Real-space renormalized dynamical mean field theory

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by  Dai Kubota

Real-space renormalized dynamical mean field theory
Dai Kubota,1 Shiro Sakai,2 and Masatoshi Imada1
1Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113-8656, Japan.
2Center for Emergent Matter Science, RIKEN, Hirosawa, Wako, Saitama 351-0198, Japan.
We propose real-space renormalized dynamical mean field theory (rr-DMFT) to deal with large
clusters in the framework of a cluster extension of the DMFT. In the rr-DMFT, large clusters are
decomposed into multiple smaller clusters through a real-space renormalization. In this work, the
renormalization effectistakenintoaccountonlyatthelowest orderwithrespect totheintercluster
6
coupling, which nonetheless reproduces exactly both the noninteracting and atomic limits. Our
1
method allows us large cluster-size calculations which are intractable with the conventional cluster
0
extensions of theDMFT with impurity solvers, such as thecontinuous-time quantum Monte Carlo
2
andexactdiagonalization methods. Webenchmarktherr-DMFTforthetwo-dimensionalHubbard

y modelonasquarelatticeatandawayfromhalffilling,wherethespatialcorrelationsplayimportant
a roles. Ourresultsonthespinstructurefactorindicatethatthegrowthoftheantiferromagneticspin
M
correlation is taken into account beyond the decomposed cluster size. We also show that the self-
energyobtained from thelarge-cluster solveris reproducedbyourmethodbetterthanthesolution

9 obtained directly for the smaller cluster. When applied to theMott metal-insulator transition, the
1 rr-DMFTisabletoreproducethereducedcriticalvaluefortheCoulombinteractioncomparableto
thelarge cluster result.
]
l
e PACSnumbers: 71.10.Fd,71.27.+a,71.30.+h,75.10.-b
-
r
t
s I. INTRODUCTION recently proposed17), when we apply an N -site cluster-
. c
t DMFT to the Hubbard model with the onsite repulsive
a
m interaction U at the inverse temperature β. In another
Strongly correlated electron systems have intensively
algorithm based on the hybridization expansion,12 its
- been studied in the past decades. Various numeri-
d cal methods have been developed to study correlated- cost grows as O[β3exp(Nc)] with increasing Nc. On the
n otherhand,intheED,thefreefermionbathofthequan-
electron models such as the Hubbard model. Neverthe-
o tumclustermodelisrepresentedbyafinitenumber(N )
c less,thecomplete understandingofthemremainsopen.1 b
of fermions. The numerical cost then exponentially in-
[ Among the methods, the dynamical mean field theory
creases as O[4(Nc+Nb)]. Thus, the accessible cluster size
(DMFT)2,3 has an advantage in calculating dynamical
2 isseverelylimitedforallthesesolvers. Furthermore,this
v properties, by taking into account full temporal fluctu- difficulty becomesmoreseriouswhenweconsiderthe or-
6 ations. Although its original single-site formalism com- bital degrees of freedom.18–21
1 pletely ignores spatial correlation effects, there are sev-
The dilemma between the accuracy and the computa-
9 eral approaches to take them into account. The ap-
5 tional cost has led several authors to propose methods
proachesincludeclusterextensionsoftheDMFT,suchas
0 to obtain better results from small clusters. These in-
thecellulardynamicalmeanfieldtheory(CDMFT),4 dy-
. clude improved interpolation schemes in the DCA22,23
1 namical cluster approximation (DCA),5 and variational
and combinations of the cluster and diagrammatic ex-
0 cluster approximation,6 which incorporate the spatial
6 tensions of the DMFT.24–26
correlation effects within the cluster. Alternatively, dia-
1 A sufficiently large cluster is required to understand
grammatic extensions7–9 have been proposed.
: strongcorrelationeffectsinlow-dimensionalsystems. For
v
i In these approaches, the original lattice model is example, the self-energy has a strong momentum de-
X mapped onto an effective quantum impurity model con- pendence in doped Mott insulators27–40 in two dimen-
r sistingofinteractingsitesandanoninteractingfermionic sions. Continuous phase transitions and regions near
a bath. The effective impurity model is solved, for exam- quantum criticalities in general require accurate treat-
ple, by the continuous-time quantum Monte Carlo (CT- ments of long-range spatial correlations. From a quan-
QMC),10–12pioneeredbyBeardandWieseetal.,13,14and
titative point of view, small cluster studies of the two-
exact diagonalization (ED)15 methods. In these cluster dimensional Hubbard model substantially overestimate
extensionsoftheDMFT,therangeofthespatialcorrela-
the critical interaction U of the Mott metal-insulator
c
tionsincorporatedintothe numericalsolutionisseverely transition (MIT)1,41,42 and the magnitude of the order
limited by the cluster size. In fact, with the increase parameter of the d-wave superconductivity.43
in the cluster size, the computational cost rapidly in-
In this article, we introduce an approachbased on the
creases, which easily makes the practical computation
CDMFT combined with the real-space renormalization,
intractable. which we call real-space renormalized dynamical mean
ThecomputationalcostoftheCT-QMCdependsonits field theory (rr-DMFT). In the rr-DMFT, a large cluster
algorithm: In the weak-coupling algorithms,10,11,16 the model is decomposed into multiple smaller-cluster prob-
costscalesasO[(N βU)3]inthe absenceofthe negative- lems which can be solvedat a considerably smaller com-
c
sign problem (while a more efficient algorithm has been putational cost than the former model, while the spatial
2
correlationsuptothesizeoftheoriginallargeclusterare action is given by
reasonablytakenintoaccountthrougharenormalization
β β
of the dynamical mean field for the smaller clusters. S =− dτ dτ′ c† (τ)G−1 (τ −τ′)c (τ′)
C Z Z iσ 0,ijσ jσ
The article is organized as follows: In Sec. II, the rr- 0 0 i,jX∈C,σ
DMFT is introduced with several practical examples of β
the decomposition of a large cluster model into multiple +U dτ n (τ)n (τ), (2)
i↑ i↓
Z
small-sizeproblems. SectionIII describesbenchmark re- 0 Xi∈C
sultsforthetwo-dimensionalHubbardmodelonasquare
where β is the inverse temperature and C represents
lattice. The short-range antiferromagnetic correlations
the cluster. The dynamical one-body part Gˆ−1(τ) is the
play key roles in this system. We show that our method 0
Weissfunction. Therr-DMFTalsostartswiththis map-
correctly takes them into account through the results
ping onto a cluster model and provides an approximate
on the spin structure factor and the description of the
way to calculate the Green’s function without using any
self-energy beyond the decomposed cluster size. The re-
C-size impurity solvers as we describe in the following.
sults obtained by this method are also favorably com-
The single-particle Green’s function G (τ) of S is
paredwiththeconventionalCDMFTresultsontheMIT, ijσ C
defined by
where the reduction of the critical strength of the onsite
cCaolcuulolamtbe arenpdulcsoiomnpiasreprtohpeerdleynrseitpyroodfuscteadt.e4s1,4w2itWheusailnsgo TrC Tτciσ(τ)c†jσ(0)e−SC
G (τ)≡ (cid:16) (cid:17). (3)
the exact diagonalization solver. Finally, Sec. IV is de- ijσ TrC(e−SC)
voted to conclusions.
Here, Tr means the trace over the degrees of freedom
C
within the cluster C, and T is the time-ordering opera-
τ
tor.
II. FORMALISM We define the actionSCex1act for a smaller cluster C1 by
Inthepresentrr-DMFTapproach,weapproximatethe e−SCex1act = TrC2e−SC, (4)
original large cluster model by mapping it onto that of TrC1e−SCex1act TrCe−SC
multiplesmallclusters,andthensolveitbyusingaquan- whereC isapartofC otherthanC . WithSexact,G
tum impurity solver for small clusters. In this paper, we 2 1 C1 ijσ
(i,j ∈C ) is obtained as
1
employthe lowest-ordercumulantexpansiontotakeinto
account the inter-cluster correlations. TrC1 Tτciσ(τ)c†jσ(0)e−SCex1act
In Sec. II.1,we elaborate how we reduce a large clus- G (τ)= (cid:16) (cid:17). (5)
ijσ
ter model to smallcluster problems througha real-space TrC1 e−SCex1act
renormalization. InSec.II.2,wedescribeanalgorithmto (cid:16) (cid:17)
systematically calculate all the elements of the Green’s Here we need only a C -size impurity solver to calculate
1
function using the mapping. It is shown that if an N- Eq. (5).
site impurity model is solvable, the Green’s function for However, Sexact generally contains n-body interaction
C1
2N-site cluster models is obtained within our method. terms (n = 1,2,...) and is not tractable with standard
By repeating this procedure,we can solvethe large clus- impurity solvers. We therefore replace Sexact with the
C1
ter problem only with the impurity solver for the small actionS ,whichisasimplequantumimpurityproblem
C1
cluster. obtained from a perturbative treatment of the coupling
between C and C . In Sec. III, we examine the validity
1 2
of this replacement by comparing the rr-DMFT results
with the CDMFT fully performed for C. At the lowest
II.1. Renormalization process order in the cumulant expansion,44 the action is written
as
We introduce the rr-DMFT for the Hubbard model, Sexact ≃S
defined by the Hamiltonian, C1 C1
β β
=− dτ dτ′ c† (τ)G˜−1 (τ −τ′)c (τ′)
H = tijc†iσcjσ −µ c†iσciσ+U ni↑ni↓. (1) Z0 Z0 i,jX∈C1,σ iσ 0,ijσ jσ
iX,j,σ Xiσ Xi β
+U dτ n (τ)n (τ), (6)
i↑ i↓
Z
Here, c (c† ) annihilates (creates) an electron with spin 0 iX∈C1
iσ iσ
σ =↑,↓ at site i. n = c† c is the density operator. with
iσ iσ iσ
The hopping amplitude between sites i and j is denoted G˜−1 (iω )=G−1 (iω )
by t . µ is the chemical potential and U(>0) is the on- 0,ijσ n 0,ijσ n
ij
siteCoulombrepulsion. TheCDMFTmapstheHubbard − G−1 (iω )G(C2)(iω )G−1 (iω ), (7)
0,ikσ n klσ n 0,ljσ n
model onto an effective quantum impurity model whose k,Xl∈C2
3
C C
1 2
j
i C C
3 4
: Traced out block
C C C C C C C C C C C C
1 2 1 2 1 2 1 2 1 2 1 2
FIG.1. (Coloronline)Schemeofstep-by-steprenormalization C C C C C C C C C C C C
3 4 3 4 3 4 3 4 3 4 3 4
to compute the Greens function Gij in rr-DMFT. The large A A B B B B
square in the left panel represents the original large cluster, 1 2 1 2 3 4
and i and j represent the sites in the cluster. Here, the gray
: Traced out block
regions,whichdoesnotincludeiandj,aretracedout,leaving
theyellow area, which is dividedintotworegions in thenext
step.
FIG. 2. (Color online) Decomposition of 2n+1-site cluster
where ω = (2n + 1)π/β is the Matsubara frequency. into four blocks Ci(i = 1,...,4). Here, Ak(k = 1,2) and
n B (k = 1,...,4) denote the groups described in the text.
Here, Gˆ(C2) is the Green’s function for the action Eakch Ci consists of 2n−1 sites.
β β
S(C2) =−Z dτZ dτ′ c†iσ(τ)G0−,i1j(τ −τ′)cjσ(τ′)
0 0 i,jX∈C2,σ means that the Green’s function for C1 is approximately
obtained by using not C-size but C - and C -size impu-
β 1 2
+U dτ ni↑(τ)ni↓(τ), (8) rity solvers only. Here, C1 and/or C2 can still be large
Z
0 iX∈C2 for impurity solvers, i.e., beyond the size tractable with
the direct use of the impurity solvers. In such a case,
namely,
we iterate the real-space renormalization step by step
Tr T c (τ)c† (0)e−S(C2) to trace out further degrees of freedom, as illustrated
G(C2)(τ)= C2(cid:16) τ iσ jσ (cid:17). (9) in Fig. 1. This figure exemplifies the calculation of the
ijσ TrC2 e−S(C2) Green’s function Gij. We implement such a renormal-
(cid:0) (cid:1) ization process for any site pairs (i,j) in the cluster C
The second term on the right-hand side of Eq. (7) rep-
to obtain all the elements of the Green’s function. In
resents the renormalization effect on the Weiss func-
the next subsection, we describe a practical algorithm
tion, which effectively takes into account the C degrees
2 for this.
of freedom. This is similar to the cavity method for
finite-size lattices, used in the nano-DMFT.45 In this
approximation, t/U is the basic small parameter. The
II.2. Algorithm to calculate Green’s function
rr-DMFT therefore reproduces the exact atomic limit.
Furthermore, the exact noninteracting limit is also re-
covered. The size of the boundary between C and C Forthe sakeofsimplicity, we restrictourselvesto two-
1 2
also controls the accuracy because the number of domi- dimensionalsystemsandsolveanN =2nN clusterwith
c s
nant terms in the renormalization is determined by the an impurity solverfor N =2 sites. It is straightforward
s
number of couplings between two clusters, which is con- to extend the algorithm to arbitrary N , clusters and
s
trolled by the size of the boundary. The renormalization dimensions.
effect on the interaction term in SC1 comes in when we First,weconsiderthecaseofn=1. TheGreen’sfunc-
consider higher-order cumulants. However, the lowest- tion for the N =4 sites is obtained as follows: We first
c
order renormalization of Eq. (7) gives results in reason- choosetwo sites, l andm, fromthe cluster andtrace out
able agreement with the CDMFT results even in the the other sites, according to the procedure described in
intermediate-coupling regime, as we show in Sec. III. Sec.II.1. Wethenobtainanactionforthetwositesland
Morespecifically,thelowest-orderapproximationgives m. The Green’s function G is calculated from this ac-
lm
a reasonable value for the Green’s function in the clus- tion with the two-site impurity solver. By iterating such
ter C = C1 + C2 while it neglects the vertex correc- acalculationforall(l,m)pairsinthe Nc-site cluster,we
tionsconnectingtheclustersC1 andC2. Thislackofthe obtain all the elements of the Green’s function.
intercluster vertex corrections limits the applicability of We proceed to a general case. Suppose that we can
the presentrr-DMFT to the systemwith enhancedlong- solve a 2n-site problem and would like to solve an impu-
rangecorrelations,e.g.,inthevicinityofquantumcritical ritymodelof2n+1 sites. We decomposethe latter model
points. The higher-order corrections from the real-space intoseveral2n-siteproblems(Fig.2)asfollows: Webreak
renormalization will mitigate the problem since they in- up2nN =2n+1-siteC intofourblocks,C (i=1,...,4),
s i
clude a part of the vertex functions beyond the size of whereeachC consistsof2n−1 sites. Theelementsofthe
i
C1. Green’s function, Gij, are categorized into six groups:
In this subsection, the action S is decomposed into
C
two cluster problems described by SC1 and S(C2). This • A1= {GC1C1,GC1C2,GC2C1,GC2C2} ,
4
• A = {G ,G ,G ,G } , to use the N -site impurity solver can be minimized by
2 C3C3 C3C4 C4C3 C4C4 s
sharing,asmuchaspossible,theactionofthesiteswhich
• B1= {GC1C3,GC3C1} , aretracedout. Forexample,weconsidertracingoutC1-
degreesoffreedominthe clusterC =C +C +C +C .
• B = {G ,G } , 1 2 3 4
2 C2C4 C4C2 The resultant action consisting of the C +C +C de-
2 3 4
greesoffreedomcanbecommonlyusedtoderiveactions,
• B = {G ,G } ,
3 C1C4 C4C1 S ,S ,andS . Thus,bysharingtheactionappearing
23 34 24
• B = {G ,G } . in each intermediate step as much as possible, we can
4 C2C3 C3C2
save the number of times invoking the impurity solver
Here, we defined GClCm as a set {Gij|i ∈ Cl ∧ and make the algorithm more efficient. This enables us
j ∈ Cm}, and omit the spin index σ for brevity. tosolvetheNc-sitequantumimpuritymodelbyusingthe
The idea behind this categorization is as follows. If N -site impurity solver O(N /N )2 times (see Appendix
s c s
we naively categorize Gij elements into six groups, A).
such as A′ = {G ,G ,G ,G } for
k CiCi CiCj CjCi CjCj Inaddition,itispossibletofurtherreducethenumber
i,j = 1,...,4 and i 6= j, they have common ele-
using the impurity solver if we utilize the symmetries of
ments. For example, {G ,G ,G ,G } and
C1C1 C1C2 C2C1 C2C2 the cluster. For instance, we can use the C4v symmetry
{GC1C1,GC1C3,GC3C1,GC3C3} share GC1C1. In order to for an Nc = L× L square cluster unless a symmetry-
reduce the total computational cost, we have removed
breaking order occurs.
suchoverlapsintheaboveclassification: Namely,wehave
excluded the diagonalelements, such as G , from the
ClCl
B groups. We have thus decomposed a problem of cal-
k III. RESULTS
culating G for i,j ∈C into six tasks of calculating the
ij
A (k = 1,2) and B (k = 1,...,4) elements, which in-
k k
In this section, we apply the rr-DMFT to the two-
clude and exclude diagonal elements, respectively.
dimensionalHubbardmodel onthe squarelattice. First,
To obtain each element of the Green’s function, we
wecomparethecomputationalcostoftherr-DMFTwith
first need to construct an effective action S consisting
ij
that of the CDMFT. Second, we calculate several phys-
of C and C degrees of freedom through the real-space
i j
ical quantities, such as the spin structure factor, self-
renormalization process described in Sec. II.1. Since S
ij
energy and density of states, to examine the accuracy
consistsofonly2n sites,itissolvablebythe assumption.
of the rr-DMFT. We also study the MIT phase bound-
For example, for the calculation of the A element, we
1 ary and show that our method offers a good approxi-
construct S and solve it.
12 mation to the CDMFT results. We use the continuous-
InconstructingS ,weneedtousetheN -siteimpurity
ij s time auxiliary-field quantum Monte Carlo method (CT-
solver2n/N timestoeliminate2n-sitedegreesoffreedom
s AUX) and, in addition, the ED in Sec. III.1 and III.5,
in the 2n+1-site problem. In practice, we first choose
for solving the impurity problem. Through this section,
N sites from the 2n sites and trace out the former by
s although the rr-DMFT is of course more advantageous
the real-space renormalization. Next we again choose
when the regular CDMFT is not tractable any more,
N sites from the rest (2n−N ) sites and trace out the
s s we basically restrict ourselves to cases where the regu-
former. By iterating suchprocesses2n/N times, we can
s lar CDMFT is feasible to allow the comparisonwith our
trace out 2n sites.
method as benchmarks.
Ingeneral,therearemanydifferentwaysto chooseN
s
sites from the 2n sites. In practice, we choose a way
to make the solved N -site cluster as compact as possi-
s
III.1. Computational cost of rr-DMFT
ble sincecorrelationsbetweenneighboringsitesaremore
important than those between distant sites. In fact, the
short-range correlations play key roles in the pseudogap Aswedescribedintheintroduction,thecomputational
regime,32,33,36,37,40 the MIT41,42 and the d-wave super- cost of the CT-QMC and ED increases rapidly with the
conductivity transition.46 As we remarked in the previ- clustersize. Since the rr-DMFTusesonlyanNs-siteim-
ous subsection, this is also justified by the following dis- puritysolver,eventhoughweneedtouseitO[(Nc/Ns)2]
cussion: Therenormalizationterms atanyordersalways times(seeAppendixA),therr-DMFTlargelyreducesthe
include the coupling terms between decomposed clusters totalcomputationalcost. InthecaseoftheCT-AUX,the
[seeEq.(7)],andthereforeitsmagnitudedepends onthe computational cost is reduced from O[Nc3] to O[Nc2Ns]
number of the coupling. This number generally becomes even in the absence of the negative-sign problem. Fur-
smallwhenwedecomposethe clusterascompactaspos- thermore, the rr-DMFT mitigates the sign problem. In
sible. We describe a detailed algorithm in Appendix B thecaseofthe ED,thecostisreducedfromO[4Nc(nb+1)]
anddemonstratethatthe resultsarenotsignificantlyaf- to O[4Ns(nb+1)(Nc/Ns)2], where nb ≡Nb/Nc.
fected by the ways of tracing out as long as we use com- Figure 3 compares the CPU time (measured with a
pact clusters. single processor) for solving an N -site quantum impu-
c
InordertoobtainalltheelementsoftheGreen’sfunc- rity problem for the two-dimensionalhalf-filled Hubbard
tion,G ,weneedtomakeiandj sweepovertheN -site model at U/t = 8. We use the ED at T = 0 and CT-
ij c
cluster. Here,itisusefultonotethatthenumberoftimes AUX with the submatrix updates47 as impurity solvers,
5
(a) (b) tosolvealwaysthewholeimpurityproblemoftheN site
104 101 c
e [s]103CDMFT rr-DFMT U time CT-AUX solvennnnroooo ssssyyyytt´´mmmm//tt mmmm==eeee 0-tttt1rrrryyyy itrhreesrpre-DctMivFeToftthheeuuseseooffthtehesycmlumsteetrrysyrmedmuectersythwehniluemin-
CPU tim111000112000 + rc1rl-0uD1stFeMr EsTyDm smolevter1yr02 Ratio of CP110010-100CC24vv sssyyymmmmmmeeettt1rrryyy01 102 brsuetseadertuoioscftfttiicohtahneleiomesfryprtmohurermititneyottprtayhrl,oecbiotClemDmispMsustttFoialTtlbioelcansaraognllevcrbeoedtsh,tr.yaeindAeulldttchihenadogtubaginyhlatttrhhhgeeee
Nc Nc rr-DMFT which solves smaller clusters.
(c) (d)
101 NotethattheslightlylargercostatN ≤16isascribed
e CT-AUX solver 1 c
m t´/t = -1 to the relatively large number of invoking the N -site
PU ti100 no symmetry n>00..86 solver in the rr-DMFT at small Nc/Ns. For instsance,
o of C10-1 sycmlumsteertr y <Sig0.4 wimhpeunriwtye ssoollvveer,awfoeunre-seidtetocluussteerthmeosdoelvlewri1th2 taimtweso-asnitde
Rati10-2 00.2 rCr-DDMMFFTT this is larger than Nc3/Ns3 = 43/23 = 8, the ratio of the
100 1N0c1 102 1 Nc10 cTohmeplautttaetriornaatliocoinsctroefasNesc-maonrde rNasp-isditlye timhapnurtihteynsuolmvebresr.
using the solver as N increases.
c
Inafrustratedcase(t′/t=−1),amorepracticalcom-
FIG. 3. (Color online) CPU time spent in solving quantum
parison for the CT-AUX results can be made by nor-
impurity models of Nc = 2×1,2×2,4×2,4×4 and 8× malizing the costs with respect to the average sign since
4 at a single core (Xeon X 5690 3.47GHz). We fix Ns =
the negative sign problem reduces the effective number
2 for the rr-DMFT and take U/t = 8 at half filling. (a)
Comparison with using ED as the impurity solver for n =2 of the Monte Carlo sampling. The renormalized ratio of
b
at T = 0 and t′/t = 0. (b) Ratio of CPU time, namely theCPUtime isshowninFig.3(c), andtheaveragesign
CPU time of rr-DMFT scaled by that of CDMFT, for CT- is plotted in Fig. 3(d). The rr-DMFT more efficiently
AUX solver. Here we take 10000 Monte Carlo samples, and reduces the CPU time for larger N , where the nega-
c
show theresults for t′/t=0 (red solid and dashed lines with tive signproblemis severefor the N -siteCDMFT while
c
diamonds)and−1(bluesolid anddashedlineswithsquares) that in the rr-DMFT stays at the same level as that of
at βt=4 with and without using thecluster symmetry (C4v N =2-site CDMFT simulations.
s
for 2×2 and 4×4 clusters and C2v for 2×1, 4×2 and At the end of this subsection, we remark on the num-
8×4clusters). Solid(dashed)linesrepresenttheresultswith
berofMonteCarlosamplingsinouractualcomputation.
(without) utilizing the cluster symmetry. In contrast to the
When the size of the quantum Monte Carlo solver is in-
rr-DMFT, the use of the cluster symmetry does not affect
creased,it is necessary to take more samples to keep the
much the total computational cost in the CDMFT. This is
because in the CDMFT it is always necessary to solve the magnitude ofstatisticalerrorsof physicalquantities at a
whole impurity problem while the symmetry can reduce the certainlevel. Namely, fora fixed numberof samples,the
numberofthesolvedimpurityproblemsintherr-DMFT.(c) N -siterr-DMFTgivesasmallerstatisticalerrorthanthe
c
′
Results of (b) for t/t = −1 normalized by the average sign N -site CDMFT since the former solves smaller cluster
c
shown in (d). problems. Forexample,inthecalculationofFig.3(b)for
t′ = 0, the relative standard errors δD/D of the double
occupancy D = hn n i/N are 0.0105 in the
and set N =2 in all the rr-DMFT simulations. i=1,...,Nc i↑ i↓ c
s 4×4-site CDMFPT and 0.00360 in the rr-DMFT (with-
In the ED, we use n =2 to represent the Weiss func-
b out the use of the cluster symmetry). In other words,
tionandsolvethe problemwiththe Lanczosmethodfol-
in order to suppress the statistical error of D to the
lowing Ref. 15. In Fig. 3(a), we measure the CPU time
same level of the rr-DMFT, the CDMFT requires about
of the CDMFT and rr-DMFT for t′/t=0 at T =0. For (0.0105/0.00360)2 ≃ 8.5 times more samples (i.e., CPU
the latter, we show two results with and without using
time). This fact shows that the rr-DMFT is practically
the cluster symmetry: We consider the C symmetry
4v more efficient than the data presented in Fig. 3(b).
for L×L clusters, and C symmetry for L×L′ clusters
2v
(L 6= L′). The results show that the numerical cost of
the rr-DMFTincreasesonly inapowerofNc andthatit III.2. Spin structure factor
is further reduced by the use of the cluster symmetry.
Next we examine the case of the CT-AUX solver for
In the two-dimensional Hubbard model, the short-
t′/t = 0 and −1 at βt = 4, where we take 10000 Monte
range antiferromagnetic correlations play an impor-
Carlo samples. For comparison, we define a ratio of the tant role in the Mott-insulating41,42,48 and pseudogap
CPU time of the rr-DMFT to that of the CDMFT and regions.28–39,49,50 The spin structure factor
show the results in Fig. 3(b). Even when the cluster
symmetry is not used, the rr-DMFT reduces the compu- 1 Nc
S(q)= hSzSzie−iq·(ri−rj), (10)
tsaytmiomneatlrcyoisnt,tehxecerpr-tDfMorFsTm,atlhleNcco’sm.puWtahteinonwalecuosset tbhee- Nc iX,j=1 i j
comesalwayssmallerthanthatbytheCDMFTincluding atq =(π,π)isausefulmeasureoftheantiferromagnetic
the case of small N . Note that in the CDMFT we need correlations. Here r is the position vector at the i-th
c i
6
(a) U/t=4
(a) U/t =4
βt=8
βt =2
t'/t=0
t´/t =0
n=0.8
S(q) n=1
S(q)
(b) U/t=8 (b) U/t =8
βt=2 βt =2
t'/t=0 t´/t =0
n=0.875 n=1
S(q)
S(q)
(0,0) (π,0) (π,π) (0,0) U/t =8
q (c)
βt =2
t´/t =-0.2
FIG. 4. (Color online) Comparisons of spin structure factor n=1
S(q)
S(q) obtained by the rr-DMFT and CDMFT for (a) U/t =
4,βt=8andn=0.8,and(b)U/t=8,βt=2andn=0.875.
Curves and symbols indicate the results of the CDMFT and
rr-DMFT, respectively. An Ns = 4-site impurity solver is
used in therr-DMFT.
(0,0) (π,0) (π,π) (0,0)
q
site.
Figure 4 compares S(q) obtained by the CDMFT for
2×2, 4×4, and 8×8 sites, and by the rr-DMFT for FIG. 5. (Color online) Comparisons of spin structure factor
N =4×4and8×8sites with the N =4-siteCT-AUX S(q) at half filling. The parameters are (a) U/t = 4,βt = 2
c s ′ ′
and t/t = 0, (b) U/t = 8,βt = 2 and t/t = 0, and (c)
solver. Here we examine the accuracy of the rr-DMFT
′
U/t=8,βt=2andt/t=−0.2. Curvesandsymbolsindicate
in (a) an intermediate-coupling (U/t = 4, βt = 8 and
the results of the CDMFT and rr-DMFT, respectively. We
n=0.8) and (b) a strong-coupling (U/t=8, βt=2 and
takeNs =4 and 16 in therr-DMFT simulations.
n = 0.875) cases, where n is the electron density. The
rr-DMFT results agree well with those of the CDMFT
(which is equivalent to the rr-DMFT with Nc = Ns) up frustrationinFig.6suppressestheantiferromagneticcor-
to 8×8 sites. In particular, the rr-DMFT reproduces relation, shifting the peak from q =(π,π) to (π,0). The
the growth of S(q = (π,π)) with the cluster size Nc. rr-DMFT again reproduces well the overall spin struc-
This result shows that our method takes the growth of tureofthe N -siteCDMFT.The S(q)peak atq =(π,0)
c
the antiferromagneticcorrelationsintoaccountatalevel grows as N increases in the CDMFT and it is also seen
c
similar to the Nc-site CDMFT. in the rr-DMFT. Notice that at βt = 4 [Fig. 6(b)] the
The results of S(q) at half filling (n = 1) are shown 8×8-site CDMFT is not available any more because of
in Fig. 5. Here, we calculate it in (a) an intermediate- the severenegativesignproblem[seeFig.3(d)]while the
coupling (U/t = 4, t′ = 0 and βt = 2), (b) a strong- 8×8-site rr-DMFT is feasible and yields a result which
coupling (U/t = 8, t′ = 0 and βt = 2) and (c) a frus- looks reasonable,i.e., smoothly connected to the higher-
trated (U/t = 8, βt = 2 and t′/t = −0.2) cases. For the temperature result [Fig. 6(a)]. Figure 6(a) shows that
rr-DMFT, we fix Nc at 8×8 and compare the Ns = 4 the rr-DMFT slightly overestimates S(q) for the 8×8
and 16 cases. While the rr-DMFT with Ns = 4 well re- cluster (basically within 10% error). It is plausible that
producesS(q)obtainedbytheCDMFT,therearevisible this originates from the increased number of coupling
differences between them. The discrepancy is improved terms (next-nearest-neighbor hoppings t′), which makes
in the Ns = 16 result since the result directly takes into the convergence of the cumulant expansion slow, and
account long-range vertex functions within the Ns-site leads to an underestimate of the spin frustration and re-
cluster. sultant fluctuations. Nevertheless, Fig. 6 demonstrates
We alsoshowa resultfor a frustratedcase(t′/t=−1) that the rr-DMFT reasonably works even when a severe
in Fig. 6. In comparison with Figs. 4 and 5, the large negative sign problem makes the calculation of the stan-
7
U/t =8, βt =2 (a) U/t=8, βt=8, t'=0, n=1
(a)
t´/t =-1, n=1
S(q)
)|
0
ω
i, (0,0) (2,2) (3,3)
r
Σ( (3,1)
| (2,0) (3,2)
U/t =8 (3,0)
(b) (1,1)
βt =4
(1,0)
(2,1)
t´/t =-1
n=1 |r|
S(q)
(b)local (c)next-nearest neighbor
8×8 CDMFT: Unavailable ω)n ω)n
i0), i1),
0, 1,
(0,0) (π,0) q (π,π) (0,0) Σr(=( Σr(=(
m m
I I
ω ω
n n
FIG. 6. (Color online) Comparisons of spin structure factor
S(q) for a frustrated case (U/t = 8 and t′/t = −1) at half
filling. Here we compute S(q) at (a) βt= 2 and (b) βt = 4.
FIG. 8. (Color online) (a) Absolute value of self-energies,
Curves and symbols indicate the results of the CDMFT and
rr-DMFT, respectively. An Ns = 4-site impurity solver is |eΣst(rM,iaωt0su)|b,aorbatafirneeqduewnictyhrprl-oDttMedFTagaanindstCrDeMal-FspTa,caetdthisetalonwce-
used in the rr-DMFT. Note that at βt = 4 the 8×8-site |r| for U/t = 8,βt = 8,t′ = 0, and n = 1. The pair of num-
CDMFT is not available because of its too large computa-
bers at each data represents r = (x,y) in unit of the lattice
tional cost.
constant. Thereareseveraldataatthesamersincethemeth-
ods break the translational symmetry. The imaginary parts
of (b)thelocal and (c) next-nearest-neighborself-energies at
dard CDMFT intractable.
the central four sites in the 4×4 cluster are plotted against
Next, we show in Fig. 7 low-temperature (βt = 8) re- theMatsubara frequency.
sultsforU/t=4athalffilling. Comparingtherr-DMFT
andCDMFTforthesameN ,wefindthattherr-DMFT
s
gives a larger value of S(q = (π,π)), which is in fact CDMFT for the same N , we see that the rr-DMFT un-
c
closer to the value of the largest-cluster (8×8) CDMFT derestimatesthevalueofS(q =(π,π))whiletheincrease
result. However, when we compare the rr-DMFT and of N certainly improves the value. This result indicates
s
that, as is expected, the rr-DMFT is not sufficient when
the system enters the critical regionwhere the finite size
effect is large or, in other words, the correlation length
U/t =4
far exceeds N . This should be attributed to the discard
s
βt =8 of the vertex corrections between decomposed small-size
t´/t =0 clusters in the rr-DMFT.
n=1
S(q)
III.3. Self-energy
In this subsection, we examine the accuracy of the rr-
DMFT for the self-energy in the real space. We use the
(0,0) (π,0) (π,π) (0,0)
q CT-AUXastheimpuritysolver. Theself-energyisanim-
portantquantityinthe strongly-correlatedsystemssince
it determines the single-particle properties, such as the
spectral weight.
FIG. 7. (Color online) Comparisons of spin structure fac-
tor S(q) for U/t = 4 and βt = 8 at half filling. Curves and While at high energy the self-energy in generaldecays
symbols indicate the results of the CDMFT and rr-DMFT, with1/ωn orevenfaster,itbehavesdifferentlyatlowen-
respectively. We take Ns = 4 and 16 in the rr-DMFT simu- ergy, depending on the properties (metallic, insulating,
lations. and pseudogap etc.) of the system. Therefore, as the
8
most severe benchmark, we first examine the accuracy
of the self-energy at the lowest Matsubara frequency ω .
0
Figure 8(a) plots the absolute value of Σ(r,iω ) against
0
the real-spacedistance|r|. We compare2×2-and4×4- metal βt=16
site CDMFT and 4×4-site rr-DMFT with N = 4, for
s
U/t=8,βt=8,t′ =0, and n=1. Here several different t'/t=0
)
2
values are obtained at the same r for each simulation, β/ n=1
since the CDMFT and rr-DMFT intrinsically break the (
G
translational symmetry of the lattice. For |r| < 1.5, we β
-
can see that the rr-DMFT gives the self-energy closer to
insulator
the4×4-siteCDMFTthantothe2×2-siteCDMFT.This
isalsoseenintheω dependenceshowninFigs. 8(b)and
n
8(c). Moreover, Fig. 8(a) shows that the rr-DMFT well
U/t
reproduces the long-range part beyond the 2 ×2 clus-
ter at least up to |r| = 3. The lowest-order approxima-
tion in the cumulant expansion, Eq. (6), works well for
FIG.9. (Coloronline)Comparisonof−βG(β/2)obtainedby
thislong-rangepartbecausethenonlocalone-bodyterms
CDMFT (solid lines) and rr-DMFT(dashed lines).
in the quantum impurity action decay with the distance
|r|. This also explains the deviation for |r| > 3, where
therr-DMFTalwaysunderestimatesthecorrelations,in-
both calculations. We find a crossover between a metal
terpolating the N -site CDMFT (giving zero self-energy
s and the Mott insulator for the N = 16-site CDMFT,
c
for this long-range part) and N -site CDMFT. Namely,
c since the hysteresis is not observed there. It is plausible
the truncation of the many-body hopping processes due
thatthefirst-orderMITlineendsatatemperaturelower
to the lowest-order approximation in Eq. (6) underesti-
than 1/β = t/16 for N =16. The 4×4 rr-DMFT with
c
mates the correlationbetween distantsites, hence giving
N =2 and 4 reproduces this feature.
s
an underestimated self-energy, while it certainly takes
Weconcludefromtheseresultsthattherr-DMFTrea-
into account the spatial correlation beyond the N -site
s sonablytakesintoaccountthespatialcorrelationbeyond
cluster.
the size of the impurity solver.
III.4. Mott metal-insulator transition
III.5. Density of states
The critical onsite interaction strength U of the MIT
c The ED solver15 directly gives real-frequency quanti-
isanothercriterionforunderstandinghowmuchthespa-
ties without use of any additional analytic continuation
tial correlations are taken into account in the method,
scheme,whichisrequiredintheCT-QMC.Asmentioned
as the previous studies with the cluster extensions of
in Sec. III.1, while the standard CDMFT with the ED
DMFT (see, e.g., Ref. [41]) have elucidated that, in the
solver requires a cost exponentially increasing with N ,
two-dimensionalHubbardmodel,theshort-rangeantifer- c
the rr-DMFT reduces the increase to a polynomial one.
romagnetic correlations considerably reduce U .
c Therefore,the combination of the rr-DMFT and the ED
We here investigate the N dependence of U within
c c is a promising way to study the real-frequency proper-
the rr-DMFT and the CDMFT at half filling. Using the
ties of large systems. In this subsection, we examine the
CT-AUX as the impurity solver, we calculate the local
accuracyoftherr-DMFTcombinedwiththeED,bycom-
Green’s function at τ =β/2, which captures the charac-
paring the result with those obtained directly with the
teristics of metals and insulators because
CDMFT.
β β ρ(ω) Figure 10 shows the density of states (DOS) ρ(ω) ob-
−βG = dω , (11)
(cid:18)2(cid:19) 2 Z cosh(βω) tained by the CDMFT and the rr-DMFT with Ns = 2-
site ED impurity solver for t′ =0 and half filling at zero
isapproximatelytheaverageofthedensityofstatesρ(ω) temperature. Wetaken =2asthenumberofbathsites
b
for |ω| . T. The results at βt = 16 are shown in Fig. 9. per impurity in all the simulations. Note that we have
For N =2 and 4, we find sudden changes of −βG(β/2) introduced a broadening factor δ = 0.05t as ω → ω+iδ
c
with varying U, which are identified with the MIT. In and shifted the chemical potential as µ → µ+U/2 to
the CDMFT results, U decreases from N = 2 to 4, fix the Fermi level at ω = 0. At U/t = 5.5, the 2×2-
c c
where the hysteresis proving the first-order transition42 site CDMFT has a gapped DOS and the rr-DMFT re-
isobserved. Therr-DMFTquantitativelyreproducesthe produces it qualitatively while the 2 × 1-site CDMFT
reduction of U with N and the hysteresis for N = 4. gives a metallic solution. This result is consistent with
c c c
The 2×2-siterr-DMFT with N =2 indicates U /t≃6, the fact that the spatial correlation reduces U as dis-
s c c
whichisclosetothe2×2-siteCDMFTresultU /t≃5.8. cussed in Sec. III.4. In Fig. 10(b), all the solutions are
c
Thisvalueissmallerthanthatofthe 2×1-siteCDMFT, gapped, where the rr-DMFT gives a gap size similar to
U /t∼7.2,despitetheuseofatwo-siteimpuritysolverin thatofthe2×2-siteCDMFTincomparisonwiththatof
c
9
hc† (τ )···c† (τ )c (τ′)c (τ′)i in the action Eq. (8):
(a) i1 1 in n j1 1 jn n
U/t=5.5 The leading correction to our method comes from the
t’/t=0 two-particlecorrelationfunctions. Thistermwillcontain
the spin-spin interaction which is related to the antifer-
ρ(ω)
romagnetic correlation.
Furthermore,whilewehavechosenthewayoftheclus-
ter decomposition according to a physical insight (see
Appendix B), it may be possible to apply an idea in the
numerical linked-cluster expansion,52,53 which fixes the
(b)
way of the cluster decomposition according to a princi-
U/t=7
ple of inclusion and exclusion,54 because both of them
t’/t=0
are based on a breakup of clusters into smaller blocks.
ρ(ω)
A potentially important application of the rr-DMFT
is to calculate the real-frequency properties, such as the
spectral function, for large clusters by using the ED
solver, which does not rely on any additional analytic
continuationscheme to obtain real-frequency properties.
ω/t Especially,our method will be useful to study the doped
Mott insulators and frustrated systems.
It is also important to study multi-orbital models for
FIG. 10. (Color online) Density of states for (a) U/t = 5.5 the understanding of the role of the orbital degrees of
and (b) U/t=7 at half filling and zero temperature. Curves freedom by extending the present rr-DMFT to multior-
indicate the results of the 2×2-site rr-DMFT (solid curves), bital systems.
2×2-site CDMFT (dashed curves) and 2×1-site CDMFT
(dot-dashed curves). Note that we use a broadening factor
δ=0.05t.
V. ACKNOWLEDGMENTS
the 2×1-site CDMFT. These results demonstrate that The rr-DMFT code partly uses the nonequilibrium
the rr-DMFT gives reasonable results even for the real- DMFT program developed by N. Tsuji in Ref. 55. D.
frequencyproperties,whichareingeneralmoresensitive K. thanks T. Misawa, T. Ohgoe and Y. Yamaji, and
to the numericalaccuracythanthe quantitiesdefined on S. S. thanks M. Ochi for helpful discussions. Some
the Matsubara-frequency axis. of the results in this article have been computed at
the facilities of the Supercomputer Center, Institute for
Solid State Physics, University of Tokyo. D. K. was
IV. CONCLUSION AND OUTLOOK financially supported by Japan Society for the Promo-
tion of Science through Program for Leading Graduate
Schools(ALPS).S.S.wassupportedbyJSPSKAKENHI
In summary, we have proposed the rr-DMFT as a
Grant No. 26800179. This work was supported by the
method to efficiently take into account the spatial corre-
ComputationalMaterials Science Initiative (CMSI), and
lationsatacomputationalcostsubstantiallysmallerthan
RIKEN Advanced Institute for Computational Science
the brute force application of the CDMFT. We have ap-
throughtheHPCISystemResearchproject(ProjectsNo.
pliedthemethodtothetwo-dimensionalHubbardmodel
hp140215 and No. hp150211).
onthesquarelattice. Wehavedemonstratedthattherr-
DMFT substantially reduces the CPU time in compar-
ison with ordinary CDMFT simulations. Furthermore,
whenweusetheCT-QMCastheimpuritysolver,therr- APPENDIX A: DETAILED ALGORITHM
DMFTsubstantiallyimprovesthenegative-signproblem,
leading to a further reduction of the CPU time. Here we describe the detailed algorithm, which solves
As benchmark tests, we have demonstrated that the an N -site quantum impurity model by using an N -site
c s
rr-DMFT well reproduces the spin structure factor, self- impurity solver O(N /N )2 times.
c s
energy, the MIT point Uc and the density of states, ob- First, we consider an Nc-site cluster and decompose
tained by the CDMFT with the same cluster-size solver. it into four N /4-site blocks C (i = 1,...,4) in accor-
c i
However, the rr-DMFT becomes insufficient when dance with Sec. II.2. We then categorize the elements
the antiferromagnetic correlation length is larger than of the Green’s function into six groups: A (k = 1,2)
k
N . One possible prescription for this is, according to and B (k = 1,...,4). In order to obtain each ele-
s k
Ref. [44, 51], to take the higher-order cumulants into ment, we calculate the action S which is givenby trac-
ij
account in the real-space renormalization process, be- ing out the degrees of freedom other than C and C
i j
cause it, in principle, partially includes the vertex cor- [for example, (i,j) = (1,2) for A ]. For A and A ,
1 1 2
rections between the broken up clusters. Of course, G and G are obtained from the actions S and
CiCj CjCi 12
they include n-particle correlation functions defined by S , respectively, where S is the action consisting of
34 ij
10
C1 C2 • {GC2C4,GC4C2},
B-type
C3 C4 • {GC1C4,GC4C1},
• {G ,G }.
C2C3 C3C2
These are again B-type problems. Hence,
C C C C C C C C M 3M
1 2 1 2 1 2 1 2
b(M,N )=4b( ,N )+ , (A.2)
s s
C C C C C C C C 2 2Ns
3 4 3 4 3 4 3 4
B B B B where3M/2N comesfromareasonsimilartoabove. In
1 2 3 4 s
this case,we trace outsites fromC =C +C +C +C
1 2 3 4
as follows: (1) We first trace out C and then C or C ,
1 2 4
to obtain S or S , respectively. (2) We first trace out
34 23
FIG. 11. (Color online) Decomposition of a cluster and re-
C andthenC orC , to obtainS orS , respectively.
sultant blocks which are used to calculate B (k = 1,...,4) 2 2 4 14 13
k If M = 2nN (n > 1) is satisfied, the solution for a
elements of Green’s function. s
and b is obtained in the following way: Let us redefine
A = a(M,N ) and B = b(M,N ), where we used the
n s n s
Ci and Cj (A-type problem). We here abbreviate a fact that a and b depend only on the ratio M/Ns. From
set {Glm|l ∈ Ci ∧ m ∈ Cj} to GCiCj. For Bk with Eq. (A.2), the recursion relation for Bn is obtained as
k =1,...,4,we calculatethe elements oftwosets G
and G from the corresponding actions S , S ,CSiCj Bn+3·2n−1 =4(Bn−1+3·2n−2). (A.3)
CjCi 13 24 14
and S23, respectively (B-type problem). We thus cal- Solving this recursion relation, since B2 = b(2Ns,Ns) =
culate all the elements of Gij for the corresponding ac- 8 is easily shown, we obtain
tionS intheA-typeproblem,whileweomitG and
ij CiCi 11 3
GCjCj in the B-type problemto avoidthe overlapin the Bn = 22n− 2n, (A.4)
computation. 4 2
In the following, we evaluate the number using the resulting in
N -siteimpuritysolverfortheN -siteimpurityproblem.
s c
2
We define a(M,Ns) and b(M,Ns) as the number apply- b(N ,N )= 11 Nc − 3Nc. (A.5)
ing the Ns-site impurity solver needed to solve the M- c s 4 (cid:18)Ns(cid:19) 2Ns
site A- and B-type problems, respectively. The A-type
problem is equivalent to an ordinary quantum impurity Similarly, using Eq. (A.1), the recursion relation for An
problemsince wecalculateallthe elements ofG within is obtained as
ij
the cluster. Because we are able to decompose a quan- 11 3n
A − 4n+ 2n
tum impurity model into two A-type and four B-type n
2 4
problems, we obtain
11 3(n−1)
=2[A − 4n−1+ 2n−1], (A.6)
n−1
M M 9M 2 4
a(M,N )=2a( ,N )+4b( ,N )+ , (A.1)
s s s
2 2 4N The solution of the recursion relation leads to
s
wherethelasttermontheright-handsiderepresentsthe Nc 22Nc Nc
a(N ,N )= −3log −17 (,A.7)
number of times of using the Ns-site solver needed to c s 4Ns (cid:18) Ns 2(cid:18)Ns(cid:19) (cid:19)
reduce the M-site problem to the six M/2 site problems
where we have used the fact that a(2N ,N )=12. As a
(namely, A through B ) as is proven below: We trace s s
1 4
result,wefindthattheN -siteimpurityproblemissolved
out the sites in C =C +C +C +C in the following c
1 2 3 4 by using the N -site impurity solver O(N /N )2 times.
three independent steps: (1) We first trace out C and s c s
1 While Eq. (A.7) is valid only for N = 2nN (n > 1),
then C or C , to obtain S or S , respectively. (2) c s
2 4 34 23 therr-DMFTalgorithmitselfisapplicabletoN 6=2nN
We first trace out C and then C or C , to obtain S c s
2 3 4 13
cases for which we expect a similar order of a(N ,N ).
or S , respectively. (3) We first trace out C and then c s
14 3
C or C , to obtain S or S , respectively. Thus, we
4 1 12 24
eliminate the M/4-site block nine times with the N -
s
APPENDIX B: TRACING OUT ORDER
site solver so that we have the last term in Eq. (A.1).
Note that, empirically, the order of tracing out does not
significantly affect the results (see Appendix B). Here, we elaborate our algorithm to determine the or-
Next, we consider the B-type problem for an M-site deroftracingoutthesites. Bycomparingitwithanother
cluster. We again decompose it into four problems (see algorithm, we show that the order does not significantly
Fig. 11). The elements G are classified into the follow- affect the final result. A basic principle of our algorithm
ij
ing four parts; is to choose a block of the sites to be traced out as com-
pactaspossiblesincetheshort-rangecorrelationisbetter
• {G ,G }, to be taken into account as much as possible.
C1C3 C3C1