Vortex Dynamics and Hall Conductivity of Hard Core Bosons

# Vortex Dynamics and Hall Conductivity of Hard Core Bosons

## Abstract

Magneto-transport of hard core bosons (HCB) is studied using an quantum spin model representation, appropriately gauged on the torus to allow for an external magnetic field. We find strong lattice effects near half filling. An effective quantum mechanical description of the vortex degrees of freedom is derived. Using semiclassical and numerical analysis we compute the vortex hopping energy , which at half filling is close to magnitude of the boson hopping energy. The critical quantum melting density of the vortex lattice is estimated at vortices per unit cell. The Hall conductance is computed from the Chern numbers of the low energy eigenstates. At zero temperature, it reverses sign abruptly at half filling. At precisely half filling, all eigenstates are doubly degenerate for any odd number of flux quanta. We prove the exact degeneracies on the torus by constructing an SU(2) algebra of point-group symmetries, associated with the center of vorticity. This result is interpreted as if each vortex carries an internal spin-half degree of freedom (’vspin’), which can manifest itself as a charge density modulation in its core. Our findings suggest interesting experimental implications for vortex motion of cold atoms in optical lattices, and magnet-transport of short coherence length superconductors.

###### pacs:
05.30.Jp, 03.75.Lm, 66.35.+a, 67.85.d

August 13, 2019

## I Introduction

Hard core bosons (HCB) are often used to describe superfluids and superconductors which are characterized by low superfluid stiffness and short coherence lengths. As such, HCB are relevant to cold atomic gases in optical lattices PS (); Zoller (), low capacitance Josephson junction arrays Ood (); Altman-JJ (); JJReview (), disordered superconducting films fisher (), and cuprate superconductors Uemura (); EK-Uemura (); Ranninger (); LB-cuprates (); PBFM (); LB2 (); MA ().

At low densities, HCB can be treated by weak coupling (Bogoliubov) perturbation theory FW (). Closer to half filling, lattice umklapp scattering and the hard core constraints become important. Recent calculations of the dynamical conductivity of HCB near half filling LA () demonstrate the breakdown of weak-scattering Drude-Boltzmann transport theory in this regime. HCB exhibit so-called ‘bad metal’ phenomenology, (i.e. large resistivity, linearly increasing in temperature). Such behavior has been often observed in unconventional superconductors BM ().

This paper also concerns dynamical correlations of HCB and their vortices near half filling. These will be exposed by including a weak orbital magnetic field in the Hamiltonian and studying the Hall effect.

Our primary results are as follows. Firstly, we apply a combination of semiclassical analysis and exact diagonalization to the gauged Hamiltonian on a finite latice on the torus. We highlight the (sometimes overlooked) fact that a uniform magnetic field of one flux quantum penetrating the surface of the torus beaks translational symmetry. As a consequence, the semiclassical vortex center is subjected to a confining potential minimized at a well defined position. Fitting the low many-body spectrum to an effective single-vortex Hamiltonian, we determine the vortex hopping rate (effective mass).

Near half filling, the vortex mass is found to be similar in magnitude to the HCB mass. This allows us to estimate the critical field for quantum melting of the vortex solid (superfluid) phase at 6.5 flux quanta per unit cell. Secondly, at half filling we find doublet degeneracies associated with an odd number of magnetic flux quanta penetrating the torus. We associate them with symmetries about the vortex position, and label the emergent degrees of freedom as ‘vortex spin’ (v-spin). Physically, these degrees of freedom correspond to the orientation of the charge density wave in the vortex cores.

Finally, we compute the Hall conductivity using thermally averaged Chern numbers. In stark contrast to continuum bosons, and to electrons in metallic bands, we find that the Hall conductivity of HCB reverses sign abruptly at half filling. The associated Hall temperature scale vanishes at half filling, signaling a possible quantum phase transition for the thermodynamic system in a magnetic field. Some of these results were briefly reported in a recent Letter LAA ().

This paper is organized as follows. In Section II the HCB Hamiltonian is introduced, with a discussion of its charge conjugation symmetry about half filling. Semiclassical approximations are derived in Section III, for the various regimes of filling. At low density, we recover the Gross-Pitaevskii theory with its Galilean invariant vortex dynamics and classical Hall effect. At half filling, the continuum limit corresponds to the anisotropic gauged non linear sigma model. Its vortices possess localized charge density waves in their cores. Section IV describes the mathematical peculiarities of the gauged torus, including translational symmetry breaking (elaborated in Appendix A). Definitions of null lines, null points and vorticity centers are provided. The point group symmetry generators and are constructed and their commutator is calculated. The proof of v-spin degeneracies at half filling is provided. Section V computes the vortex effective hamiltonian by combining semiclassical and exact diagonalization calculations. The critical field for quantum melting of the vortex lattice is deduced from our value of vortex hopping rate. Section VII computes the Hall conductance on the torus as a function of density and temperature. We conclude in section VIII and discuss experimental implications of our results in cold atoms and cuprate superconductors.

## Ii Hard Core Bosons

The conventional Bose Hubbard model for interacting bosons is

 HU = −2J∑⟨ij⟩(eiqAija†iaj+a−iqAija†jai) (1) +4V∑⟨ij⟩(ni−12)(nj−12)−\mathchar28950\relax∑ini +12U∑ini(ni−1) ,

In the hard core limit , Eq. (1) reduces to the HCB Hamiltonian as , where is the projector onto the subspace where or for each site.

We use units where . denotes a nearest neighbor link on the square lattice; the lattice constant is . is the Josephson coupling, is the boson charge and the electromagnetic gauge field on a bond. is a nearest neighbor repulsive interaction. In the HBC limit, The chemical potential corresponds to a density of half filling , with half a boson per site on average.

As is well-known, HCB operators obey an algebra corresponding to spin-:

 ~a†i = Pa†iP=S+i ~ai = PaiP=S−i ni = ~a†i~ai=Szi+12. (2)

By , HCB operators obey constrained commutation relations,

 (3)

The constraint effects of become important near half filling. is thus represented by the gauged spin-half quantum model,

 H = −2J∑⟨ij⟩(eiqAijS+iS−j+e−iqAijS−iS+j) (4) +4V∑⟨i,j⟩SziSzj−\mathchar28950\relax∑i(Szi+12).

It is widely believed that the ground state of Eq. (4) exhibits magnetic order. In the regime of , which is relevant to this paper, the ordered moment lies in the plane, . That is to say, except for the limits , the ground state of HCB exhibits long range superfluid order.

### ii.1 HCB charge conjugation symmetry

Another important distinction between the HCB Hamiltonian (4) and the finite Bose-Hubbard model of Eq. (1), is the emergence of charge conjugation symmetry in the infinite limit. One defines the unitary charge conjugation operator,

 C≡exp(i\mathchar28953\relax∑iSxi). (5)

transforms“particles” into “holes”, i.e. , and

 C†H(q\tenmibA,\mathchar28950\relax)C=H(−q\tenmibA,−\mathchar28950\relax) . (6)

At half filling (), and , the Hamiltonian is invariant under charge conjugation on any lattice structure comm:CCS ().

A consequence of (6) is that the Hall conductivity (which is linear in ) is antisymmetric in the deviation from half filling, ie

 \mathchar28955\relaxxy(n,T)=−\mathchar28955\relaxxy(1−n,T). (7)

In contrast, the superfluid stiffness and longitudinal conductivity are symmetric under .

In terms of vortex motion, (7) implies that below and above half filling vortices drift in opposite directions relative to the particle current.

## Iii Semiclassical theory

The partition function of HCB can be represented by the spin half coherent state path integral Book (); Notes (),

 Z=∫Dˆ\tenmibΩ(\mathchar28956\relax)exp(∫\mathchar28940\relax0d\mathchar28956\relax(iK−Hcl)), (8)

where

 K[ˆ\tenmibΩ,˙ˆ\tenmibΩ] ≡ 12∑i(1−cos\mathchar28946\relaxi)˙\mathchar28958\relaxi (9) Hcl[ˆ\tenmibΩ,\tenmibA] = −J∑⟨i,j⟩sin\mathchar28946\relaxisin\mathchar28946\relaxjcos(\mathchar28958\relaxi−\mathchar28958\relaxj+qAij) (10) +V∑⟨i,j⟩cos\mathchar28946\relaxicos\mathchar28946\relaxj−\mathchar28950\relax2∑icos\mathchar28946\relaxi.

are the polar angles on a sphere. The spin size plays the role of the large parameter which controls the semiclassical expansion.

In the classical (saddle point) approximation, for , the ground state superfluid stiffness is

 \mathchar28954\relaxcls = q−2∂2Hcl∂A2\tenmibr,\tenmibr+^\tenmibx∣∣∣\tenmibA=0 (11) =

which (in contrast to continuum bosons) exhibits a non-monotonic dependence on . At half filling (optimal density), is maximized. Quantum corrections enhance further by about 7% sandvik (); Troyer (). The superfluid stiffness vanishes at the Berezinskii-Kosterlitz-Thouless (BKT) BKT () transition temperature, computed to be .

The kinetic term of Eq. (8), determines the quantum dynamics. The harmonic spin-wave expansion of (10) yields a linearly dispersing phase fluctuations mode. The order parameter is suppressed to zero at all finite temperatures, in accordance with the Mermin-Wagner theorem.

### iii.1 Low density, Gross-Pitaevskii limit

For large negative values of the chemical potential , the action in Eq. (10) can be expanded around the ferromagnetic (low density) state of ,

 cos\mathchar28946\relaxi→2ni−1,sin\mathchar28946\relaxi≈2√ni(ni−1) . (12)

We define the continuous field

 \mathchar28960\relax(\tenmibxi)=√niaei\mathchar28958\relaxi, (13)

where is the lattice constant, and replace the measure by

 ∏iDcos\mathchar28946\relaxiD\mathchar28958\relaxi⟶D\mathchar28960\relax∗D\mathchar28960\relax∏i,tΘ⎛⎜⎝1−∫Vid2x\mathchar28960\relax∗\mathchar28960\relax⎞⎟⎠ , (14)

up to an unnecessary normalization constant. The Heaviside functions enforce the hard core constraint in the unit cell at each time slice. In the low density limit, these constraints are ignored, and the action (10) is expanded to leading order in , and gradients . This yields an effective Gross-Pitaevskii (GP) theory PS (),

 ZGP = ∫D\mathchar28960\relax∗D\mathchar28960\relaxexp(−SGP[\mathchar28960\relax∗,\mathchar28960\relax,\tenmibA]+…) SGP = ∫d2x∫dt[\mathchar28960\relax∗(∂t−\mathchar28950\relax)\mathchar28960\relax (15) +12m∗∣∣(−i∇−q\tenmibA)\mathchar28960\relax∣∣2+12g|\mathchar28960\relax|4]

where the effective mass and interaction parameters are given by

 m∗ = 116J g = 16(J+V). (16)

In the presence of a magnetic field , a density of vortices is produced, where is the flux quantum. The core profile function near vortex is well approximated by minimizing the GP energy, which yields PS ()

 fGP(\tenmibr)≃√nr√\mathchar28952\relax2+r2 , (17)

where is the coherence length. For , one has . The core density depletion is proportional to . Hence it decays as away from the vortex center.

In the high density limit, , the partition function can also be approximated by the same GP action (15) following a particle hole transformation (6). In this case, represents the density of holes.

By neglecting the higher order gradients, and the hard core constraints, the GP theory does not include lattice scattering effects as it is completely Galilean invariant. Consider an externally induced uniform current density

 \tenmibj=qn\tenmibvs. (18)

In the moving frame of velocity the vortices are stationary. Therefore, back in the lab frame, a purely transverse electromotive field is produced by the moving vortices,

 \tenmibE = hq^z×\tenmibjv (19) = hqnv^z×\tenmibVv = hq2nvn^\tenmibz×\tenmibj.

That is to say, in the pure GP theory, the longitudinal (dissipative) conductivity vanishes, and the Hall conductivity equals to the classical value,

 \mathchar28955\relaxxx = 0 \mathchar28955\relaxxy = (q2h)(nnv)=nqB . (20)

Spoiling Galilean invariance by the presence of nonuniform potentials, boundary conditions, or by an underlying lattice can allow vortices to tunnel between different real space positions, resulting in a longitudinal conductivity AAG (); AA ().

### iii.2 Half filling, anisotropic \mathchar28955\relax-model

Toward half filling, lattice scattering modifies the vortex structure and dynamics. At half filling , the semiclassical theory of Eq. (10) is described by the anisotropic Non Linear -Model (NLSM) haldane88 (). After a sublattice rotation all the pseudo-spin interactions are antiferromagnetic. The spins are represented by

 ˆ\tenmibΩi=\mathchar28945\relaxi^\tenmibn(\tenmibxi)√1−(\tenmibL(\tenmibxi)/S)2+\tenmibL(\tenmibxi)/S , (21)

where on the and sublattices, respectively. The Néel vector satisfies and is orthogonal to the local magnetization , i.e. .

The complex combination

 n⊥=nx+iny=|n⊥|ei\mathchar28958\relax, (22)

defines the local superfluid order parameter, and corresponds to a bipartite charge density wave (CDW) with two possible signs. Following Refs. haldane88 (); Book (); Notes (), we substitute Eq. (21) in the measure and action of (10), and expand them to quadratic order in and . Integrating out arrives at the anisotropic NLSM path integral,

 ZNLSM=∫D^\tenmibneiΥ[^\tenmibn]e−SE[^\tenmibn,\tenmibA] , (23)

where is the Euclidean action, with

 LE = 12\mathchar28959\relax⊥∣∣˙n⊥∣∣2+12\mathchar28959\relaxz˙n2z +12\mathchar28954\relaxs∣∣(∇−iq\tenmibA)n⊥∣∣2+12\mathchar28954\relaxzs(∇nz)2+m2zn2z,

and

 Υ=S∑i\mathchar28945\relaxi\mathchar28940\relax∫0d\mathchar28956\relax (1−nz)˙\mathchar28958\relaxi, (25)

is the contribution from the geometric phase. The bare coupling constants are obtained directly from :

 \mathchar28959\relax⊥=S28J,\mathchar28959\relaxz=S24(J+V), (26)

and

 \mathchar28954\relaxs=J,\mathchar28954\relaxzs=V,m2z=2(J−V) . (27)

For , the isotropic (Heisenberg) limit is at . The Néel ground state implies degeneracy between superfluid and CDW order, and the existence of two massless Goldstone modes. At finite anisotropy, , and there is one massless (phase) mode, and a gapped CDW (roton) mode at the CDW ordering wavevector .

Vortex configurations at half filling can be viewed as a localized meron (half skyrmion) of the Néel field. Since at the vortex center, and , the semiclassical vortex has a CDW in its core, as illustrated in Fig. 1.

Due to the finite anisotropy ‘mass’ , decays exponentially away from the center

 n⊥(\tenmibr) = √1−n2z(\tenmibr) ei\mathchar28958\relax(\tenmibr), nz(\tenmibr) ∼ e−r/\mathchar28952\relaxz, \mathchar28952\relaxz = √\mathchar28954\relaxzs/mz. (28)

Indeed variational calculations have previously shown that at half filling CDW ordering is found in the localized vortex core SCvortex (). In Section IV we shall show that the ‘orientation’ of the charge density wave is actually a continuous SU(2) symmetry of the quantum Hamiltonian at half filling, which we name v-spin.

Since the system is charge conjugation symmetric at half-filling, there is no net charge depletion associated with the vortex core, and thus the statistical Berry phase for exchanging two vortices is zero. In other words, the vortices exhibit mutual Bose statistics. This is to be contrasted with GP vortices at low filling. As shown in (17), GP vortices involve a large density depletion (or accumulation, above half filling), which decays slowly away from their core HaldaneWu ().

In the limit where the number of lattice sites tends to infinity, the confining potential on the vortex vanishes, and the vortex energy is periodic on the lattice. Its minima lie in plaquette centers (i.e. at dual lattice sites).

When a vortex moves between dual lattice sites, the path dependent geometric phase yields times the number of bosons enclosed by the path. At half filling, this amounts to an effective flux per dual plaquette. These phases can be incorporated in an effective hopping model by the dual lattice gauge field along the link from site to . Thus for a single vortex on the infinite lattice, one can write an effective Harper Hamiltonian,

 H∞V=−12tV∑\tenmibR,\tenmib\mathchar28945\relax(eiA\tenmibR,\tenmibR+\tenmib\mathchar28945\relaxb†\tenmibRb\tenmibR+\tenmib\mathchar28945\relax+H.c.) ∑⊙ CA\tenmibR,\tenmibR+\tenmib\mathchar28945\relax=2\mathchar28953\relax∑i∈int(C)ni, (29)

where the sum on the second line is a over a set of links comprising a closed path on the dual lattice, and is the interior of this path, which consists of a set of sites on the original lattice bounded by .

## Iv The Gauged Torus

We now return to the original HCB Hamiltonian, Eq.(4). We consider a finite square lattice, of dimensions , with sites and periodic boundary conditions in both the and directions. This toroidal geometry is convenient for the study of finite lattices as it minimizes the effects of boundaries. It also provides external control over the positions of vortices via the two Aharonov-Bohm (AB) fluxes which run along the two cycles of the torus. The lattice site positions are labelled as

 xi = 0,1,2,…,Lx−1 yi = 0,1,2,…,Ly−1. (30)

A uniform magnetic field is everywhere perpendicular to the surface, such that the total number of flux quanta penetrating the surface is , where is the flux quantum.

We construct a (piecewise differentiable) gauge field which interpolates the lattice gauge field on the surface of the torus, and obeys

 ^\tenmibz⋅∇×\tenmibA = B \tenmib\mathchar28945\relax⋅\tenmibA(\tenmibr+% 12\tenmib\mathchar28945\relax) = A\tenmibr,\tenmibr+\tenmib\mathchar28945\relax , (31)

where , . determines the magnetic fluxes which flow through vertical and horizontal circumferences of the torus. These are given by the gauge invariant Wilson loop functions,

 Wy(x) = q∮dyAy(x,y) mod2\mathchar28953\relax Wx(y) = q∮dxAx(x,y) mod2\mathchar28953\relax. (32)

The dimensionless AB parameters are defined by the Wilson loops at and ,

 \mathchar28930\relaxy = Wy(x=0) \mathchar28930\relaxx = Wx(y=0). (33)

lives on the reciprocal torus .

define the null lines on the torus. For , there is one null line in each direction , and , as depicted in Fig. 2. Their intersection is the null point , which constitutes a gauge invariant symmetry point on the torus.

 X0(\tenmibΘ) = −Lx\mathchar28930\relaxy2\mathchar28953\relax, Y0(\tenmibΘ) = +Ly\mathchar28930\relaxx2\mathchar28953\relax. (34)

The existence of a special point on the torus, demonstrates the unintuitive fact that a uniform magnetic field necessarily destroys lattice translational symmetry. This fact is closely related to the quantization of Dirac monopoles in three dimensions. We elaborate further on this fact in Appendix A. Eq. (34) shows that can be moved continuosly on the torus by changing the AB parameters  HR85 (); ABHLR88 ().

As we shall see in Sec. V, semiclassical analysis and exact diagonalizations find that the center of vorticity is located at the antipodal position of the null point on the torus,

 \tenmibRV(\tenmibΘ)=(12Lx+X0(\tenmibΘ),12Ly+Y0(\tenmibΘ)) . (35)

For larger magnetic fields, , there are null lines in each of the and directions. This introduces a set of null points which form an evenly spaced square lattice (which may or may not coincide with the original lattice sites). These are indexed by

 \tenmibRmn0(\tenmibΘ)=\tenmibR0(\tenmibΘ)+1N\mathchar28958\relax(mLx,nLy), (36)

Correspondingly there are vorticity centers,

 \tenmibRmnV(\tenmibΘ)=\tenmibRmn0(\tenmibΘ)+12(Lx,Ly) . (37)

### iv.1 Choosing a gauge

The uniform magnetic field must integrate to an integer number of flux quanta ,

 B=2\mathchar28953\relaxN\mathchar28958\relaxqLxLy (38)

The gauge field is given by

 Ax\tenmibr,\tenmibr+^\tenmibx = −mod(y−Y0,Ly)BLxH(X0,x) Ay\tenmibr,\tenmibr+^\tenmiby = mod(x−X0,Lx)B . (39)

Note that for , . The function ensures that vanishes unless is immediately to the left () of the null line . It is defined by

 H(X0,x)={10

For a continuous position we define to be the linearly interpolated gauge field between the two enclosing links in the direction.

For the null point is at , and the vorticity center is therefore at .

Our gauge choice is shown in Fig. 3. The gauge invariant content of consists of the uniform magnetic field with flux in each plaquette, and the two Wilson loop functions

 Wy(x) = xqBLy+Θy, Wx(y) = −yqBLx+Θx. (41)

### iv.2 V-spin degeneracies

In the process of calculating the Hall conductance (see Section VII), we computed the spectrum at half filling, for an even number of sites, with one total flux quantum of magnetic field. We encountered a sequence of AB fluxes , where the whole spectrum becomes two-fold degenerate. These degeneracy points are demonstrated in Fig. 4 for , for the lowest two multiplets. The level crossings indicate the existence of a non-commuting symmetry generators TH (), which act on the wavefunctions of vortices introduced by the external magnetic field. We now construct these symmetry operators and compute their commutation relations.

As discussed earlier, for a finite magnetic field (), does not possess the lattice translational symmetry. Nevertheless, with respect to the vorticity center we can define two reflection operators

 PxV(x,y) = (mod(2XV−x,Lx),y), PyV(x,y) = (x,mod(2YV−y,Ly)), (42)

which by (35) are equivalent to reflections about .

Now, by appropriately tuning using Eq. (34), the vorticity center can be chosen to coincide with a symmetry point of the square lattice, such as any lattice site, bond center or plaquette center. Reflecting the Hamiltonian about that symmetry point, leads to

 P\mathchar28939\relaxVH[\tenmibA]P\mathchar28939\relaxV = H[~\tenmibA], ~\tenmibA\mathchar28939\relax\tenmibr,\tenmibr+\tenmib\mathchar28945\relax = \tenmibAP\mathchar28939\relaxV\tenmibr,P\mathchar28939\relaxV(\tenmibr+\tenmib\mathchar28945\relax). (43)

The gauge invariant content of describes an inverted uniform magnetic field , and a reversed sign of the Wilson loop functions (32).

The reversal of the fields in can be undone, at half filling, by applying the charge conjugation transformation (5), and a pure gauge transformation . Thus, we construct two operators,

 \mathchar28933\relaxxV = UxCPxV \mathchar28933\relaxyV = UyCPyV (44)

where,

 U\mathchar28939\relax=exp(i∑\tenmibr\mathchar28959\relax\mathchar28939\relax(\tenmibr)Sz\tenmibR) , (45)

and

 \mathchar28959\relax\mathchar28939\relax(\tenmibr)=\tenmibr∫\tenmibR0d\tenmibr′⋅(\tenmibA(\tenmibr′)+~\tenmibA\mathchar28939\relax(\tenmibr′)). (46)

In the line integral we use the interpolated gauge field defined after Eq. (39). Since and describe the same magnetic fields, they obey,

 ∇×(\tenmibA+~\tenmibA\mathchar28939\relax)=0. (47)

This implies that is independent of which continuous path (of zero winding number) is chosen between and .

It is easy to verify by this construction that for all such that is a symmetry point the operators become symmetries of the Hamiltonian:

 [H[\tenmibA],\mathchar28933\relax\mathchar28939\relaxV]=0,\mathchar28939\relax=x,y. (48)

Now we calculate the commutation relation between and . This is a straightforward but slightly tedious procedure. Using the gauge choice (39), and (46),

 \mathchar28959\relaxx(\tenmibr) = 2\mathchar28953\relaxN\mathchar28958\relaxLymod(y−Y0,Ly)(1−\mathchar28942\relaxx,X0), \mathchar28959\relaxy(\tenmibr) = 0, (49)

Note that vanishes on the null lines. Multiplying the two operators yields

 \mathchar28933\relaxyV\mathchar28933\relaxxV = exp(i∑\tenmibr(\mathchar28959\relaxy−\mathchar28959\relaxx(PyV[\tenmibr]))Sz\tenmibr)PyVPxV, \mathchar28933\relaxxV\mathchar28933\relaxyV = exp(i∑\tenmibr(\mathchar28959\relaxx−\mathchar28959\relaxy(PxV[\tenmibr]))Sz\tenmibr)PyVPxV (50) = e−iΥ\mathchar28933\relaxyV\mathchar28933\relaxxV,

where we have used . The overall phase is given by the operator

 Υ = ∑\tenmibr\mathchar28961\relax\tenmibrSz\tenmibr \mathchar28961\relax\tenmibr = \mathchar28959\relaxx+\mathchar28959\relaxx(PyV[\tenmibr])−\mathchar28959\relaxy−\mathchar28959\relaxy(PxV[\tenmibr]) . (51)

It can be directly verified from (49) that

 \mathchar28961\relax\tenmibr={0\tenmibr∈null lines2\mathchar28953\relaxN\mathchar28958\relaxotherwise. (52)

Since ,

 e−iΥ=(−1)N\mathchar28958\relax(N−Nnull)=(−1)N\mathchar28958\relaxNnull. (53)

where is the number of sites which sit precisely on the two null lines. For even , and .

For odd , and odd , one obtains . Let us prove a simple lemma concerning the parity of .

Lemma: For even size lattices, if is tuned to be precisely on a lattice site, then is odd. The proof is illustrated in Fig. 5.

Proof: Since we assume that is even (to describe precise half filling), there are two cases to consider:
(i) For an even by even lattice, , if we choose on a lattice site, it is easy top see that must also sit on a lattice site. The number of sites which contribute to are the sum of lattice sites in the and directions minus the null point itself which is counted twice:

 Neenull=2m+2n−1. (54)

Hence .

(ii) For the odd by even lattice, e.g. , we choose , to be in the middle of a bond in the direction. The null line includes only the sites on the -null line which is odd:

 Neonull=2m+1 (55)

Thus, here too . Note that in both (54,55), the vorticity center is situated on lattice sites . QED.

Thus we conclude that for an odd number of fluxes , if is located precisely on any lattice site, then and anticommute.

Under these conditions, all states of must be at least two-fold degenerate. This follows the standard proof: Since

 [H,\mathchar28933\relaxxV]=0, (56)

and has eigenvalues , then each common eigenstate of and , can be labelled by . Now,

 \mathchar28933\relaxxV\mathchar28933\relaxy% V|En,1⟩=−\mathchar28933\relaxyV\mathchar28933\relaxxV|En,1⟩∝|En,−1⟩, (57)

that is to say each eigenenergy is associated with a degenerate pair of eigenstates with opposite quantum numbers of