯ St,⌬ϭϪ⑀͚ ͚ ͓rrЈrrЈ͑t f f r␣ f rЈ␣ϩt⌬arrЈf r␣ f rЈ␣͒ SBϭϪiN0 ͚ 2lijϪ ͑1Ϫij͒ . ͑103͒ ␣ ˆ ͩ 2 ͪ rrЈ i, jϭiϪ † † ϩtcrrЈbr brЈϩtsrrЈzr zrЈϩc.c.͔. ͑98͒ Here lij is defined as c Combining all the results, the approximate full partition ⌽ij 1 l ϭInt ϩ ͑104͒ function can be written as ij ͫ 2 2ͬ c ˜ ¯ with ⌽ijϭicϪ jcϩ(/2)(1Ϫij) is the gauge invariant Zϭ dfi␣dfi␣dicdis ͵ ͟i␣ phase difference across the temporal link. Int denotes the integer part. One can see that the Berry phase term for is ϱ ϱ ij ϪS absent. This is due to the fact that we have equal amplitudes ϫ ͚ ͚ ͟ ͚ ͚ e , ͑99͒ for up-up and down-down pairing in the equal spin pairing N ϭϪϱ M ϭϪϱ ͗ij͘ ϭϮ1 ϭϮ1 i i ij ij state, analogous to particle-hole symmetry in the charge sec- where ij and ij are Z2 gauge fields living on the nearest tor. neighbor links of the space-time lattice. The total action S is Gathering these terms, the final form of the action is given given by by
f c s ϭ ϩ ϩ ϩ ϩ ͑ ͒ SϭSϩS ϩS ϩS⌬ϩS0ϩSuϩSv ͑100͒ S S f Sc Ss SB Sg 105 with with
f f ¯ ˜⌬ ¯ S ϭϪi ͓¯f ͑ f Ϫ f ͔͒, S f ϭϪ ijij͓tijf i␣ f j␣ϩ t aijf i␣ f j␣ϩc.c.͔Ϫ f i␣ f i␣ , ͚ ͚ i␣ ij ij j␣ i␣ ij͚,␣ ͚i␣ i, jϭiϩˆ ␣
c cϭϪ Ϫ ϩ Ϫ ScϭϪ͚ tijij͑bi*b jϩc.c.͒, S i ͚ Ni ic jc ͑1 ij͒ , ij ˆ ͫ 2 ͬ i, jϭiϪ
s s SsϭϪ͚ tijij͑ziz jϩc.c.͒. ͑106͒ S ϭϪi ͚ M i isϪ jsϩ ͑1Ϫij͒ , ij ˆ ͫ 2 ͬ i, jϭiϪ c Here tij is ⑀tc on the spatial link and 1/4⑀u on the temporal s link. Similarly tij is ⑀ts on the spatial link and 1/4⑀v on the S⌬ϭ⑀ ͚ t⌬ijij͑aijf i␣ f j␣ϩc.c.͒, f f i, jϭiϩxˆ temporal link. tijϭ⑀t f on the spatial link and tijϭϪ1 on the ⌬ temporal link. And ˜t ϭ⑀t⌬ . The last term of Eq. ͑105͒ cor- ¯ responds to the Maxwell terms for the Z2 gauge fields, that S0ϭϪ⑀ ͚ ͚ ͓t f ijijf i␣ f j␣ i, jϭiϩxˆ ␣ we assume are generated after we integrate out excitations at high energies, ϩtcijbi*b jϩtsijzi*z jϩc.c.͔,
SgϭϪK1͚ ͟ ijϪK2͚ ͟ ijϪK3͚ ͟ ijij . 2 ᮀ ᮀ ᮀ ᮀ ᮀ ᮀ Suϭ⑀u ͑NiϪN0͒ , ͚i ͑107͒ These are the simplest terms providing dynamics of the 2 gauge fields that are consistent with the gauge symmetries, Svϭ⑀v ͑M i͒ , ͑101͒ ͚i Z2 : bi tibi ; f i␣ ti f i␣ ; ij tit jij , ˆ ˆ → → → where and x represent the time and spatial links. aij ϭarrЈ on the spatial links and zero otherwise. Z2 : zi sizi ; f i␣ si f i␣ ; ij sis jij , Using the Poisson resummation formula, one can show → → → ͑108͒ that where ti and si are Ϯ1. In the future we will call any particle that transforms under the first and the second transformations c Ϫ(SuϩS ) ͚i, jϭiϪˆ (1/2⑀u)ijcos(icϪ jc)ϪS ͚ e ϭe B, of Eq. ͑108͒ as having Z2 and Z2 charges, respectively. Ni B. Spin singlet insulating phases Ϫ(S ϩS s) ͚ ˆ (1/2⑀v) cos( Ϫ ) e v ϭe i, jϭiϪ ij is js ͑102͒ Before discussing possible spin singlet insulating phases ͚M i of the combined action ͑105͒–͑107͒ it is useful to review with properties of a pure Z2ϫZ2 gauge theory ͑107͒. Under dual-
155103-14 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103 ity transformation defined in Refs. 7,46 and 47 this model visons are present in the ground state, the coherent motion of becomes a generalized Ashkin-Teller model,48 the corresponding particles is highly frustrated and they may not be considered as elementary excitations. Only the par- ticles that are neutral with respect to the appropriate Z2 sym- SATϭϪKd1͚ viv jϪKd2͚ uiu jϪKd3͚ uiviu jv j . ͗ij͘ ͗ij͘ ͗ij͘ metry may propagate freely in a phase with condensed vi- ͑109͒ sons. And the particles that carry such Z2 charges will have to bind into neutral pairs. This is the essence of the confine- ment argument discussed in Refs. 46 and 7. Here ui and vi are Ising variables defined on the dual lattice in dϭ2ϩ1. We can identify five possible phases of Eq. When we apply the geometrical phase-confinement argu- ͑109͒: ment to the spin singlet insulating states we find the same phases as discussed in Sec. IV E. ͑i͒ fully ordered phase ͗u͘<0, ͗v͘<0, ͗uv͘<0; ͑ii͒ partially ordered phase ͗u͘<0, ͗v͘ϭ0, ͗uv͘ϭ0; • In a phase of type ͑i͒ all kinds of Z2 vortices are con- ͑iii͒ partially ordered phase ͗u͘ϭ0, ͗v͘<0, ͗uv͘ϭ0; densed. Therefore particles that carry any Z2 charges will ͑iv͒ partially ordered phase ͗u͘ϭ0, ͗v͘ϭ0, ͗uv͘<0; be bound. This is a fully confining phase where only fully ͑v͒ disordered phase ͗u͘ϭ0, ͗v͘ϭ0, ͗uv͘ϭ0. neutral composites are allowed. Holons, spinons, and neu- tral fermions are confined ͑phase CSF͒. As pointed out in Ref. 7 the Ising variables of Eq. ͑109͒ • In a phase of type ͑ii͒ we have a condensate of visons. As a result particles that carry Z2 charges are confined, but correspond to the Z2 vortices of the original gauge model. They describe gauge field configurations with plaquette particles that carry Z2 charges are liberated. Spinons are free, and holons are bound to the neutral fermions ͑phase products equal to Ϫ1, i.e., plaquettes pierced by Z2 fluxes. CFSB͒. Following Ref. 7 we call such Z2 vortices ‘‘visons.’’ In fact • In a phase of type ͑iii͒, that has a condensate of visons, we have three kinds of visons: visons that describe Z2 we have a confinement of particles with Z charges and vortices of , visons that describe Z2 vortices of , and 2 ͓͔ visons that describe a composite of and Z2 vorti- deconfinement of particles with Z2 charges. Holons are ces. The three are not independent, any one of them can be free, and spinons are bound to the neutral fermions ͑phase thought of as a composite object of the other two. However, CBSF͒. we should treat all of them on equal footing since they rep- • In a phase of type ͑iv͒ we do not have individual and resent distinct topological objects. The appearance of the visons in the ground states, but only their composites, long range order in the Ashkin-Teller model corresponds to ͓͔ visons. The geometrical phase argument becomes the condensation of visons in the original gauge model and somewhat subtle when we consider ͓͔ visons. Particles describes transition to the confining phase. From these argu- that carry either one of Z2 or Z2 charges will get a ments it follows that there are five distinct phases of the pure phase shift when they circle around such a vortex. How- gauge model in Eq. ͑107͒: one fully confining phase, three ever, particles that carry both charges acquire no phase. So, partially confining phases, and one fully deconfining phase, in a D-type phase particles that carry one of the Z2 or Z2 that correspond to the fully ordered, three partially ordered, charges are confined, but particles that carry both charges and one fully disordered phases of the Ashkin-Teller model. are deconfined. Holons and spinons are bound, and neutral ͑i͒ Fully confining phase. and visons are condensed fermions are free ͑phase CSBNF͒. simultaneously. This also implies condensation of • Finally, in a phase of type ͑v͒ we have no condensed vi- ͓͔ visons. sons, which means that all the particles are liberated. Ho- ͑ii͒ Partially confining phase. visons are condensed and lons, spinons, and neutral fermions are deconfined ͑phase and ͓͔ visons are gapped. CBSBNF͒. ͑iii͒ Partially confining phase. visons are condensed and and ͓͔ visons are gapped. C. Broken-symmetry phases ͑iv͒ Partially confining phase. ͓͔ visons are condensed ϫ and and visons are gapped. In this section we show using Z2 Z2 theory that even states with the long-range order in the model ͑105͒–͑107͒, ͑v͒ Deconfining phase. All visons are gapped. i.e., p-wave superconductors, spin singlet superconductors, nematic insulators, and nematic superconductors may differ Condensation of visons has dramatic effects on the mo- in their topological ordering and carry the remnants of the tion of spinons, holons, and neutral fermions in the model spin-charge separation that appears so dramatically in the ͑105͒–͑107͒.Wefind drastically different excitation spectra insulating phase. depending on what vortices are condensed. The reason for We begin by reviewing the case of a p-wave supercon- this is a geometrical phase factor of that particles with Z 2 ductor. charges acquire when they circle around an appropriate Z2 vortex. For example, spinons and neutral fermions get a geo- • The simplest p-wave superconductor that may be deduced metrical phase factor of when they are transported around from the model ͑105͒–͑107͒ is when holons and spinons a vison, and holons and neutral fermions get a minus sign condense simultaneously, so the system acquires finite ex- when they circle around a vison. This means that when pectation values of b and z. The geometrical phase argu-
155103-15 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
ment when applied to this system tells us that an isolated Excitations in this phase will be any pair from the set hc/2e vortex or a meron are no longer well defined exci- (hc/2e vortex, vison, , z) and b isons. tations, since they acquire a phase shift of when circling • Finally we can have a condensate of b2 and gapped around a holon or a spinon, respectively. However, if we visons. This gives unconfined hc/2e vortices, visons, bind an hc/2e vortex with a vison we find that this neutral fermions, spinons, and b isons. composite can propagate freely. The geometrical phases Of the five phases above, four of the last ones may be con- acquired by the two upon encircling a holon add up to 0 or sidered as SC* phases. 2. Equivalently, a meron, when bound to a vison, be- The construction given above for p-wave superconducting comes a well defined excitation in the presence of spinon states and spin singlet superconducting states may be gener- condensate. alized to the case of spin-nematic insulators and nematic su- • Another possible phase of a p-wave superconductor is perconductors. In those cases, just as in the two discussed when we condense holon pairs and spinons, i.e., b2 and z. above, we find five possible states. One of these is a tradi- In this phase merons are still bound to visons, however, tional version, whereas the other four are of the unconven- hc/2e vortices and visons are now deconfined. The origi- tional * variety that may be thought of as containing traces nal holons are reduced to Ising variables, which we can of quantum number separation. call b isons, following Ref. 7. They carry the leftover of The reader may be worried that we do not find hc/4e vortices disclinations in our discussion of various phases the charge symmetry, that was broken from U͑1͒ to Z2, and are well defined excitations in this phase. of p-wave superconductors. As in the previous sections we • Analogously to the previous case we can consider a situa- assumed that these excitations have been gapped out ͓see discussion after Eq. ͑79͔͒. tion with condensed b and z2. This phase will have bound hc/2e vortices and visons and liberated merons and visons. Spinons become Ising variables, z isons, that carry VI. DISTINGUISHING DIFFERENT FRACTIONALIZED PHASES the residual Z2 spin quantum numbers. • A different type of a p-wave superconductor occurs when In previous sections, we have seen how various fraction- holon pairs and holon-spinon composites condense simul- alized phases can arise in the context of Kondo lattice mod- taneously, i.e., b2 and bz acquire expectation values ͑this els and systems with a tendency towards p-wave supercon- also fixes the expectation value for z2). In such a phase ductivity or superconductivity coexisting with magnetism. spinons and holons are reduced to a single Ising variable, These phases can be described in the language of vortex and since knowing b automatically gives z. This bz ison carries skyrmion condensation or in terms of a Z2ϫZ2 gauge theory. the residual spin-charge quantum number of the system. A However, one might wonder if these results are an artifact of stable topological object in this phase may be constructed these formalisms. In particular, one can ask how these phases by taking any two of the set (hc/2e vortex, meron, vison, can be distinguished—both as a matter of principle and as a practical experimental issue—from each other and from un- vison͒. fractionalized phases. As Wen11 and, more recently, Senthil • Finally, we may have a phase with condensed holon and 4 2 2 and Fisher have emphasized recently, their ‘‘topological spinon pairs, b and z . This gives us separate b isons, z order’’—i.e., the sensitivity of the ground state to changes of isons, hc/2e vortices, merons, visons, and visons. the topology of the system—provides one means of distin- guishing fractionalized phases. The last four phases are the triplet analogs of the exotic SC* This characterization of fractionalized phases is crucial phase discussed in Ref. 7 in the case of singlet superconduct- because other heuristic definitions of fractionalized phases ors. We now consider the case of a spin-singlet supercon- can fail. To see why this is so, consider the intuitively ap- ductor. pealing statement that a fractionalized phase is distinguished • The simplest kind of a spin singlet superconductor occurs from a conventional phase by asking for the lowest energy excitation with, for instance, spin 1/2. In the conventional when we condense simultaneously holons b and visons. case, this would be an electron which also carries an electric The former ensures confinement of hc/2e vortices and charge e. In the fractionalized phases of the kind discussed visons, whereas the latter gives rise to binding of neutral above, one might expect that the corresponding excitation is fermions to spinons. a spinon which is charge neutral. However, this test for frac- • Another possibility is to have a condensate of holon pairs 2 tionalization fails if there is an attractive interaction between b and visons. This liberates hc/2e vortices and vi- the holons and spinons which binds them into an electron at sons, produces b isons that carry charge Z2 number, and low energies. This could, in principle, happen without going leaves neutral fermions bound to spinons. through a phase transition. ͑Unlike in an unfractionalized • Another option is to have a condensate of bosons b with phase, holons and spinons would still exist as unbound exci- gapped visons. This means bound hc/2e vortices and tations, but at higher energies.͒ Then, the lowest energy ex- visons, and liberated neutral fermions and spinons. citation with spin 1/2 is an electron ͑as opposed to a spinon͒ • The most intriguing phase in this series is obtained when though the system is adiabatically connected to a fractional- we condense holon pairs b2 and holon vison composites. ized phase ͑see Refs. 13, 28, and 29 for a discussion of this
155103-16 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103 effect͒. Furthermore, other tests such as the vanishing of the resulting bound state will be Bosonic. Hence, as a result of quasiparticle residue at some point of the Brillouin zone also the seemingly innocuous formation of bound states, the fail in this situation. Hence we turn to the characterization in CBSF and CFSB states appear to metamorphose into each terms of the topological properties of the system. other. Thus one is instead tempted to conclude that the CBSF Topologically ordered systems are partially characterized and CFSB phases can be adiabatically connected to each by their ground-state degeneracy on the annulus, the torus, or other. higher-genus manifolds, over and above any degeneracy This contention is supported by considering the singlet which may be due to broken symmetry. Consider the CBSF superconducting state which results if holons condense in phase. It has a twofold degenerate ground state on the annu- CBSF or if holon- vison composites condense in CFSB lus. The two ground states correspond to periodic and anti- ͑see Fig. 1͒. It is easy to see that the superconducting states periodic boundary conditions for holons and spinons as they in either case are conventional and are smoothly connected encircle the center of the annulus. In either case, electrons to a BCS state. The superconducting state can be disordered themselves have periodic boundary conditions, as they must. by vortex condensation. This will yield a fractionalized state In an unfractionalized phase, spinons and holons are con- ͑with a twofold degenerate ground state on an annulus͒ if fined within an electron so the two states are identical; the vortex pairs condense but individual vortices are uncon- excitations which could distinguish them are not part of the densed. Since the result could be either CBSF or CFSB, this spectrum. By the same reasoning, the CFSB and CSBNF appears to support the possibility that there is no phase phases also have two degenerate ground states on the annu- boundary between these phases in the part of the phase dia- lus. By extension, all of these states have ground-state de- gram near the singlet superconducting phase. generacy 4g on a genus g surface. On the other hand, However, there is a logically possible alternative, namely CBSBNF has four degenerate ground states. We can inde- that an operator which is irrelevant in the superconducting pendently choose periodic or antiperiodic boundary condi- phase and at the critical point becomes relevant at the fixed tions for the charge and spin bosons. The boundary condi- points characterizing the fractionalized phases. In that case, tions for the neutral fermions are then determined by the the actual nature of the resulting fractionalized phase de- requirement that electrons must have periodic boundary con- pends on short distance physics—the value of the coupling ditions. On a genus g surface, it has degeneracy 16g. which is formally irrelevant in the superconductor—and is These degeneracies can be interpreted in terms of the vi- not uniquely dictated by knowing that there is proliferation son spectra of the fractionalized states. The two ground states of hc/e and with hc/2e vortices remaining gapped. of CBSF on an annulus correspond to the presence or ab- Despite this caveat, a scenario in which CBSF and CFSB sence of a vison ͑i.e., a v) in the center of the annulus; the are smoothly connected to each other in the vicinity of their two ground states of CFSB correspond to the presence or transition to the superconducting state is appealing and plau- absence of a vison ͑a vЈ); the two ground states of CSBNF sible. This does not necessarily mean that CBSF and CFSB correspond to the presence or absence of a vison in the are not distinct phases. Their relationship could be similar to center of the annulus. The four ground states of CBSBNF that between a liquid and a gas, which are separated by a correspond to the presence or absence of and visons in first-order phase transition line which terminates at a critical the center of the annulus. The interpretation of these ground point, beyond which a liquid and a gas can be adiabatically states in terms of visons forms the basis for an experimental connected without crossing a phase-transition line. In Appen- probe of topological order proposed by Senthil and Fisher.19 dix C, we show that precisely such a scenario does occur in We will return to this issue later but let us, in the meantime, simpler ͑though somewhat different͒ Z2ϫZ2 gauge theory continue to pursue the question of the distinction in principle models. Thus we tentatively suggest that the first-order phase between different fractionalized phases. transition between the CBSF and CFSB phases terminates at Different states at the same level of fractionalization have a critical point. Beyond this critical point, there is no distinc- the same ground-state degeneracy; CBSF, CFSB, and tion between these phases, and it is in this region of the CSBNF all have two degenerate ground states on the annu- phase diagram that there is a phase transition to the super- lus. In order to distinguish them, we must consider their conducting phase. quantum number spectra. CSBNF does not have spin-charge separation, i.e., it is not possible to isolate a charge-0, spin- VII. FLUX-TRAPPING EXPERIMENTS 1/2 excitation at finite-energy cost. Furthermore it is possible to isolate a neutral Fermionic excitation. Both of these stand Let us now consider the practical issue of how we can in contrast to CBSF and CFSB which exhibit spin-charge identify whether a given system in an unknown phase is separation but do not support neutral Fermionic excitations. fractionalized or not and, if it is fractionalized, then what its Hence, we conclude that CSBNF is distinct from the other fractionalization pattern is. To proceed, note first that the two states despite having the same ground state degeneracy. CBSF phase contains in it the seed of superconductivity. As One might be tempted to conclude that CBSF and CFSB argued in Ref. 7, condensing the charged boson provides a are distinct because the lowest-energy charged excitation is a natural nonpairing route to superconductivity ͑of a conven- boson in one phase and a fermion in another phase. How- tional kind͒. Similarly, the CFSB phase contains in it the ever, if a holon in CBSF forms a bound state with a vison, seed of magnetism—simply condensing the spinon leads to a the resulting bound state will be Fermionic; similarly, if a conventional state with some kind of magnetic long-range spinon in CBSF forms a bound state with a vison, the order. However, it is possible to imagine a transition between
155103-17 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103 the CFSB phase and a superconductor which occurs when a Now consider a conventional BCS superconductor. This is composite formed by a holon and a vison condenses. Simi- obtained from CBSF by condensing the holon. The flux- larly, it is possible to imagine a transition between the CBSF trapping experiment performed by moving between the su- phase and a magnetic phase which occurs when a composite perconductor and CBSF gives a positive result. Now con- formed by a holon and a vison condenses. sider a modification of the experiment so that we start in the The feature of most interest to the following discussion is superconducting phase, move first to CBSF, then to simply that a direct phase transition should be possible be- CBSBNF, then back into CBSF before finally going back tween the CBSF and CFSB phases and a conventional super- into the superconductor. This again gives a positive result conductor. Upon going through such a phase transition, the since v is trapped in the annulus and it can never escape. visons of these phases acquire hc/2e units of electromag- Upon making the transition between the CBSF and CBSBNF netic flux to become the hc/2e vortices of the supercon- phases, a vЈ will be generated with probability 1/2 since the ductor. This may be exploited to devise a sensitive test for ground state of CBSF with one v will make a transition to the topological order in the CBSF phase, as argued in Refs. either of the corresponding ground states of CBSBNF with 19 and 4. equal probability. However, this vЈ will escape upon the The test proceeds as follows. Consider an annular sample transition from CBSBNF back to CBSF. Now consider a of a material which is in a conventional superconducting further modification in which we go all the way from the phase and let us suppose that we can tune the sample param- superconductor to the CFSB phase through the CBSF and eters adiabatically so that the sample makes transitions be- CBSBNF phases and then return by the same route to the tween the superconducting phase and the CBSF and CFSB superconductor. The result of this experiment will be nega- phases. Suppose that hc/2e units of electromagnetic flux are tive half of the time. This is because in going from CBSBNF trapped in the annulus when the system is in its supercon- to CFSB, the vison v condenses. Thus v which was trapped ducting phase. There must also be a vison trapped in the in the hole until the phase CBSBNF was reached can escape annulus so that the holon condensate can have periodic on moving into the CFSB phase. In going from CFSB back boundary conditions ͑without which it would cost infinite to CBSBNF, a v is generated with probability 1/2—the two energy͒: the antiperiodicity caused by the flux hc/2e is can- ground states are obtained with equal proability. This v, if it celled by the antiperiodicity due to the vison. If the system is is generated, will lead to the generation of flux hc/2e in the taken into the CBSF phase, the flux escapes since there is no superconducting state. holon condensate trapping it, but a vison will remain since it Hence, there appears to be a difference between the CBSF will cost energy ͑the vison gap͒ to unwind the antiperiodic and CFSB which can be detected in this experiment. It ap- boundary conditions of the ͑neutral͒ spinons. If the system is pears that these phases cannot be continuously connected— returned to the superconducting state, then it must generate since the probability of a negative result for the flux-trapping flux Ϯhc/2e so that the holon condensate can again have experiment of the previous paragraph must jump from 0 to periodic boundary conditions. The same analysis holds if we 1/2—at least in the vicinity of the CBSBNF phase. This can take the system into the CFSB phase except that we have to be understood in the following terms. In the CBSBNF phase, replace ‘‘holon’’ in the above description by ‘‘holon-vison there are two distinct types of visons, v and vЈ. If one or the composite.’’ On the other hand, if the system undergoes a other condensed, a transition occurs to CFSB or CBSF. The transition to an unfractionalized phase, then the vison can remaining vison in CBSF ‘‘remembers’’ that it is a v vison. escape since there are no deconfined spinons or holons Meanwhile the vison in CFSB remembers that it is a vЈ whose boundary conditions would be affected by its escape. vison. However, if we take the system far from CBSBNF so Of course, this experiment would simply be confirming that a bound state can form between a v and a holon and also the result which we arrived at in the previous section: that between a v and a spinon, then v now looks like a vЈ and the the CBSF and CFSB phases can be adiabatically continued distinction between the two phases is blurred. Combining into each other, particularly in the neighborhood of a conven- this reasoning with that of the previous section, we propose tional singlet superconducting phase. the phase diagram of Fig. 6. Let us now consider a more complicated flux-trapping experiment in which, as an intermediate step, we take the VIII. DISCUSSION system through the higher-level fractionalized phase, CBSBNF ͑see Fig. 1͒. This phase has two distinct vison ex- When electrons interact strongly, a number of interesting citations. One of these visons can be envisioned as a descen- phenomena are known to occur, including unconventional dent of the vison of the CBSF phase; we will refer to this superconductivity and magnetism. As we have seen in this as v. The other can be envisioned as a descendent of the paper, many of the physical settings which give rise to these vison of the CFSB phase or as a by-product of the further phenomena also have the potential to exhibit electron frac- fractionalization of the fermionic spinon of CBSF; we will tionalization. Different theoretical approaches, adapted to refer to this as vЈ. A direct transition from CBSBNF to the these specific systems, suggest seemingly different fraction- CBSF phase occurs when the visons vЈ condense while that alized phases. It is natural to ask if these phases are truly from CBSBNF to CFSB occurs when the visons v condense. different and, if so, what their organizing principle is. The presence of two distinct visons in the CBSBNF phase In this paper, we have pursued the idea3,4 that a crisp and distinguishes it from the CBSF and CFSB phases—indeed it coherent way of understanding quantum number fractional- will have a ground-state degeneracy of 16 on a torus. ization is provided by the concept of topological order intro-
155103-18 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
We have also discussed the possibility of quantum disordered phases in which the condensed topological objects are the hc/2e vortex–meron composites, but not hc/2e vortices or merons separately. Such phases have spinons and holons bound together but deconfined from the neutral Fermionic quasiparticles. An important issue discussed in this paper is whether one can distinguish the phases obtained by quantum disordering the spin and charge sectors of the system, for example, the phases CFSB and CBSF of the quantum disordered p-wave superconductor. The simplest choice seems to be the identi- fication of the spin excitation as a Fermionic or Bosonic particle. This, however, is not a reliable tool. In the Z2ϫZ2 FIG. 6. A schematic phase diagram indicating how the CBSF, gauge theory formulation, both spinons and holons carry Z2 CFSB, CSBNF, CBSBNF, and conventional singlet superconduct- charges, so a bound state of a Z2 vortices with either one of ing phases might fit together. The thick lines are first-order phase them ͑this can also be thought of as attaching Wilson loops transitions and the thin lines are second-order phase transitions. to the particles͒ will change its statistics from Fermionic to Bosonic or vice versa.17,42,43,52 In the deconfining phase such duced in the context of fractional quantum Hall liquids,49 anyon superfluids,50 chiral spin states,51 and short-range vortices are gapped. However, if a bound state between a Z2 resonating valence bond spin states.30 We presented two ap- charge carrying particle and a Z2 vortex forms, this bound proaches for understanding such topological order. The first state may have a lower energy than the original particle. This means that in both CFSB and CBSF phases the lowest en- one relies on the recently developed Z2 gauge theory of spin- ergy spin- or charge-carrying excitations can exist as either charge separation, originally suggested for the high-Tc cu- bosons or fermions. The subtleties discussed above lead us to prates, and generalizes it to a Z2ϫZ2 theory to include pos- sible fractionalization of spin and charge quantum numbers. consider flux-trapping experiments of the type discussed in Some of the interesting fractionalized phases are: CBSF Sec. VII. Combining all of these considerations, we outlined ͑Bosonic holons and Fermionic spinons͒, CFSB ͑Fermionic one scenario in which CBSF and CFSB phases can be sepa- holons and Bosonic spinons͒, CSBNF ͑bound Bosonic ho- rated by a first-order transition which terminates at a critical lons and spinons and neutral fermions͒, and CBSBNF point. On the other hand, one can go from CBSF to CFSB ͑Bosonic holons and spinons and neutral fermions͒. Any one through CBSBNF phase by two continuous transitions. Thus, of these phases can be further characterized by possible bro- if this scenario is correct, the relation between CBSF and ken symmetries with conventional order parameters. Each of CFSB is somewhat similar to that between liquid and gas the fractionalized phases corresponds to a different deconfin- phases. We, however, defer offering any definitive conclu- ing phase of the pure Z2ϫZ2 gauge theory and will have sion. appropriate topological Z2 vortices, visons, as finite-energy Spin charge separation in one-dimensional systems is fun- excitations. damentally different from its two-dimensional counterpart, An alternative picture of fractionalization which is also since it does not involve topological order. Another non- presented in this paper uses the language of quantum disor- trivial realization of electron number fractionalization which dered superconductors and magnets. When topological is analogous to that presented here can occur in multicom- ordering—defined by the suppression of certain defects— ponent quantum Hall systems and was discussed in Refs. 40 occurs, the Goldstone modes associated with various broken and 53. symmetries can screen the corresponding quantum numbers Another avenue for further research is the investigation of of the Fermionic quasiparticles. In this way, these quasipar- quantum-disordered states of triplet superconductors with ticles can be bleached of some or all of their quantum num- more complicated spin structures appearing in some of the bers. This may be implemented mathematically with U͑1͒ superfluid phases of 3He. We expect that these will share particle-vortex duality in both the charge and spin sectors. some features of noncollinear spin-density waves.54 Further We arrive at essentially the same picture as that of the Z2 exotic phases are likely to occur upon quantum-disordering ϫZ2 gauge theory. In those insulating phases in which hc/2e states with multiple order parameters. We have considered vortices are condensed, charge is bound to the Fermionic one of the simplest cases of this—antiferromagnetism and quasiparticles. When hc/2e vortices are gapped and hc/e superconductivity—but there are more complicated possibili- vortices are condensed, charge carrying holons can propagate ties, involving incommensurate charge and/or spin order. separately from the electrically neutral Fermionic quasiparti- In addition to the phases CBSF and CFSB that have ap- cles. In the spin sector, we can consider either meron or peared in the literature before, we proposed the possibility of skyrmion ͑which carry twice the topological charge of two additional quantum number separated phases in these merons͒ condensation, with gapped merons in the latter case. systems: phase CSBNF in which the excitations are a spin- In the former case, spin is confined to the Fermionic quasi- 1/2, charge e boson, a neutral spinless fermion and a vison particles, and in the latter case spinons will exist as indepen- and phase CBSBNF with a charge e spinless boson, a neutral dent objects, deconfined from the Fermionic quasiparticles. spin 1/2 boson, a neutral spinless fermion, and two distinct
155103-19 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103 visons, simultaneously condensed hc/e vortices and skyrmi- that the quantum disordered phases may have separated ons. quantum numbers, depending on the topological order, which The possibility of a higher SO͑5͒ symmetry which unifies can be characterized by specifying the nature of the finite- d-wave superconductivity and antiferromagnetism has been energy Z2 visons. suggested for the high-Tc cuprates and organic supercon- ductors in Ref. 55. In the Sr2RuO4 materials, a similar sym- ACKNOWLEDGMENTS metry has been proposed in Ref. 56 which combines p-wave superconductivity and ferromagnetism. An effective model E.D. and C.N. thank Aspen Center for Physics for its hos- for the coupling of quasiparticles to a fluctuating SO͑5͒ order pitality during the Winter 2000 Conference ‘‘50 Years of parameter has been derived in Ref. 57. In this model holons Condensed Matter Physics,’’ where some parts of this work and spinons are not segregated into independent quasiparti- were initiated. H.Y.K. and Y.B.K. thank ITP, University of cles from the very beginning but are naturally combined into California at Santa Barbara, where some parts of this work composite quasiparticles which transform as spinors of were performed. C.N., H.Y.K., and Y.B.K. thank Aspen Cen- SO͑5͒. Such spinors are spin doublets and carry charge e.58,59 ter for Physics for its hospitality during the summer work- There are also neutral fermions which carry no quantum shop in 2000. We also thank Aspen Center for Physics for its numbers. One can see a striking resemblance between these hospitality during the winter workshop in 2001. Useful dis- excitations and the excitations in the phase CBSBNF. This cussions with M.P.A. Fisher, E. Fradkin, S. Kivelson, and M. suggests the interesting possibility that the restoration of the Sigrist are gratefully acknowledged. This work was sup- SO͑5͒ symmetry in models with strong quantum fluctuations ported by the Harvard Society of Fellows ͑E.D.͒; NSF under manifests itself not in the existence of a bicritical point on grant Nos. DMR-9983544 ͑C.N.͒ and DMR-9983783 the phase diagram, but in the appearance of a specific form ͑Y.B.K.͒; the Alfred P. Sloan Foundation ͑C.N. and Y.B.K.͒; of quantum number separation of the electrons. A detailed the Department of Energy, supported ͑in part͒ by funds pro- discussion of quantum disordering phenomena in models vided by the University of California for the conduct of dis- with SO͑5͒ symmetry requires a detailed analysis of the non- cretionary research by Los Alamos National Laboratory Abelian Berry’s phases involved in the description of SO͑5͒ ͑H.Y.K.͒. The work of T.S. at the ITP, Santa Barbara, was spinors and will be presented in subsequent publications. supported by the NSF under Grant Nos. DMR-97-04005, The states which we have discussed, as well as the more DMR95-28578, and PHY99-07949. complicated ones alluded to above, have potential applica- tion to a variety of materials, including not only the APPENDIX A: KONDO LATTICE MODEL 22 21 cuprates, but also Sr2RuO4; heavy fermion superconduct- ors, such as CeIn ;24 and organic superconductors, such as In this appendix, we provide some of the details of the Z2 3 gauge theory reformulation of the Kondo lattice model dis- -(ET) Cu͓N(CN) ͔Cl.23 All of these compounds have 2 2 cussed in Sec. II. Consider the Hamiltonian in Eq. ͑1͒. As in magnetic ͑in come cases incommensurate͒ phases in proxim- the discussion of the pure exchange Hamiltonian, we first ity to p-wave or d-wave superconducting states. It is possible Ϫ replace the spin operator Sr by the boson operator bsr that pressure, chemical substitution, magnetic field, etc., i might drive a transition into one of the phases described here ϳe r. The exchange Hamiltonian takes the form of Eq. ͑6͒ in which the magnetism and the superconductivity are disor- and the Kondo coupling takes the form of Eq. ͑24͒. The dered by quantum fluctuations. electron hopping term is unaffected. We now change vari- Ideas presented in this paper should also apply to Bose- ables to spinon and holon operators as in Eqs. ͑7͒, ͑26͒, and Einstein condensates of spinor bosons, such as alkali atoms ͑27͒. The terms Ht ,Hk , and Hex are now given by Eqs. ͑29͒, 23Na and 87Ru which have a hyperfine spin Fϭ1. For ex- ͑25͒, and ͑9͒, respectively. In the presence of the Kondo ample, when restricted dimensionality or quantum fluctua- coupling between the local moments and the conduction tions destroy the spin ordering we expect to find condensa- electrons, the total (z component of the͒ spin at each site is tion of pairs of atoms into a global spin singlet state, and 1 when quantum fluctuations in the charge sector destroy the n ϩ c†zc . ͑A1͒ U͑1͒ phase ordering we can find states characterized by a r 2 r r spin nematic order. Some of these phenomena have been discussed in Ref. 25. We therefore define the total spinon number To summarize, we have studied the possibility of fraction- tot † z alization in systems with ordering tendencies in the charge Nr ϭ2nrϩcr cr . ͑A2͒ and spin sectors, including Kondo lattices, p-wave supercon- tot Note that Nr is conjugate to the phase r of the spinon ductors, and systems with simultaneous d-wave supercon- tot field. We will work with the operators (zr ,Nr , r , ,r) ducting and antiferromagnetic fluctuations. In the case of ↑ ↓ p-wave superconductors we find that the rich internal struc- instead of the original electron and local spin Sជ r operators. ture of their order parameter allows for the existence of the This change of variables, however, introduces some following quantum disordered phases: a charge 4e singlet redundancy—the Hilbert space of states on which the holon superconductor, a spin singlet insulator, and a spin nematic and spinon fields operate is larger than the physical set of insulator. For both the p wave superconductors and the states. This may be seen by noting that with the definition tot d-wave superconductor/antiferromagnet systems, we find above, the operator Nr must satisfy
155103-20 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
tot † z 2 Nr Ϫcr crϭeven. ͑A3͒ ⑀J † trrЈ † SIϩSIIϭ zr zr ϩ r r ϩH.c. † z 2 ͚ ͫͩ Ј 2J Ј ͪ From the definition of the holons, it follows that cr cr ͗rrЈ͘ † z † z ϭr r . Furthermore r r has the same parity as 2 † trrЈ † † . Thus we have the constraint ϩ z z Ϫ ϪH.c. . ͑A10͒ r r ͩ r rЈ 2J r rЈ ͪ ͬ NtotϪ† ϭeven. ͑A4͒ r r r We may now decouple each of these two terms with a real The Hamiltonian needs to be supplemented with this con- Hubbard-Stratanovich field to write straint to correctly represent the original model ͑before the change of variables͒. It is useful to rewrite the exchange and Kondo parts of the eϪ(SIϩSII)ϭ ͓ ͔eϪ(SϩS), ͑A11͒ Hamiltonian as follows: ͵ D D
† 2† 2 ⑀J 2 † trrЈ † HKϩHexϭJK ͑r r ϩH.c.͒ϪJ ͑zr zr ϩH.c.͒ ͚ ͚ Sϭ rr Ϫ2rr zr zr ϩ r r ϩH.c. , r ↑ ↓ ͗rr ͘ 2 ͚ Ј Јͩ Ј 2J Ј ͪ Ј ͗rrЈ͘ ͑A12͒ U 2 † z ϩ ͑NrϪ1͒ ϪU Nr͑r r͒ 4 ͚r ͚r ⑀J trr S ϭ 2 Ϫ2 z†z Ϫ Ј † ϪH.c. . U 2 ͚ rrЈ rrЈͩ r rЈ 2J r rЈ ͪ † z 2 ͗rrЈ͘ ϩ ͑r r͒ . ͑A5͒ 4 ͚r ͑A13͒
The last term is an interaction between the holons. Clearly Note that rrЈϭrЈr while rrЈϭϪrЈr . We now consider this term cannot affect issues of confinement of the holons evaluating the , integrals in a saddle-point approximation. with the spinons. We will therefore drop it for the present Looking for uniform saddle points, we write discussion. The last but one term represents an interaction between the spinon density and the holons. We again expect ͗rr ͘ϭ0 ; ͗rr ͘ϭ0 . ͑A14͒ that such an interaction is also unimportant for issues of the Ј Ј stability of fractionalized phases. We will therefore drop this Note that a nonzero value of 0 requires specifying direc- too. tions for all the links of the lattice. The saddle-point equa- We may now derive a functional-integral representation of tions are the system, proceeding as in Ref. 7. The resulting action is
SϭS ϩS ϩS . ͑A6͒ trr r B ϭ z†z ϩ Ј † ϩH.c. , ͑A15͒ 0 ͳ r rЈ 2J r rЈ ʹ Here S represents terms involving coupling along the ͑imaginary͒ time direction. This and the Berry phase SB are exactly the same as in Ref. 7. The spatial part of the action is trr ϭ z†z Ϫ Ј † ϪH.c. . ͑A16͒ 0 ͳ r rЈ 2J r rЈ ʹ SrϭSIϩSKϩSII , ͑A7͒
Note that 0 must be pure imaginary as it is the expectation S ϭϪ⑀ t z†z ϩ † ϩ† ͒ϩH.c. , I ͚ rrЈ͓ r rЈ ͑ rЈ r r rЈ ͔ value of an anti-Hermitian operator. With nonzero 0, the ͗rrЈ͘ ↑ ↑ ↓ ↓ saddle-point action therefore becomes complex—this breaks time-reversal symmetry ͑and possibly various lattice symme- † tries due to the need to specify directions to the links͒.We SKϭϩ⑀JK͚ ͑r r ϩc.c.͒, r ↑ ↓ restrict ourselves to time-reversal invariant saddle-point so- lutions, and therefore set 0ϭ0. The resulting saddle-point 2† 2 action then preserves all the global symmetries of the origi- SIIϭϪ⑀J ͚ ͑zr zr ϩH.c.͒. ͑A8͒ ͗rrЈ͘ nal model. However, it does break the local Z2 symmetry introduced by the change of variables to the holons and the We now combine the terms SI and SII and rewrite them as spinons. This can be remedied by keeping a particular set of fluctuations about the saddle point, namely those associated t 2 † rrЈ † † 4 with a change in the sign of the fields rr : Ϫ⑀J ͚ zr zrЈϩ ͑r r ϩr rЈ ͒ ϩH.c.ϩ ͑ ͒. Ј ͫ 2J Ј ↑ ↓ ↓ ͬ O ͗rrЈ͘ ↑ ͑A9͒ rrЈϭ0rrЈ ͑A17͒ The last term is a four-holon interaction which we will ig- nore on the grounds that it cannot affect issues of fractional- with rrЈϭϮ1. The rrЈ may be identified as the spatial ization. It is convenient to further rewrite the expression components of a Z2 gauge field. We thus finally arrive at the above as follows: action in Eq. ͑30͒.
155103-21 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
˜ 2 † z x y͔͒ץxϪv⌬ ͑iץϪvF iץ͓ Sϭ ͵ d xd
ϩ 2 ␥ x ͵ d xda␣͑x͒⑀ab⑀␣␥b͑x͒ 1 ϩ2v Ac͔ ץ Ϫ2AczϪv ץϩ d2xd͑†͓z 2 ͵ F x F x
† z z z z z z .xϩ2vF Ax ͔͒ץ Ϫ2A ϪvF ץ ϩ ͓ ͑B5͒ This describes the same coupling of the quasiparticle cur- rents to the fluctuations of charge and spin as in Eq. ͑50͒, FIG. 7. Order parameter for a d-wave superconductor. Gapless ជ excitations exist at kFϭ(ϮkF,0), (0,ϮkF). ˜ ϭ˜ 0ϩ 2 c Ϫ ϩ c Ϫ x Ax͒ץ A͒ Jx͑ץS f S f ͵ d xd͓J0͑ APPENDIX B: QUANTUM NUMBER SEPARATION zϪA͔͒ ͑B6͒ ץ†zϪA͒ϩJ͑z ץ†ϩJ͑z IN SYSTEMS WITH d-WAVE SUPERCONDUCTING 0 x x x AND ANTIFERROMAGNETIC FLUCTUATIONS with
Nodal fermions in a d-wave superconductor are described c † z c † J0ϭ , JxϭϪvF , by28 † z † z z J0 ϭ , Jx ϭϪvF . ͑B7͒ ˜ 2 † z z ជ ជ Quantum disordering of the superconducting and antiferro- xϩvFAxϪA nץϪA ϪvF iץ͓ S f ϭ d xd ͵ magnetic orders in Eq. ͑B6͒ may now be achieved by con- densing vortices and merons with the possibility of five ͒ ͑ ץ ϩ ជ ជ Ϫ s is v f Ax n z v⌬ ͕e ͖͑i y͔͒ , B1 phases similar to phases 3A–3E in Sec. IV D: ͑A͒ Spinons and holons confined. No quantum number where the electron operators a␣ are defined as separation. ͑B͒ Spinons unbound and holons glued to fermions. ͑C͒ Holons free and spinons bound to fermions. c ជ ជ kFϩk 11 ↑ ͑D͒ Spinons and holons bound together, decoupled from † fermions. cϪkជ Ϫkជ 21 F ↓ ͑E͒ All excitations decoupled. Free holons, spinons, neu- a␣͑kជ ͒ϭ ϭ ͑B2͒ 12 ckជ ϩkជ tral fermions. F ↓ † ͫ 22ͬ ϪcϪជ Ϫជ ͫ kF k ͬ APPENDIX C: ADIABATIC CONTINUATION ↑ BETWEEN DIFFERENT PHASES OF A Z2ÃZ2 and the coordinate system was rotated in such a way that the GAUGE MODEL WITH MATTER FIELDS. x axis goes along the nodal direction that we are considering In the pure Z ϫZ gauge theory there are five phases that ͑see Fig. 7͒. 2 2 are distinct and separated by phase transitions. A question Antiferromagnetic fluctuations are introduced via that we address in this section is whether this distinction survives in the presence of matter fields.
2 2 † Safϭ d d kd qnជ qជ c ជ ជ ជ ␣ckជ ϩkជϩqជ ϩH.c. ͵ ͵ ϪkFϩk␣ F 1. Toy models Let us begin with a simple model, ϭ 2 ជ ␥ ជ ͵ d xd n͑x͒a␣͑x͒⑀ab⑀␣␥b͑x͒, ͑B3͒ SϭϪK1͚ ͟ ijϪK2͚ ͟ ijϪ͚ ijijviv j , ជ ᮀ ᮀ ᮀ ᮀ ij where nϭ(cos ,sin ). ͑C1͒ Spin and charge may again be decoupled by rotating the fermions as in Eq. ͑48͒, where viϭϮ1 is an Ising matter field. To construct the full phase digram of this model we consider several limiting cases. When ϭ0 we have two independent gauge fields, iz/2 iz/2 a␣ϭe e a , ͑B4͒ each of which has confining and deconfining phases. When both K1 and K2 are small we have a confining phase for both with the result and , that has no extra degeneracy on topologically non-
155103-22 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
FIG. 8. Phase diagram of Eq. ͑C1͒. Plane ABCD corresponds to ϭ0, EFGH to ϭϱ, ADHE to K1ϭ0, BCFG to K1ϭϱ, ABFE to K2 ϭ0, and DCGH to K2ϭϱ. The pure gauge model (ϭ0) has only two partially confining phases (2Ј and 2Љ) because not all the possible lattice Maxwell terms are present. They are sepa- rated by one first-order or two second-order phase transitions for small , but may be continu- ously connected for larger .
trivial manifolds and is labeled 1 in Fig. 8. When K1 is large confining phases which appeared to be distinct for ϭ0 and K2 is small we have a phase that is confining for , but ͑phases 2Ј and 2Љ in the ABCD plane͒ may be continuously deconfining for ͑phase 2Ј in Fig. 8͒. There is an analogous connected through a path that takes advantage of the finite  phase for K2 large and K1 small ͑phase 2Љ in Fig. 8͒ which is region of the phase diagram. It is important to realize that confining for and deconfining for . When both K’s are our argument for the existence of a path connecting phases large, we have a fully deconfining phase with degeneracy 4 2Ј and 2Љ does not depend on the details of how the phase on a cylinder. When K1ϭϱ͑BCGF plane in Fig. 8͒ there are boundaries in Fig. 8 are connected. One can always find a no frustrated plaquettes for , so we can choose a gauge path which begins in phase 2Ј, approaches face ABFE, goes where all ijϭ1. The model is then the same as in Ref. 46 up to EFGH, crosses to EHDA, and finally comes down to and its phase diagram can be easily constructed. When K1 2Љ without crossing the phase boundaries ͑this path does not ϭ0 ͑ADHE plane in Fig. 8͒ we find that integrating out ij have to actually be on any of the faces and it may be suffi- and vi only adds a constant to the action for and does not cient to be in their vicinity͒. It is interesting to note that in affect the confinement-deconfinement transition which takes the cross section DBFH in Fig. 8 the phase diagram looks place for the same value of K2, regardless of the value of . similar to a liquid-gas phase diagram, where the two phases When ϭϱ͑EFGH plane in Fig. 8͒ we must have 2Ј and 2Љ may be separated by a first-order transition or ͟ ᮀijijϭ1 on every plaquette, so we can choose a gauge continuously connected arond the critical point which termi- where ijijϭ1 on every link. The fields and are iden- nates the first-order line. tical and there are only two phases, a confining ͑phase 1͒ and The real reason why phases 2Ј and 2Љ of the gauge theory a deconfining ͑phase 2͒ with the transition determined by ͑C1͒ may be connected to each other is that both of them are K1ϩK2. The full phase diagram may now be obtained by related to the Higgs phase ͑phase 2͒ for the vi matter field. connecting the lines on the faces of the cube in Fig. 8. It is This may be explained by noting that in such a Higgs phase immediately clear from this picture that the two partially visons of either or are forbidden, but their composite is
FIG. 9. One scenario for the phase diagram of Eq. ͑C2͒ with two kinds of superfluid phases SF1 and SF2. Shaded figure shows a phase boundary of the superfluid and insulating phases. Diagonal shading corresponds to the boundary of SF1 and horizontal shading to a boundary of SF2. There is a continuous path to go between partially confin- ing phases 2Ј and 2Љ without crossing the phase boundaries.
155103-23 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
logical excitation: a bound state of and visons that does not interact with the matter field. This has interesting impli- cation that we have hc/2e vortices that are bound to either or visons, and the two kinds of vortices are distinct. We do not at this stage know what the generic phase diagram for Eq. ͑C2͒ in the K1K2 cube is. One possibility FIG. 10. Another scenario for a phase diagram of Eq. ͑C2͒ when is shown in Fig. 9. As in the Ising case there is a way to the point M is exactly on the superfluid-insulator phase boundary. connect phases 2Ј and 2Љ continuously by going to finite . As before, the shaded figure shows a phase boundary of the super- There is another qualitatively different phase diagram fluid and insulating phases with diagonal and horizontal striping ͑without introducing new phases͒ where the point M is at the that correspond to SF1 and SF2, respectively. In this case there is no phase boundary with the superfluid phase. This will remove continuous path connecting phases 2Ј and 2Љ without crossing the the possibility of a continuous path between phases 2Ј and phase boundaries. 2Љ ͑see Fig. 10͒. At this point we are unable to make a definite comment not, so this phase should be related to the phases where these on of the validity of either of the scenarios shown in Fig. 9 or visons are condensed separately ͑but not simultaneously͒. Fig. 10. We note, however, that this issue is amenable to Note that this argument may no longer apply if the matter study by numerical or other means. Thus future work should field carries a quantum number, and a Higgs phase breaks be able to settle this satisfactorily. some continuous symmetry. Let us, for example, explore the Another important model to consider is one in which the model where the matter field is an XY order parameter, matter field is Fermionic. An appropriate model is
SϭϪK1 ijϪK2 ij ͚ᮀ ͟ᮀ ͚ᮀ ͟ᮀ SϭϪK1 ijϪK2 ijϪ ijiji j , ͚ᮀ ͟ᮀ ͚ᮀ ͟ᮀ ͚ij ͑C3͒ Ϫ ijijcos͑iϪ j͒. ͑C2͒ ͚ij where the ’s are real fermions. Following the same kind of In this case, the Higgs phase has superfluid order, and there- arguments as before we find the phase diagram shown in Fig. fore is fundamentally distinct from confined insulating 11. There is no Higgs phase for the fermions which leads to phases. Thus we can no longer easily claim the equivalence phases 2Ј and 2Љ being distinct even for finite . of the two phases in which either or ͑though not both͒ fields are confining. We can again attempt to construct a phase diagram fol- 2. Full Action lowing construction on each of the outside faces of the cube. Let us now consider the action ͑105͒ and ask how many The ABCD plane is the same as in Fig 8. The BCGF plane truly distinct phases it has. The phase space of this model is (K1ϭϱ) will now have three phases: a confining and a de- large and an explicit construction of the full phase diagram is confining phases without broken XY symmetry ͑phases 2Љ difficult. We note, however, that the charge sector of the 60 and 4͒, and a phase with broken XY symmetry. When K1 theory is precisely the same as Eq. ͑C2͒. Consequently, if in ϭ0 ͑ADEH plane͒ we have four phases. This is obvious Eq. C2, the two phases 2 and 2Ј are smoothly connected to from the fact that when we integrate out ij we find that ij each other, they will necessarily be so for the full action as and cos i are decoupled from each other, and we have sepa- well. If on the other hand, in Eq. ͑C2͒, the two phases are rate order-disorder and confinement-deconfinement transi- distinct, then that is evidence ͑though not proof͒ that they are tions. We therefore find two superfluid phases SF1 and SF2 distinct in the full theory as well. Thus unambiguous deter- that differ in their degeneracy on the nontrivial manifolds ͑on mination of the phase diagram of Eq. ͑C2͒ will shed consid- the cylinder it is 1 for SF1 and 2 for SF2). The origin of this erable light on the important conceptual issue of whether extra degeneracy for SF2 is that it has a finite energy topo- CBSF and CFSB are distinct or not.
FIG. 11. Phase diagram of Eq. ͑C3͒. When the matter field is Fermionic distinction between phases 2Ј and 2Љ survives for all .
155103-24 FRACTIONALIZATION PATTERNS IN STRONGLY . . . PHYSICAL REVIEW B 65 155103
While the distinction between CBSF and CFSB, if any, is 2 ϩ d k † † subtle, it is very clear that they are both distinct from O ϭ c ͑k͒c ͑ϪkϩQ͒. ͑D2͒ ͵ 2 ↑ ↓ CSBNF. One cannot find any vison attachment that would ͑2͒ map the spectrum of CSBNF to either CBSF or CFSB, which In other words, the triplet px density wave and the p-wave proves rigorously that it is a phase fundamentally distinct superconductor are related in precisely the same way as a from the other two. charge-density wave and an s-wave superconductor. In par- To summarize the discussion in this section, we argued ticular, Hamiltonians with short-ranged interactions can be that studies, numerical or otherwise, of simple models of the constructed for which both states are exactly degenerate; form of Eq. ͑C2͒ should be extremely useful in deciding on such Hamiltonians could describe a critical point between the issue of whether CBSF and CFSB are distinct quantum these two states. phases. One can, however, prove rigorously that CBSF and In the triplet px density wave state, there is no spin mo- CFSB are fundmentally different from the other partially ment ͑at any wave vector͒, since the right-hand side of Eq. confining phase of Eq. ͑105͒ CSBNF. The other phases of ͑D1͒ vanishes upon integration over kជ. However, the spin- Eq. ͑105͒: the fully confining phase ͑CSF͒ and the fully de- nematic order parameter, which may be calculated from Eq. confining phase ͑CBSBNF͒, will be distinct from any of the ͑D1͒, is nonvanishing: partially confining ones and from each other as may be seen 1 1 from their degeneracy on nontrivial manifolds. S S Ϫ ␦ S2 ϭ ͉⌽ជ ͉2diag͑2/3,Ϫ1/3,Ϫ1/3͒, ͳ i j 3 ij ʹ 2 Q ͑D3͒ APPENDIX D: UNFRACTIONALIZED PHASES where Si is the ith component of the total spin of the system. In this paper, we have, for the most part, focused on states Hence this is the natural spin-nematic state which results in which the electron is fractionalized. However, even the when a p-wave superconductor is quantum disordered by transitions which do not lead to electron fractionalization are flux-hc/2e vortex condensation. The possibility of spin nem- rather interesting. One would ordinarily assume that strong atic states in the context of high-Tc cuprates has been pro- quantum fluctuations will completely disorder a p-wave su- posed in Ref. 62. perconductor. However, as we pointed out in Sec. IV B, if When the spin degrees of freedom are disordered, but the hc/4e vortex- disclination composites are gapped, then the charge remains ordered, the triplet p-wave superconducting spin symmetry can be restored without affecting the charge; order parameter and the spin-nematic order parameter van- alternatively, the superconductivity can be destroyed without ishes; only the charge-4e order parameter is left. The con- affecting the spin ordering. densation of merons causes the Fermionic quasiparticles to Let us consider, first, what happens when flux hc/2e vor- be confined to spin. Once the merons have condensed, the tices condense, but no other topological defects condense. topological quantum number in the spin sector is no longer well defined, so the flux hc/4e vortex- disclination com- Then the charged degrees of freedom are disordered, but the posites become simple flux hc/4e vortices, as we would ex- spin nematic order parameter should be undisturbed. Follow- pect for a charge-4e superconductor. Said differently, the ing the arguments of Ref. 61, a possible unfractionalized spin meron condensate screens the spin topological charge of the nematic insulating state is ͑in the notation of Ref. 61͒ a trip- flux hc/4e vortex- disclination composites, thereby making let p density wave: x rendering them simple flux hc/4e vortices. Remarkably, by ␣ quantum disordering the spin sector of a p-wave supercon- ͗␣†͑kϩQ,t͒ ͑k,t͒͘ϭ⌽ជ ជ sin k a. ͑D1͒  Q•  x ductor, we have changed the charge of its order parameter, This state is related to the p-wave superconducting state by a which may, for example, lead to some unusual critical behav- ‘‘rotation’’ generated by ior of the superconducting transition.
1 P.W. Anderson, Science 235, 1196 ͑1987͒. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 ͑1991͒. 2 D.S. Rokhsar and S. Kivelson, Phys. Rev. Lett. 61, 2376 ͑1988͒. 9 R. Jalabert and S. Sachdev, Phys. Rev. B 44, 686 ͑1991͒. 3 X.G. Wen, Phys. Rev. B 44, 2664 ͑1991͒. 10 S. Sachdev and M. Vojta, J. Phys. Soc. Jpn. 69, Suppl. B, 1 4 T. Senthil and Matthew P.A. Fisher, Phys. Rev. B 63, 134521 ͑2000͒. ͑2001͒. 11 X.G. Wen and Q. Niu, Phys. Rev. B 41, 9377 ͑1990͒; X.G. Wen, 5 D.P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 ͑1988͒. ibid. 44, 2664 ͑1991͒. 6 B. Shraiman and E. Siggia, Phys. Rev. Lett. 62, 1564 ͑1989͒; C. 12 S.A. Kivelson and D.S. Rokhsar, Phys. Rev. B 41, 11 693 ͑1990͒. Jayaprakash, H.R. Krishnamurthy, and S. Sarker, Phys. Rev. B 13 L. Balents, M.P.A. Fisher, and C. Nayak, Phys. Rev. B 60, 1654 40, 2610 ͑1989͒; K. Feinsberg, P. Hedega˚rd, and M. Brix Ped- ͑1999͒. ersen, ibid. 40, 850 ͑1989͒; C.L. Kane, P.A. Lee, T.K. Ng, B. 14 P.W. Anderson, Mater. Res. Bull. 8, 153 ͑1973͒; P. Fazekas and Chakraborty, and N. Read, ibid. 41, 2653 ͑1990͒. P.W. Anderson, Philos. Mag. 30, 23 ͑1974͒. 7 T. Senthil and Matthew P.A. Fisher, Phys. Rev. B 62, 7850 15 S.A. Kivelson, D.S. Rokhsar, and J.P. Sethna, Phys. Rev. B 35, ͑2000͒. 8865 ͑1987͒. 8 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 ͑1991͒; S. 16 F.D.M. Haldane and H. Levine, Phys. Rev. B 40, 7340 ͑1989͒.
155103-25 DEMLER, NAYAK, KEE, KIM, AND SENTHIL PHYSICAL REVIEW B 65 155103
17 N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 ͑1989͒. Zee, Int. J. Mod. Phys. B 4, 437 ͑1990͒. 18 S. A. Kivelson, Phys. Rev. B 39, 259 ͑1989͒. 38 D.H. Lee and C.L. Kane, Phys. Rev. Lett. 64, 1313 ͑1990͒. 19 T. Senthil and Matthew P.A. Fisher, Phys. Rev. Lett. 86, 292 39 K. Moon et al., Phys. Rev. B 51, 5138 ͑1995͒. ͑2001͒. 40 E. Demler, C. Nayak, and S. Das Sarma, Phys. Rev. Lett. 86, 20 A.J. Leggett, Rev. Mod. Phys. 47, 331 ͑1975͒. 1853 ͑2001͒. 21 Y. Maeno et al., Nature ͑London͒ 372, 532 ͑1994͒. 41 S. Sachdev, Phys. Rev. B 49, 6770 ͑1994͒. 22 J.G. Bednorz and K.A. Muller, Z. Phys. B: Condens. Matter 64, 42 T.K. Ng, Phys. Rev. Lett. 82, 3504 ͑1999͒. 189 ͑1986͒. 43 T.K. Ng, cond-mat/9911099 ͑unpublished͒. 23 R.H. MacKenzie, Science 65, 820 ͑1997͒; S. Lefebvre et al., 44 C. Nayak, Phys. Rev. B 62, R6135 ͑2000͒. cond-mat/0004455 ͑unpublished͒. 45 In principle, we should also consider nonsinglet charge-4e super- 24 4e G. Stewart, Rev. Mod. Phys. 56, 755 ͑1984͒. conductors, with ⌬ <0, Q<0 but vanishing ⌬␣ . These occur 25 E. Demler and F. Zhou, cond-mat/0104409 ͑unpublished͒. when the flux hc/4e vortex- disclination composite becomes 26 See, for example, Brandon Carter, astro-ph/0010109 ͑unpub- unstable and the flux hc/4e vortex and the disclination exist lished͒. as separate stable excitations. These states will be discussed in 27 T. Senthil and M.P.A. Fisher, J. Phys. A 34, L119 ͑2001͒. subsequent publications. 28 L. Balents, M. Fisher, and C. Nayak, Int. J. Mod. Phys. B 12, 46 E. Fradkin and S.H. Shenker, Phys. Rev. D 19, 3682 ͑1979͒. 1033 ͑1998͒. 47 F. Wegner, J. Math. Phys. 12, 2259 ͑1971͒. 29 L. Balents, M. Fisher, and C. Nayak, Phys. Rev. B 61, 6307 48 Phase Transitions and Critical Phenomena, edited by C. Domb ͑2000͒. and M. Green ͑Academic Press, New York, 1977͒. 30 G. Baskaran, Z. Zou, and P.W. Anderson, Solid State Commun. 49 X.G. Wen, Adv. Phys. 44, 406 ͑1995͒. 63, 973 ͑1987͒; J.B. Marston and I. Affleck, Phys. Rev. B 39,11 50 A. Fetter et al., Phys. Rev. B 39, 9679 ͑1989͒; W. Chen et al., J. 538 ͑1989͒; G. Baskaran and P.W. Anderson, ibid. 37, 580 Mod. Phys. 3, 1001 ͑1989͒; X.G. Wen and A. Zee, Phys. Rev. B ͑1988͒; G. Kotliar and Liu Jialin, ibid. 38, 5142 ͑1988͒; N. Read 44, 274 ͑1991͒. and B. Chakraborty, ibid. 40, 7133 ͑1989͒; L.B. Ioffe and A.I. 51 V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59, 2095 ͑1987͒; Larkin, ibid. 39, 8988 ͑1989͒; P.A. Lee and N. Nagaosa, ibid. 46, X.G. Wen, Phys. Rev. B 40, 7387 ͑1990͒; X.G. Wen et al., ibid. 5621 ͑1992͒; A.M. Tikofsky and R.B. Laughlin, ibid. 50, 10 165 44, 274 ͑1991͒. ͑1994͒; B.L. Altshuler, L.B. Ioffe, and A.J. Millis, ibid. 53, 415 52 F. Wilczek, Phys. Rev. Lett. 48, 1144 ͑1982͒. ͑1996͒; P.A. Lee and X.G. Wen, J. Phys. Chem. Solids 59, 1723 53 Y.B. Kim, C. Nayak, E. Demler, N. Read, and S. Das Sarma, ͑1998͒. Phys. Rev. B 63, 205315 ͑2001͒. 31 E. Fradkin and S.A. Kivelson, Mod. Phys. Lett. B 4, 225 ͑1990͒. 54 A.V. Chubukov, S. Sachdev, and T. Senthil, Nucl. Phys. B 426, 32 S.L. Sondhi and R. Moessner, Phys. Rev. Lett. 86, 1881 ͑2000͒; 601 ͑1994͒. R. Moessner, S.L. Sondhi, and E. Fradkin, cond-mat/0103396 55 S. C. Zhang, Science 275, 1089 ͑1997͒. ͑unpublished͒. 56 S. Murakami, N. Nagaosa, and M. Sigrist, Phys. Rev. Lett. 82, 33 C. Nayak and K. Shtengel, cond-mat/0010242, Phys. Rev. B ͑to 2939 ͑1999͒. be published͒. 57 E. Demler and S.-C. Zhang, Ann. Phys. ͑N.Y.͒ 271, 83 ͑1999͒. 34 D. Vollhardt and P. Wolfe, The Superfluid Phases of Helium 3 58 S. Rabello, H. Kohno, E. Demler, and S.-C. Zhang, Phys. Rev. ͑Taylor and Francis, New York, 1990͒. Lett. 80, 3586 ͑1998͒. 35 A.M. Polyakov, Gauge Fields and Strings ͑Harwood Academic 59 D.J. Scalapino, S.C. Zhang, and W. Hanke, Phys. Rev. B 58, 443 Publishers, New York 1987͒. ͑1998͒. 36 A. Auerbach, Interacting Electrons and Quantum Magnetism 60 P.E. Lammert, D.S. Rokhsar, and J. Toner, Phys. Rev. Lett. 70, ͑Springer-Verlag, New York, 1994͒. 1650 ͑1993͒. 37 C. Dasgupta and B. Halperin, Phys. Rev. Lett. 47, 1556 ͑1981͒; 61 C. Nayak, Phys. Rev. B 62, 4880 ͑2000͒. M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 ͑1989͒; 62 L.P. Gorkov and A.V. Sokol, JETP Lett. 52, 504 ͑1990͒;V. M.E. Peskin, Ann. Phys. ͑N.Y.͒ 113, 122 ͑1978͒; P.R. Thomas Barzykin, L.P. Gorkov, and A.V. Sokol, Europhys. Lett. 15, 869 and M. Stone, Nucl. Phys. B 144, 513 ͑1978͒; X.G. Wen and A. ͑1991͒.
155103-26