Aspects of Non-Abelian Many Body Physics Citation Bradford, Kent B. (1997) Aspects of Non-Abelian Many Body Physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/rn7x-3882. https://resolver.caltech.edu/CaltechTHESIS:11192019-171459529 Abstract The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Preskill, John P. Thesis Committee: Preskill, John P. (chair) Politzer, Hugh David Wise, Mark B. Weichman, Peter B. Defense Date: 23 May 1997 Other Numbering System: Other Numbering System Name Other Numbering System ID UMI 9800345 Record Number: CaltechTHESIS:11192019-171459529 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11192019-171459529 DOI: 10.7907/rn7x-3882 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 13579 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 20 Nov 2019 01:32 Last Modified: 19 Apr 2021 22:32 Thesis Files Preview PDF - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

Aspects of Non-Abelian Many Body Physics Thesis by Kent B. Bradford In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1997 (Submitted May 23, 1997) 11 ACKNOWLEDGEMENTS First and foremost, I would like to acknowledge my advisor, John Preskill, with­ out whose guidance and support this project would not have been possible. I also benefited greatly from discussions with John Hartman, Patrick McGraw, and Daniel Gottesman. My brother Mark gave me invaluable guidance after I started writing code early in the project. I am deeply indebted to my parents, who gave me the support, both financial and moral, to persevere in my graduate studies. I would like to thank Darla for her kindness, warmth, and sense of humor. There aren’t words to express the affection I feel for my sister Renata, whose heroism in the face of extreme adversity has touched me more than she could possibly know. I have known Chris Springfield longer than any other person in Southern Califor­ nia, having had the privelege of studying with him during our undergraduate days at the Colorado School of Mines. I have not known Dave Santiago quite as long, but such is the nature of our friendship that I feel like I have known him ail my life. This thesis is dedicated with love to Corinne Gilliam, who has made the last year of my Caltech career the happiest, and whom I will cherish always. iii Abstract The general formulation of quantum statistical mechanics hints at interesting gen­ eralizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 -f 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S3 discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a pe­ riodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z2 charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace. IV Contents 1 2 hιtroduction 1 1.1 Non-abelian Vortices......................................................................................... 2 1.1.1 Electric Charge..................................................................................... 4 1.1.2 Vortex-Vortex Interactions................................................................. 4 1.1. 3 Cheshire Charge .................................................................................. 7 1. 2 Braid Statisticc................................................................................................... 8 1. 3 Exotιc Statistics in Condensed Matter Systems.......................................... 11 Algebraic Aspects of the Many-Vortex Problem 13 2 .1 Preliminary Definitions..................................................................................... 13 2.2 The Sector Group................................................................................................ 14 2.3 Sectors Of The Three Vortex Problem ....................................................... 15 2.4 Sectors Of The Four Vortex Problem........................................................... 19 2.4.1Φt = (1233................................................................................................ 19 2.4.2 Φt=e....................................................................................................... 21 Many-vortex Sectors......................................................................................... 22 2.5 3 A Non-abelian Vortex Lattice Gas 27 3.1 The Patching Convention .............................................................................. 27 3.2 The Multi-vortex Hilbert Space..................................................................... 29 3.2 .1 Position Space...................................................................................... 29 3.2.2 The Internal Space............................................................................... 30 3.3 The Lattîce Hamiltonian.................................................................................. 31 3.4 Translation Symmetry on the Cylindds....................................................... 32 3.5 Ground State Structure 36 .................................................................................. v 4 39 Summary and Conclusions A Proofs of the Lemmas in Section 2.5 41 B Numeric,α Ciaculation of Subspaces and Sector Groups 49 B.0.1 Singular Value Decomposition.......................................................... 49 B.0.2 Sector Group Calculatioos................................................................. 49 B.0.3 Calculation of the Bosonic Subspace 51 ............................................. C The Modified Léaiczos Method 52 C.l Useful TecllISsqιles................................................................................................ 53 C.2 Implementation Detaail...................................................................................... 56 Bibliography 57 vî List of Figures 1.1 A standard path to define the flux of an isolated vortex........................... 5 1.2 A set of standard paths....................................................................................... 5 1.3 (a) The vortices a and b before the exchange, (b) The paths <α, and 0' deform to the correct paths after the exchange............................................ 3.1 Ordering convention for vortices on the lattice. xq is the b^^e^p^ii^ni for flux measurements................................................................................................. 3.2 6 28 Three particles on a 3 × 4 laatice: the dashed llne ls the Inneenational Dateline..................................................................................................................... 33 Vil List of Tables 2.1 Characters of the two "extra” representations of Td connained in the charged subspace: u = exp(⅜πi)..................................................................... 19 2.2 Character table of Z3: φ = exp((πi).............................................................. 19 2.3 table of S3............................................................................................ 21 3 .1 Uncharged-sector g. s. eneιrges........................................................................... 38 1 Chapter 1 Introduction Generalizations of Bose and Fermi statistics in 2 -f- 1 dimensions have been studied for many years. Most familiar is anyon statistics, in which the wave function acquires an arbitrary phase upon the interchange of two indistinguishable anyons [24]. Anyon statistics are exhibited in nature by the quasiparticles of the Fractional Quantum Hall Effect (FQHE) [21]. The behavior of many-anyon systems may be probed through Mean Field Theory[3] or exact numerical analysis of finite-size systems[10, 11]. This thesis is concerned with a broader generalization of quantum statistics, namely non-abelian statistics [16, 32, 8]. Specifically, we consider collections of non­ abelian vortices[4, 28, 25, 23], particles which are labeled with a quantum number called flux (the reason for this terminology will be made clear in Section 1.1) that takes its value in a non-abelian group. Non-abelian particles call for a generalized notion of indistinguishability. The flux associated with a particular particle in the many-body system depends on its history. There is at present no Mean Field Theory for many-body non-abelian systems, and even the problem of three bodies is not well understood[25]. In this chapter, we review the basic properties of non-abelian vortices and their interactions, most notably the holonomy interaction and Cheshire charge. We briefly discuss the general framework of quantum statistical mechanics, in particular braid statistics in 2 + 1 dimensions. In Chapter 2, we discuss the algebraic properties of many-vortex systems. The model that is the basis for the calculations in this thesis is introduced, followed by the definition of a sector group. We then discuss the three- and four-vortex sectors of the model in some detail, followed by a generalization to n-vortex sectors. In Chapter 3 , we introduce a lattice-gas model suitable for numerical studies of non-abelian systems. The ground state structure of three and four vortex systems on the lattice is discussed. 2 Chapter 4 contains a summary and concluding remarks, as well as possible future directions for work in this field. 1.1 Non-abelian Vortices Non-abelian vortices arise as topological defects in spontaneously broken non-abelian gauge theories in (2 + 1) dimensions. Consider a model with a scalar Higgs field Φ coupled to a non-abelian gauge field with gauge group G[4, 5]. With the hermitian generators Ta of G in the representation of the Higgs field and gauge coupling s, the gauge-covariant derivative operator is Dμ = ∂fi + îsA“Tc. The Lagrangian is C= - + (DμΦ)'DfiΦ + F(Φ) (1.1) containing kinetic terms for the gauge and Higgs fields and the invariant potential for the Higgs. In the Higgs phase, V(Φ) is minimized for a non-vanishing Higgs field, hence Φ acquires a vacuum expectation value (vev) (Φ) and the symmetry is broken to the symmetry group H of the Higgs vev. Due to the gauge symmetry, any element of G acting on a vev produces a vev, and any element of H acts trivially on a vev by definition, so the vacuum manifold is the coset space G∕ H. We now look for time-independent finite-energy solutions to the field equations of Eqn. 1.1. The requirement that the energy be finite implies that the fields must approach the vacuum configuration at spatial infinity. Therefore, the gauge potentials A“ must approach a pure gauge and the Higgs field at infinity must take its values in the vacuum manifold. The field equation for Φ is DμΦ = 0, which can be integrated to yield Φ(r') = a(r', r, C)Φ(r) where a(r,, r, C) = Pe×p ^'s J J a (1.2) (1.3) Here P denotes path-ordering, and C is a path from r to r'. Consider a circle of very 3 large radius parameterized by an angle since the Higgs field is tovariastly constant, we have {Φ(0))= α(*){Φ(0)) (1-4) Sisgle-valuedness of the Higgs condensate requires that a(2n) ∈ H. We can now determine the conditions under which stable topological defects exist. a(θ) is a map from Si to the vacuum manifold, so there are stable defects if the fundamental group πi(G∕H) is sos-trivial. Any path for which α(2ii) 6 H is closed in the vacuum manifold, and hence is an element of πι(G∕H). If G is not simply- connected, we consider the universal covering group G and the lift H of the unbroken gauge group into the covering group; then ii(G∕H) = π(G∕H). Two closed paths for which a(2π) are hi and ft2 are homotopy if and only if hi and /i2 axe in the same connected component of H ; therefore, ki(G∕H) = π0(H). If H is a discrete group, every connected component of H consists of a single group element, so we can endow π⅛ with a group structure; π0(H) = H. To summarize, if G is spontaneously broken to a discrete group i∕, there will be topological defects characterized by a magnetic flux taking its value in H 1. A sos-abelian vortex is a topological defect in (2 + 1) dimensions associated with a sos-abelian discrete unbroken gauge group H. Is the case where H is discrete, all of the gauge bosons pick up a mass of order eu, where v sets the scale of the Higgs vev. For processes that take place at energies well below v, we can take v to be infinite; the vortices are effectively poist-like, and there are so long-range gauge fields. That does sot mean that there are so long-range interactions between the vortices; we shall see in Sec. 1.1.2 that vortices interact topologically through a sort of generalized Aharonov-Bohm interaction.* 'The identification of the group element a(2îr) with a magnetic flux follows from the abelian case G = I'll), in which case a(2ιr) = exp(ιeΦβ), where Φβ is the magnetic flux linked by the loop. 4 1.1.1 Electric Charge The defects discussed in the previous section can carry electric charge as well as magnetic flux. The electric charge of a dyon (a flux/charge composite) is specified by the transformation properties of the state under a global gauge transformation: a state with definite charge is associated with an irreducible representation of the global gauge group. In particular, a state with zero charge always transforms according to the trivial representation. If the unbroken gauge group is non-abelian, in general the globally defined gauge group in the presence of a vortex is a subgroup of the local unbroken gauge group H [6, 1]. This can be seen as follows; consider a circle around the vortex, parameterized by an angle 9. At every 9 there is a subgroup H(9) C G that stabilizes the Higgs condensate at 0; all of the H(β) are isomorphic to the same abstract group H, but the embedding of H in G varies smoothly with θ 2. 3 Let Sa(0) be a basis for the generators of S∕γ(0). Then Sa(θ) = a(θ)Sa(0}a~1(9) (1.5) For a transformation to be globally well defined, it is necessary for α(2τr) to com­ mute with Sa(O): therefore, the global gauge group consists of all elements of H that commute with the vortex flux α(2π∙), i.e., the global gauge group is the normalizer 3 N(a(2π}) of the magnetic flux. 1.1.2 Vortex-Vortex Interactions We now consider the problem of patching single vortices together into a multi-vortex configuration. This will involve establishing a number of conventions to uniquely specify the state of the multi-vortex system[4, 8]. Start with an isolated vortex. First, we select an arbitrary basepoint xo and a standard path C that encircles the vortex in a definite sense (cf. Fig. 1.1). The flux 2In other, words, H{θ) is a section of a G principle fiber bundle over Si. 3The normalizer of a group element is the set of all elements of the group that commute with the given element: N(a) = {h 6 H :ha — ah}. φ ο Figure 1.1: A standard path to define the flux of an isolated vortex. 3 Figure 1.2: A set of standard paths. of the enclosed vortex is a(C, xo) = P exp (y j 6H A ■ dx (1.6) Since the gauge connection is flat everywhere outside the vortex core, the flux linked by any loop that can be smoothly deformed to C without crossing the vortex is the same. If there are multiple vortices, it is necessary to choose a set of standard paths that encircle the vortices in a definite order (Fig. 1.2). The total flux of the multi­ vortex configuration is specified by the product (in the group H) of the individual fluxes, in the order they are encircled. For example, the total flux in Fig. 1.2 is a(Cι,x0)a(C2,XQ)a(C3, xy The importance of assigning a definite order to the single­ particle fluxes in the multi-vortex confi{g.lΓatron is readily apparent if the group H is non-abelian. The fact that the gauge connection is flat outside of curvature singularities at 6 (a) (b) Figure 1.3: (a) The vortices a and b before the exchange, (b) The paths a' and 0' deform to the correct paths after the exchange. the vortex cores implies that the locations of the vortices must be excised from the space. Consider n vortices on a connected, oriented two-dimensional space Σ, asd let ∑(n) be the space Σ with holes at the vortex positions. It is clear that our choice of ba-sep^nt asd standard paths on ∑(n) amounts to a homomorphism p from the fundamental group of ∑(n) to the group H[23]: p : 7r1(Σ(s),x0) ∣→ H (1.7) Is general, there will be a number of topologically inequivalent sets of standard paths to specify the individual vortex fluxes (and the total flux). This is the patching ambiguity. By prescribing a set of standard paths, we are fixing a convention to resolve this ambiguity; it is somewhat analogous to the procedure of fixing a gauge. Osce we choose a set of standard paths, we must continue to measure fluxes relative to this choice. This has an important physical consequence if the vortices are free to move os ∑(n), as we shall sow see. In Fig. 1.3, a vortex whose flux is a £ H relative to the standard path a is about to trade places in a counterclockwise sense with a vortex whose flux is b relative to the path 0. In order to specify the particle fluxes after the exchange, it is necessary to construct paths that will deform to standard paths after the exchange, without crossing the core of asy vortex. These paths are labeled a' and 0'. Homotopically, we see that β' ~ a, a, ~ a ∣∫3q (1.S) Hence we come to the conclusion that, after the exchange, the particle whose flux originally waa b now hss flua aba~ 1, which is distinct (müsss a and 6 commuee) . Tiis long range effect is eeraeed tha “l^c^ll^nc^I^^ tt ss essentially a non-abelian variant of the Aharonov-Bohm effect4. The holonomy interaction can be expressed in terms of the braid operator R: if we express the original state of Fig. 1.3(a) as |a,6), the braid operator performs a counterclockwise exchange: R∣a,b) = ∣aba~1,a) (1-9) Clearly, a clockwise exchange is achieved with the operator R~l. The total flux a 6 of the state is preserved by the braid operation. A global gauge transformation h acts on a vortex flux by conjugation: a — hah~l. It may seem, then, that a flux should merely be labeled by a conjugacy class. This is not true, however; a b = hah~l ^ hbh~l, so vortices distinct in one gauge are distinct in every gauge. One may conclude that the conjugacy classes in H form degenerate multiplets. 1.1.3 Cheshire Charge Non-abelian topological defects have a curious property; it is possible for them to carry electric charge that is nonlocalizable, i.e., it is not associated with a physical, gauge invariant charge density localized on the object. This sort of charge, which is called Cheshire charge, is a global, topological effect. A simple example will suffice to illustrate the concept of Cheshire charge. Consider an unbroken gauge group H = S3, and the two-vortex state |( 12), (23)). The vortices are assumed to be pure fluxes, not dyons, so individually they carry no electric charge. 4It is distinguished from the original Aharonov-Bohm effect in that the holonomy interaction is classical, not intrinsically quantum mechanical. 8 The total flux of this state is (321). The globally defined gauge group is the centralizer of (321) in S3 (c∕. Sec. 1.1.1), which is just the Z$ subgroup {e, (123), (321)}. Under a global gauge transformation by the element (123), the state transforms to |(23), (31)). The latter state is transformed to ∣(31),(12)) by a (123) gauge transformation. We may now combine these states into three states that axe invariant under the global gauge group {4 ~ exp(2Zπ-∕3))r W = ⅛12)'<23)> + ∣(23), (31)) + ∣(31), (12))) (1.10) |4) = ~^(I12))(23)) +<11(23), (31)) + *⅛),(12))) (1.11) |c) = i-(L1∣(i2), 223)) + ^2∣2^3), (31)) + ≠∣(31), (12))) (1.12) Observe that state |a) transforms according to the trivial representation of Zβ, and hence is uncharged, whereas states |6) and |c) transform as the nontrivial represen­ tations of Z3, and therefore carry electric charge. This charge is not localized on either or both of the particles, but is a global property of the two-vortex state; this is Cheshire charge. 1.2 Braid Statistics Here, we briefly discuss the possible varieties of quantum statistics possible in two spatial dimensions[33]. The quantum statistics of indistinguishable particles can be expressed in a gen­ eral way as follows; consider n indistinguishable particles, moving on a manifold M (typically Rd, where d is the spatial dimension). If the particles cannot coincide (like fermions or hard-core bosons), the classical configuration space of the system is [Mn — Z^n], where Dn is the subspace of Mn where two or more particles have the same coordinates. In quantum mechanics, it is necessa^ to identify all configurations that differ by a permutation of the coordinates of Indistinguishable particles, so the 9 configuration space is Cn = [Mn - Dn]∕Sn, (1.13) where Sn is the permutation group on n objects. In the path integral formalism of quantum statistical mechanics, one sums over all histories cosnecting a given initial and final state (if we are calculating a thermal average, for example, the initial and final states are identified). If the configuration space Cn is not simply connected, the path integral decomposes into a sum of path integrals over disjoint sectors, corresponding to the homotopy classes of Cn. The terms is the sum corresponding to different homotopy classes seed sot be weighted equally, as long as overall unitarity is respected. If ose defines a set of “exchange operators” that carry the final configuration of the system around a closed path in Cs, clearly the different homotopy sectors are mixed by the operators, and if the sectors are weighted differently, the amplitude is transformed. By considering applying two such exchanges in turn, it is clear that the exchange operators generate a unitary representation of the fundamental group 7Γι(Cs). For M = Rd, d > 3, t1(C∏) — Sn, and exotic statistics are sot allowed. However, for d — 2, τ∙i(Cπ) = f?s, the braid group os n strands. Bn is generated by (n — 1) braid generators σ,∙, which obey the Yasg-Baxter relations: σia-j = o-jC-,, ∣i — j\ > 2 (1.14) = σ,∙+iσ,∙c∙,∙+ι, 1 <i<n-2 (1.15) These are the only defining relations for Bn, which is as infinite, discrete, sos-abelias group. Unitary, nsesdimessiosal representations of Bn exist, is which each generator is represented by the same phase, <,∙ = etθ, 1 < i < n —1 (these are the only abelias representations consistent with the Yasg-Baxter relations). The phase θ is arbitrary, asd hesce interpolates between Bose asd Fermi statistics. Systems of particles that transform as a nsesdimessinnal representation of Bn with 6 ^ 0, t are called anyons; asyos statistics is knows to be relevant is certain two-dimensional Quantum Hall 10 systems (the Fractional Quantum Hall Effect, FQHE). Non-abelian vortices transform according to non-abelian representations of Bn. If one considers an arbitrary three-vortex state |a,6, c), along with two braid operators Ri, R2 that braid the first and second pair, respectively, it is easy to demonstrate that R]R2Ri = R2R1R2· For n particles, there are n — 1 braid operators H,∙, and all of the Yang-Baxter relations are easily verified. In systems with non-commuting vortices, multi-dimensional irreducible representations of Bn arise. These systems are said to obey non-abelian statistics. As emphasized by Lo and Preskill[25], non-abelian vortices may be treated in the above general framework, if one extends the usual concept of “ineistinguishhbility.” Two vortices are considered indistinguishable if their fluxes lie in the same conjugacy class of H, and they are in the same representation of the normalizer of their respective fluxes5 {i.e, they have the same charge). In other words, vortices are indistinguishable if it is possible (through topological interactions with other vortices) for an exchange amplitude to interfere with a direct amplitude. Two vortices a and 6, with the same charge but fluxes belonging to distinct elements of the same conjugacy class, are not identical, in that a 6 vortex will not be annihilated by an a anti-vortex, but they are treated as indistinguishable for the purposes of statistics. As shown by Bais et a∕.[5], the classification of indistinguishable vortices is inti­ mately related to the representation theory of the quasi-triangular Hopf algebra, or “quantum double” [7, 13]. The quantum double D{H) associated with the group H is an algebra of order ∣H∣2 generated by the operators (using the notation of Refs. [25, 23]) P∏g, h,g 6 H: here, g performs a global gauge transformation, and Ph projects out the total flux h. The projection operators obey PhPg = fh,gPh, aPha~1 = Paha~' (1-16) and hence {Pha)(P3b) = δh,aga-i(Phab) (1-17) 5It is easily shown that the normalizers of two group elements in the same class are isomorphic. 11 Following Ref. [5], let aC be a conjugacy class of H, and °Γ the a-th irreducible representation of the normalizer aN of aC in H. A flux/charge sector is labeled AC,a Γ). The flux/charge sectors are irreducible representations of the quantum double D(H)[5, 23, 25]. In fact, the sectors form a complete set of irreducible rep­ resentations of D(H). The particles belonging to a sector should be considered a degenerate multiplet[5]. It should be clear from the proceeding discussion that two vortices are to be considered indistinguishable if and only if they belong to the same irreducible representation of D(H)[25]. 1.3 Exotic Statistics in Condensed Matter Systems Thin-layer condensed matter systems, such as two-dimensional inversion layers or MOSFETS, provide essentially two-dimensional systems in which braid statistics may be realized. Indeed, it is strongly believed that quasiparticles with fractional statistics arise in systems that exhibit the Fractional Quantum Hall Effect[21]. The incompressible Laughlin wavefunctîond Φm for N electrons is Φm(2,,...,2jv) = ⅛-a)mexp{-∑¾ ∏ l<j<k<N (1.18) j=l 4*0 Here zj are the complex coordinates of the electrons, m is an odd integer, and lq is the magnetic length. This state is a good approximation to the ground state of a Quantum Hall system at filling factor v = 1∕m. There are excitations around this state called quasiholes. The state with a quasihole at position z0 is, in terms of the Laughlin state Φm, N ^m )(20,2l,∙..,2v) = Π (¾-∙ε0)Φm(2l,..∙,2Υ) ( 1-19 ) (j=1) The quasihole has fractional charge e∕m, where e is the electron charge[14, 22]. Con­ sidering multi-quasihole solutions, we may expect that the Aharonov-Bohm effect on a fractionally charged excitation slowly dragged around another quasihole will yield 12 fractional statistics^]. Arovas, Schrieffer asd Wilczek demonstrated this using a Berry phase calculation [2]. For non-abelian statistics to be possible, there must be a set of degnerate ex­ citations, that transform among each other by nos-trivial matrices under adiabatic exchange. Moore and Read[26] have argued, using conformal field theory, that quasi­ hole excitations of the paired Pfaffian Hail state obey sos-abelias statistics. Recent advances in evaluating the degneracies of multi-quaeihnle paired Pfaffian sta^t^^^[2T, 30] have strengthened this conclusios. 13 Chapter 2 Algebraic Aspects of the Many-Vortex Problem In this chapter, we discuss the structure of the braid algebra, acting on many-vortex systems. Because of the extreme mathematical complexity of the problem, we will for the most part restrict attention to a specific model, described below. This model is simple enough to be somewhat tractable, yet still exhibits most of the interesting features of the general non-abelian problem. Specifically, we will take the unbroken gauge group to be S3, the permutation group on three objects. The indii^nducLl uartialei wül we taken en be ekm^ennt of t hh two-cycle class of S,3, wi3h zero chargo (howgver, canVnrlS'cbtons wito mwie tman om vortex may have a total charge, due to Cheshire charge). Unless otherwise stated, the results in this thesis apply to this S3 model. 2.1 Preliminary Definitions We may assume that a convention has been prescribed that resolves the patching ambiguity for multi-vortex states described in Sec. 1.1.2. Therefore, there exist states in which a definite flux may be assigned to each particle, so a typical n-vortex state is ∣a,b, .. .,k) a,b,.. .,k ∈ H (2.1) The ordering prescribed by the patching convention allows one to specify the total flux Φi associated with the state 2.1, Φi = ab. ..k ∈ H (2.2) 14 Is as s-vortex sector, we may define n — 1 braid operators Ht∙, « «+1 β,∣α,...,∕, * i+l .. ,k) = ∣a,..., fgf 1, f ,... ,k) (2.3) It is easy to verify that the braid operators obey the Yasg-Baxter relations: RiRj = RjRi, (2.4) \i — j I >2 (2.5) RiRi+ιRi = fR+iRRi+i It is also convenient to define the operator C(g), which conjugates the entire state by g— C(g)∣a, b,...,k) = ∖gag~1, gbg 1,..., gkg 1) g€H (2.6) If g is in the centralizer of the total flux Φt, C(g) implements a global gauge transformation. The conjugation operator commutes with all of the braid operators: m R] = o 2.2 (2.7) The Sector Group Having specified the unbroken gauge group to be H = Sß, and the fluxes of the individual vortices, a sector is determined by the number of vortices s, the total flux Φt € S3, asd the total charge, which is the representation of the centralizer of the total flux, N(Φt). If, following Ref. [5], we denote the a-th representation of N(Φt) by 0Γ(W(Φt)), a flux/charge sector cas be specified as (n,Φt,a Γ(√V(Φi))). On a gives s-vortex sector, it is a straightforward exercise to calculate the braid matrices Ri (I < i < n — 1) associated with the actios of the generators σ,∙ of Bn os the many-vortex states. These matrices themselves generate a group, which we will define to be the sector group K(n,Φt,a T). The sector (n, Φt,αr T(iV(φi))) is a (generically reducible) representation space of its corresponding sector group. 15 It is important to note the differences between the sector group K and the braid group Bn. The braid group for a sector is determined entirely by the number of particles (and the topology of the underlying configuration space), whereas the sector group also depends on the unbroken gauge group H, and the total flux and total charge of the sector. While the braid group is an infinite, discrete group, we shall see that the sector group for a finite number of vortices is of finite order. The generators R{ of the sector group obey the Yang-Baxter relations determined by their parent braid group, plus additional defining relations imposed by the struc­ ture of the underlying model (and required by the fact that K is finite). One defining relation is determined by considering the repeated application of a braid generator to a two-vortex state. Either the fluxes associated with the individual vortices commute, in which case R acts trivially on the state, or the fluxes do not commute, and the fluxes associated with the vortices cycle through the conjugacy class of H to which they belong. Since the conjugacy class has a finite number of elements, the original state is restored after some finite number k of braidings. Thus the defining relation may be written Rl- = e (for(l <i<n — 1)). In the S3 model underlying this thesis, the single particle fluxes take their values in the two-cycle class of S3, which has three elements, hence k = 3 for this model: E% = e (2.8) An additional defining relation for K that involves all of the generators for a given sector will be introduced in Sec. 2.5. 2.3 Sectors Of The Three Vortex Problem The product of three two-cycles in S3 is again a two-cycle, so there are three possible values of the total flux Φt. Since it is always possible to arrange for the total flux to be any given element in a conjugacy class of H by an appropriate gauge choice, the three possible total fluxes are equivalent, and we can arbitrarily take Φt to be (31). 16 The centralizer of (31) is {e, (31)} = Z2, which has two irreducible representations, so there are two possible charge states, the uncharged state (corresponding to the trivial representation 1Γ(Z2)), and the charged state, (corresponding to the non-trivial representation 2Γ(Z2)). There are 9 states of three two-cycle vortices that have Φt = (31). This 9­ dimensional space decomposes under the action of the global gauge group into a ö-dimensional uncharged space and a 4-dlmnnsionhl charged space. The Uncharged Sector One state in the uncharged sector may be immediately singled out: W = ∣(∙31), (31), (31)) (2∙9) This state is deedy tHivia undee any bsatding, ss it bbts>l∏s imdee vvotex enchnagg. The other fonu ssates in the uncharged sector are, in arbitrary order, |1> = ⅛(H122, (62), (31»+ I (1Î3), ( 23), ( 33») (2.10) ∣2) = (30, (23»+ I (22» m, K») (2.U) |3) = -^(ICT,^)^»+ I SK» , K2)) (2.12) |4) = ⅛|(31)((12),(12)2+∣S33(,(23),(23») (2.13) Applying the braid operators R∣ and R2 to the states above simply permute the states, so it is clear that the sector group for this sector ∕<(3, (31),1 Γ) C S4. With the states numbered as above, we can represent R-l and R2 as follows: Rx = (1)(234) R2 = (4)(132) (2.14) 17 Since Hi and R2 are both even permutations, we see that the sector group is a subgroup of A4, the even permutations on four objects. In fact, it is quite simple to verify that the sector group is A4: K(3,(31)∕Γ) = A4 (2.15) There is another bosonic state in this sector, namely l⅛) = j(11) + |2) + |3) + ∣4)) (2.16) The remaining 3-dimensional subspace in the uncharged sector corresponds to the 3-dimsnsiosal irreducible representation of A4, and hence is truly non-abelian in nature. The Charged Sector The charged sector is spanned by the states |1) = ⅛∣112), (12), (31)) -K23), (23), (31))) (2.17) |2) = a=(∣112), (31), (23)) -(23), (31), (12))) (2.18) |3) =-^=1[(23), (12),(23)) — ∣112),(23), 112))) (2.19) |4) = -^α(31),^23),(23))-KJ1)ÄM (2.20) Due to the minus signs in the states above, one cannot say that Hi and R2 are simply permutations in S4, because a braid operator acting on a state may produce a sign change as well as a permutation. The signs of the states above were chosen so that Hi does not produce any such sign flips, so that it is possible to write Hi = (1)(234) as before. However, it is no longer possible to write H2 as a permutation. The action of H on the charged subspace is IS ft2∣l) = -∣3) • R* ∣2) = ∣ι> ■ Ri∣3) = -12) ⅛∣4) = ∣4) (2.21) If we restrict ourselves to ordinary representations, we discover that the braid operators generate the group Td, the double group of the point group of the tetra­ hedron (the point group of the tetrahedron T = A4). But it is also possible to view the representation generated by the R's as a projective representation of A4 (a projec­ tive representation differs from an ordinary representation is that the representatives corresponding to group elements <1 and g2 obey p(gi)p(g2) = w(l,,2)p(gιg2), where u(l,2) is a phase factor - for the case at hand, oj = ±1). Since the latter possibility proves to be more easily geseralizable, that is the point of view we will take: K(3, (31),2 T) = A4 (projectively realized) (2.22) In the uncharged sector, we discovered that there were was a two-dimessiosal invariant subspace that obeys Bose statistics. That there are so bosonic states is the charged sector cas be demonstrated by the following argument (which is generalized is Sec. 2.5): Consider the effect of the operator RγR2 on a three-vortex state: R∣R2∖a,b,c) = ∣ΦtcΦ71,α,6) (Φt = abc) (2.23) Applying R1R2 two more times conjugates the original state by the total flux: (R1t^2)3μ^,^√^) = j<tβΦ→, Φt6Φt-l,ΦtcΦt-1) (2.24) 19 5Γ 6Γ Cl 2 2 C3 -u? c2 U -ω2 U C4 Cs 0 0 -2 -2 c6 —U U>2 C7 U2 —U Table 2.1: Characters of the two "extra” representations of Td contained in the charged subspace: u> = exp(^πi) 1Γ 2Γ 3Γ Ci 1 1 1 C2 c3 1 1 φ Φ2 φ> φ Table 2.2: Character table of Z3: φ> = exp(∣7τi) Since Φt ∈ N(Φt)∙, this is a global gauge transformation by the total flux. If the global gauge group is a cyclic group generated by the total flux (as is the case for this sector), every non-trivial global gauge transformation can be effected by a braiding. Therefore, a bosonic state, which is trivial under all braidings, must also be trivial under all gauge transformations, hence it must be in the uncharged sector. The charged subspace is reducible when considered as an ordinary representation of TD. The space decomposes into 5Γ φ6 Γ, whose characters are listed in Table 2.1. It is clear from the characters that these two representations are related by an auto­ morphism. 2.4 Sectors Of The Four Vortex Problem The product of four two-cycles is either a three-cycle or the identity. Just as in the case of two-cycle total flux, the physics can’t depend on which three-cycle is chosen, so we will arbitrarily take the total flux to be (123) in that sector. 2.4.1 Φi = (123) There are 27 states of four two-cycle vortices with Φi = (123). The centralizer of (123) is {e, (123), (321)} = Z3. The 27 states are divided equally among the three inequivalent irreducible representations of Z3 (table 2.2). 20 Φt = (123), Uncharged sector A typical state in the uncharged sector is |1) = -i=(|(12(l((2)(((2)((23)) +1(23), (23), (23),(31)) + |(31), (31), (31), (12))) (2.25) As in the uncharged three-vortex sector, the braid generators may be represented as permutations on the states. In some basis, the braid generators for this sector may be written as Rr = (1)(2)(3)(465)(798) R2 = (1)(6)(9)(275)(348) Rz = (4)(5)(6)(132)(789) (2.26) These are even permutations, so the sector group is a subgroup of Ag. Direct computation (using a computer, cf. App. B) shows that A(4, (123),1 T) is a proper subgroup of Ag, of order 216. There is a bosonic state in this sector, namely IB) = j(H> + 12) + |3) + 14) + |5) + |6) + 17) + |8) + |9)) (2.27) Unlike the uncharged three-vortex sector, there is only one bosonic state in this sector. Φt = (123), Charged sectors Table 2.2 shows that the and 3Γ sectors give essentially the same physics. The braid operators in these sectors cannot be simply represented by permuta­ tions, since unremovable phases arise in the braiding. The phases are cube roots of unity, and the charged sectors can be considered to be Z$ projective representations of the sector group of the uncharged sector. There are no bosonic states in this sector. 21 Ci 1r 2f 1 3r 2 1 C2 1 -1 0 C3 1 1 -1 Table 2.3: Character table of S3. 2.4.2 Φi = e The centralizer of the trivial flux is all of S3 (Table 2.3). There are 27 states of four two-cycle vortices with trivial total flux. Φf = e, Uncharged Sector There are 5 states in this sector. Ose bosonic state may be written down immediately: ∣Bi) = -^=1∣(12),(12),(12),(12),) + ∣(23), (23))(23‰ (23), ) + ∣(31), (31), (31),(31 )>) (2.28) It is possible to write the other four states is such a way that the braid generators are as follows: Ri = (1)(234) R2 = (4)(132) R3 = Ri (2.29) Remarkably, there are only two distinct braid generators in this sector, asd com­ parison with En∏. 2.14 staws that tee pacee of eecoor (4 , e,l T) is isomropht c oo tee space of sector (3, (31)∕ T). In particular, we note that the secoor group K(4, e,1 T) ≥ K{3, (31),1 T) S A4 asd that the bososic subspace is two dimensional. (2.30) 22 Φt = s, Charge 2Γ There are 4 states in the non-trivial one dimensional irreducible representation of S3. Φust as in the uncharged sector, there are only two distinct braid generators, and the space is isomorphic to the charged three-vortex space: Ri — R3 K( 4, e,2 r(S3)) — K{ 3, OU? Γ(Z2)) (2.31) Φi = e, Charge 3Γ This subspace is 18 dimensional. Unlike the other charged sectors considered so far, it has a two dimensional bosonic subspace. The sector group is order 216, like the sector group for the uncharged, Φt — (123) sector. 2.5 Many-vortex Sectors First we will present several formulae and lemmas that are valid for an arbitrary number of vortices. Proofs are given in App. A. The dimensions of the various sectors of the S3 vortex model are given in Lemma 1: Lemma 1 For an odd number n of vortices, the dimensions of the sectors are dim(n, (31),1 Γ(⅞)) — 1(3” + 1) (2.32) Γ(Z2)) — 1(3” - 1) (2.33) dim(n, For an even vortex number n, the sectors with trivial total flux have dimensions dim(^z, e,1 T(53)) — + 1) (2.34) dim(z^, e,2 Γ(53)) — ^(3^2-1) (2.35) dim(n, e ,3 ∏S3)) — 2(3S"2) (2.36) 23 The sectors with even vortex number n, three-cycle flux have dimensions * dim(n, (123),1 T(Z3)) = dim(n, (123),2 T(Z3)) = dim(n, (123),3 T(Z3)) = 37"2 (2.37) The following two lemmas are more general than Lemma 1, in that they are not specific to the S3 model. Lemma 2 is the additional defining relation alluded to in Section 2.2, and Lemma 3 follows immediately from Lemma 2 Lemma 2 In any n vortex sector with trivial total charge, and in sectors with trivial total flux and arbitrary charge, the generators of the sector group obey the relation (R1R2...Rn.l)n=e (2.38) In ant/ sector with n vortices, the generators of the sector group must obey (RlR2...Rn-l)n-no = e (2.39) where no is the order of the global gauge group for that sector. Lemma 3 In any sector whose global gauge group is a cyclic group generated by the total flux, there are no bosonic states in sectors with nontrivial charge. The next two lemmas apply to the S3 vortex model. They generalize the results of the preceeding sections. Lemma 4 In any d-dimensional uncharged sector of the S3 vortex model, the sector group is a subgroup (not necessarily proper) of the alternating group on d objects, that is to say, K(n, Φi,1T) C Ad d = dim(n, Φi,1 r) (2.40) 24 Lemma 5 In any sector whose global gauge group is a cyclic group Z,m generated by the total flux, the space of any non-trivially charged sector is a Z/v projective representation of the sector group of the corresponding uncharged sector. In Sec. 2.4.2, it was noted that the representation of the sector group on os the uncharged, trivial total flux four vortex sector is isomorphic to the representation of the sector group os the uncharged three vortex sector with Φt = (31), and likewise there is as isomorphism between the three vortex 2Γ(Z2) sector and the four vortex, trivial flux 2Γ(5,a) sector. Using Lemma 1, ose sees that the dimension of the un­ charged, Φi = (31) n-vortex sector is always equal to the dimension of the uncharged. φt = e (n + l)-vortex sector, and likewise for the corresponding charged sectors. It turns out that the isomorphism between sector group representations discovered is Sec. 2.4.2 generalizes for the S3 model: Theorem 1 The representation space, with respect to the sector group, of the un­ charged, Φi = (31), n-vortex sector (n odd) is isomorphic to the sector group repre­ sentation space of the uncharged, Φi = e, (n + 1)- vortex sector: that is to say, the corresponding sector groups are isomorphic: K(n, (3I),1T(Z2)) S K(n + 1, √ T(⅞)) n odd (2.41) and the representations (typically reducible) of the sector group on these sections are equivalent. A corresponding isomorphism exists between the charged, Φt = (31), n-vortex representation space and the non-trivial one dimensional charged (2r(S3)), Φt = e (n + 1 )-vortex space. The fisal lemma concerns bosonic subsoaccs. The Bose states is sos-abelias vortex sectors are particularly interesting from a physics etasdpoist, for one expects that the ground state of any sector that has a bosonic subspace will be a Bose state. Lemma 6 Let bose(n, Φt,a T) denote the bosonic subspace of the given sector. For any uncharged sector with an odd number of vortices, and for any uncharged sector 25 with an even number of vortices and trivial total flux, dim^ose^, (31)/ Γ)) > 2 n odd, (2.42) drm(brse(n) e∕ Γ)) ≥ 2 n even., (2.43) dim^ose^, s,^)) > 2 (2.44) n even For an uncharged sector with an even number of vortices and three-cycle total flux, dim^ose^, (123),1 T)) ≥ 1 n even. (2.45) Theorem 1 and Lemma 3 combined show that there are no bosonic states in the non-trivial one-dimensional charged states (^(Sa)) with trivial total flux. Lemma 6 only yields lower bounds on the bosonic dimensions of the uncharged sectors. The results of the numerical procedures described in App. B may help shed light on sectors with more vortices. One finds, for n < 7 vortices, that The bosonic dimension of trivial flux sectors with non-abelian charge 3r(53) is two. • The bosonic dimension of uncharged sectors with Φt — (31) and with Φi — s is exactly two. • The bosonic dimension of uncharged sectors with Φt — (123) is exactly one. It seems likely, therefore, that the bounds in Lemma 6 are saturated. The sector groups grow in size quite rapidly with increasing numbers of vortices. The order of the sector group in the uncharged five vortex sector is at least 25,920, and the order in the seven vortex uncharged sector is greater than 100,000. This is to be expected, since Lemma 1 combined with Lemma 4 show that the groups which contain the sector groups for uncharged sector grow combinatorial^ in the dimension of the sector, which in turn grows exponentially. There can also be a wide gap in the size of sector groups for different sectors with the same number of vortices, as Theorem 1 shows. 26 The sector group for the trivial-flux, non-abelian charge sector with four particles is isomorphic to the sector group of the four vortex zero charge, three-cycle flux sector. This is not surprising, since they have the same number of generators and the same defining relations (including the defining relation implied by Lemma 2). We haven’t proved that no new defining relations occur for larger vortex numbers, so we can’t be certain that the corresponding sector groups will be isomorphic for any number of particles, but it seems likely. In addition, the characters of the representation of the sector group on the non-abelian charge are twice the characters of the three-cycle flux sector. Lemma 1 shows that the non-abelian charge sector is always twice as large as the uncharged three-cycle flux sector for any number of vortices; furthermore, the lower bound on bosonic dimension for the non-abelian charge sectors is again twice the uncharged three-cycle flux lower bound. It is likely, perhaps, that that the non-abelian charge sectors are always isomorphic to two copies of the corresponding uncharged, three-cycle flux sectors. 27 Chapter 3 A Non-abelian Vortex Lattice Gas In this chapter, we investigate the ground state properties of a lattice gas of non­ abelian vortices. In doing so, we are extending the work of Canright and Girvis[10, II] on abelian asyons to nos-abelian particles. They considered identical asyoss on a two-dimensional lattice, with Hamiltonian H = -t 52[c∣cj∙ exp(iφij) + h.c.] + u ^2 Wj (.∙o) (m) (3.1) Here, fij is the anyon phase (with a possible contribution from as externally applied magnetic field), and u is a constant that sets the strength of a nearest-neighbor interaction. The lattice has periodic boundary conditions in one direction and hard walls in the other direction, so topologically it is equivalent to a cylinder. The ground state was iteratively determined using the modified Lanczos techsique]15, 12]. 3.1 The Patching Convention In order to define many-vortex states, it is necessary to define a convention for patch­ ing the single vortex states together. Essentially, this defines as ordering os the single particle fluxes, to specify a unique total flux. On the lattice, we define the following ordering: vortices live os the vertices of the lattice, which may be specified by a row number and a column number. Vortices are numbered starting at the bottom of the leftmost column, moving to the top of the column, then starting again at the bottom of the next column. In other words, a particle has a higher number than another if the number of its column is larger, or if they are is the same column and the number of its row is greater. This order 28 Figure 3.1: Ordering convention for vortices on the lattice, xo is the basepoint for flux measurements. determines the order in which the single particle fluxes fill the multi-particle state, and also the value of the total flux. The meaning of this convention is show in Fig.3.1 for three vortices with single particle fluxes a, 6, and c. The basepoint xo anchors the path C that defines the total flux. This path can be deformed to the path C1C2C3, travelling path C3 first, then Ci, then Ci. This specifies the state and total flux ∣a,6,c) Φi = abc (3.2) The vector potential of a vortex is taken to vanish everywhere except for a delta­ function singularity on a string that starts on the core of the vortex and ends at infinity (or a wall). The ordering convention for vortices in the same column determines how the strings must be deformed to maintain the proper order. The total flux associated with the loop C may be determined by the order in which the strings intersect the loop. As the particles hop around the lattice, the order changes, but the total flux remains constant due to the action of the braid operators. If the particle with flux c in Fig.3.1 hops to the left across the string of the 6 vortex, it becomes conjugated by 6’s flux; this is a counterclockwise braiding on the last pair of particles: ∣a,6, c) t→ i?2|a, 6, c) = ∣a,6c6 1,6) (3.3) 4 29 If the same particle hops to the left again, its string sweeps across the vortex with flux a. leading to a braiding of the first pair of particles in a clockwise sense: ∣a,bcZΓ1,b) h- R1"1 ∖<^, bcb~∖b) = 3.2 (bcb~l)-'a(bcb'l), b) (3.4) The Multi-vortex HiΓb<ert Space Having defined a patching convention, the Hilbert space factors into an internal space, specifying the vortex fluxes, and the position space. A typical three-vortex state on a 2 x 3 lattice may be written \ip) — ∣ internal) ($ ∣poss.) — |a, 6, c) (?) 1 0 0 0 1 1 (3-5) Here, a, 6, c ∈ H are the single particle fluxes, a 1 denotes an occupied lattice site and a 0 denotes an unoccupied site. The patching convention unambiguously specifies which single particle flux is associated with which lattice site. A basis for the Hilbert space of γ vortices with a given total flux on a specified lattice may be constructed using states like Eqn. 3.5. This space can further be decomposed into sectors irreducible under the global guage group and any spatial symmetries in the problem (translation symmetry for periodic boundary conditions is discussed in Sec. 3.4). 3.2.1 Position Space Here we will define several operators that act on the position part of the states. A lattice site is specified by a 2d vector γ — (γ, c), where r and c are the row and column of the site. We define creation/annihilation operators a*(r) and c(γ), respectively, at every site. The vortices have hard cores, so the operators obey (0t(r))2 — («(P))» — 0 (3.6) 30 As usual, one defines a number operator n(r): n(r) = at(r)a(r) (3.7) The ordering operator Ω(r) is defined in terms of the patching convention of Sec. 3.1. Acting on a state like Eqn. 3.5, it yields the order a flux at site r would take is the internal space. It does this by counting the number of vortices that precede that site according to the patching convention. i!(r) = Ω((r',c')) = £ £ n((rj,c∙)) + ∑ n(ri,c') + 1 n c,<c' ri<r' 3.2.2 (3.8) The Internal Space Several operators that act on the internal space have bees introduced is previous chapters. These are the braid operators Ri, which implement a counterclockwise braid os the z’-th pair of vortices, and the conjugation operator C(h), which conjugates the entire state by h € H. Is addition, it is useful to define the operators 2∣(z) and z∕()), f ∈ H, which create and destroy, respectively, a flux f at the z-th position in thee vortex internal state: x in ∙ i+i n+i, zj(i)∖a,.. .,h,---,k) = (a,.·.,/, h,..., k) t «+1 n i (3.9) n— 1 Zf(i ∣^, -∙.,9, h,.-,k) = 6/,ala ∙ ..,A, ∙) (3.10) The operator A∕l(i) conjugates only the flux is position i by the group element heff: A√i)∣α, ...,g,...,k) = ∣a,..., hgh~1,... ,k) (3.11) Using the ordering operator from Sec. 3.2.1, it is sow possible to define operators c^(r) and c∕(r), acting on both the position and internal space, that create/destroy 31 a particle of flux f at position r on the lattice: c/( r) = z/(Q(r))a(r) cj(r) = 2*il(r))θt(r) 3.3 (3.12) The Lattice Hamiltonian We may now construct a Hamiltonian analogous to Eqn. 3.1 for non-abelian vortices on a lattice. The vortices are allowed to hop to nearest-neighbor sites, leading to a kinetic term analogous to the first term of Eqn. 3.1. There is no explicit interaction term, but the kinetic term must take into account the statistical interactions of the vortices, i.e., the braiding properties. To construct the Hamiltonian, consider its effect on an n-particle flux/position eigenstate (like Eqn. 3.5): W = ∖<^iu92,-∙∙,an) ® ∖posn.) (3.13) Say a vootex at site si = (rt∙, c,∙) and flux flu(rl) hops tp the unoccupied site one unit to the left r,∙ = (r,-,ct∙ — 1). wwo things apppen: first the moving article is successively conjugated by the fluxes of all the vortices below it in column c,, then all of the vortices above the vortex in column (c,∙ — 1) are conjugated by the resultant flux. It is convenient to define operators that count particles above or below a lattice site r = (r, c): n«(r) = YL n(r.,c) r,> r nt(r) =∑n(r,∙,c) f<r (3.14) The total flux that conjugates the moving vortex is then f^(r,∙) = flrΩ(r,)-n6(r,) ∙ --<ya(r )-≡ ∏ ge € H )=Ω(γ·,-1 (3.15) 32 The product is the product in the group H. Eqn. 3.15 defines the ordered product [I, which is takes to order terms right to left. The operator that conjugates the vortices braided by the moving vortex is Ω(ri)-n6(r,)-no(Γj)-1 A(r,∙,rj∙) = ∏ Λm(A(r,-)5Ω(rt)A(r,∙)^1) (3.16) m=Γ(Γt)-Γ()cpi·-l Is terms of these operators, the Hamiltonian is 1 H — -<∑∑(cl(Γlc⅛(rι^^1(Γr·)Λ(Γ.·,Γr·)Cα(Γt·) + h.c.) <m> c (3.17) As alternate form of the Hamiltonian may be constructed using the braid op­ erators. Is terms of the ordered product, define two braiding operators i51(ri) and B2(γ,∙,γj∙)= Ω(r,-n6(∏) B,( γ.) — Ri (3.ls) Λ√ (3.19) ∏ (—nirO-i Γ(γϊ)—Γ6(γ,)—γο(ρ)—1 Bohr,) — ∏ m=Ω(r, )-st(ιri-l One cas sow consider the braiding process to be a sequence in which ose removes a particle from the position part of the state, applies the appropriate braiding operators to the internal state, asd then creates the particle is the sew site is the position space. H = -t ^2,(a*('rj')B2(ri, Γj·)Bi(r)·)a(r,·) + h.c.) (3.20) (∙j> 3.4 Translation Symmetry on the Cylinder We periodically identify the lattice is ose dimension (call it the x direction) asd erect hard walls is the other directios, so the configuration space is a cylinder. Typically, ose would expect that the trasslatios mvαriasee in the x direction enables ose to 1It is implicit that k and A are the identity for up and down hops, i.e., for hops in the same column. 33 Figure 3.2: Three particles on a 3 x 4 lattice: the dashed line is the International Dateline. blnck-dihgnshlize the Hamiltonian by a discrete Fourier transform is x, i.e., by going to momentum space in the x direction. The spatial symmetry decomposes the multi­ vortex space into blocks labeled by the wave vector kr, where kx ranges over the Brillouin Zone for the lattice. It turns out that the translation symmetry is modified for son-hbclihs vortices, as we shall sow elucidate. Topologically, the cylinder is homeomorphic to an annulus. The columns of the lattice are spokes on the annulus, and the rows are concentric circles. We chose a basepoint x⅛ for the flux definition, and select a column as a starting point, numbering the columns from 1 to Nc. Fig. 3.2 shows the case of three particles os a 3 × 4 lattice. The total flux of this state is Φt = abc, by our patching convention. The particle with flux a is in column 1, and by hopping ose unit clockwise it can reach column 4. In doing so, it goes from being the first vortex by our patching convention to the last, so apparently ∣a,b, c) ι→ ∣∈, c, a) (3.21) But because these are sos-abelian particles, is general abc ^ bca, so it would appear that the total flux changes. Since the total flux is conserved, this is sot possible. This hoohrent dilemma is resolved by taking care to use only the paths prescribed by the patching convention to measure flux. The standard paths to define single particle fluxes reach only as far as column Nc, they are not allowed to wrap back 34 around to column 1. Consider a particle with flux a, the first particle in the internal state, sitting in column 1. The combined flux of the rest of the particles is α^lΦi. Take the path C2 from the basepoint to encircle all of the particles except the first, and take Cχ to be the path encircling particle 1 that deforms to the proper path after particle 1 has jumped from the first to the last column, becoming the last particle. Paths Ci and C2 are indicated in Fig. 3.2. The flux bounded by Cχ is equal to the flux that the hopping particle will have after it jumps from the first to the last particle. Let B be a path that encircles the central hole of the annulus in a clockwise sense: by considering the gauge strings of the vortices (c/. Sec. 3.1) we see that the flux linked by B is the total flux Φi. Let Co be a standard path that encloses vortex a. before the hopping. Then it is easy to see that, under homotopy, Ci S B'1C0B (3.22) In other words, the flux linked by Ci is ΦΓ1aΦt, so the hopping from column 1 to column Nc yields ∣a, 6, c) i-> ∣6, c, Φt-1aΦt) (3.23) and the total flux is preserved. Obviously, it )i possible for the last particle to hop ccooss the ‘‘Internaononal Dateline”2(a line seodrrtiag cooumn 1 and cohlmn Nc) aad become the frst particle. The operator F that performs the last-to-first rotation on the state, conserving total flux, may be expressed in terms of the braid operators: F = RlR2...⅛l-i (3.24) F∖a,b,..,k) = ∣ ΦtkΦfl,a,b,...) (3.25) The first-to-last rotation is performed by F~l Because acting on a state with F is equivalent to a braiding, F is trivial on any bosonic state. It is simply a rotation on trivial flux sectors. 2Credit for coining this term goes to John Preskill. 35 To analyze the effect of the International Dateline on the translational symmetry of the vortex gas, it is instructive to consider a simple system: indistinguishable hard­ core bosons os a nne-dimeneioshl ring with N sites. Define the operator T which translates every particle one site to the right; if T commutes with the Hamiltonian, there is a translation symmetry that is isomorphic to Z/v- The Brillouin Zone consists of the wave vectors ⅛ == jjÿ -π, l<j<N (3.2∈) The projection operator that projects onto the j-th momentum state is N-1 Vj = ∑ eimk>Trn m=0 (3.27) If we consider the same space, with indistinguishable vortices instead of identical bosons, we must consider a different translation operator T, which translates every particle to the right one lattice spacing and applies the operator F to the state every time a particle hops across the International Dateline. Applying T N times leads to a gauge transformation by the total flux of the state: f,Ι≠> = C(Φ1)M) (3.28) If the total flux is son-trivial, and the order of the total flux in the global gauge group is p, then we see that the translation symmetry group is sow isomorphic to Z^.v)· For example, consider the sectors of the S3 vortex model with an odd number of particles, so that the total flux is a two-cycle and the global gauge group is Z2 generated by the total flux. The translation symmetry group is Z2f, with Brillouin Zone wave vectors kj = Î1 ≤i ≤nv (3.29) On the uncharged sector, C(Φi) acts trivially, so under hhe tanslLti^o^n symmetry, hhe 36 states transform as the Z,v subgroup of Zj corresponding to wave vectors with j odd or even, depending on whether N is odd or eves. (Eqs. 3.28 shows that the effective translational symmetry group will always be Z/v on as uncharged sector). On the charged sector, the translation symmetry will be Zj furthermore, the states in this sector will have momenta kj with j even or odd, depending on N odd or even. The other projection operators will arrihilate the charged states. An immediate consequence is that there are no tronslatiorally invariant states in charged sectors with Z2 global gauge group, i.e., there are so zero-momentum states in these sectors. An analogous situation may be found is as ordinary conducting cylinder. If there is a magnetic flux through the cylinder that is not an integer multiple of the flux quantum, the momentum in the angular coordinate (in the direction that is periodically identified) is shifted by some fractional amount. For the vortices os a cylinder, the International Dateline is a brasch cut which alters the boundary conditions in the x direction, shifting the momentum spectrum3. 3.5 Ground State Structure Having constructed a Hamiltonian with the required braiding properties, it is now possible to numerically calculate the ground state for small numbers of particles on a finite lattice. The numerical techniques used are described in Appendix C. Actually, one cas predict the qualitative structure for various vortex sectors by simply considering the braiding and translation properties of the system. For any sector that contains a bosonic subspace, we can expect that the bosonic states will have the lowest g.s. energy. There should be a finite gap between the bosonic subspace and the Γrn-abelrcΓ subspace. The ground state energies should be zero-momentum (kx — 0) states, unless the sector does not contain zero-momentum states, as is the case for son-trivial Z2 charge sectors. Consider first the sector (3, (31),1Γ(Z() ) on a 2 x 3 lattice. As discussed in Sec. 3.4, the translation symmetry group for this (uncharged) sector is isomorphic to 3∑ would like to thank John Preskill for pointing out this analogy. 37 Zz, with Brillouin Zone vectors kx = 0, ±^f ∙ There are two bosonic states in this section so the ground state energy (with kx = 0) is doubly degenerate, and equal to —5.60 ± 0.005 (in some units). The lowest energy in the non-abelian subspace (corresponding to the threo-dlmonsional irreducible representation of the Ai sector group) for kx = 0 is —4.56 ±0.01 (in the same units), illustrating the anticipated gap between the bosonic and non-abelian subspaces. The ground state degeneracy of the uncharged sector is somewhat mysterious. It is easily shown that a set of degenerate states of a Hamiltonian forms a representation of the symmetry group of that Hamiltonian. In the usual case, the representation is irreducible, so that the degeneracy can be clearly interpreted in terms of the symme­ try of the physical system. When the representation is reducible, the degeneracy is termed accidental, and it is usually lifted by an arbitrarily small perturbation to the Hamiltonian. In the case of the sector (3, (31 ),l T), the two-dimensional ground state subspace is reducible, splitting into two trivial representations of the sector group. Additionally, we note that this degeneracy is very robust: it is unaffected by any perturbation that acts only on the position space, and moreover it is unaffected by any perturbation involving products of the braid operators (since these operators act trivially on bosonic states). Furthermore, this degeneracy is pervasive, as shown by Lemma 3. When accidental degeneracies occur with such regularity, one usually suspects that there is a larger, “hidden” symmetry group of the Hamiltonian4. There may be a hidden symmetry of Eqn. 3.20, such that the distinct bosonic states in sectors where the g.s. degeneracy occurs form irreducible representations under the larger symmetry group. Moving on to the non-trivial charge (3, (31),1 Γ(Z2) ) sector, we note from Sec. 3.4 that the translation group is Z&, with odd-j representations realized on the charged sector, so the “effective” Brillouin Zone is kx = r, ±^. Also, in Sec. 2.3, we discovered that under the sector group, the chaxged sector splits into two automorphic irreducible 4The classic example of this is the explanation of the ‘extra’ degeneracy of the hydrogen atom spectrum in terms of the SU(2) x 5CZ(2) symmetry of the Hamiltonian, corresponding classically to the conservation of the Runge-Lenz vector in a 1/r potential. 38 n 3 4 4 5 Φt (31) e (123) (31) Fbose -5.606 ± 0.005 -8.801 ± 0.003 —8.801 ± 0.003 -8.800 ± 0.003 Fsos—ab -4.56 ± 0.01 -6.91 ± 0.04 -7.99 ± 0.03 -8.17 ±006 AE/n 0.35 0.47 0.20 0.13 Table 3.1: Uncharged-sector g.s. energies. representations, so there will again be at least a two-fold ground state degeneracy. It turns out that the actual ground state energy of —3.8 ±0.1 occurs for kx — ir, so the ground state is indeed doubly degenerate. For the trivial flux sectors of the four vortices os a 3 x 3 lattice, the translation symmetry group is always isomorphic to Z3, and the lowest energies will be found is the translationally invariant (zero-momentum) blocks. The g.s. energy for four bosons on a 3 x 3 lattice is —8.801 ± 0.003, which is the g.s. energy for both the uncharged and the non-abeliar charged (3Γ(S3)) sectors, both of which are doubly degenerate. The g.s. energy is the (Γ(S'3) sector, which doesn’t contais a bose state, is —4.23 ±0.13. The g.s. energies of the son-abelian subspaces of the uncharged asd non-abelias charged sectors axe —6.91 ±0.04 and —7.97 ± 0.12, respectively. The uncharged, Φt = (123) sector has one bososic state, and the g.s. energy of the nrn-abelicΓ subspace of this sector is —7.99 ± 0.03. Since this is the same energy as the nonicbslran charged trivial flux sector (within numerical uncertainties), it is consistent with the conjecture that the 3T(S'3) sector is isomorphic to two copies of the uncharged, three-cycle flux sector. In Table 3.1, the bososic asd non-abelian ground state energies for the uncharged three through five vortex sectors are listed. The last column gives the per-particle energy gap, where ΔE = Esos-ab - Ebθse∙ 39 Chapter 4 Summary and Conclusions A general description of the behavior of large systems of indistinguishable sos-abelias vortices remains an extremely difficult problem. However, we have made incremental progress in discerning some features, at least in ose representative model. The braid group essentially acts as a finite group os sectors with a finite number of particles. The generators of the associated finite group, which is in a many-to-ose homomorphism with the corresponding braid group, necessarily obey the Yang-Baxter relations, plus additional defining relations related to the underlying physical model. One of these additional relations, Lemma 2 , is a necessary condition for the sector group to be finite (whether it is sufficient in general is unknown). The size of the sector group grows very rapidly in the number of particles. The dimension of the vortex sectors also grows exponentially, and one would expect that the dimensions of the irreducible representations of the braid group contained is the sectors will also grow rapidly. Thus excited states should be highly degenerate. However, this huge degeneracy doesn’t seem to occur is the bosonic eubspaces that appear is the uncharged and non-abelian charged sectors of the S3 vortex model, at least for reasonable numbers of particles. For the Hamiltonian studied is this thesis, a kinetic term encoding the braiding properties of the vortices, the bosonic subspaces (where they occur) will contain the ground states of the lattice gas, so the ground state degeneracy doesn’t seem to be growing at a large rate. However, in a number of subephces, the ground state is (at least) doubly degener­ ate. These degeneracies don’t result from any obvious symmetry of the Hamiltonian, so they hist at some deeper symmetry. One expects, for the kinetic Hamiltonian, that the ground states of the uncharged and non-abelian charged sectors will be a Bose gas, with an energy gap separating the Bose states from the non-abelian states. This is sot the case for charge states associated with son-trivial one-dimensional representations of the global gauge group. 40 since Bose states do not occur in these sectors. The calculation of the ground state energies is technically difficult. We define a patching convention for the multi-vortex states, and (at least partially) decompose the internal space of a sector into invariant subspaces under the sector group. The periodic boundary conditions enable us to exploit the translational symmetry of the Hamiltonian, but one must be careful to properly account for the “International Date­ line.” The ground state energies are calculated iteratively, using a modified Lanczos technique. The dimension of the total Hilbert space (internal and position) grows extremely rapidly with particle number, slowing convergence drastically. We have succeeded in calculating g.s. energies for up to five vortices, and the calculations appear to confirm expectations. Future Directions One obvious extension to the work in this thesis would be to analyze other rela­ tively simple models, in an attempt to discern universal features of the non-abelian many body problem. In addition, we can consider more complicated models, such as models with charged vortices, or stable vortices from several different thnjegaty classes. Flux quantization is an effective probe for pairing and collective behavior[9]. Our geometry allows for the introduction of an external flux, which essentially amount to a non-trivial boundary condition for vortices traversing the cylinder. However, we can only choose external fluxes from a discrete set. One might expect that vortices would become dalt-ohtrolated in such a way that they commute with the external flux, and that sectors with an odd number of vortices would be frustrated. We have written code to calculate g.s. energies with an external flux, but have not yet achieved reasonable convergence for more than three vortices. 41 Appendix A Proofs of the Lemmas in Section 2.5 Lemma 1 For an odd number n of vortices, the dimensions of the sectors are dimhW,' Γ(Z2)) — i(3” + l) d⅛(κ(3())(Γ(Z()) — ∣(3"-1) For an even vortex number n, the sectors with trivial total flux have dimensions dim(n, e∕ Γ(S,3)) = j(3"→ + l) dim(n,e,( T(Ss)) — ^"2 — 1) dim(n, s,3 Γ(⅞)) — 2(3^) The sectors with even vortex number n, three-cycle flux have dimensions dim(s, (123),1 T(Z3)) — dim(s, (^^ T(Z3)) — dim(s, (123),3 T(Z3)) — 3s-2 Proof: Consider the case s odd. There are 3s states of n two-cycle vortices. For odd n, these states have a total flux which is a two-cycle, so there are 3n^l states with φt — (31). We may construct the matrices that represent the well-defined global gauge group,, {e, (31)}. Since distinct two-cycles of S3 do not commute, the only state which is invariant under conjugation by the total flux (31) is the stats |(31), (311),.., (31)). There is only one son-zero element on the diagonal of the matrix representative of (31), so we can compute the characters of the representation of the Z3 global gauge 42 group on this space, namely, x(e) = 3"-1, x((31)) = l The orthogonality theorem for group characters states that the number of times np the p-th irreducible representation of a group G appears in a given representation is np — — 53 Xp(9M9) (A-l) nG g€G where no is the order of the group, the χp are the characters of the p-th irreducible representation, and the χ are the characters of the reducible representation. Using the character table of Z2 and the characters of the vortex sector above, one immediately arrives at the first two equations in Lemma 1. All of the other formulae are obtained by an analogous calculation. Lemma 2 In any n vortex sector with trivial total charge, and in any sector with trivial total Jinx and arbitrary charge, the generators of the sector group obey the relation (R1R2 ■ ∙ ∙ Rn~i)n — e In any sector with n vortices, the generators of the sector group must obey (RlR2...Rn-l)nn = e where no is the order of the global gauge group for that sector. Proof: Consider the operator (RiR2 ∙ ■ . R-n-ι) acting on as n-vortex state: (RιR2...Rn-ι)∖a,b, ...,k) = ∣ΦjfΦi l,at,0,...) 43 where Φi = ab.. .k. Applying this operator n times yields (RR ... R∏-ι)n∖α, b,..,k) = ∣ΦiaΦf~l, Φi6ΦΓ1,.. ,, ΦMΦf~1) This is a global gauge transformation of the state by the total flux. Since the sector has zero total charge, the operator must act trivially on any state. The second relation is a consequence of the fact that the order of an element of a finite group must divide the order of the group. Lemma 3 In any sector whose global gauge group is a cyclic group generated by the total flux, there are no bosonic states in sectors with nontrivial charge. Proof: Lemma 2 shows that a global gauge transformation by the total flux can al­ ways be achieved by a braiding. In a sector where the global gauge group is generated by the total flux, all global gauge transforms are equivalent to a braiding. Therefore any state that is trivial under all braidings is in an uncharged sector. Lemma 4 In any d-dimensional uncharged sector of the S3 vortex model, the sector group is a subgroup (not necessarily proper) of the alternating group on d objects, that is to say, K(n,Φt,lT) C Ad d = dim(n, Φί/Π Proof: Consider an initial basis of the n-vortex space with given total flux, consisting of states which have a definite flux at every point, i.e., states of the form ∣a, 6,..., k), Φf = ab.. .k. The operator that projects onto the uncharged subspace is V= ∑ C(h) hζN(Φt) (A.2) This operator adds together all of the states of the initial basis that are related by a global gauge transformation, with equal weight. Label the linearly independent states 44 obtained by applying the projection operator to the isitial basis |1), |2), .... |d). Because the braid operators commute with global gauge transformations, all of the braid operators simply permute the d uncharged states. The relation i?? — s for the braid operators of the S3 model implies that the braid operators must be products of one-cycles asd disjoint three-cycles, therefore they must be even permutations, hence tf(n,Φt,1Γ) C Ad. Lemma 5 In any sector whose global gauge group is a cyclic group Zp generated by the total flux, the space of any non-trivially charged sector is a Zn projective representation of the sector group of the corresponding uncharged sector. Proof: The group Zjv has N isequivalest irreducible representations. The projection operator for the j-th irreducible representation is Vj — ∑ e,w>C(Φ',), *7 — 1≤7≤V (A.3) l=0 where we define Φ° — s. Applying the projection operator to the initial flux eigenstate basis, ose obtains d linearly independest vectors as a basis for the 7-th charge state, labeled ∣1), ..., |d). For asy j > 1, it is clearly possible to put these states is a one-to-ose correspondence with the uncharged j — 1 states, in which each charged state contains the same flux eigenstate basis members as the corresponding uncharged state, but weighted by phase factors that are N-th roots of unity. The braid operators os the charged sector connect the same states as the corresponding operator on the uncharged sector, modulo these phases. The usremovability of the phases for the charged sectors follows from Lemma 2 Theorem 1 The representation space, with respect to the sector group, of the un­ charged, Φi — (31), n-vortex sector (n odd) is isomorphic to the sector group repre­ sentation space of the uncharged, Φi — s, (τs + 1)- vortex sector: that is to say, the corresponding sector groups are isomorphic: A(n,(31),1 Γ(Z2)) — K{n + 1, e∕ Γ(S3)) n odd 45 and the representations (typically reducible) of the sector group on these sections are equivalent. A corresponding isomorphism exists between the charged, Φt = (31). n-vortex representation space and the non-trivial one dimensional charged (2Γ(?)))· Φt = e (e + 1)-vortex space. Proof: Let the states |i) 1 < i ≤ d be the basis states for the sector (n, (31),1 Γ(Z2)), and the operators Ri, 1 < i < n — 1 the braid operators calculated using this basis. From the states |i), obtain the states |i') by applying the operator C((23)), and the states |z") by applying the operator C((12)), K' = C(223))∣Z) (A.4) |Z")=^122^)|t) (A-5) It should be clear that the ∣z') and |i") are bases for the uncharged, n-vortex sectors with total flux (12) and (23), respectively, and that the braid operators os each of these sectors are identical to the braid operators on the (31) flux sector. Now construct a basis for the uncharged, (n + 1)-vortex trivial flux sector by the following procedure: take a state |i), append a flux (31), take a state |z'), append a flux (12), take a state |z’"), append a flux (23), and add the states together: |e- = Iz"^)^ + ∣z'(122)f + ∣z"( 223)f (A.6) It is obvious from this construction that (i) the d states ∣et∙) form a btsis for ohe sectas s o + 1, e,1 Γ(^ )) -f-1 Lemma 1)( an)) (ii) thh first o — 1 braid opeiatoos Ra or this sector are identical to the taaid operators on the sector (n,(31),1 Γ(Z2)). Because there is one adaitinnhl vortex in the trivial flux sector, there is one more braid operator, Rn. In order for the sector groups to be isomorphic, this extra operator must be ‘redundant,’ i.e., it must be possible to obtain this operator by taking products of 46 the first n — 1 braid operators. This is always possible, as we shall now see. Lemma 2. applied to the sector (n, (31),1 Γ(Z2)), says that (RlR2...Rn-l)n = e (A.7) By construction , this is also true in the n + 1 vortex sector. Lemma 2 applied to (n + 1,o∕ Γ(S3)) tells us that (RlR2...Rn)n+1 =e (A.8) Using the Yang-Baxter relations Eqn. 2.4, it is possible to establish, by induction, the formula RnRn-l ■ ■ ∙ Rn-i{R∖R2 ∙ ∙ ∙ Rn) = {RlR∙2 ■ ∙ ∙ Rn-l)Rn ■ ■ ■ Rn-i-l (A.9) Applying Eqn. A.9 repeatedly to Eqn. A.8 yields (i?l . . . Rn-i)nRnRn-l ∙ ■ ■ R∖R∖R2 ∙ ∙ ∙ Rn = Rn ∙.∙ RιRι∙.. Rn = o (by Eqn. A.77 (A.10) A straightforward group manipulation puts this in tho form Rn = ^¾∙∙.λγ⅛1∙∙∙ ⅛ (A.ii) Now wo multiply both sides from tho left with R„, and use tho S3 model defining relation Eqn. 2.8: R^ = o = Rn{Rn-l . . . RlRl . . . Rn-l) 1 Rn = Rn-l ∙ ■ ∙ R∖Rl . . ∙ Rn-1 (A.12) Tho group generated by Ri,..., Rn is the same as tho group generated by Ri c.., Rn-ι 47 and the theorem is proved. The statement concera(ng the charged s^c^Ct^r^^ follows from an analogous construction, applied to the charged n-vortex Φf = (31) sector. Eqn. A. 12 for n = 3 is R3 = R2R∖R2 (A.13) Using the defining relations, it is possible to show that R2fi%R2 = Ri. on tho throe- vortex uncharged sebsdace( providing a check on Eqn. 2.29. Lemma 6 Let bosn(n,Φt,“Γ ) dnnote the bosonic subsaace of the given sector. For any uncharged sector with an odd number of vortices, and for any uncharged seotor with an even number of vortices and trivial total flux, dim(boso(n, (31)^^) ≥ 2 n odd, dim^ose^, o^T)) ≥ 2 n even, dim(bose(n, o,3 T)) ≥ 2 For an uncharged sector with an even number of vortices and three-cycle total flux, dim(boso(n, (123),^)) ≥ 1 n even. Proof: Tho proof is by construction. An uncharged sector with an even number of vortices and throe-cyclo total flux is spanned by states labeled ∣l), ..., ∣d), and the braid operators simply permute the states. The state ∣J3) = ∣l) + ... + ∣d) (A.14) On uncharged odd vortox sectors with Φt = (31), there is a state ∣Bα) = |(3^l)(^ll)(··((31)) (A.15) which is bosonic, henco it decouplos from the other d—1 states, which can bo combined as in Eqn. A.14 to form another bosonic state. On uncharged ovon vortex sectors with 48 Φt — e, the ‘extra’ bososic stats is |&> — ∣(1^, (12),..., (12)) + ∣(23), (23),... ,(23)) + ∣(31), (31),..., (31)) (A.16) The (ussormalized) projection operator for the 3T sectors with Φi — e is V3 — 2C(e) -C((123)) — C((321)) Applying this in turn to the states ∣(12), (12),,.., (12)), ∣(2∙3), (23)),.., (23)), and ∣131), (331, ).., (31)) yields the bosonic states 1B,> — 21(12), (12)),.., (12))-1(23) ,(23),.. ,,G^3t)> - 1(31), (311),....(31)) ∣½) — 2123), (23‰,. ., (23)) — ∣(12},(12),.. ,,H2)>-∣(31), (311),..., (31)) ∣⅛) = 2131), C^^1),. ., (3())-|(12),((2),.. ,,112)>- ∣(23), (23)),. ∙,(23)) But, e.g., ∣B1) + ∣F2) — — ∣-β3), so these states represent only two linearly independent bososic states. 49 Appendix B Numerical Calculation of Subspaces and Sector Groups For more than three or four vortices, hand calculation of braid matrices, charge sectors, etc. become impractical. This Appendix describes the numerical techniques used to facilitate the process. B.0.1 Singular Value Decomposition The Singular Value Decomposition (SVD) is ae extremely effective techniqe for com­ puting nrtenentmal bases of subepαces. It is based oe the fact that aey real, m x n matrix A can be factored in the form A = UΣVt (B.l) where U is an m x n column-orthogonal matrix, Σ is an n x e diagonal matrix with eon-negative values on the diagonal, and Γ is ae e x n orthogonal matrix (for pronf· see Ref. [18]). The columns of Γ whose corresponding values on the diagonal of Σ are zero form an nrthnnntmal basis for the nullspace of A, and the rows of U whose corresponding diagonal values in Σ are nonzero form ae otthnentmal basis for the range of A. The code I wrote to perform the SVD is essentially a translation of the Algol routine is Ref. [17] to ANSI/ISO C, incorporating a few elements of the routine is Ref. [29]. B.0.2 Sector Group Calculations It is straightforward to encode the multiplication table for a finite group. Gives an no x no array (no is the order of the group) representing the multiplication table of a finite group {e.g, S3) and a set of basis elements, {e.g., the two-cycle class), it is •50 then not difficult to find all sets of n elements of the basis sot that combine to give some specified total flux. Tho states thus generated are not in a definite charge secter, i.e., they art in a reducible representation of the centralizer of tho total flux. If the uncharged sebsdaoo is desired, one can specify the non-trivial elements of the global gauge group; tho program selects a state and conjugates it by all non-trivial elements of tho global gauge group, generating the unchanged states. The braid operators can now bo calculated on the uncharged sector. By Lemma 4, tho braid operators on the uncharged sector are permutations of the states, so the code represents the R as permutations. If a non-trivial charge sector is required, the code makes use of the projection operator for irreducible representations, Λ = —- ∑ x;(9)T(i) (B.2) n° hCNO/o) Here dp is the dimension of tho p-th representation, the Xp art the characters of tho representation, and T(g) is the matrix representing g in the reducible representation. To find the subsdate for a given charge state, the appropriate projection operator is calculated, and then passed to the SVD routine. The rows of the resulting U ( cf. Sec. B.0.1) whose singular values are non-zero art selected as a basis for tho given charge sector, the columns of V whoso singular values vanish form a basis for tho orthogonal complement. Using these vectors, an orthogonal transformation matrix 5 is constructed, which block-diagonalizes the braid matrices, one block corresponding to the desired charge sector. After the braid operators on a given charge sector are calculated, the sector group may be computed in a brute force manner by starting with the braid operators, systematically multiplying the group olomonts with each other, storing new elements as they are created, until the group closes. In this way, a multiplication table for the sector group is generated. After tho multiplication table is complete, the classes can be computed. The elements may bo stored and manipulated either as matrices in tho representation of tho group on tho sector space, or, if the sector is uncharged, 51 they can be represented as permutations. The former possibility has the advantage that the characters of the sector group representation may be computed, but it is much more costly ie terms of storage requirements and computation time, so it is impractical for sectors with more than four vortices. B.0.3 Calculation of the Bosonic Subspace Once the braid matrices for a given sector are computed, it is possible (and desirable) to extract the bosonic subspace. A state is bosonic if and only if it is invariant under all of the braid operators of the sector. Consequently, the bosonic subspace of a given sector bose(n, Φt,a Γ(V^(Φi))) can be determined as n—1 bosein, Φt,a Γ) = ∩ null {Ri - I) (B.3) ι=1 Here T is the identity matrix, aed sull() denotes the nullspace of the gives matrix. The SVD is useful to extract the bosonic subsphce. The n — 1 braid matrices are ‘stacked’ into ose larger matrix (whose dimensions are (n - 1 )d x d, if the braid matrices are d x d), which is passed to the SVD routine. Aey element of the sullspace of this large matrix is is the intersection of the nullspa^ces of all of the braid matrices l. As ie the calculation of charge sectors, a, basis for the bosonic subsoace and its orthogonal complement is built from the U and Γ matrices returned from the SVD routine, which are in turn used to construct an orthogonal trasefnrmatinn that block- diagonalizes the matrices. 1If one only desires to find the nullspace intersection of two matrices there is a more efficient algorithm [18, pp. 602-603], but for more than two matrices, the procedure above calculates the bosonic space in one fell swoop •52 Appendix C The Modified Lanczos Method The modified Lanczos method is an efficient iterative algorithm to find the ground state of a given Hamiltonias[Γ2]. The algorithm is as follows: given a trial state p, calculate the expectation value of the Hamiltonian H; (C.l) eo — Let the state p measure the dfffersnce betwesn the trias stat e s,dd sn eiemstate of H; Hp - eθipo — bp (C.2) where 6 is chosen so that p ss properly normalized. From the trial state p asd the state p (which is orthogonal to p). construct the state p: P = p + apo T-H^ (C.3) where a is a variational parameter. Calculating the expectation value of H is the state p as a function of or, one finds the value of a that minimizes this expectation value. The state pl is a closer approximation to the true ground state of H, and el — (ip1∣H∣≠ι) is an improved estimate of the ground state energy. This method cas be iterated, with p serving as the trial state for the next iteration. To summarize the formulae for ose iteration (all expectation values are taken with respect to the trial state p): - m2 b= J = (-ff3)-3(ff)(⅞ι)+2(ff)3 263 (C.4) (C.∙5) •53 a = f-jp + 1 (C.6) Q = (H)+ab (C.7) and the state ≠ι is given by Eqn. C.3. C.l Useful Techniques This section describes the actual implementation of the Laeczos method, adapting standard techniques^, 31, 19]. A basis for the position states is developed as follows. Since every site is occu­ pied by either ose or no vortices, the position states may be represented as binary ietegers[15]. If there are Nτ- rows and Nc columns, the lowest-order Nc bits represent the first row, the next Nc bits the second row, and so on. For example, with three vortices on a 2 x 3 lattice, the state 1 0 0 0 1 (C.8) 1 is represented by the number (1 + 2) + (32) = 35. The internal states are also represented by a number: the smallest number of high-order bits required by the dimension of the internal space are reserved for this purpose. Is this way, the internal and position states are combined to give a single, unique integer. The (modified) translation symmetry of the lattice vortices is used to block- diagonalize the Hamiltonian. To begin, consider the case of a trivial flux sector, so that the translation symmetry in the x direction is unmodified, and we need only consider the position part of the state. All of the position eigenstates that can be transformed into one another by shifting every particle to the left or right a certain eumber of sites will be combined into states with definite total i-momestum. For ex­ ample, the tthnslαtionally invariant (zero-momentum) state associated with Eqn. C.8 54 is _1_ √3 1 0 0 ∖ ∕ + 011/ 0 ∖ ∕+ 101/ 0 1 (C.9) Clearly, it is only necessary to store one of the terms of the state to uniquely specify the state. By an arbitrary convention, we store the position state whose binary representation is a maximum as a representative of the entire transitionally invariant state. For example, the entire state Eqn. C.9 would be represented by the number 35. Higher momentum states axe represented analogously. To generate a state with x-momontum kj, apply the projection operator Nc-l Vj = 53 c-imk>τm m=0 (C.10) to the position eigenstate that is the representative (i.e., the maximum integer) for the momentum state. Note that by our phase convention, the theffiiOent of the representative state is real. The operators in the Hamiltonian correspond to bitwise operations on the integer representatives. The integers generated by moving bits around won’t necessarily be proper representatives of the new states generated; every time a state is generated, it is rotated to produce the maximum possible number, which is stored as the repre­ sentative for the state. The code also keeps track of how many shifts are necessary to produce the representative; for non-zero momentum, the state must be multiplied by a factor exp(iskj), for s shifts, due to our phase convention. Now complications arise if the sector doesn’t have trivial total flux, due to the modification of the translational symmetry discussed in Sec. 3.4. Now the operator F must be applied to the internal state every time every time a particle is rotated through the International Dateline. For example, consider the uncharged sector of the S3 throe vortex problem, in which one finds throe internal states that transform into each other under the action of F. Labeling the internal states ∣1), |2), and |3), F acts as the throe-cycle (123). The transitionally invariant state analogous to Eqn. C.9 is 55 ∕ _1_ |1)® y/5 V 0 o \ 1 1 ) + |2) ® 0 11/ 1 1 o \ l i ) + |3) ® 101/ 0 1 ∖λ ) 110/ y 0 0 (C.11) Only the first term is stored as a representative. If the bit in the top left corner jumps one space to the left, the state generated is 0 0 1 1 1 |3) ® 0 (C.12) The internal state changes because the first vortex becomes last, so the operator F~γ acts on the state. The new state is not a valid representative, it must be rotated twice to the left to be maximized. In doing so, another particle hops to the left across the International Dateline, so F~l acts again. The representative of the new state generated by the upper left bit hopping is finally determined to be |2) ® 1 0 0 1 0 (C.13) 1 Had the state been in a higher momentum block, the phase exp(2fΛy) would have been generated. In calculating expectation values, care must be taken in evaluating the norms of the states[15]. For example, the norm of the state represented by Eqn. C.12 is 3, but the norm of the state represented (in position space) by 0 0 0 1 1 (C.14) 1 is 32 = 9 in a trivial flux sector, or 3 again if in an uncharged sector whose states are not invariant under F. In a non-trivial charge sector, further complications arise. For example, as noted in Sec. 3.4, in a non-trivial Z2 charge sector, the translation group is Z2vc∙ The non­ 56 vanishing states are the odd-momentum states, and there is a further factor of (2) 2 in the norm. C.2 Implementation Details All of the code described in this appendix and App. B was written in ethnahrd ANSI/ISO C. 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