Arun Paramekanti, Leon Balents and Matthew P. A. Fisher- Ring exchange, the exciton Bose liquid, and...

8/3/2019 Arun Paramekanti, Leon Balents and Matthew P. A. Fisher- Ring exchange, the exciton Bose liquid, and bosonizatio

1/27

Ring exchange, the exciton Bose liquid, and bosonization in two dimensions

Arun Paramekanti,1,2 Leon Balents,2 and Matthew P. A. Fisher11 Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030

2 Department of Physics, University of California, Santa Barbara, California 93106-4030

Received 7 March 2002; published 20 August 2002

Motivated by the high-Tc cuprates, we consider a model of bosonic Cooper pairs moving on a square lattice

via ring exchange. We show that this model offers a natural middle ground between a conventional antiferro-magnetic Mott insulator and the fully deconfined fractionalized phase that underlies the spin-charge separation

scenario for high-Tc superconductivity. We show that such ring models sustain a stable critical phase in two

dimensions, the exciton Bose liquid EBL. The EBL is a compressible state, with gapless but uncondensed

boson and vortex excitations, power-law superconducting and charge-ordering correlations, and broad spec-

tral functions. We characterize the EBL with the aid of an exact plaquette duality transformation, which

motivates a universal low-energy description of the EBL. This description is in terms of a pair of dual bosonic

phase fields, and is a direct analog of the well known one-dimensional bosonization approach. We verify the

validity of the low-energy description by numerical simulations of the ring model in its exact dual form. The

relevance to the high-Tc superconductors and a variety of extensions to other systems are discussed, including

the bosonization of a two-dimensional fermionic ring model.

DOI: 10.1103/PhysRevB.66.054526 PACS numbers: 75.10.Jm, 71.10.Hf, 71.10.Pm, 74.20.Mn

I. INTRODUCTION

Despite intense experimental effort exploring the phase

diagram of the cuprates, the nature of the pseudogap regime

remains mysterious. In the very underdoped normal statethere are strong experimental hints of local pairing and su-perconducting correlations,1 3 despite the absence of phasecoherent superconductivity. Within this picture,4 the pseudo-gap regime supports a pair field with nonzero amplitude butwith strong phase fluctuations which disrupt the supercon-ductivity. A theoretical approach then requires disorderingthe superconductivity by unbinding and proliferating vorti-

ces. Unfortunately, such an approach will likely face a mostworrisome dilemma. Proliferation and condensation of singlehc/2e vortices necessarily leads to a confined insulator,5

which should show sharp features in the electron spectralfunction in apparent conflict with ARPES experiments.3 Onthe other hand, if pairs of hc/2e vortices condense,5,6 thepseudo-gap phase must necessarily support gapped unpairedvortex excitations7 9 visons,10,11 which have yet to beobserved.12,13 Is there an alternative possibility? In this paperwe find and explore a truly remarkable quantum fluid phaseof two-dimensional 2D bosons, which we refer to as anexciton Bose liquid EBL, which answers this question inthe affirmative. The EBL phase supports gapless charge ex-citations, but is not superconductingthe Cooper pairs arenot condensed. The hc/2e vortices are likewise gapless anduncondensed. Being a stable quantum fluid with no brokenspatial or internal symmetries, the EBL phase has manyproperties reminiscent of a 2D Fermi liquida 1D locus ofgapless excitations a Bose surface, a finite compressibil-ity, an almost T-linear specific heat with logarithmic correc-tions.

Neutron scattering measurements in the undoped cuprateshave revealed a zone boundary magnon dispersion that canbest be accounted for by presuming the presence of appre-ciable four-spin exchange processes.14 Recent theoretical

work points to the importance of such ring exchange pro-cesses driving fractionalization and spin-chargeseparation.6,15,16 Motivated by these considerations, we focuson a class of model Hamiltonians describing a square latticeof 2D bosons Cooper pairs with appreciable ring exchange.It is hoped that such microscopic models will be appropriatein describing the charge sector in the underdoped cuprates.When the ring exchange processes are sufficiently strong, wefind that the EBL phase is the stable quantum ground state.

Lattice models of interacting bosons in two dimensions,such as the Bose Hubbard model, have been studied quiteexhaustively during the past several decades,17,18 primarilyas models for Josephson junction arrays and superconducting

films but also in the context of quantum magnets with aneasy-plane U1 symmetry and most recently in the contextof trapped Bose condensates moving in an optical lattice.19

Typically, the ground-state phase diagram consists of a su-perconducting phase and one or more insulating states.17 Atfractional boson densities commensurate with the lattice, sayat half filling, the insulating behavior is driven by a sponta-neous breaking of translational symmetry. When accessedfrom the superconductor, such insulating states can be fruit-fully viewed as a condensation of elementary vortices.20 Veryrecent work on a Kagome lattice boson model with ringexchange16 which arises in the context of frustrated quan-tum magnets has revealed the existence of a more exotic,

fully gapped insulating state with no broken symmetrieswhatsoever. In this state the charged excitations are frac-tionalized, carrying one half of the bosons charge,21 andthere are also gapped vortex excitations called visons. Here,we study the simpler system of bosons with ring exchangemoving on a 2D square lattice. Our central finding is theexistence of a critical gapless quantum phase of 2D bosonswhich is stable over a particular but generic parameter range.This phase is in some respects very similar to the familiargapless Luttinger liquid phase of interacting bosons and Fer-mions moving in one spatial dimension.2224 Indeed, to de-scribe this phase we introduce a 2D generalization of

PHYSICAL REVIEW B 66, 054526 2002

0163-1829/2002/665/05452627 /$20.00 2002 The American Physical Societ66 054526-1


2/27

bosonization. Specifically, we introduce a new duality map-ping that transforms a square lattice model of 2D bosons ina rotor representation with ring exchange, into a 2D theoryof vortices hopping on the sites of the 2D dual lattice.Then, based on the symmetries we construct a low-energyeffective description, in terms of the two dual phases of theboson and vortex creation operators. The EBL phase has aGaussian fixed point description in terms of these fields. In

addition, there are various nonquadratic interactions, whichscale to zero in the EBL. Instabilities towards superconduct-ing and insulating states are triggered when one or more ofthese interactions become relevant, closely analogous to 2kFinstabilities in a 1D Luttinger liquid.

Since this paper is quite long, the remainder of Introduc-tion is devoted to a brief summary of the ring model Hamil-tonian and the bosonized description of the EBL. The Hamil-tonian we focus on describes 2D bosons in a rotorrepresentation:

HU

2 r n r n2K

rcosxyr, 1

with

xyrrrxryrxy . 2

Here r-label sites of the 2D square lattice and r and n r arecanonically conjugate variables,

r ,n rir,r . 3

Representing the phase of the boson wave function, r istaken to be 2 periodic, rr2, so that the eigenval-ues of the conjugate boson number operator nr are integers.The mean boson density is set by n, and we will primarily

focus on the case with half filling: n

1

2 . The argument ofthe cosine in the second term is a lattice second derivative,involving the four sites around each elementary squareplaquette. This term hops two bosons on opposite corners ofa plaquette, clockwise or counterclockwise around theplaquette, and is an X-Y analog of the more familiar SU2invariant four-site ring exchange term for spin one half op-erators which arises in the context of solid 3-He Refs. 25and 26. Alternatively, we may view the bosons as par-ticles and the vacancies as holes over a uniform back-ground density. In this case, the ring-exchange process forbosons on a plaquette is just the propagation of a particle-hole pair exciton from one edge of the plaquette to theopposite edge. This motivates us to call the critical liquidphase of this model as the exciton Bose liquid.

In addition to the conventional spatial, particle-hole, andtotal number conservation symmetries see Sec. III, this ringHamiltonian has an infinite set of other symmetries. Specifi-cally, the dynamics of the Hamiltonian conserves the numberof bosons on each row and on each column of the 2D squarelatticea total of 2L associated symmetries for an L by Lsystem. This is fewer than a gauge theory, which has anextensive number of local symmetries, but these symmetrieswill nevertheless play a crucial role in constraining the dy-namics of the model and stabilizing a new phase.

We will also consider adding a near-neighbor Boson hop-ping term,

Htt r,r

cosrr, 4

where the summation is taken over near-neighbor sites on the

square lattice. This term breaks the 2L U1 symmetries cor-responding to conserved boson numbers on each row andcolumn, leaving only the globally conserved total bosonnumber.

The remarkable result of this paper is that, over a particu-lar but generic parameter range, ring models of this typesustain the EBL phase. The effective description of the EBLis in terms of low-energy coarse grained variables and an additional field see Sec. III, which is related vian n1xy to the long-wavelength density, and furthercan be used to construct a vortex creation operator ve i. To fix notation, we define the n r ,r ,r operators onthe sites of the original square lattice, with (x,y ) coordinates

taking half-integer values, and r with Nr ,r operators tobe defined later on the dual square lattice with integer (x ,y)coordinates. The , fields are governed by an approxi-

mately Gaussian Euclidean effective action S 0

dL,

with LL0L1, and is imaginary time, 1/kBT. TheGaussian part of the Lagrangian is

L0H0,r

i

rxyr , 5

where the sum is over sites r, rrx/2y /2 on the dualand original lattices, respectively, and the effective Hamil-

tonian is

H0k K k2 xyk2U k22 xyk2 . 6

Here the momentum (k) integral is taken over the Brillouinzone kx , ky, and (xy) k denotes the Fourier trans-form of xyr and similarly for xy). The functionsK(k),U(k) are nonvanishing finite periodic, and analytic.Their values along the kx and ky axes parametrize the Boseliquid, much as the effective mass and Fermi liquid param-eters do in a Fermi liquid. Experts will note a strong simi-

larity to the bosonized effective action for a one-dimensionalLuttinger liquid,23,24 which is explored in Sec. III. Like in aLuttinger liquid, there are additional nonquadratic termsin the action. It is sufficient to keep only those of the formL1L1L1, with

L1r

q,s

tqm,ncosqrrs , 7

with r on the original lattice, and

PARAMEKANTI, BALENTS, AND FISHER PHYSICAL REVIEW B 66, 054526 2002

054526-2


3/27


4/27

column of the 2D square lattice. Specifically, these symme-tries imply an invariance of the energy and action under

rrxx yy , 15

for arbitrary functions x(x) and y(y ). This invariancedictates that in the Harmonic spin-wave form the energymust vanish whenever kx0 or ky0.

Since the plaquette term involves a lattice second deriva-tive rather than an ordinary lattice gradient, however, it isclear that phase () fluctuations will be large in the ringmodel, and the spin-wave expansion is suspect. Indeed, evenin the classical limit U0, one can readily see that vor-tex configurations in which r winds by 2 around someplaquette e.g., for a vortex with center at xy0, xy/2(x)(y )(x)(y )3/2(x)(y ),where (z) is the Heavyside step function are finite in en-ergy rather than logarithmic as in an ordinary X-Y model.Further, double vortex configurations in which this wind-ing is 4 can be smoothly deformed into zero-energy con-formations e.g., xy(x)(y). This suggests

that for nonzeroU

there will be substantial vorticity in thelow-lying states.To address the legitimacy of the spin-wave expansion, it

is necessary to account for the periodicity of the cosine po-tential. This is most readily accessed by transforming to adual form, just as one transforms to a dual bosonized repre-sentation for a system of 1D particles.24 Here we introduce anew 2D quantum duality transformation27 that is specificallytailored for the ring model.

B. Plaquette duality

We consider dual fields living on the sites of the duallattice, denoted r and Nr , which are canonically conjugate

variables:

Nr ,rir,r . 16

Here we take Nr to be 2periodic so that r/ has integereigenvalues. The reason for this unusual choice of normal-ization should become apparent below. For notational easewe are denoting the sites of both the original and dual squarelattices as r. In analogy to Eq. 2, it is convenient to intro-duce an operator defined on the plaquettes of the dual latticei.e., sites of the original lattice as

xyrrrxryrxy . 17

These two new dual fields are related to the original fields bythe relations

Nrxyr , xyrnr , 18

where rrx/2y /2. One can check that with Nr and ras conjugate fields, the original variables satisfy the requiredcommutation relations.

To interpret the new variables, consider now a vortex cen-tered on some site (x ,y) of the dual lattice the core.Classically, for the four sites on plaquette of the originallattice surrounding this site,

e ixa/2,yb/2 aib2 Nv

, 19

where a ,b1, and Nv

denotes the number of vorticesvorticity on this plaquette. Comparing to Eq. 18, onefinds NN

vmod 2, so N can be interpreted as a vortex

number operator, modulo two. Since and N are canonicallyconjugate, the operators ei perform canonical transla-tions of N, and hence can be regarded as vortex creationand annihilation operators, respectively. Note that since Z, e 2i1, consistent with the ambiguity in NN

vun-

der even integer shifts. Clearly, periodicity of in the origi-nal variables is encoded in the discreteness of in the dualdescription.

The plaquette Hamiltonian, when reexpressed in the dualvariables, reads

HKr

cosNrU

22r xyrn2. 20

We will for the time being neglect the tunneling term Ht,

which will be returned to later. Note the strong similaritybetween the dual and original forms. To bring this out moreclearly, and for the numerical simulations to be consideredshortly, it is useful to go to a path integral formulation. Inparticular, consider the partition function

ZTreH. 21

Expanding Z in the usual Trotter fashion with a time slice0, using the discrete basis of eigenstates of r ,one finds Appendix A

Z r()

exp

L , 22with the Lagrangian

Lr

2

ln 2K

r2 U22 xyrn2 ,23

where rr()r() /. The dependence of thetime derivative term in Eq. 23 is familiar from the time-continuum limit relating, e.g., the (d1)-dimensional clas-sical and d-dimensional quantum transverse-field Ising mod-els. Note the strong similarity of Eq. 23 to Eq. 12, whichemphasizes their nearly self-dual nature.

The formulation in Eqs. 22-23 is quite convenient fornumerical simulations. For the simulations, we define an

integer-valued height field r() through

xyrnr(B)xyr, 24

where n r(B)1(1)xy /2 is the background staggered

density, chosen such that r() obeys periodic boundary con-ditions. In this language, we obtain a generalized anisotrop-ic solid-on-solid model in 21 dimensions. The correlationfunctions for the fields are easily reexpressed and numeri-cally evaluated in terms of these height variables, using


054526-4


5/27

Monte Carlo methods. An indication of the sort of resultsobtained is shown in Fig. 1 which presents the phase diagramof this model as a function of U/K and , based on anevaluation of the density correlations. For the simulations,we worked on a LxLxL lattice, with various systemsizes indicated in the paper. We used a Metropolis algorithm

with a single-site update, r()r()1, which corre-sponds to a ring-exchange move for the boson density. We

checked for equilibration of various quantities, and averagedthe data over 104 106 sweeps of the lattice depending on thecorrelation function and location in the phase diagram. No-tice that in the time continuum limit 0, the simulationsreveal two phases as the dimensionless ratio K/U is varied,separating an ordered charge-density wave state at large Ufrom the EBL phase when the ring exchange term is large.

Before obtaining an effective low-energy description, it isconvenient to rewrite the dual partition function, Eq. 22 interms ofcontinuous variables using the Poisson summationformula

Z

d

r

sr()

exp

r,

2isr

r

s r ] exp

L . 25In Eq. 25, we have included a parameter 1, which soft-ens the discrete- constraint exactly imposed for 0.Carrying out the sum over the integer-valued s r() variables,one finds

ZZ0 dreS0S1, 26

where Z0 is a constant divergent for 0) ,

S 0

L, 27

S 1r

q1

v2qcos 2qr 28

and the v2q are O(1) coefficients whose precise values arenot important. It is convenient to drop the constant and in-troduce infinitesimal source fields h, h , for the density andenergy density, respectively,

Z h,h dreS0S1Sh, 29with

S hr,

hr

xyrnh r

r2 , 30

where rrx/2y/2.

III. EFFECTIVE MODEL

We now make an educated but perhaps bold guess as tothe nature and existence of a low-energy effective descrip-tion. In the spirit of the renormalization group and effective-field theory, we imagine defining a temporally coarse-grainedvariable in which the high-frequency modes of are av-eraged over

rrfnxy , 31

where the square brackets indicate an average over fast

high-frequency modes, and we have for convenience re-moved the mean curvature in by a nonfluctuating spa-tially dependent shift. Provided that we average only overhigh-frequency modes, we expect that the resulting effectivedescription ofwill remain local both in space and imagi-nary time. Inspection of Eq. 26 shows that the dual micro-scopic action for can be written as a sum of a quadratic part( ,rL) and nonquadratic corrections. We postulate thatthe low-energy effective action for is a sum of a renormal-ized quadratic term similar to L and small nonquadraticcorrections. Although we have not implemented a detailedrenormalization group treatment, the latter statement is tan-tamount to declaring that the system is controlled by a stable

Gaussian fixed point. Obviously, such a fixed point can haveat best a large finite basin of attraction, and we will return tothe problem of determining the region of stability of theGaussian theory in Sec. V.

A. Symmetries

Formally, the above postulate of the effective-field theorydescription can be formulated as the statement that the gen-erating functional can be approximated for low-frequencysource fields by Z h,h Z0Z h

,h , where Z0 is an ir-relevant constant, and

FIG. 1. The phase diagram of the time-discretized dual model as

a function of the two coupling constants, showing a compressible

EBL phase and an incompressible phase that we identify as a ,charge-density wave. The compressibility was obtained from the

k0 extrapolation of the density susceptibility nn (k0,

0) on8832 lattices, with some checks made on larger system sizes.

As seen from the inset, the critical U/K stays finite in the time

continuum limit K0 indicating a phase transition at U/K2.4

0.4 in the quantum model.

RING EXCHANGE, THE EXCITON BOSE LIQUID, AND . . . PHYSICAL REVIEW B 66, 054526 2002

054526-5


6/27

Z dreS0[]S1[]Sh[], 32where S0 is a renormalized quadratic form, S1 containssmall renormalized nonquadratic perturbations, and Sh is lin-ear in h, h we drop higher-order terms in the infinitesimalsources. In general, we expect all three parts of the action totake the most general form possible consistent with locality,analyticity since only high-frequency modes are averagedover, and the symmetries of the problem, which should befound from the microscopic partition function, Eq. 29.These are

1 Row/column symmetry: x,y()x,y() mx(x)my(y ), where mx(x) and my(y ) are arbitrary integer-valued functions. This corresponds to conservation of thenumber of vortices modulo 2 on each row and column, adual version of the conserved boson number on each row/column of the original lattice.

2 Space and time translations: r()rR(0),where R is an arbitrary lattice vector and 0 is an arbitraryreal number. Under this translation, the sources must also be

translated, h r/()hrR/ (0).3 Reflections across a row or column containing a site

of the dual lattice (or bonds of the original lattice): x,yss sx ,sy , where s ,s1. The sources transform as

hx,y ss h sx ,sy

, hx,y

hsx ,sy

.

4 Reflections across a row or column containingbonds of the dual lattice (or sites of the original lattice):This is not independent, and can be obtained from acomposition of a translation and a site reflection. Butfor completeness, x,yss1/2s(x1/2),1/2s(y1/2) ,

hx,y ss h 1/2s(x1/2),1/2s(y1/2)

,

hx,y h1/2s(x1/2),1/2s(y1/2)

.

5 Time reversal: r()r(

), h r/

()h r/

(

).6 Four fold rotations: x,yy,x , hx1/2,y1/2

hy1/2,x1/2 , hx,y

hy,x .

7 Particle-hole symmetry: This is an invariance only forinteger or half-integer densities (2 nZ). For such values,

the symmetry operation is xy 2nx yxy , hrhr

,

h rh r

.

B. Gaussian action

The form of S is dictated by these symmetries and therelation between the microscopic and coarse-grained fields,Eq. 31. Note that the shift by nxy implies that is not a

scalar. Consider first S0. By space and time translationalsymmetry, it is diagonal in momentum and frequency space,

S01

2

k

M k, k,2, 33

where kd2k/(2)2,

d/(2), and the mo-

mentum integral is taken over the Brillouin zone kx , ky. By row/column translational symmetry,

M kx0,ky ,0 0 for all ky , 34

and similarly for kxky . The latter condition implies theexistence of gapless excitations along the kx and ky axes inmomentum space. These are the only such gapless statesmandated by symmetry, and we expect the low-energy phys-ics to be dominated therefore by momenta near these axesand by low frequencies. Quite generally, the kernel M canbe expanded near the zero modes at kx0, and at lowestorder takes the form

M k,A ky 2B kykx

2 , 35

where the expansion functions A(ky) and B(ky) are even and2 periodic in ky . A similar expansion is of course possiblearound ky0, with the identical expansion coefficientsdue to rotational symmetry. Analyticity at k0 further im-plies B(0)0. A convenient representation of M, whichsatisfies these requirements, is

M k,2Ek

2

2K k, 36

with mode energy,

Ek4K kU ksin kx/2sin ky/2 . 37

Here, U(k) and K(k) are positive 2 periodic functions thatcharacterize the EBL phase, but it is only their behavior forkx1 or ky1 that determine the universal low-energyproperties of the theory.

C. Interaction terms

Consider next the interaction terms, S1dL1. Localityrequires that they couple combinations ofr fields only withnearby points r. Hence we consider a successive sequence ofterms, L1mL1; m , coupling fields at a total ofm distinct

points. We expect that the L1; m becomes exponentially in-creasingly small with increasing m. First consider the single-site terms, which are highly constrained, particularly bytranslational invariance, under which xynxy transformsas a scalar. Generally,

L1; 1xy

q1

2qcos2qqQxy

q2

K q1

2q , 38where Q2n, and has the physical meaning of the small-est reciprocal lattice vector of a one-dimensional lattice with

density n

like 2kF for a charge-density wave. The secondset of terms involve more time-derivatives than the analo-gous quadratic interactions in S0, and hence are negligible at

low frequency, so we will take K q10 in the following. The

2q terms will play an important role in Sec. V. Note that,except when the density takes special commensurate valuessuch as the interesting case n1/2), they are strongly os-cillatory in space.

Next consider two-site terms. Similar to the Kq1 terms in

Eq. 38, a variety of spatial lattice and time derivativeinvariants are possible, but are negligible relative to S0, so


054526-6


7/27

we do not include them here. More interesting are sinusoidalterms, which must take the form

L1; 2 xy ;m,n

qq

w 2q,2q ,mn

cos2qxy2qxm,yn

Qqxyqxm yn w 2q,2q,mn

cos2qxy

2qxm,ynQqx yq

xm yn , 39

where rotational and reflection invariance require w 2q,2q,nm

w 2q,2q,mn

, and without loss of generality we may also

impose w 2q,2q,nm

w 2q,2q,mn

. Like the 2q terms in Eq.

38, most of the operators in Eq. 39 are highly oscillatory

for generic n. An important exception arises in w 2q,2q,mn for

rational densities nz/z, where z,z are integers. Thenthese terms are nonoscillatory if zmq and znq are z timesintegers. We consider in particular the case z1, for whichm ,q and n ,q can be chosen as all possible factors of zl intotwo integers, with arbitrary integer l. Keeping only theseterms, one has

L1; 2xy

mqnqzl

w 2qm ,ncos2qxyxm,yn ,

40

where, w 2qm,nw 2q

n,m by rotational invariance. Since e i actsas a vortex creation operator, the terms in Eq. 40 can bethought of as hopping 2q vortices a distance s(m ,n) onthe 2D dual lattice.

Finally, consider ShdLh . As for L1, we expect thatthe largest contributions will occur in terms involving rfields at a small number of distinct points r. For simplicity,we keep only the most local of these. Employing the sym-metries above, one finds

Lhxy

hx1/2,y1/2 c 0xyxyq1

c2q xy sin2qxy

qQxy hxy c 0 xy2 q1

c2q cos2qxy

qQxy , 41where c2q

/ are constants determined by the high-energy

physics. Here we have kept only the single-plaquette termsin the charge density, and the single-site terms in the energydensity. Note that the operators multiplying h involve sinesrather than cosines, which is a consequence of particle-holeand reflection symmetries. The form in Eq. 41 can be de-rived perturbatively in v2q using an explicit one-step coarse-graining procedure.32 The quantitative reliability of this con-structive procedure is, however, limited to small bare v2q , so

for the problem of interest the constants c 2q/ should be

viewed as phenomenological parameters. Equation 41should be interpreted analogously to the nontrivial bosoniza-tion rules for, e.g., the charge density and other operators inone dimension. Very generally, a low-energy effective action

must be accompanied by rules specifying the operator con-tent of various physical fields. In analogy to the notationsused there, one may rewrite Eq. 41 as a scaling equalitybetween the microscopic and coarse-grained fields, as in Eq.9.

D. Self-dual formBosonization analogy

Having established the form of the effective action for the variables, we are also interested in computing quantitiesinvolving the boson phase . Since the effective theory isperturbative in S1, it is straightforward to reintroduce us-ing Gaussian integration. In particular, the quadratic actionS0 in Eq. 33 with the kernel in Eq. 36 can be transformedinto the form shown in Eq. 5 in Introduction using aHubbard-Stratonovich transformation,

exp 12k1

2K k k,

2

dexp

1

2k

K kxyk2

exp

r

i

rxyr , 42

where rrx/2y /2 as usual, and (xy)k denotes theFourier transform of xyr . After an integration by parts inthe last term on the right-hand side above, the full Gaussianpart of the action takes a particularly transparent form, S 0L0, with Lagrangian L0 defined in Eqs. 5, 6.

The partition function is now represented as a path inte-gral over both sets of low-energy fields, r and r , in anappealing self-dual form. The Gaussian theory above gives afixed-point description of the EBL phase. When augmented

by the operator content of the fields, as in Eq. 9, togetherwith the irrelevant nonlinear interaction terms in Eqs. 38,39, it gives a complete description of the universal proper-ties of the EBL phase, as detailed in the following section.

It will sometimes be convenient to integrate out the dualfield , leaving a Gaussian action just in terms of the phaseof the boson wave function,

S01

2

k

M k, k,2, 43

with kernel,

M k,

2

Ek2

U k, 44

and with Ek as given in Eq. 37. Notice that this form isidentical to the Gaussian fixed-point theory in terms of inEq. 33, except that U(k) has replaced 2K(k) in the de-nominator of Eq. 44.

The above self-dual representation makes particularly ap-parent the close analogy between this theory and bosoniza-tion theory in one-dimension.24 In particular, underlying boththeories is a pair of dual scalar fields, one the phase of thewave function and the other related to the density. The Lut-


054526-7


8/27

tinger liquid fixed point is a Gaussian theory in the twofields, and has a form that is nearly identical to the EBLaction above,

L1dH1di

xx, 45

with Gaussian Hamiltonian,

H1dx K2 x2 U22 x2 . 46

Notice that the long-wavelength particle density in bosoniza-tion theory, nx/, has simply been replaced by a lat-tice second derivative in our (21) -dimensional theory: nxy/. The commutation relations between the two dualfields has also been modified, as is apparent by inspecting theBerrys phase term involving both fields in the aboveLagrangians. The expressions in Eq. 9 relating the bareboson density and energy density to the low-energy fields area generalization of the more familiar bosonization expres-

sions, where, for example, the boson density near 2kF isproportional to cos2.A key strength of bosonization in 1D is that it allows one

to study the instabilities of the Luttinger liquid towards vari-ous types of ordered phases, such as a charge-density wavestate. Similarly, the above Gaussian representation of theEBL fixed point is particularly suitable for studying instabili-ties both towards insulating states with broken translationalsymmetry and towards a superfluid. But before undertakingthis analysis, we explore the EBL phase in some detail.

IV. THE EXCITON BOSE LIQUID PHASE

We now turn to the physical properties of the EBL phase,

using the effective low-energy theory developed in the pre-vious section. For the present, we will assume stability of theEBL, so that all physical quantities are accessible via pertur-bation theory in S1) around the Gaussian theory for ,.The validity of this assumption is discussed in Sec. V.

A. Thermodynamic properties

Consider first the thermodynamic properties of the EBL.The simplest is the compressibility n/ . This is trivi-ally given from the Gaussian theory, and one finds 1/U(k0). Numerically, the compressibility may be ob-tained from an extrapolation of the density-density correla-tion function,

nn k,n r d 1

2xyrxy00 e

inikx,

47

from which nn (k0,0). The compressibility de-termined in this way from the simulations were used to de-fine the phase diagram in Fig. 1

Also interesting is the specific heat. Since the EBL isessentially a free boson theory, this is determined completelyfrom the density of states for the collective free boson modes

EkEEk, 48

where the boson energy is

Ek k4sin kx2 sinky

2 , 49

and (k)

U(k)K(k). Like an ordinary Fermi liquidmetal, the density of states gets a large finite contributionfrom the low-energy modes near the Bose surface alongthe coordinate axes in momentum space. Unlike a Fermi liq-uid, however, there is a weak logarithmic divergence associ-ated with the crossing point at kxky0. In particular,

E1

20 ln1/C0O, 50

where 0(0,0), E/0 and

C04 ln 20

dk

sin k/2 0

k,01 . 51

Note that the constant term in the density of states dependsupon the full form of the dispersion (k,0)(0,k) allalong the Bose surface, while the logarithmic term dependsonly upon the behavior at k0. The specific heat is then

Cv

T

4

0

dxx 2 Tx

sinh2x/2, 52

which to the same accuracy gives

Cv

T

420I0 ln 0T C0I0I1 , 53

and In0

dx x2ln(1/x) n/sinh2(x /2), or I042/3, I1

2( 32E)2/38(2 )4.643, where (z) is the Rie-

mann function, and E0.5772 is Eulers constant.

B. Boson correlation functions

It is interesting to contrast the large density of states forcollective excitations with the tunneling density of states forthe original bosons, which we will see is strongly sup-pressed. In particular, consider

G r,eir()ei0(0 ) . 54

By symmetry, since the boson number is conserved on each

row and column, G(r,) can be nonzero only for r0. Tofurther determine the behavior of G in the EBL, we mustrelate the microscopic Bose creation/annihilation operatorsto the low-energy modes. By symmetry,

e ire ir1A cos2r2nxy . 55

We expect the low-energy properties to be dominated by thefirst term, i.e., simply replacing . Since is a Gaussianvariable,

G r,eF()r,0 , 56


054526-8


9/27

with

F12 000

2k

U k

2Ek1eEk.

57

Since Ek vanishes linearly both as kx0 and as ky0,inspection of Eq. 57 shows that F() has a double loga-

rithmic divergence as . Indeed, a careful calculationshows that

FF2

2ln 0

2F1

ln0 58

for 01, with

F2

1

22U0

K0, 59

F1

1

2

0

dkU k,0K k,0

1

2 sink/2U0

K0

1

k

1

22U0

K02 ln E , 60

where U0U(0,0), K0K(0,0). The ln20 behavior of

F() at large implies that G() decays faster than anypower law. This translates into a similar singular behavior forthe tunneling density of states. Writing the boson Greensfunction in a spectral representation,

G 0,0

dEtunEeE, 61

one finds using a simple saddle-point analysis that the abovebehavior at large requires

tunEexp 2 ln20

E ln

0

E lnln0

E ,

62

where F2, F1

1F2

ln F2

.Thus, although the EBL possesses a large set of gapless

modes, the density of states for adding a boson into the sys-tem vanishes at low energy. Again, this is analogous to aLuttinger liquid in 1D;24 the conservation of particles perrow or column means that the added particle affects all theparticles in a particular row or column, leading to a sup-pressed amplitude for such tunneling events. Remarkably,the tunneling density of states actually vanishes fasterthan ina Luttinger liquid, indeed faster than any power law in en-ergy. This behavior can be roughly understood as arisingfrom two orthogonality catastrophes occurring simulta-neously in the row and column in which the boson is addedor removed.

It is also instructive to consider the boson four-point cor-relation function, for simplicity at equal times. Due to therow/column symmetries this is nonvanishing only when thefour points sit at the corners of a rectangle:

G(4 )

x,y e ioo ()eixo ()eioy ()e ixy () . 63

Upon replacing with the low-energy field , this can bereadily evaluated using the Gaussian EBL action. For any fixed y, one finds power-law behavior in x and vice versa

since G(4 )(y ,x)G

(4 )(x ,y ),

G(4 )

x,y

1

x (y) for x1, 64

with

y 1

2

0

dkU0,kK0,k

sin2 ky/2

sin k/2. 65

Note that Eq. 64 requires only x 1 and places no restric-tion on y. Hence it is also obtained when both arguments arelarge, and thus

G(4 )

x ,y exp

1

2U0

K0

lnx lny C lnx lny

,

66

as x ,y , with

C1

2U0

K0Eln

1

2

0

U 0,kK 0,k

1

2 sink/2U0

K0

1

k . 67

C. Vortex correlation functions

The exact plaquette duality makes it possible to define anumber of characteristic vortex correlators in terms of thefield e i. As for the boson correlator above, the behavior inthe EBL depends upon the expression for the vortex operatorin the low-energy variables. By symmetry, we expect theleading terms involving the smallest exponentials of) tobe

e irReAe inxy e ir, 68

where in general A is complex. Note that, since / is aninteger, the microscopic vortex field satisfies exp iexpi,but this is not true for . In the special case of n1/2,particle-hole symmetry further implies A is real,

whence

e ir n1/22A cosr 2 xy . 69With this relation in hand, let us consider first the vortex

two-point function,

G r,eir()ei0(0 ) . 70

As for the boson correlator G, the vortex two-point func-tions vanishes unless r0 due to the dual row and column


054526-9


10/27

symmetries. Evaluating it using Eq. 68 and the Gaussianaction, one finds similar results to the boson tunneling den-sity of states,

G 0,exp F2

2ln 0

2F1

ln 0 , 71where

F2

1

2K0

U0 , 72

F1

0

dkK k,0U k,0

1

2 sin k/2K0

U0

1

k 1

2K0

U0

2 lnE. 73

In fact, these coefficients are obtained directly from Eq. 59using the duality transformation K(k)U(k)/2. In Fig. 2we show numerical results for the two-point vortex correla-tion function obtained from the quantum Monte Carlo simu-lation in the parameter regime corresponding to the EBLphase. The downward curvature of the data is consistent with

a decay more rapid than a power law, and as shown in theinset can be fit to the form in Eq. 71 with U0 /K01.

Next consider the vortex four-point function, which due tothe dual row/column symmetry is nonvanishing only whenthe four points sit at the corners of a rectangle,

G(4 )x,y e ioo ()e ixo ()e ioy ()e ixy () . 74

This can be reexpressed in terms of the low-energy field using Eq. 68, and then evaluated with the Gaussian EBLaction. For any fixed y, one finds power-law behavior in x,

G(4 )x ,y

cosnxy

x v(y)for x1, 75

with

v

y 0

dkK0,kU0,k

sin2 k y/2

sin k/2. 76

Interestingly, one can see from Eq. 75 that for the case n

1/2, the four boson correlator vanishes exactly wheneverx,y is odd as a consequence of particle-hole symmetry. Thisbehavior, and the associated power-law correlations, areshown in Fig. 3. In the limit when both x ,y, the vortexfour-point correlator vanishes faster than any power law,

G(4 )x,y cosnxy expK0

U0 lnx lny

Cv

lnxlny , 77as x ,y , with

CvK0

U0Eln

0

K0,kU0,k

1

2 sink/2K0

U0

1

k . 78

With the exception of the cos(nxy) prefactor in Eq. 77, allthe results in this section can be obtained from those in theprevious one by the duality transformation and

U(k)2K(k).

FIG. 2. Vortex correlation function G(0,)ei0()ei0(0 ) in

the EBL phase. The correlation function decays faster than a power

law exp(14K0 /U0ln 0

2) at long times see text. Inset

shows this decay in a finite-size scaling plot (L4 to L32) of

ln G

(0,) versus ln for L

/2, from which we extract U0

/K01, nearly its bare value for these parameters.

FIG. 3. The four-point vortex correlation function G(4 )(x

1,y ,0), defined in the text, evaluated in the EBL phase. The

correlation function vanishes for odd values of y due to particle-

hole symmetry at n1/2. The expected power-law decay of the

envelope is shown in the inset as a finite-size scaling plot, from

which we extract the exponent v(1 )1.90.1, close to its barevalue for U/K1.


054526-10


11/27

D. Collective correlation functions

Next, let us consider the correlations of two-particleoperators such as the boson and energy densities. As above,we require the operator correspondences between the micro-scopic and effective variables. These were already workedout in Sec. III, and summarized in Eqs. 9 and 10 in In-troduction. For simplicity, we focus on the most interestingcase of n1/2, for which Q, and furthermore keep onlythe lowest nontrivial harmonics with q1. In this limit, Eqs.9 and 10 reduce to

nx1/2,y1/2c 0xyxy

c 2

a,b0,11 (xa)(yb)ab

sin2xa,yb , 79

xy c0 xy

2c 2

1 xy cos2xy . 80

First consider the density-density correlation function,

nn r,nx1/2,y1/2n 1/2,1/20 . 81

Substituting for n using Eq. 79, one obtains three contri-butions to nn :

nn r, c02nn

00 r, c 22nn

22 r,c0

c 2nn

20 r,.82

The cross term nn20 is negligible. The first contribution,

nn00 (r,) is just the correlator between coarse-grained

densities xyxy . This term is nonzero in the Gaussiantheory, and gives a smooth function of r, with a power-law

behavior at large arguments. For instance, at equal times andlarge x , y 1,

nn00

r,0K0U0

1

x 2y 2. 83

More generally, nn00 has a smooth Fourier transform,

nn00 q,n

2

U k

Ek2

n2Ek

2. 84

Perturbative corrections from S1 to nn00 do not modify this

qualitative behavior.The remaining contribution to the density-density cor-

relator comes from the sin 2 terms in Eq. 79. Naively, thiscontribution is ultralocal and hence uninteresting, i.e., van-ishes unless x , y 1, as a consequence of the fact that thediscrete row/column symmetries are promoted to continuousones at the Gaussian level. One may interpret this as mean-ing that the vorticity on each row or column is conservedexactly in the Gaussian model. This conclusion, however, isincorrect once the nonquadratic corrections in S1 are takeninto account, since expanding factors of2(1)

xy cos 2cansupply vorticity in units of 2 to a particular site. Hence,

nn22 becomes nontrivial at second order in 2. Provided

x , y 1, the appropriate second-order perturbative expres-sion is

nn22 r,

22

8 abcd0,1 1 (xca)(ydb)abcd

Cxca,ydb(4 ) , 85

where

Cx,y(4 )

d1d2exp2i[oo (0)xy ()x0(1)0y(2)] 0 , 86

where the expectation value indicates an average calculatedin the Gaussian theory. At large distances, x ,y 1, oneexpects Cxca,ydb

(4 ) ()Cx,y(4 )() , independent of

a,b,c,d. In this limit, only the prefactor in Eq. 85 dependsupon a ,b ,c ,d, and the sums can be carried out explicitly. Todo so, it is convenient to employ the identity

1 xy12 11

x1 y1 xy, 87

valid for integer x,y . Applying this identity to Eq. 85, onlythe last term survives the sum, and gives

nn22

r,22

1 xyCx,y(4 )

, 88

for x ,y 1. This indicates the presence of (,) correla-tions in the boson density. More generally, if only x 1 buty 1 but still of O(1), one finds

nn22

r,2

2

41 xy

bd0,1

11 ydbCx,ydb(4 ) ,

89

so at any fixed y the density-density correlator oscillates atwave vector as a function of x and vice-versa by rota-tional symmetry. To establish the range of these correla-

tions, we must consider Cx,y(4 )() in some detail. Using prop-

erties of Gaussian fields, we have

Cx,y(4 )

d1d2expcx,y,1 ,2 , 90where

cx,y,1 ,22xy 000 x01

0y2 20 . 91

The calculation and analysis ofcxy is somewhat involved,and is described in detail in Appendix B. There we derive auseful approximation for cxy which captures the behavior inall the relevant limits ( x1,01 ,02 , and y x ),


054526-11


12/27

cxy 2vy lnx2y

22

K0U0

F 01y F02

y , 92where y 0 but in general depends weakly upon the fullform of K(0,ky),U(0,ky), and y and the crossover functionF(X) satisfies

F X 0 X1ln2X X1.

93

Using Eqs. 90, 92, 93, we find

Cxy(4 )

x 1 y0

2

e2( K0 /U0)1/4

x2y222v(y),

94

and of course the same behavior with xy .

This power-law behavior of Cxy(4 ) , and hence nn (r,)

translates into singularities in the static structure factor,

S nn kxy

ei(kxxkyy)nn r,0 , 95

for wave vectors near ,. In particular, let kq,and consider the limit qx1 with qy fixed. In this limit thesingular behavior of the structure factor is dominated bylarge x but y of O(1) . Hence we may apply Eq. 89 towrite

Snn qqx1 2

2 odd y

cos qyy1

cos qy x eiqxxCxy(4 ) 0 . 96

From the power-law behavior in Eq. 94, one can readily seethat the Fourier transform in x leads to singularities for smallqx . Indeed, if any v(y )1/4 for any odd y, nn diverges asqx0 at fixed qy , while for 1/4v3/4, the structurefactor remains finite but has a divergent second derivative atqx0,

Snn qS nn0

q odd y

Ay qy sgny qx

y,

97

where S nn0 (q) is a smooth function, y14v(y), andAy

(qy)22cos qyy(1cos qy) is a positive amplitude peaked

at qy0. Hence the behavior for small qx is dominated bythe minimum over odd y) value of

v(y ) maximum ofy).

The zero-frequency susceptibility, nn (k)nn (k,n0) has a similar but stronger divergence due to the extratime integration,

nn qnn0 q

odd yAy

qy sgny qxy,

98

where nn0 (q) is another smooth function, Ay

(qy)Ay(qy),

and y24v(y )1y signals the stronger diver-gence. The difference in exponents implies that if Snn (q) has a divergent slope at qx0, nn (q) itself diver-gences there. Conversely, if nn (q) has a slope diver-gence at qx0 occurs for 1/2v3/4), the static struc-ture factor does not. Numerical results for the densitysusceptibility in the EBL phase are shown in Fig. 4, andreveal a singular cusplike behavior at wave vector . Thisform is consistent with that predicted by Eq. 98 with a

maximum value 1yma x0.

Now consider the energy-energy correlation function,

r,xy xy 0 c , 99

where the c subscript indicates the cumulant expectationvalue. As for the density-density correlator, we can employEq. 80 to expand as

r, c02

00 r, c22

22 r,c 0

c 2

20 r,.

100

As for the density case, 00 has a smooth Fourier transform,

and 20 is negligible. We focus on

22 , which for x ,y 0) to second order in 2 is

22

22

81 xy Cxy

(4 ). 101

Using Eq. 87, 94, one straightforwardly sees that there are

singularities in 22 (k) as kx(ky)0 and kx(ky). In par-

ticular,

FIG. 4. The density susceptibility nn (k,0) in the EBL

phase, close to the phase boundary, along 0,02,2. The cusp

at , is due to subdominant power-law correlations in the EBL

phase, which may be viewed as arising from corrections to thedensity operator in the Gaussian theory as discussed in the paper.

Inset shows the gray-scale plot of the susceptibility over the entire

Brillouin zone, centered on , dark regions indicate large sus-

ceptibility, and the kx0 and ky0 lines are omitted from the

gray-scale plot for clarity. The singular cusp is visible along the

indicated 0,02,2 direction.


054526-12


13/27

S kkx1S

0 k even y

Ay ky sgny kx

y,

102

kkx1

0 k

even yAy

ky sgny kx

y,

103

and

S qx ,ky qx1 S

0 qx ,ky

odd y

Ay ky sgny qx

y,

104

qx ,ky qx1

0 qx ,ky

odd yA

y

ky

sgny

q

x

y,

105

where Ay(ky)Ay

(ky)cos kyy. Similar formulas for ky0, are obtained by rotation. Note that because Ay

(ky)

cos kyy is negative for ky and y odd, the energy-energycorrelator has a singular dip near k and singular peaksnear k(0,0),(,0), and (0,). Numerical results for theenergy susceptibility in the EBL phase are shown in Fig. 5.Notice the singular cusplike peaks at wave vectors (0,) and(,0), as predicted in Eq. 105. This is in qualitative agree-

ment with a large (,0),(0,) energy susceptibility seen indirect quantum Monte Carlo simulations of an S1/2 easy-axis ring-exchange model.28

E. Electrical conductivity

We finally consider the electrical conductivity in the EBL.To obtain an expression for the current operator requires cou-

pling in a vector potential. The usual minimal coupling pre-scription for a lattice model of bosons would have one re-place a discrete lattice derivative as

rrrrA r , 106

with x, y, and define the current operator by differentiat-ing with respect to the vector potential. For the boson ringmodel this prescription is ambiguous due to the second de-rivative form, and a family of gauge inequivalent forms arepossible,

xyrxyrxA ryyA r

x , 107

where xfrfrxfr denotes a discrete derivative. Gaugeinvariance requires 1, whence this can be reexpressedas,

xyrxyrxA ry1 , 108

where xAyyA

x is the gauge invariant flux throughthe plaquette. If one derives the boson ring model by startingwith a model of electrons reformulated in terms of a Z2gauge field coupled to spinons and chargeons as detailed inAppendix D, one arrives at an appealing symmetric form forthe ring term with 1/2.

Henceforth we focus on the zero wave-vector conductiv-

ity . In this case, the above ambiguity is irrelevant, sincea spatially uniform vector potential does not enter for anyvalue of. But the vector potential will still of course enterinto the boson hopping term in Eq. 4, and upon differentia-tion generates the usual current operator,

Ir tsinrr. 109

When we coarse grain the theory, we should write down thecurrent operator in terms of the slow field ,

IrcItsinrr , 110

with cI being a dimensionless constant. The other contribu-tions will include terms that hop a single boson several lat-tice spacings and terms that hop several bosonsgenerallyall local terms allowed by the symmetries. Such terms willgenerically be subdominant at low frequencies, and so weretain only the leading contribution.

With the current operator in hand it is straightforward toobtain the conductivity from the usual Kubo expression:

1Re in, 111

with

FIG. 5. The energy susceptibility (k,0) in the EBL

phase, close to the phase boundary, along 0,00,2. The cusp

at 0, is due to subdominant power-law correlations in the EBL

phase which may be viewed as arising from corrections to the

energy-density operator in the Gaussian theory as discussed in thepaper. Inset shows the gray-scale plot of the susceptibility over the

entire Brillouin zone, centered on , dark regions indicating

large susceptibility. The singular features are most clearly visible

as peaks dark spots at the 0, and ,0 points and as a dip

white spot near ,.


054526-13


14/27

in 1

n d

reinIr

yI0y 0 . 112

We have dropped the diamagnetic contribution since it willnot contribute to the real part of the conductivity. Evaluatingthe correlator using the Gaussian EBL action gives at zerotemperature,

IryI0y0 0t2y ,0x22

, 113

with scaling dimension (1)/2 where (y ) is given ex-plicitly in terms of the Bose liquid parameters in Eq. 65.Performing the time integration and spatial summation givesa power-law singular contribution,

n in At21 int()n

22re g in ,

114

where A is a positive constant and re g is analytic in itsargument and thus does not contribute to the real part of theconductivity. Analytic continuation to real frequencies givesa singular contribution to the complex conductivity of the

form

At21 int() i23 i2, 115

with real 2(). We note that causality places strong con-straints on the phase angle29 from the singular contribution,consistent with the above form. Finally, taking the real partgives the optical conductivity,

1At2sin 23. 116

Notice that the amplitude of this contribution vanishes forinteger . For these special cases, the higher-order contribu-tions to the current operator should be kept, and will contrib-ute a similar form but generally with larger scaling dimen-

sions.As we discuss in the next section, stability of the EBL

phase to boson hopping requires that 2, implying an op-tical conductivity vanishing rapidly at low frequencies,1()

with 1. For 2, the EBL will be unstableto superconductivity, but for small boson hopping amplitudethe transition temperature would be low. In that case, theoptical conductivity above Tc might still be well describedby the above power-law form.

One may also directly calculate the nonlinear dc conduc-tivity, (Ey ,T)Iy/Ey as a function of Ey and T. Thisis mathematically complicated, but formally quite similar tothe perturbative calculations of tunneling conductance be-

tween parallel Luttinger liquids, as carried out, e.g., in Ref.30. We forgo this calculation here for the sake of brevity,quoting only the resulting scaling form

Ey ,TAt2T23GEy /T, 117

where G(E)1 as E0, and F(E)E23 for E1. Thisimplies in particular that the dc linear response conductiv-ity at nonzero temperature behaves as (T)At2T23.These results are for the pure ring model, but will doubtlessbe altered somewhat in the presence of impurity scattering.This we leave for future study.

V. INSTABILITIES OF THE EXCITON BOSE LIQUID

There are two classes of perturbations that one can add tothe EBL fixed point described by the quadratic action S0)that can potentially destabilize the phase. The first are termsinvolving the hopping of bosons. When relevant, such bosonhopping terms stiffen the phase fluctuations of the bosons,leading to off-diagonal long-range order and a superconduct-

ing state. As we shall see, the perturbative relevance of theboson hopping terms is determined by the Bose liquid pa-rameters that enter into the fixed-point action that character-izes the EBL phase. There are regions of parameters whereall such boson hopping terms are irrelevant and the EBLphase is stable to such superconducting perturbations, as wedetail in Sec. V A below.

The second class of perturbations involve hopping or mo-tion of vortices, conveniently expressed in the dual represen-tation. When relevant, these perturbations signal a condensa-tion of vortices which typically leads to a breaking oftranslational symmetry and drives the system into an incom-pressible insulating state. The presence of these instabilitiesand the precise form of the translational symmetry breakingdepend sensitively on the boson density, generally requiringboson densities commensurate with the underlying lattice. InSec. V B below we focus on half filling (n1/2), and studythe nature of the resulting commensurate insulating states.

If the boson density is incommensurate with the lattice, onthe other hand, small vortex-hopping terms are unimportant.Provided the Bose liquid parameters are in the regime whereboson hopping is likewise irrelevant, the EBL exists as acompletely stable critical phase. The gapless EBL is then a2D analog of the stable 1D Luttinger liquid.

A. Boson hopping and superconductivity

Consider then the stability of the EBL phase in the pres-ence of boson hopping operators. To be general, we considerprocesses where q bosons hop along a displacement vector s:

Ltr

q,s

tqs cos qrrs. 118

To assess the perturbative relevance of such processes, wecompute the two-point function of the tunneling operators,

Tqs

rcos qrrs , 119

using the Gaussian action for the EBL. The row/columnsymmetries in the EBL phase greatly constrain the spatialdependence of these correlators. We consider first the specialclass of tunneling operators that hop bosons along the x or yaxes with s(m ,0) or s(0,m). For this class of operatorsone finds a power-law decay both in time and in one spatialdimension:

Tqm y r,Tq

my 0,00y ,0x2v

22qm , 120

with scaling dimension,


054526-14


15/27

qmq 2

2

dky

2U0,ky

K0,ky

sin2 mky/2

sin ky/2

q 2

2 m .

121

Notice that these scaling dimensions vary as q2,and will generally increase slowly qm q

2/ ( 22 ) U0 /K0lnm) with hopping distance m. As expected, thesescaling dimensions increase with U/K.

For the remaining boson tunneling operators with vector sconnecting two different rows and two different columns, thecorrelator is further constrained by the row/column symme-try, being spatially local,

Tqs

r,Tqs

0,00Gr,0 for sxsy0, 122

with G()2qs

.General renormalization group reasoning implies that op-

erators are irrelevant about a given fixed point when the as-sociated scaling dimension exceeds the space-time dimen-sion, Dd1, with dynamical exponent z1. But dueto the constrained form of the above correlators, which only

exhibits a power-law decay in a reduced set of space-timedimensions, D re d , one expects that the condition for irrel-evance should be modified to be D re d . Thus, when

q,m2 and qs1, one expects that all of the boson-

hopping operators should be unimportant as one scales downin energy, and the EBL phase will be stable. This can alwaysbe achieved in principle by increasing the ratio of U/K.

It is tempting to strengthen this expectation by construct-ing an explicit renormalization group RG transformation,but this is somewhat problematic due to the peculiar exis-tence of zero-energy states at both kx0 and ky0 in theEBL phase. One could try to integrate out gapped modesaway from the zero-energy cross in the Brillouin zone,

and then successively integrate out shells of modes pinch-ing down onto the cross. A difficulty arises, however, in therescaling transformation of the momenta, because the rangeof both kx and kythe interval ,is not invariant.Similar but less severe difficulties are encountered when onetries to implement a momentum shell RG procedure for a 2DFermi surface, due to the possible modifications of the shape/size of the Fermi surface. It might be possible to circumventthis difficulty along the lines of Shankar,31 or by an RGprocedure in frequency space.32 Some insight can be gleanedby ignoring the zero modes along the kx0 axis, and con-sidering a 1D RG transformation where the integration isover a shell of momenta in kx for all ky, and fre-

quency, and then rescaling both kx and i.e with z

1) butnot ky . The resulting perturbative linearized RG equationfor the Boson hopping amplitude in the y direction, tq,mtq

my is then simply

ltq,m1zq,mtq,m , 123

with z1 and q,m the scaling dimension given explicitlyabove. This argument indeed supports the expectation thatweak boson hopping will be irrelevant provided q,mD re d2. In the absence of a fully controlled 2D RG pro-cedure, we verify this conclusion by resorting to perturbation

theory. In Appendix C, we do just this by formally comput-ing corrections to the two-point function, e ir()eir(0 )perturbatively in powers of the boson hopping tq1,m1. Sta-bility of the EBL phase requires that the long-time behaviorbe unmodified from that calculated within the Gaussiantheory. Carrying out this expansion to second order in t1,1 ,we show explicitly that this is in fact the case provided1,12, thereby confirming the expected result. One also

obtains the same condition for perturbative stability using aself-consistent Gaussian variational action.33 This is similarto the case of the sine-Gordon model in 11 dimensions,where such an approach reproduces the exact RG result forthe location of the critical point where vorticity becomes arelevant perturbation.

When the scaling dimension of a single boson-hoppingterm is sufficiently small, 1m2, one expects the EBLphase to be unstable to a superfluid state in which the bosonscondense, with e i0.34 If the original lattice bosons aresupposed to represent the Cooper pairs say in a model of thecuprates, then this will of course be the superconductingphase.

B. Vortex hopping and insulating states

1. Stability

We next consider the effects of the various nonlinear in-teractions involving the vortex operators in Eq. 38 and39, which can potentially destabilize the EBL phase. At ageneric density, for which n is irrational, all these cosineterms are oscillatory, and, if weak, cannot lead to any long-wavelength divergences. Hence we expect that away fromcommensurate densities, and provided qm2 for all q ,m sothat boson tunneling is irrelevant, the EBL is a stable zero-temperature phase of matter.

For rational densities, some of the vortex operators will benonoscillatory, and must be considered more carefully. Inparticular, for very commensurate boson densities i.e., nequal to a small-denominator rational fraction, it is not apriori obvious whether the EBL can even in principle bestable to both boson hopping and vortex operators. To dem-onstrate the issues, we present a stability analysis for thespecial set of rational fillings, n1/z , where z is a positiveinteger.

At n1/z, the onsite terms take the form,

Lv

r

q1

2qcos 2q 2qz x y . 124Note that although for qj z, with integer j1, these termsare spatially varying, they are not wholly oscillatory, in thesense that they all have a nonzero spatial average for con-stant , e.g., for prime z, and qj z, cos(2q/zxy)xy1/z . Hence even for qj z, they cannot be argued awaysimply on the grounds that they are oscillatory. Instead, toassess the importance of these perturbations, we again con-sider the two-point function of these operators evaluatedwith the Gaussian EBL action,

G2q r,e i2qr()ei2q0(0 )0r,0F, 125


054526-15


16/27

with F()exp(q2K0 /U0ln2()) at large times. Due to

the dual row/column symmetry this correlator is spatiallylocal, and is also short ranged in time, vanishing fasterthan any power law. The associated scaling dimension is thusinfinite, and these operators are strongly irrelevant and willnot destabilize the EBL. However, as we shall see, they willplay an important role in determining which of the variousinsulating phases is selected when the EBL is driven unstable

by the two-site terms.When the two-site terms in Eq. 39 are weak, we can

drop those that are spatially oscillating and focus on the restwhich take the form of vortex hopping and creation terms,

Lw r,a,

w a,Oa, r, 126

with operators, Oa,(r)cos(2qr2qrs), where alabels the various values of integers q ,q and the hoppingvector s. Again, to establish the perturbative relevance ofsuch terms in the EBL phase, we evaluate the two-point cor-relators with the Gaussian fixed-point action. For the opera-

torsO

, the dual row/column symmetry is especially re-strictive and we find that the associated correlator is onceagain spatially local

Oa, r,Oa, 0,00Fr,0 , 127

and short ranged in time with F()expca,qln2()

with constant c a,q), so that like the onsite terms, these op-erators cannot destabilize the EBL. A similar behavior isfound for O , except for the special class of operators withqq, which correspond physically to 2q vortices hoppingalong a vector s. Of these, except for the special case withs(m ,0) and s(0,n), the two-point function is again spa-tially local, although it is now a power law in time, F()

2

. While such operators could potentially destabilizethe EBL if the power 1, they will generally be less sin-gular than the remaining class of operators with vortices hop-ping along the x or y axis,

Lww 2q,mxy

qmjz

cos2qxyxm,y

cos2qxyx,ym . 128

The operators with qmj z with integer j) are truly oscil-latory i.e., have zero spatial average for constant ) at n1/z and have been dropped. We expect the coefficientsw 2q,m to be positive since these operators will be generated

at second order from the irrelevant onsite terms; w 2q,m2q

2 .For this last class of operators the two-point function is

-correlated in one-spatial direction but a power law in theother and in time,

O 2q,my r,O 2q,m

y 0,00y ,0x2v

22 2q ,mv

,129

where O 2q,my (x,y )cos2q(xyx,ym). The associated

scaling dimension is given in terms of the Bose liquid pa-rameters,

2q,mv2q2

0

dkyK0,ky U0,ky sin2 mky/2

sin ky/22q2

v m .

130

As for the boson-hopping operators, the vortex-hopping scal-ing dimensions vary as q 2, and also increase slowly withincreasing hopping distance m. In general, we expect that forlarge z the most relevant of these will be w 2,z , with scaling

dimension 2,zv 2v(z)2K0 /U0lnz, so for a suffi-

ciently large z this can be rendered arbitrarily large, andhence all the vortex-hopping terms can be made irrelevant for any U(k), K(k). Thus for sufficiently large z, such that2q,mD re d2 for all qmj z, there is certainly a domainof stability for the EBL phase i.e., where the Bose liquidparameters are further tuned to make all hopping terms irrel-evant.

For small z, however, this is not clear. Indeed, it isstraightforward to show that for z2 i.e., half filling, n1/2), a choice of uniform functions U(k)U0 ,K(k)K0does not lead to a stable regime. In particular, in this case,the leading vortex-hopping operator has scaling dimension

2,2v

2v(2 )4K0 /U0, while the leading boson-hoppingoperator has scaling dimension 11(1/

2)U0 /K0. Hence22

v 114/2, which implies min(11 ,22

v )2/2 , sothat at least one or the other operator is relevant. We have notdetermined whether a stable EBL phase might be possible athalf filling when U(k), K(k) are momentum-dependent.

2. Instabilities

As indicated above, for small z, either a vortex- or boson-hopping instability may be inevitable. In any case, it is inter-esting to study the nature of the state resulting from relevantvortex-hopping terms. Here, we briefly study the nature ofthe resulting phase for half filling, taking into account the

presence of the two most potentially relevant operators, withq1,m2 and q2,m1. In general, at half filling, thereare Bose liquid parameters for which both w 22 and t1,1 arerelevant, and more complex behavior may well occur in thisregime. We will, however, neglect boson hopping com-pletely, as appropriate for large U/K.

To this end, let us assume that w 2,2 is the most relevant

operator, with 2,2v2. We also assume that w 4,1 is the next

most relevant it could be irrelevant, but still the least irrel-evant remaining operator. Then, provided all bare couplingsare small, we imagine integrating down in energy until w 2,2becomes comparable to the Gaussian terms in S0. This re-

quires w 2,22,2v

U(,0)2, which always occurs for suffi-

ciently small with 2,2v2. At this point it seems appro-

priate to minimize the potential w 2,2cos(2xy2x2,y)cos(2xy2x,y2), simultaneously with S0. This minimi-zation somewhat underestimates fluctuation effects, whichwill be commented upon later. The most general form forxy which minimizes the w2,2 term and keeps the Gaussianaction small can be written as

r0 r1 x1 r1

y2 rax

2

b y

2,

131


054526-16


17/27

where 0/1/2(r) are slowly varying functions of space. In-serting this expression for into the action including S0 andthe w 2,2 term only gives

S0eff

r 12K0

02

1

2K1

22

2

A

2 0 2A

2 x12A

2 y12

A

2x2

2

A

2y2

2 , 132where KK(,0), A8U(,0)/

2, Aw 222,2v

w2,2

2/(22,2v

)U(,0) 2,2

v

/(22,2v

), where is the reduced

low-energy momentum cutoff. Note that we are not rescalingany fields or coordinates in this schematic RG treatment, justtaking into account the fluctuation corrections to w 2,2 . Thusthe low-energy continuous fluctuations around the minimaare described by three massless fields j (j0,1,2), which

unlike in the absence of w 2,2 have an ordinary relativis-tic dispersion. In addition to these continuous variables, thediscrete degeneracy of distinct minima allowed by periodic-ity of the cosine is indexed by the integers a ,b .

Having taken into account w 2,2 already in Eq. 132, wenext include the effects of a weak, renormalized w 4,1 cou-

pling, that after renormalization becomes of order w 4,1R

w 4,14,1v

w 4,1 w 2,2/U(,0) 4,1v

/(22,2v

). If all the barenonlinear couplings are small, then after renormalization,this will be the largest remaining term, which is why we treatit next. Inserting the decomposition of Eq. 131 into thefour-vortex-hopping term, one obtains

S1effw 4,1

R

rcos 81cos 82, 133assuming small gradients of 1/2 , as mandated by Eq. 132.Equations 132, 133 describe a 3D height model for1 ,2. As is well known in such models, the fluctuations ofthe free scalar fields 1 ,2 are bounded, so that the cosineterms in Eq. 133 are always relevant and pin these fields tointeger multiples of/4, which we can take, without loss ofgenerality, to be zero. Thus the net effect of w 4,1 is to leaveonly 0 as a low-energy mode.

Setting therefore 120 in the pinned flatphase, we have

e 2ix,ye 2i01 axby . 134

Here, since the fluctuations of 0 are likewise bounded, wehave replaced it by its average zero mode value 00 .This average can be determined by minimizing the onsiteLagrangian in Eq. 124. In principle, the parameters in Eq.124 should also be renormalized, e.g., 2q2q2qexpq2cqln

2w2,2/U(,0) . This gives,

e 2i01 ab , 135

for 42/8 with 20 assumed and

e 2i01 ab 2/84i12/84 2 , 136

for 42/8.The spatial orderings that are implied by these mean-field

solutions follow readily from the expressions in Eqs. 9,10 relating the bare boson density and plaquette energydensity to the low-energy field . In the former caseabove, the boson density is uniform since sin(2qx,y)0,whereas there is a plaquette energy density wave with,

x,yc 21 (xa)(yb). 137

The four states with a ,b0,1 correspond to the fourplaquette density wave states of the original lattice bosonmodel in which one out of every four plaquettes is resonating

more strongly.When 42/8, on the other hand, since sin(2)0 the

plaquette energy density wave states with reduced order c 2

(2/84)c 2) are coexisting with a charge-density-wave

CDW state ordered at (,),

nx1/2,y1/2c 212/84

21 xy. 138

In the limit in which 4, the amplitude of theplaquette density wave vanishes, leaving only CDW order.We believe that an absence of a pure CDW state more gen-erally is an artifact of the mean-field treatment being em-ployed, which ignores all fluctuations in the field. In par-ticular, one can imagine domain walls forming betweendomains of different plaquette-density-wave order, whichcost an energy Ewallw 2,2(2 /4)

2 for small w 2,2 and large4 . When this energy is small, fluctuations will undoubt-edly disorder the plaquette-density-wave order, leaving thepure CDW.

Our quantum Monte Carlo simulations on the boson ringmodel in the dual representation reveal that in the large Ulimit with U/K(U/K) c2.5, the EBL phase is unstable tothe formation of a CDW state. This is apparent in Fig. 6which shows that the density structure factor peak at (,)grows as L 2, indicating the presence of long-ranged CDW

FIG. 6. Finite-size scaling of the density structure factor peak at

(,), deep within the incompressible phase. The staggered mag-

netization m clearly tends to a finite value as 1/L20, indicating a

nonzero (,) charge-density-wave order parameter.


054526-17


18/27

order. This order is, however, weak, with a staggered mag-netization m0.09 see Fig. 6, much smaller than the clas-sical value m0.5. It is also instructive to examine the vor-tex two-point function in the CDW state, which is shown inFig. 7. Notice that the vortex correlation function is longranged indicating a vortex condensation (e i0 ) , a sexpected in such a conventional insulating state.

Both the plaquette- and charge-density-wave states will beinsulators, with a charge gap. This follows since in bothcases the field is pinned by the cosine potentials and isnot fluctuating. Since the bare boson density is nxy/,adding a particle at the origin can be achieved by shiftingxy xy for all x ,y0. This will cost a finite amount of

energy coming from the plaquette at the origin when thefield is pinned.

VI. EXACT WAVE FUNCTION

In this section we consider a variant of the boson ringHamiltonian which allows us to obtain an exact zero-energywave function when the couplings are carefully tuned. Wethen perturb away from the soluble point, using the exactwave function to compute properties of the adjacent quantumphase. Specifically, we find a translationally invariant fluidphase with a finite compressibility, behavior consistent withboth a superfluid and the EBL. But a calculation of the boson

tunneling gap in a finite system shows a 1/L scaling as ex-pected for the EBL, and inconsistent with the 1/L 2 depen-dence of a 2D superfluid. Thus, we confidently conclude thatthe EBL phase exists over a finite portion of the phase dia-gram adjacent to the soluble point.

The model we consider is a hard-core version of theboson-ring model, in which only zero or one boson is al-lowed per site, b b0,1. In the usual way we represent thehard-core bosons as Pauli matrices,

b , 139

b, 140

nbb1

21z, 141

where is the standard vector of Pauli matrices, and

12 (

xiy). The Hamiltonian consists of two terms,

H1/2xy

J4x,y x1,y

x1,y1 x,y1

H.c.

u4 Pflipx ,y , 142

where

Pflipx,y 1

16 1 ,1 1x1/2/2,y1/2/2z .

143

The first term with J40 in Eq. 142 is the hard-core analogof the ring-hopping term proportional to K in the rotor

model, Eq. 1. The operator Pflip(x ,y) is a projection opera-tor onto the two flippable configurations of the squareplaquette whose lower-left corner is at the site (x ,y). Foru40, this term competes with the ring term by disfavoringconfigurations with flippable plaquettes.

Remarkably, the ground state ofH1/2 can be found exactlyfor the special Rokhsar-Kivelson RK point u 4J4, follow-ing a general construction in the spirit of the Rokhsar-Kivelson point of the square lattice quantum dimer model,36

and employed more recently in Ref. 16 for a similar spinmodel on the Kagome lattice. The solution can be seen byrewriting H1/2 as

H1/2xy Pflipx,y J4 1 u,v0,1 xu,yvx

v

,144where vJ4u 4 measures the deviation from RK point. Forv0, an obvious zero-energy ground state of H1/2 is thefully polarized state

0x,y

xyx1 . 145

A useful alternate representation follows by rewriting x

1(1/2) (z1 z1), and expanding out thedirect product,

01

2N/2

xy1xy

xyzxy , 146

which demonstrates that 0 is a uniform real superpositionof all configurations in the Sz boson number basis.

This state, however, has uncertain z boson number, andso can be decomposed into many distinct ground states by

projection. In particular, at a given average xyz 2nxy

1m , we may project onto spatially uniform states ac-cording to

FIG. 7. Vortex correlation function G(0,) in the incompress-

ible CDW phase. The correlation function saturates as seen from the

finite-size scaling plot in the inset indicating that the vortices con-

dense, leading to a conventional insulating state.


054526-18


19/27

0;m1

Z

x,y1

L

Px m Py m 0 , 147

where Z is a normalization constant, since the row/columnprojection operators,

Px m 02

d2

exp ixy xyz m ,

148

Py m 0

2d

2expiy

xxy

zm ,

commute with H1/2 and amongst themselves, reflecting theconservation of boson number on each row and column bythe ring dynamics.

Of course, other nonuniform ground states may be ob-tained by choosing m differently on different rows and col-umns. This vast degeneracy signals a pathology of the RK

point, which is in fact at the boundary of a first-order transi-tion to a phase-separated frozen regime, for v0. Wetherefore focus instead on the behavior for infinitesimal v0, which splits this degeneracy. For boson densities nearhalf filling, i.e., m1/2, we expect that the uniform stateswill be favored energetically as v is increased to slightlypositive values, since these states have in this density regimemore flippable plaquettes see below. For larger m , thisassumption is certainly violated, however, since, e.g., at verylow boson density the ring moves clearly do not connect allpossible configurations. At m0, however, the set of uni-form configurations does form a single ergodic componentunder the ring move as can be straightforwardly shownnumerically35 and probably argued analytically. So we ex-pect the set of uniform states to be an adequate descriptionfor small m .

To make contact with the EBL fixed-point description, wecalculate some energetic properties for infinitesimal positivev using first-order perturbation theory. From the splitting ofthe different projected states at O(v), we calculate twoquantities: 1 the ground-state energy density (m)E(m)/L 2 as a function of boson density n(m1)/2,from which we obtain the compressibility (m), and 2 thesingle-particle gap 1(m ,L) for a finite-size LL system,essentially the addition energy for a boson onto a particularrow and column in the grand canonical ensemble, definedmore precisely below.

Consider first the ground-state energy density. For theprojected state at magnetization m, the first-order energyshift is

E m vxy

0;m Pflipx,y 0;m

v

Zxy

0 Pflipx,y x,y1

L

Px m Py m 0 ,149

where the latter equality follows from the fact that P flip isdiagonal in the z basis and hence commutes with the pro-

jection operators. Using Eq. 145, one readily sees that theenergy shift can be rewritten in a form reminiscent of clas-sical statistical mechanics,

E m vxy

P flipx,y ; xy , 150

where

Oxy 1

Z xy1O

x,y1

L

Px m; Py m;.

151

Here P flip , Px , and Py are the classical functions obtained

from Pflip , Px , and Py , respectively, by replacing the opera-

tor xyz by xy , and Z is chosen such that 1 xy 1.

We see that Eq. 151 simply defines an expectation valuefor an infinite-temperature Ising model with a constrainedmagnetization on each row and column. By proceeding alongsimilar lines, one can calculate the energy shift for a statewith an extra boson on a single particular row and column,and hence obtain 1. Amusingly, both problems can besolved exactly using a saddle-point technique see AppendixE. The results demonstrate that the constraints are nearlyirrelevant, and the lattice gas behaves nearly as its uncon-strained counterpart in the large system limit. In particular,the energy density, as L, becomes

m v

8 1m 2 2. 152

This can be easily understood by assuming that each site is

completely independent, and that the only effect of the con-straint is to determine the relative probabilities of the two-spin states, which may be understood to be p (1m)/2,p (1m)/2. On a given four-site plaquette, onlythe two of 16 total configurations in which the spin alter-nates around the plaquette are flippable. Hence the average

flippability per plaquette is 2p 2p

2(1m 2)2 /8, in agree-

ment with the result above.The single-particle gap is somewhat less intuitive. To

be precise and avoid ambiguities due to an unspecifiedchemical potential, we define

1 m 12 E1 m;1 2E1 m;0 E1 m ,1 ,

153

where E1(m;) is the ground-state energy of a state in whichthe number of bosons on row and column xy1 is in-creased by relative to the uniform state at magnetization m,

0;m ,1

ZPx1 m2/L Py1 m2/L

x,y2

L

Px m Py m 0, 154


054526-19


20/27

and E1(m;)vxy 0;m , Pflip(x ,y) 0;m ,. One finds

1 m 2v

L1m 2O1/L2 . 155

These two quantities can be compared to the generalpredictions expected from our theory of the EBL phase.Consider first the compressibility. From Eq. 152, one has

d/dn2d/dmvm(1m 2), and hence

dn

d

1

2v

1

13m2. 156

Note that the compressibility diverges and becomes negative

for 1/3 m 1, indicative of an instability of the uniformstate well away from half filling. For m1/3, however, is finite, consistent with the EBL phase. Indeed, from ourgeneral harmonic description of the EBL phase as in Eq. 6,we have 1U(k0).

In order to rule out a superfluid phase, we now computethe single-particle gap, 1(L), using the EBL Gaussian

fixed-point theory to show that it varies as 1/L as in Eq.155. This gap can be extracted from the spatially localcorrelator,

Geir()eir(0 ), 157

evaluated in a finite L L system. From the spectral repre-sentation, one has, at zero temperature,

1L lim

ln G

. 158

Performing the Gaussian integral using the effective action inEq. 43, one has

ln G

d

2

1

L2

k

U k

2Ek2

1ei,

159

where the wave-vector sum is over inequivalent values withkx ,ky(2/L)Z in the first Brillouin zone, and with themode energy Ek given explicitly in Eq. 37. As , the integral in Eq. 159 is dominated by those terms for whichEk0. This occurs along the Bose surface, i.e., for kx0 orky0. In the finite-size system there are 2L12L suchpoints for L1), hence

ln G 0,2U

L d

2

1

21ei, 160

where

U1

2 dkx

2U kx,0

dky

2U0,ky , 161

is the Bose surface average of U(k). Thus, we finally have

ln G 0,U

L, 162

giving the 1/L behavior for the single-particle gap

1U

L. 163

In a 2D superfluid, the single-particle gap is much smaller,vanishing as 1/L 2 in the large-L limit.

The agreement of the finite compressibility and 1/L scal-

ing of 1 between the soluble model and our fixed-pointtheory of the EBL strongly argues that the ground state of

H1/2 is the Bose liquid for 0v1 and m1/3. If thispostulate is correct, certain combinations of the Bose liquidfunction U(k) can be obtained explicitly for the solublemodel. In particular, we find

U k0 2v 13m 2 , 164

U2v1m 2 . 165

VII. DISCUSSION

A. Fractionalization, the Z 2 gauge theory, and the high-Tccuprates

In this paper we have described a remarkable phase ofquantum matter, the EBL, which we argue occurs in a classof square-lattice boson-ring models with X-Y symmetry.Having done so, it is reasonable to reflect upon the context inwhich such models are physically appropriate and the conse-quences perhaps observable. As discussed briefly in Intro-duction, the primary motivation for these models comes fromthe high-temperature cuprate superconductors. Both the re-markably high critical temperatures and the strange behav-iors in the normal state of these materials motivated anumber of theorists early on to the radical suggestion that

spin-charge separation might underly these peculiarities.3740

In this picture, the electron charge, liberated both from itsspin and its Fermi statistics as a chargon or antiholon,is relatively free to Bose condense to form a superconductingstate. At higher temperatures, or when the material is under-doped, the chargons and spin-carrying fermionic spinonsform an unconventional fluid, which would no doubt behavevery differently from a normal metal.

Subsequent theoretical and consequent experimental workhas both advanced the status of these ideas and posed a se-vere challenge to their applicability to the cuprates. A recentformulation6,10 of interacting electrons in terms of spinonsand chargons minimally coupled to a Z2 gauge field hashelped authors to support the concept of electron fractional-ization with a

Arun Paramekanti, Leon Balents and Matthew P. A. Fisher- Ring exchange, the exciton Bose liquid, and...

Documents

Transcript of Arun Paramekanti, Leon Balents and Matthew P. A. Fisher- Ring exchange, the exciton Bose liquid, and...