Slave rotor theory of the Mott transition in the Hubbard model: a new mean field theory and a new variational wave function

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by  Tao Li

Slave rotor theory of the Mott transition in the Hubbard model: a new mean field
theory and a new variational wave function
Tao Li1, Tomonori Shirakawa2,4, Kazuhiro Seki2,3 and Seiji Yunoki2,3,4
1Department of Physics, Renmin University of China, Beijing 100872, P.R.China
2Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
3Computational Materials Science Research Team,
RIKEN Advanced Institute for Computational Science (AICS), Kobe Hyogo 650-0047, Japan
4Computational Quantum Matter Research Team,
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
6 (Dated: January 12, 2016)
1
A new mean field theory is proposed for theMott transition in theHubbardmodel based on the
0
slaverotor representation of theelectron operator. This theory providesa betterdescription of the
2
roleofthelongrangechargecorrelationintheMottinsulatingstateandoffersagoodestimationof

n thecriticalcorrelationstrengthfortheMotttransition. Wehaveconstructedanewvariationalwave
a functionfor theMott insulatingstatebased onthisnewslaverotor meanfieldtheory. Wefindthis
J
new variational wave function outperforms the conventional Jastrow type wave function with long

1 range charge correlator in the Mott insulating state. It predicts a continuous Mott transition with
1 non-divergent quasiparticle mass at the transition point. We also show that the commonly used

on-site mean field decoupling for the slave rotor corresponds to the Gutzwiller approximation for
]
the Gutzwiller projected wave function with only on-site charge correlator, which can not describe
l
e theMott transition in any finitedimensional system.
-
r
t PACSnumbers:
s
.
t
a
The study of Mott transition and the possible quan- site) and the doublon(doubly occupied site) in the Mott
m
tum spin liquid ground state around the Mott transi- insulating state. While physically appealing, there is no
- tion point in Hubbard type models has attracted a lot microscopicjustificationfor the use of the heuristic form
d
n of attention in the strongly correlated electron system ofthe Jastrowfactor,exceptinone dimension, whenthe
o community1. The organic compound with the formula collectivechargefluctuationistheonlyimportantcorrec-
c κ−(ET) Cu (CN) forming a triangular lattice is par- tionatlowenergybeyondthefreefermiongroundstate9.
2 2 3
[ ticular interesting in this respect2–4. It is generally be- At the same time, a very large system is needed to see

1 lieved that the multi-spin exchange process in the inter- clearly the signature of Mott transition in the Jastrow
v mediate correlation regime may be crucial for the stabi- wavefunction, whichis governedby the long wavelength
8 lization of the quantum spin liquid ground state5. Mea- physics.
7 surementsonκ−(ET) Cu (CN) seemstosupportsuch Theslaverotorrepresentationofthe electronoperator
2 2 3
3 anexpectation3,4. Morerecently,thepossibilityofquan- is aneconomicwayto describe the chargedegree offree-
2
tum spin liquid groundstate aroundthe Mott transition domofthe Hubbardmodel10,11 andiswidelyusedinthe
0
. hasalsobeenhotlydiscussedforHubbardmodelswitha studyoftheMotttransitionoftheHubbardsystems10–14.
1 Dirac-type electron dispersion6,7. Differentestimationsonthe criticalvalueU ofthe Mott
c
0
ThestudyofthespinliquidstateinHubbardmodelsis transition have been made using various kind of mean
6
more complicated than in Heisenberg models as a result field treatment of the slave rotor. In the simplest on-
1
: ofthe addedcomplexityofthe chargedegreeoffreedom. site meanfieldapproximation,the Motttransitionis ap-
v
Depending onthe valueofU forchargegapopening,the proached when both the coherent weight and the band
i
X Hubbardmodelcanexhibitdifferentphysicsintheinter- width of the quasiparticle excitation vanish. This is be-
r mediate coupling regime. The spin liquid phase is pos- lievedtobe true only inthe limitofinfinite dimensions ,
a sible only when the charge gap opens before other sym- when the spatial correlation is irrelevant. This problem
metry breaking transition. Thus, anaccurate estimation can be solved by a large-N treatment of the U(1) charge
of the Mott transition point is important in the study of rotor,inwhichtheunitaryconstraintonthechargerotor
spin liquid phase in Hubbard models. The Jastrow-type is relaxed to a requirement on the average. However, it
variationalwavefunction is acommonly usedwayto de- is not clear to what extent the approximations adopted
scribe such an non-symmetry breaking transition. The ontherotorarerelevantforelectronintheoriginalHub-
Jastrowwavefunction is composedof the productof the bard model, since the mean field theory breaks the U(1)
Slater determinant |Ψ i for the free electronanda Jas- gauge symmetry inherent in the slave rotor representa-
FS
trow factor describing the charge correlation in the sys- tion. Especially, it is not clear what is the relation be-
tem. PreviousstudiesfindthattodescribetheMotttran- tweenthevariousmeanfieldtheorieswiththevariational
sition from the metallic phase, the Jastrow factor must approaches.
belongranged8. Inparticular,thelongrangechargecor- The purpose of this paper is to bridge the variational
relationisresponsibleforthebindingoftheholon(empty approach and the slave rotor theory. In this paper, we
2
propose a new mean field theory for the Hubbard model and
in the slave rotor representation. The new mean field
U
treats better the long range charge correlation and pre- H =−J ei(θi−θj)+ L2, (6)
r 2 i
dicts a continuous Mott transition for Hubbard model
<i,j>,σ i
X X
with non-divergent quasiparticle mass at the transition
point. We then construct explicitly the variational wave respectively, where tf = t < ei(θi−θj) > and J = t <
function related to this slave rotor mean field theory by f† f >. After such a decoupling, the spinon part
σ i,σ j,σ
enforcing the rotor constraint on the slave rotor mean becomesafreefermionsystemsandhasthesameHamil-
fieldgroundstate. Thesoconstructedwavefunctioncan tPonian as the free electron. It can then be shownthat at
be written as the product of a permanent for the charge thezerotemperatureJ = K ,inwhichK istheabsolute
ZN
sector and a Slater determinant for the spin sector. We value of the ground state energy of the non-interacting
have performed variational Monte Carlo simulation on system on the same lattice, Z and N are the coordinate
this new wave function and find that it outperforms the number and the total number of sites of the lattice re-
Jastrow wave function with long range correlator in the spectively.
Mottinsulatingstate,althoughitinvolvestwovariational The rotor part is still nontrivial and further approxi-
parameters. At the same time, we find that the widely mation is needed to solve it. In the commonly used on-
used on-site mean field approximation for the slave ro- site mean field approximation, one takes x=<eiθi > as
tor generates the simple Gutzwiller wave function of the asite-independentconstant. ThedecoupledrotorHamil-
form gD|ΨFSi , which is not suitable for the description tonian then becomes the sum of independent rotors
of Mott transition in finite dimensional systems.
Intheslaverotorrepresentation,theelectronoperator U
H =−Kx cosθ + L2. (7)
is written as θ i 2 i
i i
X X
c =e−iθf . (1)
i,σ i,σ This model can exhibit two phases depending on the
Here e−iθ is the lowering operator of a U(1) rotor that valueof KU. Inthe smallU limit, therotorwillbreakthe
U(1) rotational symmetry and x will be nonzero. This
describes the charge degree of freedom of the electron,
correspondstoametallicphaseinwhichboththespinon
and f is the fermionic spinon operator that describes
i,σ
band width and the quasiparticle weight are renormal-
thespindegreeoffreedomoftheelectron. Torecoverthe
ized by a factor x. On the other hand, when U is large
correct Hilbert space and the algebras among c and
i,σ
enough, the rotor will recover the U(1) rotational sym-
c† , the rotor and the spinon degree of freedom should
i,σ metry and x will become zero. In such a case, both the
be subjected to the following constraint
spinon band width and the quasiparticle weight will be
L = f† f −1. (2) zero. The critical value Uc for such a transition can be
i i,σ i,σ obtainedfromthe self-consitentequationx=<eiθ> and
σ
X is given by U = 2K. On the triangular lattice this cor-
Here L is the angular momentum of the slave rotor on responds to U ≈4.7t.
i c
site i and is conjugate to phase variable θ . The on-site mean field treatment can provide a rough
i
WeconsidertheMotttransitionoftheHubbardmodel understanding of the Mott transition on the triangular
of the form lattice. However,thecriticalvalueU fortheMotttransi-
c
tionpredictedbyitismuchsmallerthanthatisgenerally
H =−t c†i,σcj,σ+U ni,↑ni,↓, (3) expected. The divergenceof the spinon effective mass at
<i,j>,σ i the Mott transition is also not generally accepted for a
X X
finite dimensional system. Another way to see the in-
here n = c† c , n = c† c . In the slave rotor
i,↑ i,↑ i,↑ i,↓ i,↓ i,↓ sufficiency of the on-site mean field approximation is to
representation, up to a constant and a shift in chemical
study the related variational wave function, which can
potential, the model can be written as
be constructed by enforcing the rotor constraint on the
mean field ground state. The mean field wave function
U
H =−t fi†,σfj,σei(θi−θj)+ 2 L2i. (4) of the system in the rotor representationis given by
<i,j>,σ i
X X
Herewehaveexploitedtherotorconstrainttorewritethe |MF>= φ |m i |f −FS>, (8)
mi i
interactiontermasthekinetic energyoftheslaverotors. Yi Xmi !
The form of the Hamiltonian is inviting to decouple the
spinondegreeoffreedomandtherotordegreeoffreedom. in which |f −FSi denotes the spinon Fermi sea, φmi is
ThisresultsinthefollowingeffectiveHamiltonianforthe therotorwavefunctiononsitei. Thetruewavefunction
rotor and the spinon of the system is givenby |Ψ>= iPi|MFi, in which Pi
is the projector enforcing the rotor constraint Eq.(2) on
H =−t f† f (5) site i. Since the spinon occupatQion number on a given
s f i,σ j,σ
site can only be 1,0 or 2, the rotor angular momentum
<i,j>,σ
X
3
m can only be 0, 1 or −1. In such a case, |Ψi reduces in which n = b† b , s+ = b+ b +b+ b , s− =
i i,α iα i,α i i,1 i,0 i,0 i−1 i
to the usual Gutzwiller projected wave function of the b+ b +b+ b .
form gD|ΨFSi, with the factor g given by g = φ±1/φ0. i,−In1tih,0e larig,0eUi,1limit, mostsites will be singly occupied
HereD isnumberofdoublyoccupiedsitesinthesystem. and we expect both n and n to be small. We thus
−1 1
As is well known, the simple Gutzwiller projected wave assumeb condensesandtreatitasac-numbertobede-
0
functioncannotdescribetheMotttransitioninanyfinite terminedself-consistentlyfromtheconstraint. Therotor
dimension. Onlyinthelimitofinfinitedimensions,when Hamiltonian then becomes
the Gutzwiller projector can be treated exactly with the
Gutzwiller approximation, the Gutzwiller wave function Hr = − Jη (b+i,1bj,1+b+i,−1bj,−1+h.c.)
can predict a Mott transition at finite U. At the same <i,j>
X
time, the divergence of spinon effective mass is realized − Jη (b+ b+ +b+ b+ +h.c.)
only in the limit of infinite dimensions, when all spatial i,1 j,−1 i,−1 j,1
<i,j>
correlation can be neglected. We thus conclude the on- X
U
siteslaverotormeanfieldapproximationisequivalentto + (n +n ). (11)
i,1 i,−1
the commonly used Gutzwiller approximation and can 2
i
X
not describe the Mott transition in finite dimensions.
The reason for the failure of the simple Gutzwiller inwhichη =|hb0i|2. Herewenotethatthecondensation
wavefunctiontodescribetheMotttransitioninfinitedi- ofb0 doesnotbreakthe U(1)rotationalsymmetryofthe
mension is that the long range charge correlation,which rotor since it carries zero charge.
has been proved to be crucial for a correct theory of The above Hamiltonian can be diagonalized to give
Mott transition, is not properly accounted for in such
H = ǫ (β+ β +β+ β )+const. (12)
a local treatment. The main effect of such long range r k k,1 k,1 k,−1 k,−1
chargecorrelationis to introduce attractionbetween the Xk
holon(emptysite)andthedoublon(doublyoccupiedsite) Hereǫ = ξ2−∆2,inwhichξ = U−ZJηγ and∆ =
in the singly occupied background. In the Mott insulat- k k k k 2 k k
ZJηγ . γ = 1 eik·δ, where δ denotes the vectors
ing state, the holon and doublon will be bounded to- k pk Z δ
connecting nearest neighboring sites on the lattice. The
gether because of such attraction. For smaller U, the P
value of η can be determined self-consistently from the
holon-doublon pair will disassociate and the system will
constraint Eq.(9) and is given by
become metallic. To restore such long range charge cor-
relationinthevariationalwavefunction,variouskindsof 1 ξ
η =<b†b >=2− k. (13)
charge correlator have been proposed. In particular, the 0 0 N ǫ
k
long range Jastrow factor has been extensively adopted Xk
in the study of the Mott transition in Hubbard mod- The minimum energy for charge excitation is given by
els. However, the Jastrow factor is constructed from a ǫ . The Mott transition occurs when this gap closes.
k=0
two-bodyconsiderationandisheuristicinnature. Inthe Fromthisrequirement,wefindtheU forMotttransition
c
following,weproposeanew chargecorrelatorbasedona isgivenbyU =4ZJη . Hereη isthevalueofηatwhich
c c c
newtypeofslaverotormeanfieldtheory. Thenewmean the gap closes and is given by
fieldtheoryovercomesthedrawbacksoftheon-siterotor
mean field theory and predicts a non-divergent effective 1 1−γk/2
η =2− . (14)
c
mass at the Mott transition. The Uc predicted by the N k (1−γk/2)2−γk2
new theory is also more reasonable. X
To formulate the new mean field theory, we introduce For a given lattice, ηc is apmathematical constant. On
a boson representation of the rotor degree of freedom. the triangular lattice, this constant is found to be ηc ≈
As a result of the rotor constraint, the angular momen- 0.8971. Thus the Uc for the Mott transition on the tri-
tum of the rotorcanonly be 0 or ±1. Here we introduce angularlattice is Uc ≈3.5884K ≈8.43t. This is a better
threebosonoperatorsb andb torepresentthesethree estimation than that obtained by the local mean field
0 ±1
states. The three bosons are thus subjected to the con- treatment we discussed above15.
straint of Wenowconstructthevariationalwavefunctionrelated
to the new mean field theory. The mean field ground
b†b =1, (9)
α α state of the system is the product of the paired ground
Xα state of b1 and b−1 boson and the Fermi sea for the
inwhichα=0,±1. We note that the three bosonscarry spinon,
different charges. More specifically, b carries charge α
i,α
relative to the singly occupied background. With these |MF>=ePi,jW(i,j)b†i,1b†j,−1|0i|f −FS>, (15)
boson operators, the rotor Hamiltonian can be written
in which |0i is the boson vacum,
as
U 1 ∆
Hr =−J (s+i s−j +s−i s+j )+ 2 (ni,1+ni,−1), (10) W(i,j)= N ξ +kǫ eik·(Ri−Rj) (16)
<Xi,j> Xi Xk k k
4
is the pair wavefunction for the b boson. In principle,
±1
we should also consider the condensate of the b boson
0
in the above mean field wave function. However, the 5
b condensate only contributes a multiplicative constant
0
to the wave function, which only depends on the total
4
number of singly occupied sites. Using the constraint
Eq.(9),thismultiplicativeconstantcanbe writteninthe
3
formofaGutzwillerfactorgDandbeabsorbedinEq.(15)
=0
bymultiplyingW(i,j)withafactorg. Thetrueelectron k
2
wave fucntion of the system is obtained by enforcing the
rotor constraint Eq.(2) and is given by
1
|Ψi= P |MF>=Perm[gW]|Ψ i. (17)
i FS
0
i 10 12 14 16 18 20
Y
U/t
Here |Ψ i is the Fermi sea state of the free electron,
FS
Perm[gW] is the permanent of the matrix gW. W is a
matrix of dimension D with its matrix element given by FIG. 1: The evolution of the charge gap with U/t. The red
W(i,j),whichisthepairwavefunctionbetweendoublon dotrepresenttheextrapolatedvalueofthegapclosingpoint.
at site i and holon at site j.
We have performed variational Monte Carlo study on
theabovewavefunctiononthetriangularlattice. Asthe

computation of permanent of a matrix is exponentially -10
expensive in its dimension, our calculation is limited to Jastrow wave function
thelargeU regime,whenthenumberofdoublonissmall. y (t) -20 permanent wave function
In our calculation, we have used a 12×12 cluster with g
er
periodic-antiperiodicboundarycondition. Therearetwo en -30
variational parameters, namely g and λ = 2ZUJη, to be ate
optimized in our wavefunction. To determine the Uc for d st -40
Motttransition,we use the meanfield expressionfor the un
o
charge gap, which is given by ǫk=0 = 2Uλ (λ−1)2−1. gr -50
The gap closes when λ approaches 2 from above. When
p
λ>2,thepairwavefunctionW(i,j)isshortrangedand
-60
the holon and the doublon are bounded together. Fig.1 6 8 10 12 14 16 18 20
plot the evolution of the charge gap with U/t. The gap U/t
exhibits a linear dependence on U/t close to the Mott
transition point. We note that the same behavior is also
FIG. 2: The ground state energy as calculated from the Jas-
predicted by the large-N theory. From a linear extrapo-
trow wave function and the permanent wave function on a
lation,wefindthechargegapclosesaroundU/t≈10.75. 12×12 cluster. In the former case, v(i−j) at all distances
Thisresultishigherthanthe generallyacceptedvalueof are optimized.
7 to 8.
Finally, we compare our result with that produced
by the wave function with long range Jastrow factor. phase,althoughitinvolvesmuchsmallernumberof(only
The Jastrow wave function can be written as |Ψi = two) variational parameters.
ePi,jv(i−j)ninj|ΨFSi. Here ni and nj denotes the elec- In summary, we have proposed a new mean field the-
tron number at site i and site j, v(i −j) is the varia- ory for the Mott transition in the Hubbard model based
tional parameter introduced to control the charge corre- on the slave rotor representation. The new theory can
lation between these two sites16. The comparison of the better capture the long range charge correlation, which
ground state energies on a 12×12 cluster predicted by is known to be crucial for a correct theory of Mott tran-
both theories is shown in Fig.2. For the Jastrow wave sition. The new theory predicts a continuous Mott tran-
function, we have optimized the variational parameter sition on the triangular lattice around U = 8.43t with
v(i − j) at all distances. The shoulder in the ground non-divergentquasiparticle mass at the transition point.
state energy around U/t=12 is a precursor of the Mott The variational wave function corresponding to the new
transitioninthethermodynamiclimit. Toseeclearlythe mean field theory has the form of the product of the
signature of Mott transition in the Jastrow wave func- permanent for the charge degree of freedom and the de-
tion, much larger lattice is needed. Compared with the terminant of the spin degree of freedom. The new wave
Jastrowwavefunction,thepermanentwavefunctionpro- functionisfoundtoworkbetter thanthefully optimized
posed in this paper is obviously better in the insulating Jastrowwavefunction in the insulating phase. However,
5
the U predicted from such a wave function is still sig- subject of a future paper.
c
nificantly larger than the generally accepted value. We
think this deficiency shouldbe attributedto the spinde- The computations have been done using the Magic-
greeoffreedom, forwhichthe superexchangeinteraction II Supercomputing facility in Shanghai supercomput-
in the large U regime is not fully accounted for by the ing center, the RIKEN Integrated Cluster of Clusters
Slater determinant wave function. It should be noted (RICC) facility and the RIKEN supercomputer sys-
thatsuchsuperexchangeeffectcanbeincorporatedinthe tem(HOKUSAI Great Wave). Tao Li is supported by
wave function through backflow correction in the Slater NSFC Grant No. 11034012 and Research Funds of Ren-
determinant17,18. Wefindthebackflowcorrectioncanin- min University of China. This work has been also sup-
deedimprovesignificantlytheestimateofU inourwave ported in part by RIKEN iTHES Project and Molecular
c
function. Adetaileddiscussiononthispointisleftasthe Systems.
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