Table Of ContentExotic Gapless Mott Insulators of Bosons on Multi-Leg Ladders
Matthew S. Block,1 Ryan V. Mishmash,1 Ribhu K. Kaul,2 D. N. Sheng,3
Olexei I. Motrunich,4 and Matthew P. A. Fisher1,4
1Department of Physics, University of California, Santa Barbara, California 93106, USA
2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA
3Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
4Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
(Dated: January 31, 2011)
Wepresentevidenceforanexoticgaplessinsulatingphaseofhard-corebosonsonmulti-legladders
with a density commensurate with the number of legs. In particular, we study in detail a model of
bosonsmovingwithdirecthoppingandfrustratingringexchangeona3-legladderatν =1/3filling.
1 For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless
1 modesandpowerlawtransversedensitycorrelationsatincommensuratewavevectors. Wepropose
0 a determinantal wave function for this phase and find excellent comparison between variational
2 Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian,
thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions
n
of our results to other N-leg systems and to N-layer two-dimensional structures.
a
J
PACSnumbers: 71.10.Hf,71.10.Pm,75.10.Jm
8
2
Creating a complete catalogue of the possible zero- even though the phase is an incompressible insulator
]
l temperature phases of interacting bosons in low- (Fig. 1). In particular, we will construct a trial wave
e
- dimensional lattice systems is a fundamental open prob- function for this phase, suitable for use with variational
r
t lemincondensedmatterphysics[1]. Thelastfewdecades Monte Carlo (VMC) calculations, and provide detailed
s
have seen a number of proposals for new entries in this comparisonstonumericalresultsobtainedusingtheden-
.
t
a catalogue: the gamut of phases now runs from the well- sity matrix renormalization group (DMRG) method to
m studied “solids” and “superfluids” to exotic possibilities argue for the existence of such an exotic phase in a spe-
- such as variants of a “Bose metal” [2–5]. Equally chal-
d lenging is the identification of parent Hamiltonians that
n realize each of the entries as a ground state. This un- y J
o y
derstandingwillclearlyaidinthesearchfornovelmany-
c J
[ body physics in, for example, synthesized materials [6] ⊥
and ultracold atomic gases in optical lattices [7].
3 K
v Weakly interacting bosons generally condense forming
5
asuperfluidatlowtemperatures. However,forstrongin-
0 teractions at noninteger commensurate densities, bosons x x
1
4 naturally break translational symmetries of the lattice, d1 d2
. crystallizing into a solid. It is also possible for these ky=0 ky=0
08 two orders to coexist, resulting in the formation of a ky=±2π/3 ky=±2π/3
0 “supersolid” for the bosons. At integer densities, there
1 is also the possibility of a featureless “Mott insulator”
: that has a nondegenerate ground state with a gap to
v
i all excitations. These phases have been demonstrated
X
to exist convincingly in numerical simulations of lattice !1 0 1 !1 0 1
r models[8–10]. Statesthatbreaktimereversalsymmetry kx/π kx/π
a
(“chiral spin liquid” [11]) or lattice rotational symmetry
FIG. 1: (color online) Top: Illustration of the 3-leg ladder
(“nematic” [12]) have also been proposed. A possibility geometrywithperiodicboundaryconditionsinboththexand
distinct from all of these symmetry-breaking phases is a ydirections. ThehoppingstrengthisJ inthexdirectionand
Bose metal, i.e., a critical quantum liquid of bosons that J⊥ in the y direction; the ring term K moves a pair of hard-
does not break any symmetries yet has many gapless ex- core bosons on diagonal corners of a plaquette to the other
diagonal corners (if empty). Bottom: The d and d bands
citations. Such Bose metal phases have been studied in 1 2
for the 3-leg system at ν = 1/3 are shown for the gapless
detailpreviously, whereinthesystemsstudiedwerecom-
Mottinsulatorstudiedhere. Thereareexactlyenoughbosons
pressible with respect to the addition of bosons [3–5].
so that the d fermions fill up one band completely. The
2
In this Letter, we will propose and study a unique bold curves indicate occupied momentum states, and note
that there are two degenerate bands at k =±2π/3.
phase of matter that has gapless collective excitations y
2
cific microscopic lattice model of bosons. We will focus
our study here on the square lattice 3-leg ladder (see
Fig. 1) at boson filling factor ν = 1/3 as a prototypical
exampleofthephysicsofinterest. Eventhoughourulti-
mategoalistoaccessnoveltwo-dimensional(2D)phases,
the more tractable ladder systems are very intriguing in
their own right, and in our concluding remarks, we will
propose clear extensions of our ideas that are aimed to-
wards two dimensions.
Wave function.—A wave function of bosons can be
constructed by taking the product of two fermionic wave
functions and evaluating them at the same coordinates
(Gutzwiller projection): FIG. 2: (color online) Summary of our numerical results:
phase diagram in the K/J −J /J plane of the 3-leg lad-
⊥
Ψ (r ,...,r )=Ψ (r ,...,r )Ψ (r ,...,r ), (1) der at ν = 1/3 filling as determined by DMRG and VMC
b 1 M d1 1 M d2 1 M calculations.
which corresponds in terms of operators to a “parton”
constructionb† =d†1d†2withtheconstraintd†1d1 =d†2d2 = fromastrong-couplinganalysisoftheU(1)gaugetheory
b b[4,5]. Inthecaseofinteresthere,wheretimereversal
†
for the d and d fermions: within this framework, one
symmetryispresentonaquasi-one-dimensionalladder,it 1 2
canreverseengineerEq.(2)asapotentialparentHamil-
isnaturaltoputeachfermioninaSlaterdeterminantby
tonian for ground states of the form given by Eq. (1)
occupying one-dimensional (1D) bands. We stress that
(see Ref. [4]). Indeed, it was found that taking d (d )
thed andd fermionsneednothaveidenticalbandfilling 1 2
1 2
to hop more easily along x (y) and then increasing this
configurations.
anisotropy effectively increases K/J. At ν =1/3 on the
Forthe3-legladdersystem(Fig.1),therewillbethree
3-leg ladder, one hence expects that as the K term in-
bands of d and d fermions. At exactly ν = 1/3 filling,
1 2
creases,thethreed bandsshouldbecomealmostdegen-
aninterestingpossibilityarisesthatisspecialtothiscom- 1
erate, while the d bands should become almost disper-
mensurate density: there are enough fermions to exactly 2
sionlessandsplitapart. Thisinturnnaturallycausesthe
fill one d band completely, but still have up to three
2
d fermionstogointoabandinsulatingstateandthe d
partiallyfilledd bands. Theresultingprojectedstateof 2 1
1
fermions to go into a metallic state (see Fig. 1). This
bosons is an exotic “gapless Mott insulator” (GMI): the
is exactly the situation discussed above in our descrip-
d state is a band insulator while the d state is a band
2 1
tion of the GMI phase, and is our primary motivation to
metal. It is this phase that we will explore in detail in
search the phase diagram of the model, Eq. (2), for the
this Letter.
GMI phase.
Model.—The microscopic model for which we find the
3-legladder.—WehenceforthconsiderEq.(2)onthe3-
GMI phase consists of hard-core bosons hopping on a
legladderatfillingfactorν =1/3. WhenJ K,J ,the
square lattice with a four-site ring-exchange interaction: (cid:29) ⊥
ground state is a superfluid (SF) with quasi-long-range
(cid:88) (cid:88) order. On the other hand, when J K,J, one boson
H = J (b b +h.c.) J (b b +h.c.)
JK − †r r+xˆ − ⊥ †r r+yˆ on each rung goes into a zero y-m⊥om(cid:29)entum state, and
r r
(cid:88) the system is a conventional rung Mott insulator with a
+ K (b†rbr+xˆb†r+xˆ+yˆbr+yˆ+h.c.), (2) unique ground state and a gap to all excitations. The
r
region where we expect the GMI phase is when the K
where b destroys a boson at site r. The J and J terms term dominates.
r
⊥
are the usual hopping terms for bosons in the directions As we demonstrate below, we do in fact have exten-
parallel and perpendicular to the chains, respectively. sive numerical evidence for the existence of an exotic
The ring term K is the interaction that makes the phase GMI phase in the expected region of the phase diagram.
diagram interesting. When K > 0, which is the case Specifically, in addition to the SF and fully gapped rung
studied here, the model has a sign problem thereby ren- Mott state, we find a gapless Mott insulator with two
dering quantum Monte Carlo calculations inapplicable. 1Dgaplessmodes. OurresultsaresummarizedinFig.2.
The K term causes a pair of bosons to hop without Wefirstsearchedthephasediagramusingthevariational
center-of-mass motion, as illustrated in Fig. 1. In this wavefunctions: projectedd andd bandswithdifferent
1 2
sense, it frustrates the attempt of the simple hopping bandfillings(alongwithvariationalexponentsonthetwo
term to cause the bosons to maximize their kinetic en- Slater determinants to give additional variational free-
ergy by condensing and yet at the same time disfavors dom [5]) and Jastrow wave functions for the superfluid.
the formation of a simple solid. A more compelling (al- We then also scanned the phase diagram with DMRG
beitmoreabstract)motivationtostudythismodelcomes at the points marked as circles, triangles, and squares in
3
Fig. 2, keeping between 2000 and 4000 states per block
to ensure converged results. Figure 2 shows the phase 0.6
diagramofK/J vs. J /J withcoloredregions(roughly) 0.5
⊥ )
distinguishing the various phases. These regions gener- y
q0.4
ally coincide with the VMC state that minimizes the en- ,x
ergyateachpoint,althoughsomeadjustmentshavebeen (qb0.3
n
madetoincorporatestrongconclusionsfromtheDMRG. 0.2 DMRG: VMC:
TheGMIphasehasthreepartiallyoccupiedd1bandsand 0.1 qy=0 qy=0
oesnte. fTulhlyefirullnedg dM2obttanpdh—astehiiss imsothdeelepdhawseellofinmtahine VinMterC- !01 qy=±!2π0/.53 0 0.5qy=±2π/31
q /π
x
calculations by a projected wave function in which both
thed andd fermionsfullyfillthek =0band. Thereis
also a1SF ph2ase and two unidentifiedyregions (see below) 0.02
0.5 0.01
marked as white. 0
We shall focus here on establishing the existence and )y0.4 !1 0 1
q
properties of the GMI phase; these points are marked ,x0.3
q
by squares (green) in Fig. 2. A comparison between (b
D0.2 DMRG: VMC:
VMCandDMRGcalculationsoftheFouriertransformed qy =0 qy =0
single-boson Green’s function and density-density corre- 0.1 qy =±2π/3 qy =±2π/3
lation function, for a representative point in the GMI
phase,isshowninFig.3fora3-legsystemoflengthL = !01 !0.5 0 0.5 1
x q /π
24. Specifically, we plot the boson momentum distribu- x
tion nb(q) ≡ 1/(3Lx)(cid:80)r,r(cid:48)(cid:104)b†rbr(cid:48)(cid:105)eiq·(r−r(cid:48)) a(cid:80)nd density-
density structure factor Db(q) ≡ 1/(3Lx) r,r(cid:48)(cid:104)[nr − d1 :
ν][nr(cid:48) −ν](cid:105)eiq·(r−r(cid:48)), where nr ≡b†rbr.
In the proposed GMI phase, since the d fermion
2
is gapped, there is a gap to any excitation that car- FIG. 3: (color online) Comparison of DMRG and VMC cal-
ries nonzero boson number. This implies a single-boson culations of the boson momentum distribution nb(q) (top
Green’s function that decays exponentially in real space panel) and density-density structure factor Db(q) (middle
panel; inset is a zoom of q = 0 curves) in the GMI phase
and thus a boson momentum distribution with no singu- y
of the 3-leg ladder at the point K/J = 2.7 and J /J = 1.0.
lar features; the latter is clearly demonstrated in the top ⊥
The lack of singular features in n (q) and q2 dependence of
b x
panelofFig. 3inboththeDMRGandVMC.Becauseof
D (q ,q = 0) around q = 0, both highlight the insulat-
b x y x
the fully filled d2 band, the VMC state contains exactly ing nature of the GMI phase. However, the density-density
one boson on each rung and hence gives completely flat structure factor still “sees” the gaplessness of the phase due
nb(qx,qy = 0, 2π/3), while the true ground state (as tothepartiallyfilledd1 bands[4],hencethesingularfeatures
obtained by D±MRG) does contain some short-distance in Db(qx,qy = ±2π/3). These singularities can be identi-
fied with “2k ” wave vectors in the d fermion’s band filling
fluctuations. Within the GMI, the q = 0 mode occupa- F 1
configuration (bottom panel) in which the k =0 band con-
tion,n (q=0),doesnotincreasewithincreasingsystem y
b tains 12 fermions and the k = ±2π/3 bands each contain 6
y
sizeintheDMRG,andafinite-sizestudyoftheone-and
fermions. Ofcourse,thed fermionconfiguration(notshown)
2
two-boson gaps indicates that these gaps attain rather consists only of one fully filled k =0 band.
y
large, finitevalues inthe L limit[13]. Allof these
x
→∞
findings are consistent with the insulating properties of
the GMI phase.
shown in the bottom panel of Fig. 3. In this case, the
Measurement of the density-density correlation func- mainsingularitiesinD (q)at(q ,q )=( 3π/4, 2π/3)
b x y
± ±
tion gives compelling evidence for the presence of the can be identified with the “2k ” wave vectors associ-
F
GMI phase, in particular, its gapless nature and incom- ated with moving a d fermion from the k = 0 band
1 y
pressibility. In the VMC wave function, D (q) contains to the k = 2π/3 bands (going across the bands); the
b y
±
contributions from gapless particle-hole excitations aris- other singularities can similarly be identified with other
ing from partially filled bands of either fermion species “2k ” moves that are allowed. In particular, singular-
F
(d only in the case of the GMI) and hence has singular ities at q = 0 are suppressed by d : the VMC wave
1 y 2
structure at various momenta [4]. Indeed, the DMRG function has exactly one boson on each rung which pro-
calculation of the density-density structure factor dis- duces D (q ,q = 0) = 0. The DMRG calculation, on
b x y
plays singular features that are impressively well repro- theotherhand,givessmallbutfiniteD (q ,q =0)with
b x y
duced by the corresponding VMC calculation in which q2 dependence around q = 0. This result directly im-
x x
the lowest energy state has a band filling configuration plies that the system is an incompressible insulator and
4
larger region above the GMI. The corresponding DMRG
2
pointsaremarkedasquestionmarksandsquares(green)
3
GMI with a question mark, respectively. In the latter region,
1.5
VMC favors a GMI ground state, but the features of the
c 2 SF GMIintheDMRGfailtopersist,e.g.,thesingularitiesin
1
x) Db(qx,qy =±2π/3)smoothenandthecalculatedcentral
( chargedropsbelowc 2. Ananalysisofthespectralgap
S
0.5 (cid:39)
1 versusL usingexactdiagonalizationandDMRGreveals
0 10 20 x
x that the system is either gapless or has a very small gap
0
0 1 2 3 4 5 6 in this region [13], without any obvious ordering.
K/J
Extensions.—A natural extension of our findings is to
the general N-leg ladder at ν = m/N, with integer m.
FIG. 4: (color online) Effective central charge c as inferred
In such systems, a GMI phase analogous to the one dis-
from scaling of the entanglement entropy versus subsystem
cussed above is a clear ground state candidate. We leave
size S(x). For the 3-leg ladder, we fit the von Neumann en-
tanglement entropy of leftmost contiguous blocks of size 3x for future work a numerical investigation using DMRG
to the scaling form S(x) = (c/3)ln[(L /π)sin(πx/L )]+A toassessthestabilityofGMIphasesinthesesystems;al-
x x
toextractc[15]. Inthemainplot,cisplottedversusK/J at thoughwecanreportthatonthe2-legladderatν =1/2
J⊥/J = 1.0 for a system of length Lx = 24; the jump from the GMI is likely unstable in our model [14], a result
c (cid:39) 1 to c (cid:39) 2 at the SF-GMI boundary is striking. In the
which partially motivated the present study.
inset, we plot S(x) in both the SF (K/J = 1.0) and GMI
Anotherinterestingextensionofourresultsistostruc-
(K/J = 2.7) phases; the points are the DMRG data, while
the curves are the fits to the scaling form. tures that consist of N layers of 2D systems at a general
filling of ν =m/N, e.g., a three-layer system at ν =1/3.
Insuchasystem,thed fermionscanexactlyfillmofthe
2
N 2D bands resulting in a gapless Mott insulating state.
rules out the possibility of a superfluid, either conven-
Theoretically, this state is stable and can exist in two
tional or paired. These results are general for the entire
dimensions. We expect that an appropriate model that
GMI phase in Fig. 2, except that the locations of the
realizesthisphysicswouldhavesubstantialringexchange
singularities in D (q ,q = 2π/3) can be changed by
b x y
± on the plaquettes between the layers, as this interaction
tuning K/J and/or J /J to alter the resulting d band
1
filling configuration. ⊥ willgivethenecessarydisparityinhoppingtokeepthed1
system metallic while forcing the d system into a band
2
A crucial aspect of the GMI phase is the expectation
insulating state. Thus, we are hopeful that an appropri-
that it has two 1D gapless modes at low energy: at the
ate JK model will realize this quasi-2D analogue.
mean-field level, Fig. 1, there are naively three 1D gap-
This work was supported by the NSF under Grants
less modes corresponding to the three partially filled d
1
No. DMR-0529399 (M.S.B., R.V.M., and M.P.A.F.) and
bands. Wemust,however,enforceaconstraintsuchthat
No. DMR-0907145 (O.I.M.); the DOE under Grant No.
d†1(r)d1(r) = d†2(r)d2(r) to realize the physical bosons. DE-FG02-06ER46305 (D.N.S.); and Microsoft Station Q
This is accomplished by means of a gauge field along
(R.V.M. and R.K.K.).
withcorresponding(opposite)gaugechargesattachedto
each flavor of parton. In the strong-coupling limit, one
linear combination of the original gapless modes is ren-
deredmassiveleavingonlytwogaplessmodes[5,14]. We
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c 1asexpectedforasinglegaplessbosondensitymode
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235116 (2007).
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mental/10.1103/PhysRevLett.106.046402forananalysis
