Table Of ContentBose Metals and Insulators on Multi-Leg Ladders with Ring Exchange
Ryan V. Mishmash,1 Matthew S. Block,1,2 Ribhu K. Kaul,2 D. N. Sheng,3
Olexei I. Motrunich,4 and Matthew P. A. Fisher1
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 11, 2012)
We establish compelling evidence for the existence of new quasi-one-dimensional descendants of
the d-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant
bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and
2 M. P. A. Fisher, Phys. Rev. B 75, 235116 (2007)]. In particular, motivated by a strong-coupling
1 analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the
0 N-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-
2 leg systems where we have identified two novel phases: a compressible gapless Bose metal on the
four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former
n
is conducting along the ladder and has five gapless modes, one more than the number of legs.
a
This represents a significant step forward in establishing the potential stability of the DBL in two
J
dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is
0
insulating along the ladder but has two gapless modes and incommensurate power law transverse
1
density-densitycorrelations. Whilewehavealreadypresentedresultsonthislatterphaseelsewhere
[M. S. Block et al., Phys. Rev. Lett. 106, 046402 (2011)], we will expand upon those results in
]
l this work. In both cases, we can understand the nature of the phase using slave-particle-inspired
e
variationalwavefunctionsconsistingofaproductoftwodistinctSlaterdeterminants,theproperties
-
r ofwhichcompareimpressivelywelltoadensitymatrixrenormalizationgroupsolutionofthemodel
t
s Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the
. universallow-energyphysicswithabosonizationanalysisoftheappropriatequasi-1Dgaugetheory.
t
a Wewillbrieflydiscussthepotentialrelevanceofthesefindingstohigh-temperaturesuperconductors,
m cold atomic gases, and frustrated quantum magnets.
-
d PACSnumbers: 71.10.Hf,71.10.Pm,75.10.Jm
n
o
c I. INTRODUCTION uid theory, but that still have correlations with singu-
[ larities residing on one-dimensional (1D) surfaces in mo-
2 One of the recent challenges in condensed matter mentum space. Perhaps the most prominent example
v are states with a spinon Fermi surface, so-called “spin
physics has been to understand quantum phases char-
7 Bosemetals”, whichhavebothalonghistoryinthecon-
acterized by singular surfaces in momentum space. The
0 text of the cuprates11 and also a renewed interest in the
canonical example of such a phase is the free Fermi gas
6
context of the organic materials κ-(ET) Cu (CN) and
4 (or more generally a Fermi liquid), where the singu- 2 2 3
EtMe Sb[Pd(dmit) ] .12–15 Here, due to the presence of
. lar surface is simply the Fermi surface. Ironically, de- 3 2 2
0 an emergent gauge field when going beyond mean field,
spite our immense theoretical understanding of the free
11 Fermi gas1 and the interacting Fermi liquid,2 a con- a Fermi-liquid-like quasiparticle description is inapplica-
ble, thus making fully controlled analytics very challeng-
1 trolled and unbiased numerical demonstration of such
ing and further promoting the importance of numerics.
: a phase in an interacting microscopic model of itinerant
v Not surprisingly though, numerics suffers from the same
fermions moving in two or more dimensions is still ex-
i
X tremely difficult. The main roadblocks in this numerical difficultiesasbefore: (1)thesignproblemislikelyanec-
essary condition for realistic parent Hamiltonians, and
r pursuit are (1) the infamous sign problem in quantum
a Monte Carlo simulations3 and (2) the anomalously large (2) the beyond boundary law scaling of spatial entan-
glement is likely the same as that encountered for free
amount of spatial entanglement present in states with a
Fermi surface,4–7 thereby rendering recently developed fermions.16
sign problem-free tensor network state approaches inad- We focus here on a closely related quantum phase,
equate with current techniques.8–10 Although the most the “d-wave Bose liquid” (DBL). The DBL is an exotic
obvious and familiar, the free Fermi gas and Fermi liq- quantum phase of uncondensed itinerant bosons moving
uid are not the only examples of phases with singular on the two-dimensional square lattice first considered in
surfaces in momentum space. Ref.17. LikestateswithaspinonFermisea,eventhough
Sincethediscoveryofthecupratesuperconductors,in- themicroscopicdegreesoffreedomarebosonic,theDBL
terest has emerged in novel two-dimensional (2D) quan- toohasasetofgaplessexcitationsresidingon1Dsurfaces
tum phases that fall outside the paradigm of Fermi liq- in momentum space, i.e., “Bose surfaces,” and so access-
2
ing the DBL in 2D using fully controlled techniques, ei- boson operator as a product of two fermionic partons:
thernumericallyoranalytically,isnomoretractablethan
for the spinon Fermi sea. However, we make progress b†(r)=d†1(r)d†2(r), (2)
studying the DBL in a controlled way by continuing
to employ the heretofore fruitful philosophy14,15,18,19 of where the hard-core boson Hilbert space is recovered by
buildingapictureofsucha2Dphasethroughasequence requiringthatthedensitiesofthetwopartons,ρdα(r),be
of controlled quasi-1D ladder studies. In fact, we believe the same at each lattice site, i.e., ρd1(r)=ρd2(r)=ρ(r),
that the very presence of singular surfaces in momen- whereρ(r)isthebosondensityatsiter. Withinagauge
tum space actually renders ladder studies more informa- theory framework, the mean-field picture of the phase
tive and allows us to circumvent some of the usual nu- consists of two independent species of fermions hopping
merical and analytical difficulties. When placed on the onthesquarelatticewithanisotropichopping; thephys-
N-leg ladder, a given 2D phase with a continuous set ical Hilbert space is then obtained by strongly coupling
of gapless modes residing on a 1D surface should enter the d1 and d2 fermions with opposite gauge charge to
a phase that is a distinctive multi-mode quasi-1D de- an emergent U(1) gauge field.17,18 If one takes the d1
scendant of the parent 2D phase, with a number of 1D (d2) fermion to hop preferentially in the xˆ (yˆ) direction,
gapless modes that grows linearly with N (used inter- then the corresponding wave function [Eq. (1) with Ψd1
changeably with Ly in this paper: N = Ly).20 In such (Ψd2) being a filled Fermi sea compressed in the xˆ (yˆ)
quasi-1D geometries, both numerics and analytics are direction] has a characteristic d-wave sign structure;17
on much stronger footing: in principle, potential parent that is, the sign of the wave function goes through a
Hamiltonians can be solved numercially with the den- sequence, + + , upon taking one particle around an-
− −
sity matrix renormalization group (DMRG),21–23 which otherandhencethelabel“d-waveBoseliquid.” Perhaps
can then be supplemented with variational Monte Carlo more importantly, as alluded to above, this projected
(VMC) calculations24 using appropriate projected trial wave function has power law singularities of various mo-
wave functions to help map out the phase diagram. De- mentum space correlators, e.g., the boson momentum
spite the fact that the DBL is a strong-coupling phase distribution and density-density structure factor, resid-
that cannot be characterized perturbatively in terms of ingon1Dsurfacesinmomentumspace.17 Thesesurfaces
the original bosons, the slave-particle approach on the are perhaps the most distinguishing feature of the DBL
laddersaccessesthephaseinanovelwaythatistractable and are crucial to our identification of quasi-1D descen-
viaagaugetheorythatcanbetreatedusingconventional dantsofthephaseontheN-legladder. Astrong-coupling
bosonization techniques. analysis17 of the aforementioned gauge theory motivates
Guidedbythislineofattack,weaimtoaccomplishtwo a simple microscopic parent Hamiltonian which can po-
maingoalsinthispaper: (1)establishtheexistenceofan tentially harbor the DBL phase. This Hamiltonian con-
exotic metallic DBL phase, i.e., a “Bose metal,” on the sists of usual nearest-neighbor hard-core boson hopping
four-leg ladder, which we will argue is both a close rela- supplemented with an explicit frustrated four-site ring-
tiveoftheproposed2DphaseofRef.17andanontrivial exchange interaction (see also Sec. IIC):
extensionofthequasi-1DdescendantDBLphasediscov-
ered on the two-leg ladder in Ref. 18; and (2) elaborate H =HJ +HK, (3)
on our previous work in Ref. 19, which argued for the
H = J (b b +H.c.), (4)
existence of a novel, but fundamentally quasi-1D, gap- J − †r r+µˆ
r;µˆ=xˆ,yˆ
less Mott insulating phase on the three-leg ladder. We (cid:88)
emphasize that while we have already presented high- HK =K (b†rbr+xˆb†r+xˆ+yˆbr+yˆ+H.c.), (5)
lights of the latter three-leg gapless Mott insulator in a r
(cid:88)
previous work,19 the former four-leg gapless Bose metal
with J,K > 0. We focus on this model Hamiltonian,
constitutes a new and very exciting result. These two
the so-called “J-K model,” extensively in this paper.
seemingly disparate phases are actually close relatives:
For K = 0, we expect a generic superfluid with the
they share a common parton gauge theory description,
bosons condensed at q = 0 and off-diagonal long-range
can bemodeled within the same class ofprojected varia-
order. It is in the regime where the ring-exchange term
tionalwavefunctions,and,finally,manifestthemselvesin
contributes appreciably to the overall Hamiltonian that
the same microscopic model, all of which we will demon-
the d-wave Bose liquid is expected to onset. In the
strate in this work.
strong-coupling limit of the lattice gauge theory for
The model wave function for the DBL is obtained by
the DBL, the relative strength of the ring term to the
taking aproduct oftwo distinctSlater determinants and
hopping term, K/J, increases with increasing hopping
evaluatingthematidenticalcoordinates(Gutzwillerpro-
anisotropy between the d and d partons. Thus, we
jection): 1 2
expect the ring term to potentially stabilize the DBL
Ψ (r ,...,r )=Ψ (r ,...,r )Ψ (r ,...,r ), (1) phase, and, as in Ref. 18 on the two-leg ladder, we do
b 1 Nb d1 1 Nb d2 1 Nb in fact remarkably find evidence for such a scenario on
where N is the total number of bosons. In the language three- and four-leg ladder systems.
b
of operators, this corresponds to writing the (hard-core)
3
To conclude this section, we now discuss several lines suchasystemcouldenteruponaddinglocalattractivein-
of motivation behind this work. Aside from being in- teractions? This question was first asked by Feiguin and
trinsically interesting in its own right, we believe the FisherinRef.26,andalthoughthereareseveralpossibil-
DBL phase is potentially relevant to modern-day experi- ities within a BCS treatment, the most interesting sce-
ments and othertheoreticalpursuitsina numberofcon- nario involves formed Cooper pairs entering a “metallic”
texts, including high-temperature superconductors, ul- d-wave Bose liquid state instead of Bose condensing at a
tracold atomic gases, and frustrated quantum magnets. finitesetofmomenta. Suchaphaseisnotaccessibleina
We start by focusing on the first case, where we believe mean-fieldBCSanalysis;however,itcanbearguedtobe
the ideas behind the DBL can be used to describe the a reasonable outcome by observing that when deriving
charge sector of a particular conducting, non-Fermi liq- an effective boson Hamiltonian within perturbation the-
uid phase of itinerant electrons, i.e., a “strange metal,” ory at strong coupling, a ring term identical to the one
which may possibly be related to the infamous strange we consider in the pure boson context is generated for
metal phase of the cuprates. Specifically, we have in increased hopping anisotropy between the two fermion
mind the following scenario. In the spirit of the slave- species.26 Itispreciselythisringtermthatdrivesourbo-
boson treatment of the t-J model,11 one can (excluding son system into a “metallic” DBL phase (see Sec. IIIB
site double occupancy) decompose the electron creation and Ref. 18). Thus, such a “Cooper pair Bose metal”
operator as a product of a slave boson (“chargon”) and (CPBM) may exist at strong hopping anisotropy and in-
a fermionic spinon: c (r) = b (r)f (r). This leads to a termediatetostrongattractiveinteractions. Evidencefor
†σ † σ†
gauge theory formulation in which the spinons and slave such a phase was in fact recently found in a genuine at-
bosons are coupled to an emergent gauge field. While it tractiveHubbardmodelonthetwo-legladderinRef.29,
is natural in this context for the spinons to form a Fermi and our results here on three and four legs in the pure
sea(seeabove),thebehavioroftheslavebosonholdsthe boson context may warrant further future studies of the
key in determining the properties of the resulting elec- CPBM, both theoretically and experimentally.
tronic phase: if the slave bosons condense, a traditional
A final line of motivation to study the DBL involves
Fermi liquid phase is obtained, whereas if they do not,
thinking of the hard-core boson ring model [see Eq. (3)]
we may say that the phase is a “non-Fermi liquid.” An
as the easy-plane limit of an SU(2) invariant spin-1/2
example of how the latter case can be achieved involves
model with four-spin cyclic ring exchange (see, for ex-
further decomposing the slave boson into two fermionic
ample, Ref. 30). In such a model, at zero magnetic field
partons just as we did for the real boson in Eq. (2):
(half-filling in the boson language), exact diagonaliza-
b†(r) = d†1(r)d†2(r), where again d1 (d2) is taken to fill tion (ED) indicates, among other things, the presence
a Fermi see compressed in the xˆ (yˆ) direction, and now of a spin-nematic uniaxial magnet intervening between
there are two emergent gauge fields needed to enforce a four-sublattice biaxial N´eel state and a fully gapped
thephysicalelectronicHilbertspace. Thissetupsetsthe SU(2) symmetric valence bond solid.31 The same model
stage for a theory of a “d-wave metal” phase. While we has also been studied on the two-leg ladder.32 Using the
willpresentworkonthisphaseelsewhere,bothin2Dand samemulti-legladderapproachweemployhere,itwould
on the two-leg ladder, we believe scaling up the picture be interesting to study this model in the presence of a
of the charge sector to many legs (as we do in this pa- Zeeman field, which would correspond in the boson lan-
per) will be an informative endeavor toward an eventual guage to a finite chemical potential. Because the 2D
theoretical understanding of the 2D “d-wave metal.” DBL is expected to be generically stable only at densi-
ties slightly below half-filling (and likely not present at
A direct experimental realization of the DBL can po-
exactlyhalf-filling),17 itisconceivablethatsuchamodel
tentially be achieved in systems of ultracold quantum
in a sector of non-zero net magnetization could enter a
gases. Although it has been proposed that the boson
spin liquid phase related to our DBL. Thus, understand-
ring-exchange model [see Eq. (3)] can be engineered di-
rectlyinacoldatomsystem,25 perhapsthemostfeasible ingthephysicsoftheDBLandtheU(1)symmetricring-
exchange model of Eq. (3) represents a first step in un-
scenarioinvolvesengineeringapairofmismatchedFermi
derstanding this putative spin liquid phase.
surfaces in a two-component Fermi gas by introducing
a hopping anisotropy between the two species.26 Such Therestofthepaperisorganizedasfollows. SectionII
anisotropy can be achieved with a spin-dependent (more describes the basic machinery of our work, including the
precisely “hyperfine-state-dependent”) optical lattice, a gauge theory description (Sec. IIA) and the construc-
setup that was experimentally first demonstrated sev- tion of the variational wave functions (Sec. IIB); this
eral years ago.27,28 In the noninteracting limit, we then sectionalsoestablishesourmicroscopicmodel(Sec.IIC)
have a situation very similar to the mean-field descrip- and provides definitions of the physical measurements
tion of the DBL phase as discussed above, i.e., a system we consider (Sec. IID). Section III addresses in detail
oftwoindependentspeciesoffermionscharacterizedbya the results of our study on the four-leg ladder, including
hopping anisotropy. However, a fundamental difference most prominently the detection and characterization of
is that we are now talking about real fermionic atoms, theDBL[4,2]gaplessBosemetalphase. SectionIVgives
as opposed to fermionic partons modeling real hard-core a similar analysis of our three-leg study, giving special
bosons [see Eq. (2)]. What are the potential phases that attention to the DBL[3,0] gapless Mott insulator phase.
4
ThisphasewasthefocusofourrecentLetter,Ref.19. We d1d1 d2d2
first summarize the results of Ref. 19 and then present
new, additional evidence and arguments in support of
the stability of the gapless Mott insulating phase. Fi-
noualrlyp,lainnsSfeocr.fVut,uwreewdoisrckuassndouerxtceonnsciolunssioonfsthaendidesoamsperoe-f !k!k kF(±k1F(3±π1/34π)/4)kF(±k1F(π±/14π)/4) kF(±k2F(π±/24π)/4)
sentedinthispaper. AppendixApresentsourbosonized
solutionstothegaugetheoriesforboththeDBL[4,2]and
theDBL[3,0]aswellasastabilityanalysistakingintoac- kkxxkx kxkkxx
counttheeffectsofshort-rangeinteractions. AppendixB ky ky
3π/4
contains results on the three-leg ladder at incommensu-
rate densities and the successes and failures of the DBL
π/4
framework in these systems. Specifically, we find strong
evidenceforaDBL[3,1]phasefordensitiesρ<1/3,while kx kx
π/4
for ρ > 1/3 no DBL phase appears to exist; instead we −
have identified a phase consistent with a three-leg de-
3π/4
scendent of the “bond-chiral superfluid” phase predicted − π π π π
− −
inarecentspin-waveanalysisofthe2DJ-K model.33 In
Appendix C we discuss the situation on the two-leg lad- FIG. 1: (color online). (top panel) Band filling for the
DBL[4,2] showing the Fermi wave vectors for the right mov-
der at half-filling and, in particular, why we fail to find
ing d and d partons. (bottom panel) An overhead view of
a phase analogous to the DBL[3,0] in that system. 1 2
theoccupiedmomentumstatesfortheDBL[4,2],highlighting
the 2D nature of the phase; it is clear that the d Fermi sea
1
is compressed along xˆ, while the d Fermi sea is compressed
2
II. PRELIMINARIES
along yˆ.
A. Gauge theory description: the DBL on the
N-leg ladder
where P = R/L = +/ and denotes the right- and left-
−
movers. Note that it is not necessary, in general, for
Whatfollowsisageneralizationofwhathasbeendone either parton to occupy all N bands; therefore, k in the
y
intheappendixofRef.18forthetwo-legladder. Herewe sum runs only over the partially filled bands for each
will summarize the approach and state the key results. flavor α. If the d partons partially occupy n bands and
1
More details can be found in Appendix A. the d partons partially occupy m bands, we denote the
2
Wecandescribethed-waveBoseliquidstatebyfirstre- resulting state as DBL[n,m]. Since the two flavors of
expressingthebosonicoperatorsasproductsoffermionic partons are interchangeable, we can always choose N
≥
partons as in Eq. (2). On the N-leg ladder, each of the n m. With this convention, the d (d ) partons are
1 2
≥
partons has the freedom to fill at most N 1D bands in most easily associated with those that hop preferentially
momentumspacecorrespondingtotheN transversemo- in the xˆ (yˆ) direction as described in the introduction.
menta;e.g.,choosingperiodicboundaryconditionsinthe Finally, if we sum up the Fermi wave vectors for each
yˆ direction, we have ky = 2jπ/N for j = 0,...,N 1. parton, we recover the boson density ρ:
−
From our mean-field understanding of the state, we ex-
pect the partons to occupy contiguous strips centered k(ky) = k(ky) =Nπρ. (7)
F1 F2
about k = 0 in each band; such strips can best be
describedx as “Fermi segments” and the edges of each (cid:88)ky (cid:88)ky
segment as “Fermi points,” the locations of which are
Continuing with our gauge theory description, we as-
the Fermi wave vectors. These points are the locations
sign equal and opposite gauge charges to the d and d
1 2
where partons can gaplessly be added or removed from
partons and turn on an appropriate U(1) gauge field; in
the system and lead naturally to the concept of right-
quasi-1D, this behaves like a conventional electric field
and left-movers (see, for example, Fig. 1). Labeling the
at long distances attracting the oppositely charged par-
right-movers’ momenta as k(ky), where α = 1,2 refers tonstowardoneanother. Inthelimitofstrongcoupling,
Fα
to the two flavors of partons and k labels the band, we the partons are bound together on each site such that
y
cantakethecontinuumlimitinthelongitudinaldirection d†1(r)d1(r) = d†2(r)d2(r) = b†(r)b(r), thus realizing the
(xˆ) and then linearize the dispersions around each Fermi physical bosons. While the mean-field treatment would
point, capturing all the relevant low energy physics, and predictthattheDBL[n,m]possessesn+mgaplessmodes
approximately decompose the partons as follows: (since the partons are completely free in this case), the
effect of the gauge field is to render massive the overall
d†α(x,y)∼ exp iP kF(kαy)x+ky(y−1) dα(kPy)†(x), charge mode (see Appendix A) reducing this number by
k(cid:88)y,P (cid:104) (cid:16) (cid:17)(cid:105) one. Hence, a critical feature of the DBL[n,m] is the
(6) existence of n+m 1 gapless 1D modes.
−
5
The gauge theory also offers an explanation for the gauge currents will again be enhanced. Hence, we ex-
singularbehaviorobservedatincommensuratewavevec- pect enhanced singularities in the density-density struc-
tors in momentum-space correlators such as the boson ture factor at wave vectors (k(ky)+k(ky(cid:48)),k +k ). As
momentumdistributionfunction,n (q),andthedensity- ± Fα Fα y y(cid:48)
densitystructurefactor,Db(q),fortbheDBL(seeSec.IID an example, the fermion bilinear d1(kRy)†d1(kLy(cid:48)) corresponds
forexplicitdefinitionsofthesequantities). Considerfirst tosuchanenhancement. Thegaugetheorypredictsthat
n (q), the Fourier transform of the boson Green’s func- the non-oscillatory, zero-momentum term in the boson
b
tion, G (r). In the mean-field treatment, GMF(r) = density-densitycorrelatorwillbeunaffectedbythegauge
GMF(r)Gb MF(r)/ρ; that is, simply the product obf the in- interactions, i.e., this term will simply remain 1/x2, and
didv1idual fde2rmionic parton Green’s functions. Therefore, the other “2k ” wave vectors, (k(ky) k(ky(cid:48)),k k ),
GMF will oscillate at various (Pk(ky) + P k(ky(cid:48)),Pk + will always beFsuppressed relati±ve tFoαth−e mFeαan fiyel−d. yA(cid:48) s
b F1 (cid:48) F2 y
P k ) wave vectors and decay as 1/x2; these momenta in the case of the boson Green’s function, considering
(cid:48) y(cid:48)
correspond to the gapless addition/removal of a boson, the momentum space density-density structure factor,
which involves creating/destroying a d1 and a d2 parton Db(q), allows for a more organized view of those wave
at certain Fermi points. When gauge field effects are vectors that are enhanced by the presence of the gauge
taken into account, we expect these power laws to be al- field.
teredforthedifferentwavevectorswithsomeoscillations The gauge theory predictions involving the number of
dying off more quickly and some more slowly relative to gapless modes and the locations of singularities allow us
the mean field. In Appendix A, we present a complete to identify the DBL[n,m] if and when it shows up as the
low-energytheoryfortheDBLphasesonladders,which, ground state of a given model. We have proposed such
in principle, allows the exact calculation of all power law a model in Eq. (3) and will discuss it in greater detail
exponents. However, these exponents are also affected below in Sec. IIC.
byshort-rangeinteractionsand,duetothelargenumber Bysolvingthegaugetheorymodelusingbosonization,
of parameters in the theory, we do not attempt to do onecanrecasttheDBL[n,m]asaninteractingLuttinger
these calculations explicitly. The gauge theory suggests liquidwithn+m 11Dmodeswithpotentiallydistinct
−
a mechanism of Amperean enhancement whereby added dispersion velocities and corresponding, nontrivial Lut-
(or removed) partons with opposite group velocities are tingerparameterscharacterizingallpowerlawexponents.
favoredsincetheiroppositegaugechargesgenerateparal- This solution, along with a stability analysis, appears in
lel currents, which will subsequently attract one another Appendix A.
thereby satisfying the gauge constraint and realizing the
physical bosons. Assuming that the gauge interactions
dominate in our theory, we can rely on the intuition of B. Wave function: DBL states on the N-leg ladder
this Amperean rule despite the effects of the short-range
interactions mentioned above. Thus we expect enhanced While the DMRG method can readily be employed on
singularities in the momentum distribution function at any quasi-1D model to extract out measurements of the
wave vectors (k(ky) k(ky(cid:48)),k k ). As an example, groundstatecorrelationfunctions, atbest, onewouldbe
± F1 − F2 y − y(cid:48)
the fermion bilinear d(ky)†d(ky(cid:48))† corresponds to such an able to say that these functions decay as power laws and
1R 2L oscillate at incommensurate wave vectors if the ground
enhancement. These “Bose points” are the quasi-1D fin-
state were indeed some DBL[n,m]. To make sense of
gerprints of the “Bose surfaces” of the parent 2D DBL
the locations of these wave vectors, we have developed
phase; that is, the finite transverse momenta of the lad-
a more direct test for the presence of this phase using
der slice through the 1D singular curves present in full
variational wave functions and VMC. These wave func-
2D resulting in these points of singular behavior. It can
tions can be thought of as crude representations of the
be shown within the context of the gauge theory that
DBL[n,m] phase as described by the bosonized parton
the other wave vectors, ±(kF(k1y)+kF(k2y(cid:48)),ky +ky(cid:48)), are al- gauge theory in the previous section. Following the mo-
ways suppressed relative to the mean field. In momen- tivationofourmean-fieldpartondescription,westartby
tum space, nb(q) shows clearly those wave vectors that writing the bosonic wave function as a product of two
are enhanced with sharp peaks or kinks in the curve. Slaterdeterminantswhereintheorbitalsoccupiedbythe
Turning now to the density-density correlator in real- partons correspond to the band filling prescription laid
space, this function can be approximated in the mean- out in the previous section (see Fig. 1). But such a wave
field treatment as the average of the individual parton function without any further constraints would simply
correlators: DMF(r) [DMF(r) + DMF(r)]/2, and is correspond to that of two flavors of noninteracting spin-
b ≈ d1 d2
therefore expected to oscillate at various “2k ” wave less fermions. We must project this wave function into
F
vectors and decay as 1/x2. These can be thought of as the space of the physical bosons; i.e., the set of positions
particle-hole excitations between bands of the same fla- of the partons must be equivalent in both determinants
vor of parton. When the gauge field is turned on, the [seeEq.(1)]. ThisGutzwillerprojectionofthewavefunc-
powerlawexponentswillbemodifiedandtheAmperean tion plays the role of the strongly-coupled gauge field in
rule predicts that excitations corresponding to aligned the gauge theory. This sort of variational approach has
6
d
1
d d
2 2
d
1
FIG.2: (coloronline). SchematicpictureofhowtheanisotropyoftheFermiseasinthepartonrepresentationisdrivenbythe
ring-exchangeterminthemodel,Eq.(11). Ifoneallowsthed andd partonstovirtuallyhopalongtheirpreferreddirections,
1 2
an effective ring exchange for the bosons is generated. Because we are considering partons with fermionic statistics, the sign
of the ring-exchange interaction is rendered positive [cf. Eq. (13)]; in contrast, bosonic partons would generate negative ring
exchange.34,35 The resulting sign can intuitively be understood by noting that the ring-exchange process involves one crossing
of the partons, as depicted in the second leftmost plaquette shown above, so that the ring-exchange energy is minimized by
having a positive (negative) coefficient for fermionic (bosonic) partons.
beenrathersuccessfulin1Dsystems,suchasfortheorig- of unity.14 Varying these exponents can also be seen as
inal studies of the t-J model.36–41 augmenting the pure Gutzwiller-projected Slater deter-
The variational parameters in this wave function are minant wave function with a logarithmic Jastrow-type
simplyhowonedefinestheshapeoftheFermiseas,built factor.
up from 1D segments for d and d (see Fig. 1). To be Under the Gutzwiller projection, different mean-field
1 2
truly“d-wave”innature, theremustbesomeanisotropy statescanleadtothesamephysicalwavefunction(gauge
in this filling such as in the DBL[2,1], which was identi- redundancy). Foroursystemandourpracticalpurposes,
fied and characterized in Ref. 18, or the DBL[4,2] and thiscanbesummedupasfollows: ifoneshiftstheFermi
DBL[3,0] states described later in this paper and in sea of d by momentum Q and simultaneously shifts the
1
Ref.19. Butthereexistsmuchflexibilityinhowonedoes Fermi sea of d by momentum Q, the wave function
2
−
this and many nontrivial fillings are possible in general, remains unchanged. The most significant consequence
each corresponding to different locations of the singular of this property is that there exists some arbitrariness
features in the momentum-space correlators. Indeed, it in the choice of parton boundary conditions. Since we
is the agreement of these locations between the DMRG are going to be considering a finite system with periodic
results and those of the most energetically competitive boundary conditions in both directions for the physical
VMCstatesthatformsthemostcompellingevidencefor bosons, we actually have the option of choosing the par-
the existence of the DBL[n,m] in a given model. ton boundary conditions, in each direction separately, to
Additionalvariationalfreedomcanbeaddedbyscaling bebothperiodicorbothantiperiodic. Eitherselectionis
the magnitude of the determinants in the boson wave mappabletotheotherbyvirtueofthisshiftingproperty,
function; this is one of the fruitful methods borrowed and so we will often make this choice based on what is
from the 1D t-J studies mentioned above. Explicitly, if most conceptually clear or aesthetically appealing.
we start by writing schematically
Ψb = G[(detD1) (detD2)], (8) C. Microscopic model: hard-core bosons with
P ×
frustrating ring exchange
where D and D are the two matrices populated with
1 2
eiqi(α)·rj, the set {q(iα)} is the Fermi sea for dα, and PG In the introduction, we stated the model of primary
denotes the Gutzwiller projection, then we can replace
interest in Eqs. (3)-(5). The unfrustrated version of this
each determinant as follows:
model(K 0),withbothhard-coreandsoft-corebosons
≤
detD detD at and away from half-filling, has a long history in the
detDα = detDα α detDα pα α , (9) past decade as a proposed candidate for harboring a de-
| | detD →| | detD
α α
| | | | confined quantum critical point and/or an exotic quan-
such that tum phase dubbed the “exciton Bose liquid” (EBL),42 a
relative of our DBL. However, for this particular model,
detD1 detD2 both of these scenarios have been largely ruled out in a
Ψ = . (10)
b PG(cid:34)(cid:32) detD1 1−p1(cid:33)×(cid:32) detD2 1−p2(cid:33)(cid:35) sequenceofquantumMonteCarlostudies,43–47 although
| | | | recent work has shown that the EBL can be stabilized
The exponents p and p can now be varied; this allows if one supplements a K-only model, Eq. (5), with ring
1 2
one to tune the m+n 1 Luttinger parameters men- exchange on 1 2 and 2 1 plaquettes.34,48
− × ×
tionedintheprevioussection. Weexpectthatthe“bare” Our focus here is on the explicitly frustrated (K > 0)
Gutzwiller wave function, i.e., p = p = 1, corresponds case. Thisring-exchangetermanditsroleinmanifesting
1 2
to all Luttinger parameters fixed at their trivial values the proposed DBL phase has a simple, intuitive physical
7
explanation, which is visualized in Fig. 2. where ρ(r) = b (r)b(r) and ρ N /(L L ) (with no
† b x y
≡
We can modify Eq. (3) somewhat to allow for argument)istheaveragebosondensity. Also, itsFourier
anisotropic hopping of the bosons; that is, transform,thedensity-densitystructurefactor,isdefined
as
H =H +H , (11)
hop K 1
D (q) D (r,r)eiq(r r(cid:48)) = δρ δρ .
Hhop =−J (b†rbr+xˆ+H.c.)−J⊥ (b†rbr+yˆ+H.c.), b ≡ LxLy r,r(cid:48) b (cid:48) · − (cid:104) −q q(cid:105)
r r (cid:88)
(cid:88) (cid:88) (17)
(12)
We also found it useful to consider in some circum-
HK =K (b†rbr+xˆb†r+xˆ+yˆbr+yˆ+H.c.). (13) stances current-current correlations on our ladder sys-
r tems. To this end, we define the current operator as
(cid:88)
Now there are two dimensionless parameters, K/J and Jµˆ(r) i b (r+µˆ)b(r) b (r)b(r+µˆ) , (18)
J /J, in addition to the boson density ρ. Thus, for b ≡ † − †
a⊥fixed value of the density, we can explore a two- where µ=xˆ,yˆ,(cid:2)and the current-current corre(cid:3)lator as
dimensional phase diagram in search of DBL phases.
Our model lives on an N-leg square ladder wherein Cµˆ,νˆ(r,r) Jµˆ(r)Jνˆ(r) . (19)
we can define lattice coordinates: x = 1,...,L , where b (cid:48) ≡ b b (cid:48)
x
(cid:68) (cid:69)
L is the length of the chains, and y = 1,...,L , where
x y The associated structure factor is simply the Fourier
L =N,thenumberoflegs. TheringtermintheHamil-
y transform:
tonianappliestoallelementarysquareplaquettesonthe
ladder. Weusetheconventioninallnumericalworkthat Cµˆ,νˆ(q) 1 Cµˆ,νˆ(r,r)eiq(r r(cid:48)). (20)
each hopping term for a given pair of sites and each ring b ≡ L L b (cid:48) · −
x y r,r(cid:48)
term for a given plaquette is counted precisely once in (cid:88)
the sums. In this work, we have only considered the yˆ-yˆ and xˆ-xˆ
correlations, i.e., Cyˆ,yˆ(q) and Cxˆ,xˆ(q).
b b
As mentioned earlier, having a means by which to de-
D. Measurements termine the number of gapless 1D modes is a critical di-
agnosticfordetectingaDBL[n,m]state. Wecanreferto
We examine many quantities in this work and will de- conformal field theory (CFT), the results of which sug-
fine them explicitly in this section. The first and most gestarelationshipbetweenthenumberofgaplessmodes
obvious is the ground state energy. This can be probed and the central charge, a quantity that can be extracted
directly with DMRG and ED (for small system sizes). from the scaling form of the entanglement entropy. For
With VMC, we instead compute the trial energy of the asubsystemsizeX inasystemofoveralllengthL with
x
variational wave function using our model [see Eq. (11)]. periodic boundary conditions, the von Neumann entan-
Minimizing this quantity with respect to the variational glement entropy S is given by the scaling form49
parameters (the band fillings and the exponents on the
determinants), allows us to find the most competitive c Lx πX
S(X,L )= log sin +A, (21)
x
VMC state at each value of K/J and J /J and plot 3 π L
out a VMC phase diagram, which can then⊥be compared (cid:18) x (cid:19)
where A is a nonuniversal constant independent of the
against DMRG results (see Secs. IIIA and IVA).
subsystemlengthandcisthecentralchargethatweseek.
We examine three main correlators to characterize the
Inourladderstudies,weonlyconsidercleanverticalcuts
ground state for a given set of model parameters. First
throughallN =L chainssothattheleftblockcontains
is the single-particle Green’s function: y
NX sites(X outofL totalrungs). Despitethefactthat
x
G (r,r) b (r)b(r) , (14) the bosonization analysis leads to a theory of n+m 1
b (cid:48) ≡ † (cid:48) freebosonicmodeswithgenerallydifferentvelocitiessu−ch
and its Fourier transform, t(cid:10)he boson (cid:11)momentum distri- that the full system is not conformally invariant, we still
bution function: expect the overall measure of gaplessness to be insen-
sitive to this detail: since we are extracting c from the
1
n (q) G (r,r)eiq(r r(cid:48)) = b b , (15) groundstatewavefunctiononly,whichhasnoknowledge
b ≡ LxLy r,r(cid:48) b (cid:48) · − †q q of mode velocities, the scaling form of S should not de-
(cid:88) (cid:10) (cid:11) pendonthevelocities. Thisisconsistentwiththeknown
whereL isthenumberoflegsoftheladder(interchange- “LlogL”scalingforfreefermions4–7andprojectedFermi
y
able with N in this paper) and L is the number of sites sea states16 in 2D, in which conformal invariance is not
x
in each chain of the ladder. present. By measuring S using DMRG while varying X,
Next, we define the boson density-density correlator: we can attempt to fit this form to the data and extract
the constant fit parameters c and A. Doing so requires
D (r,r) [ρ(r) ρ][ρ(r) ρ] , (16) highly converged DMRG data on large system sizes, L ,
b (cid:48) (cid:48) x
≡(cid:104) − − (cid:105)
8
in order to get reliable estimates of the central charge. tions by construction. However, such weak instabilities
Since this task becomes increasingly more computation- should not be expected a priori, and for the DBL[4,2]
ally challenging with greater spatial entanglement, our and DBL[3,0] phases, we have done the best we can to
DBL[n,m]phasescanbeparticularlydifficulttoanalyze rule out such scenarios by going to reasonably long sys-
asc=n+m 1growslarge. Forexample,whilewewere tems using both fully periodic and cylindrical boundary
−
successfulinmeasuringacentralchargeofapproximately conditions.
two for the DBL[3,0] (see Sec. IVB), we exhausted the At this point, it is convenient to mention a subtlety
computationalresourcesfortheDBL[4,2](seeSec.IIIB) regarding the DMRG and how it handles ground states
where we expect c=5. withnon-zeromomentum. Tobeginwith,itisnotpossi-
Finally, we also considered comparisons of the ground ble to directly extract the ground state momentum from
state momentum at various points in the phase diagram theDMRGstatessincethereal-spaceblockingconstruc-
fortheDBL[3,0]. Welookedatsmallsystemsizeswhere tion necessarily breaks translational invariance. If this
finite-size effects are expected to be significant and ex- momentum is indeed zero or an integer multiple of π,
tractedthetotalmomentumfromEDcalculations,which there is no ambiguity in our DMRG wave function and
is easily inferred from ED results since the diagonaliza- the choice of real- versus complex-valued wave functions
tion is carried out in blocks diagonal in the momentum is irrelevant. If, however, this is not the case and a given
quantum number. We then compared these results to groundstatehasmomentumQ, thenitnecessarilyhasa
the ground state momenta of the VMC states, which are time-reversedpartnerwithmomentum Q. TheDMRG
−
alsotrivialtocomputesinceweareusingcomplex-valued ground state is thus some real-valued combination of
orbitals, eiqr: one simply sums up all of the individual these two states. While the lattice-space measurements
·
momenta of the partons, which fill definite momentum discussedabovemaydependslightlyonthedetailsofthis
orbitals (see Figs. 5 and 6, bottom panel). combination, the differences become less significant with
increased system size. In all the VMC/DMRG compar-
The DMRG calculations presented in this paper were
isonsbelow,wechoosepointsinthephasediagramwhere
performed with anywhere between D 1000-9000 states
(cid:39) Q=0 so as to completely avoid this ambiguity.
per block and at least 6 finite-size sweeps, where each
With the preliminary details having now been fully
“sweep” traverses the L L sites of the lattice twice.
x y discussed, we are now prepared to give detailed descrip-
The accuracy of the results is strongly dependent on the
tionsoftheresultsforthemodel,Eq.(11),onthree-and
phase being studied, the chosen system size and bound-
four-leg ladders.
ary conditions, as well as the physical quantities being
measured. For example, within the four-leg DBL[4,2]
phasediscussedinSec.IIIB,themomentumspacecorre-
latorsonwhichwefocus,n (q)andD (q),areconverged III. GAPLESS BOSE METAL PHASE ON THE
b b
toarelativeerroroflessthan10 2,andthegroundstate FOUR-LEG LADDER
−
energy to a relative error on the order of 10 4; the den-
−
sitymatrixtruncationerrorisontheorderof10 5. Inall HerewepresentresultsofourVMCandDMRGstudy
−
otheridentifiedphases,theaccuracyofourresultsisbet- of the model, Eq. (11), on the four-leg ladder (N =
ter;e.g.,withinthefour-legsuperfluidphase(seeFig.4), L = 4) with periodic boundary conditions in both the
y
the truncation error is on the order of 10 8 keeping only transverseandlongitudinaldirections. Naturally,wefirst
−
D = 2000 states, and in the three-leg DBL[3,0] phase searched for metallic DBL[n,m] phases on the three-leg
of Sec. IVB the truncation error is on the order of 10 7 ladderbutwereunsuccessfulinfindinganythatwereex-
−
keepingD =4000states. Thesequotederrorsareforthe tensible to 2D, most likely due to the non-bipartiteness
caseoffullyperiodicboundaryconditions,whichwepre- ofthelattice. Wedidfindsomeotherinterestingresults,
ferduetotheincommensuratenatureoftheDBLphases. which are discussed in Sec. IV and Appendix B. Here,
The entanglement entropy is typically the last quantity we focus on L 4 system sizes with L = 12,18,24.
x x
×
toconverge,andobtaininghighlyconvergedentropydata WeexpectqualitativelydifferentDBL[n,m]statesinthe
inthemulti-modecriticalsystemsweencounterisanex- two naturally defined density regimes: 0 < ρ < 1/4 and
tremely challenging numerical task. We were able to ob- 1/4 < ρ < 1/2. For small boson densities in the former
tainsuchconvergedentropydata(usedfordetermination case, we find that the system readily phase separates as
ofthecentralcharge, c)withinthesuperfluid, DBL[3,0], the ring coupling is increased. For this reason, and that
and DBL[3,1] phases, but were unfortunately unable to we would like to find metallic DBL[n,m] phases that are
do so within the DBL[4,2] phase. Finally, we note that extensible to 2D, we considered densities in the latter
insuch(potentially)“verycritical”quasi-1Dphases,itis regime. While the behavior is expected to be generic for
notpossibletogotoverylongsystems(L (cid:38)100,say,at ρbetween1/4and1/2,forconcretenesswewilluseboson
x
fixedL =N =3,4)withtheDMRGtodefinitivelyrule density ρ = 5/12 for all three system sizes. The results
y
out eventual small gaps and corresponding long (finite) are summarized in the phase diagram, Fig. 3, which we
correlations lengths; this is further compounded by the shall discuss first. The differently colored regions are de-
factthatthefinitebonddimensionmatrixproductstates termined from a VMC study on the 18 4 system while
×
producedbyDMRGgiveexponentiallydecayingcorrela- the DMRG points are from all three system sizes. Fol-
9
DMRG:
20
q =0
y
q = π/2
y
±
q =π
y
15
)
y
q
,
x
(q10
b
n
5 20
×
FIG. 3: (color online). Phase diagram of the four-leg system
at boson density ρ = 5/12, using system sizes 12×4 with 0
N =20 bosons, 18×4 with N =30 bosons, and (for some !1 !0.5 0 0.5 1
b b
q /π
points) 24×4 with N =40 bosons. The colored regions are x
b
delineatedusingVMCdata. Thegrayregionindicatesphase
separationandisdelineatedschematicallybyconsideringthe
bosondensityinrealspacewiththeDMRGapproach. DMRG
pointsforthesuperfluid,DBL[4,2],andphaseseparationare 0.3
indicatedbywhitecircles,greensquares,andgraydiamonds,
respectively. Finally, in the region labeled “[4,4],” the VMC 0.25
finds a DBL state with four equally occupied bands for both
the d1 and d2 partons, although this region is likely just a ) 0.2
y
superfluid. q
,
x
(q0.15
b
D
lowing that, we shall give an in depth analysis of the
0.1
DMRG:
DBL[4,2] phase.
q =0
y
0.05 q = π/2
y
±
q =π
y
A. The four-leg phase diagram at ρ=5/12 0
!1 !0.5 0 0.5 1
q /π
x
For small K/J, the DMRG confirms the existence of
a quasi-1D version of the generic superfluid described FIG. 4: (color online). Boson momentum distribution func-
earlier (points marked with circles in Fig. 3) exhibiting tion(toppanel)anddensity-densitystructurefactor(bottom
quasi-long range order with the bosons “condensed” at panel) for the superfluid point at J /J = 1, K/J = 1 for
⊥
q = 0 (see Fig. 4, top panel) and a central charge of a 24×4 system size with N = 40 (ρ = 5/12), as obtained
b
c 1 corresponding to the single gapless mode, as de- by DMRG. The values of nb(q) at qy = ±π/2,π have been
ter(cid:39)mined by measuring the entanglement entropy. We scaledupbyafactorof20. Theq=0condensationisreadily
apparentinn (q),asisthe|q |behaviorofD (q)nearq =0
modelthisphaseintheVMCcalculationsusingasimple b x b x
at q =0.
Jastrow wave function that simulates the inter-particle y
repulsion using a constant potential for nearest neigh-
bors and a power law in the separation distance in all
at incommensurate wave vectors in the boson momen-
other cases. The Jastrow wave function contains three
tum distribution function and the density-density struc-
floating point variational parameters that are optimized
ture factor (see Figs. 5 and 6 for characteristic points).
byminimizingthetrialenergyoverafinelygrainedmesh
ThepreciselocationsofthesefeaturesevolveasK/J and
of the phase diagram. The phase diagram indicates that
J /J arevariedwithintheDBLregionofthephasedia-
this modeling is largely successful as it reproduces rea-
gr⊥am(seeFigs.5and6). Thisisconsistentwiththepar-
sonablywellthephaseboundarytotheDBL[4,2]regime.
tonpicturewherelargerK/J leadstogreateranisotropy
The density-density structure factor for a typical point
in the Fermi seas and hence the number of d partons in
in the superfluid phase is shown in Fig. 4, bottom panel. 1
Tishcelecahralyrapcrteesreisntti.c|qx|behavioraroundqx =0forqy =0 tehvoeluktyiobnanodf tNhde(1kyp)ea→k lNocb/a4tiofnors.alIlnkyth,ewehxicthremdraivlesstatthee,
K/AJlo>ng1t.7h5evcuiataJ⊥fir=st-Jor,dtherepDhBaLse[4t,r2a]nosintsioetns.fTorhrisouisghdley- Nisd(d1kiyv)is=ibNleb/b4y,fwouhric,hnc(aqn)obnelycobmeetsruinlyderpeaenlidzeedntwohfeqn Ninb
b y
termined using DMRG (points marked with squares in the variational wave function since there is a conserved
Fig.3)bylookingfortheappearanceofsingularfeatures numberofbosonsineachchain(seeFig.6). WithVMC,
10
we use the wave functions as described in Sec. IIB to We scale up the most competitive VMC states at these
model the DBL. At the onset, we perform an exhaustive points from the 12 4 system by multiplying all band
×
search over all possible, unique band filling configura- occupation numbers by two in the hopes of accurately
tions, the only constraint being that the filled orbitals capturing the behavior of the phase on a 24 4 system
×
form a contiguous strip centered at k = 0 within each withN =40. Thesamepointsonthislargersystemsize
x b
band. Note that there are cases where the band occu- are studied with DMRG and the comparison of these re-
pation numbers are such that no choice of parton lon- sults is shown in Figs. 5 and 6. We perform this scaling
gitudinal boundary conditions results in all bands being procedure to avoid doing a full VMC energetics study
symmetrically filled. In such cases, the resulting wave on the 24 4 system where the pool of potential trial
×
function has a non-zero total momentum in the xˆ direc- wavefunctionsissufficientlylargeastorenderthestudy
tion and all possible, unique resolutions of the “leftover” intractable. Naturally, the VMC method is capable of
particles are considered in the search. This first analy- calculating correlators for a single state on system sizes
sisisperformedwiththe“bare”wavefunction, i.e., with much greater than this, while the DMRG approach is
the exponents on the determinants, p and p , set equal reaching its computational limit.
1 2
to one. Using these results, a pool of the most competi- WenowdescribeindetailthestatefoundatJ /J =1
tivebareVMCstateswithintheparameterregimeshown and K/J = 2.5 (see Fig. 5) as a representativ⊥e of the
in Fig. 3 is formed and then a second analysis stage is DBL[4,2] phase and give an explanation for the loca-
performedvaryingtheseexponentsover0 p ,p 1. tions of the singularities in the structure factors consis-
1 2
≤{ }≤
ThebestDBLstateiscomparedenergeticallytothebest tent with the gauge theory predictions. First, we will
Jastrow state at each point in a finely grained mesh over take antiperiodic boundary conditions for both partons
thephasediagramandthisishowtheVMCphasebound- in both the xˆ and yˆ directions so that the bands will
ary is determined. be filled symmetrically; we remind that this is consistent
For still larger values of the ring coupling, roughly withthephysicalbosonsobeyingperiodicboundarycon-
K/J > 4.25, the DMRG data suggest spatial phase ditions in both directions. The band filling situation is
separation into regions of zero density and density 1/2 displayed visually in Fig. 5, bottom panel; clearly, this
(points marked with diamonds in Fig. 3). Specifically, state has zero total momentum. For the d partons, the
1
our DMRG generally “gets stuck” in a nonuniform state k = π/4 bands each have 12 orbitals filled, while the
y
±
with the (1 2ρ)L centermost rungs at near zero den- k = 3π/4 bands each have 8 orbitals filled. This leads
− x y ±
sityandtheremaining2ρLx outermostrungsat1/2den- to Fermi wave vectors of k(±π/4) = 12π/24 = π/2 and
F1
sity;thereisacorrespondingsharpfeatureinthedensity- k(±3π/4) = 8π/24 = π/3. (We point out that there is
densitystructurefactor,D (q),atq= (2π/L ,0). The F1
b ± x a technicality here: in order to be consistent with the
actualgroundstateispresumablyanequalsuperposition
continuum limit, it is best to define the Fermi wave
of such states with the center of the low-density region
vectors as halfway between the last filled orbital and
at all different L rungs of the ladder, to which we could
x the first unoccupied orbital.) For the d partons, only
neverhopetoequilibratewiththeDMRG.Thistendency 2
the k = π/4 bands are occupied and have 20 par-
to increase the local density is expected since the ring y ±
ticles each; hence the relevant Fermi wave vectors are
term induces an effective attraction of the bosons. It
is apparently the case that the hopping terms are suffi- kF(±2π/4) =20π/24=5π/6. One can verify explicitly that
the sum rule of Eq. (7) is satisfied for both parton fla-
cient to stabilize phases, such as the superfluid and the
vors. The optimal variational exponents on the determi-
DBL[4,2], over a large region of the phase diagram for
nants for the corresponding point in the 12 4 system
smaller values of K. ×
are p = p = 0.8, which are close to the bare values
Throughout our phase diagrams (see Fig. 3 above and 1 2
of unity; these exponents have been carried over to the
Figs. 7 and 14 below), question marks denote a lack of
scaled up 24 4 state.
surety in the DMRG data for specifying a phase. In ×
Looking first to the boson momentum distribution
most cases, however, these points are located near phase
function (Fig. 5, top panel), we see that two of the
boundariesonwhichweplacelimitedfocusinthispaper.
predicted enhanced momenta show up: (k(±π/4)
± F2 −
k(±π/4),0) = (π/3,0), (k(±π/4) k(∓3π/4),π) =
F1 ± ± F2 − F1
B. The DBL[4,2] phase (π/2,π). The other two that are predicted to be en-
±
hanced are present in the VMC data, although weak,
and appear to be smoothed out in the DMRG data:
We now turn our attention to the DBL[4,2] phase it-
self (see Fig. 1), which is a conducting, metal-like phase ±(kF(±2π/4)−kF(∓1π/4),±π/2)=±(π/3,±π/2),±(kF(±2π/4)−
with one more gapless 1D mode than the number of legs k(±3π/4), π/2) = (π/2, π/2). Note that due to the
F1 ∓ ± ∓
ontheladder. Thisphaseshowsnosignsoforderingand inversion symmetry of the lattice, the k = π/2 bands
y
±
breaksnosymmetries;inparticular,itrespectstheinver- are always degenerate in all cases. While the amplitudes
sion symmetry of the lattice. We choose two character- of the peaks in the VMC data are not quite right, the
istic points deep within this region of the phase diagram agreement of the singular locations is striking. The fact
for further analysis: J /J = 1,0.1 at fixed K/J = 2.5. that the coincident singularities at q = 0,π are quite
y
⊥
