Table Of ContentInterlaced coarse-graining for the dynamic cluster approximation
P. Staar
IBM Research – Zurich, CH-8803 Ru¨schlikon, Switzerland
M. Jiang, U.R. H¨ahner, and T.C.S. Schulthess
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
T.A. Maier
Computer Science and Mathematics Division and Center for Nanophase Materials Sciences,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
(Dated: January 18, 2016)
6 The dynamical cluster approximation (DCA) and its DCA+ extension use coarse-graining of the
1
momentumspacetoreducethecomplexityofquantummany-bodyproblems,therebymappingthe
0
bulklatticetoaclusterembeddedinadynamicalmean-fieldhost. Here,weintroduceanewformof
2
an interlaced coarse-graining and compare it with the traditional coarse-graining. While it gives a
n morelocalizedself-energyforagivenclustersize,weshowthatitleadstomorecontrolledresultswith
a weaker cluster shape and smoother cluster size dependence, which converge to the results obtained
J from the standard coarse-graining with increasing cluster size. Most importantly, the new coarse-
5 grainingreducestheseverityofthefermionicsignproblemoftheunderlyingquantumMonteCarlo
1 clustersolverandthusallowsforcalculationsonlargerclusters. Thisenablesthetreatmentoflonger-
rangedcorrelationsthanthoseaccessiblewiththestandardcoarse-grainingandthuscanallowforthe
] evaluationoftheexactinfiniteclustersizeresultviafinitesizescaling. Asademonstration,westudy
l
e the hole-doped two-dimensional Hubbard model and show that the interlaced coarse-graining in
- combinationwiththeextendedDCA+ algorithmpermitsthedeterminationofthesuperconducting
r
t Tc on cluster sizes for which the results can be fit with a Kosterlitz-Thouless scaling law.
s
.
t PACSnumbers:
a
m
- I. INTRODUCTION patch, and 0 otherwise. One then coarse-graines the
d
Green’s function
n
co strMonugclhy coorfreltahteedneulemcterroincalmawteorrikals iins bathseed oanreeaxacotf G¯(K,iωn)= NNc (cid:88)φK(k)iω +µ−ε −1ΣDCA(k,iω )
[ calculations that determine the state of a finite size k n k n
lattice and regard this state as an approximation (2)
1
tosetupaneffectiveclusterproblem,inwhichthecluster
v of the thermodynamic limit. The dynamic cluster
8 approximation (DCA)1,2 uses a similar philosophy, in self-energy Σc(K,iωn) = Σc[G0(K,iωn)] is calculated as
3 which the bulk lattice problem is represented by a afunctionalofthecorrespondingbareclusterpropagator
8 finite number of cluster degrees of freedom. But in G0(K,iωn) = [G¯−1(K,iωn) + Σc(K,iωn)]−1. The
3 approximation in Eq. (1) of the lattice self-energy as a
contrast to finite size calculations, the DCA uses coarse-
0 piecewiseconstantcontinuationoftheclusterself-energy
graining to retain information about the bulk degrees of
.
1 freedom not represented on the cluster. This leads to an leads to discontinuities between the patches and in some
0 cases to strong finite size effects, manifested as a strong
approximation of the thermodynamic limit, in which the
6 dependence on the cluster shape and size5.
bulkproblemisreplacedbyafinitesizeclusterembedded
1
In order to weaken these effects, the DCA method
inamean-fieldhostthatisdesignedtorepresenttherest
:
v of the system. This approximation makes the problem was recently extended through the inclusion of a lattice
i self-energy with continuous momentum dependence5.
X tractable so it can be solved with exact methods such as
quantum Monte Carlo3. This extended DCA+algorithm is obtained by reversing
r
Eq. (1) to give a relation
a Tosetuptheclusterproblem,onestartsbydividingthe
first Brillouin zone (BZ) into N patches, each of which
is represented by a cluster mocmentum K (see Fig. 1, Σ (K,iω )= Nc (cid:88)φ (k)ΣDCA+(k,iω ) (3)
c n K n
N
top left, for an example of a 16-site cluster)2. One then k
assumes that the self-energy is well approximated by a
coarse-grained self-energy2,4 between the cluster self-energy Σc(K,iωn) and the
DCA+ lattice self-energy ΣDCA+(k,iω ). As discussed
(cid:88) n
ΣDCA(k,iωn)= φK(k)Σc(K,iωn). (1) in Ref.5, ΣDCA+(k,iω ) with continuous momentum
n
K dependence is then determined from a deconvolution
Here, Σ (K,iω ) is the self-energy of the N -site cluster of Eq. (3) after the cluster self-energy Σ (K,iω ) is
c n c c n
and the patch function φ (k) = 1 for k inside the Kth interpolated between the cluster K momenta. It was
K
2
shown that the DCA+ algorithm reduces the cluster
shape and size dependence of the DCA self-energy, and,
in addition, weakens the fermion sign problem6 of the
underlying QMC cluster solver.
One usually defines the patches as the Brillouin zones
ofthesuperlattice(seeFig.1left)andsetsφ (k)=1or
K
0forkinsideoroutsidetheKthpatch,respectively2. But
thechoiceofcoarse-grainingpatchfunctionsφ (k)inthe
K
DCA and DCA+ is not unique. In Ref.7, for example,
Gull et al. used a star-like patch geometry for a 4-site
clustertodeformthecentralpatchinordertocapturean
important part of the Fermi surface. As we will discuss,
there is a set of constraints that must be satisfied by the
patching. But these constraints leave ample freedom in
choosing different shapes of the coarse-graining patches
and different forms of the functions φ (k).
K
Here, we introduce a new interlaced coarse-graining,
study its effects on the self-energy of a single-band
Hubbard model and compare the results with the
standard coarse-graining. For small cluster sizes, we
find that the interlaced coarse-graining leads to a more
localized self-energy with less dependence on the shape
of the cluster. In the infinite cluster size limit, it
gives results that converge with those obtained from
the standard coarse-graining. As an important benefit,
it significantly reduces the QMC fermion sign problem, FIG. 1: The location of the cluster momenta K and the
enabling calculations with larger cluster sizes. As shape of the patches for a 16A cluster (top) and 16B cluster
an example, we show results for the superconducting (bottom). The standard coarse-graining uses the Brillouin
transition temperature, for which the interlaced coarse- zone of the superlattice as the patches (left), while the new
coarse-graining uses patches, in which regions assigned to
grainingprovidesaccesstolargeenoughclusters, sothat
neighboring K points are interleaved.
T can be converged.
c
coarse-graining φ(0) and new patching φ(2) for the case
II. INTERLACED COARSE-GRAINING of another 16-site cluster with different shape, the 16B
cluster.
As noted, the patching must satisfy a number of AsdiscussedinRef.[5],Eq.(2)maybeinterpretedasa
constraints8. First,allpatchesmusthaveequalsize. This convolutionofthelatticeGreen’sfunctionG(k,iωn)with
ensures that the algebra of the operators of the effective the patch function φK(k), which may also be written as
cluster model obey the usual fermionic algebra. Second, φ(k−K). Thus, the patch function essentially acts as a
the patch functions should satisfy an orthonormality filter. Since it is used to map the lattice problem onto
condition, i.e. NNc (cid:80)kφK(k)φK(cid:48)(k) = δKK(cid:48), so that the cluster problem, it should pass the contribution to
different patches do not overlap or, in other words, at G(k,iωn) that is localized on the cluster, and cut off
any momentum k, there is exactly one K for which contributions outside the cluster. Therefore, it is useful
φ (k) is nonzero. Finally, the patches should have the to investigate the coarse-graining in real space, where,
K
samesymmetryasthecluster,sothatthecoarse-grained after Fourier-transforming Eq. (2), one has
Green’sfunctionandself-energyhavethesamesymmetry (cid:88)
G¯(X,iω )= φ(X+x)G(X+x,iω ). (4)
as the cluster. n n
For the regular 4×4 cluster labeled as 16A, the x
standard choice of the coarse-graining patches, which Here, a vector r=X+x to a site in the real space bulk
we label φ(0)(k), is shown in Fig. 1 in the top left latticeisbrokenupintoavectorXwithintherealspace
panel. Inthetoprightpanel,weintroduceanewstriped cluster and a vector x to the location of a cluster in the
coarse-graining scheme defined by the patch functions bulk lattice, and φ(X+x) is the Fourier-transform to
φ(2)(k), in which patches from neighboring K-points are realspaceofthepatchfunctionφ(k). AsnotedbyHettler
K
interleaved. The generation of these patches is detailed et al.8, for a square cluster of size L×L, one has for the
in the Appendix and the label (2) indicates the number standard coarse-graining
of stripes (see Appendix). Obviously, these patches
2 (cid:20) (cid:21)
satisfy the constraints of equal volume, orthonormality φ(0)(r=X+x)=(cid:89) sin[π(xl+Xl)/L] , (5)
and symmetry. The bottom panels show the standard π(xl+Xl)/L
l=1
3
L = 6 neighboring clusters can contribute with either positive
1.2 or negative weights, depending on the range r. This
1.0 sin(πx/L) can even lead to an overestimation of the short-range
πx/L
0.8 φ(0) correlations within the cluster. The φ(2) coarse-graining,
on the other hand, is always likely to underestimate the
) 0.6 φ(2)
0 non-local correlations and thus is more controlled. As
x, 0.4 L/2 the cluster size increases, both approaches will give the
(
φ 0.2 same G¯(r) once L/2 is sufficiently large relative to the
0.0 length-scale over which G(r) vanishes.
0.2
−
0.4
− 0 2 4 6 8 10 12 III. APPLICATION TO THE 2D HUBBARD
x MODEL
L = 8
1.2 Next we study the effects of these differences in the
1.0 sin(πx/L) coarse-grainingonthemomentumdependenceoftheself-
πx/L
energy. The Hubbard model that we study has a nearest
0.8 φ(0)
neighbor hopping t, a next-nearest neighbor hopping t(cid:48)
0) 0.6 φ(2) and a Coulomb repulsion U, and its Hamiltonian is
x, 0.4 L/2
φ( 0.2 H =(cid:88)t c† c +U(cid:88)n n . (6)
ij iσ jσ i↑ i↓
0.0 ij,σ i
0.2
−0.4 Here, c(iσ†) (creates) destroys an electron with spin σ on
− 0 2 4 6 8 10 12 site i and n = c† c is the corresponding number
iσ iσ iσ
x operator. To solve the effective cluster problem in the
DCA and DCA+, we use the continuous-time auxiliary-
fieldQMCalgorithmdevelopedbyGulletal.9 withhigh-
FIG. 2: (Color online) Fourier-transform of the patch efficiency updates10.
functions φ(0)(k) and φ(2)(k) to real space for L×L clusters ThetoppanelofFig.3showsresultsfortheimaginary
withL=6(top)andL=8(bottom)plottedversusr=(r,0).
part of the self-energy, ImΣ(K,iω ), for K = (π,0)
φ(2)(r)fallsoffmorerapidlywithr thanφ(0)(r)andremains n
and (π/2,π/2) obtained for the 16A cluster. Here, we
close to 0 for r≥L. have set t(cid:48) = −0.15t, U = 7t and the site filling (cid:104)n(cid:105) =
0.942 and temperature T = 0.125t. For the standard
coarse-graining φ(0), one observes a large difference in
where x (X ) is the lth component of the vector x (X). the low frequency behavior of ImΣ(K,iω ) between
l l n
Fig. 2 shows the r dependence of φ(0)(r) along the x- K = (π,0) and (π/2,π/2), which has been observed in
direction r=(r,0) for an L×L cluster with L=6 (top) earlier DCA calculations (see e.g. the work in Ref.7).
and 8 (bottom). The sinusoidal dependence of φ(0)(r) The interlaced φ(2) coarse-graining, in contrast, gives a
with the 1/r envelope from Eq. (5) can be seen. In self-energy with much less momentum dependence. As
the same figures, we also plot the r-dependence of the expectedfromtheplotsinFig.2andtheirdiscussion,the
new interlaced patch function φ(2)(r). One sees that φ(2) patching gives a more local coarse-grained Green’s
φ(2)(r) falls off more rapidly with r than the standard function G¯(r) and thus a more local self-energy Σ [G¯]
c
patchfunctionφ(0)(r). Thismaybeunderstoodfromthe with less momentum dependence.
fact that the φ(2) coarse-graining averages over a more The bottom panel of Fig. 3 shows results for
extended momentum region and thus leads to a more ImΣ(K,iω ) with K = (π,0) for both the 16A and
n
local result. In addition, for distances r > L, φ(2)(r) the 16B clusters. Even though these clusters have
stays close to 0, while φ(0)(r) gives a significant negative the same size, the standard φ(0) coarse-graining gives
contribution to the coarse-grained average. When the results that vary significantly between the two clusters,
lattice Green’s function G(r) is short-ranged and drops withqualitativelydifferentbehaviorinthelowfrequency
to zero for r ≥ L/2, only the x = 0 term contributes region. In contrast, the φ(2)coarse-graining gives almost
to G¯(X) in the coarse-graining sum in Eq. (4), and identical results for these two cluster shapes. Again,
hence G¯(X) = φ(X)G(X). In this case, the standard this can be understood from the fact that the interlaced
φ(0) coarse-graining gives a better approximation, since φ(2) coarse-graining gives a more local self-energy with
φ(0)(r) is closer to 1 for r ≤ L/2 and thus gives a weaker k dependence, which thus is less affected by the
G¯(r) that is closer to the “real” G(r). When G(r) location and shape of the coarse-graining patches.
is longer-ranged, however, the φ(0) coarse-graining is Asnoted,oneexpectsthatthisdifferenceintheresults
less controlled, since longer-ranged contributions from from different forms of the coarse-graining will decrease
4
with increasing cluster size. To show this, we plot in
1.0
− Fig. 4 ImΣ(K,πT) with K = (π,0) for the half-filled
1.2 (cid:104)n(cid:105) = 1 model with t(cid:48) = 0 and U = 8t for the φ(0)
−
) and φ(2) patching. Also shown in this figure are large
ωn −1.4 cluster results obtained on a finite size lattice with the
i
K, 1.6 determinantal QMC (DQMC) algorithm11. One sees
Σ( − K=(π,0),φ(0) that for small cluster sizes, the standard φ(0) coarse-
m −1.8 K=(π,0),φ(2) graininggivesmuchbetterresultsthatconvergefasterto
I 2.0 K=(π/2,π/2),φ(0) the exact large Nc limit, while the φ(2) coarse-graining
− underestimates the correlations. Again, this is expected
K=(π/2,π/2),φ(2)
2.2 fromthedifferencesinrealspacerbehavioroftheφ(0)(r)
−
0 2 4 6 8 10 andφ(2)(r)showninFig.2. Withincreasingclustersize,
iω however, both curves converge to the same large cluster
n
DQMC result.
1.0
− Next we turn to the effects of the coarse-graining on
the fermion sign problem of the underlying QMC solver.
1.2
− For the doped (cid:104)n(cid:105) =(cid:54) 1 Hubbard model in Eq. (6), the
)
ωn −1.4 sign problem is found to become exponentially worse
i
K, 1.6 with increasing lattice size, decreasing temperature and
Σ( − K=(π,0),16A,φ(0) increasing U12. QMC simulations of the doped model
m −1.8 K=(π,0),16A,φ(2) are therefore limited to small lattices, high temperatures
I or weak coupling U. QMC simulations within the
−2.0 K=(π,0),16B,φ(0) framework of the DCA1,2,8 have been found to have a
K=(π,0),16B,φ(2)
2.2 much less severe sign problem than QMC simulations
− 0 2 4 6 8 10 of finite size lattices3. Lacking a rigorous mathematical
iω argument, the DCA improvement of the sign problem
n
was attributed to the action of the mean-field host on
thecluster3. ThishasallowedQMCcalculationsatlower
FIG. 3: (Color online) Imaginary part of the self-energy for
temperaturesandlargerU thanthoseaccessiblebyfinite
different patching φ(0) and φ(2) for (a) the 16A cluster for
sizeQMCsimulations2. Furtherprogresswasmadewith
K=(π,0) and (π/2,π/2) and (b) the 16A and 16B clusters
for K = (π,0). The parameters are t(cid:48) = −0.15t, U = 7t, theintroductionoftheDCA+method5,whichwasfound
(cid:104)n(cid:105)=0.942 and T =0.125. to exhibit an additional reduction of the sign problem.
This was ascribed to the removal of artificial long-range
correlations, which arise in the DCA because of the
discontinuities in the self-energy, through a continuous
lattice self-energy in the DCA+5.
0
0
DCA,φ(0)
5 DCA,φ(2) −5 1.00
− DQMC 10
−
15
Σ − 0.75
m 10 20
I − − 0.00 0.05 0.10 0.15
mc DCA+,φ(2)
−15 σq 0.50 DCA,φ(2)
0.25 DCA+,φ(0)
20
− 0.00 0.01 0.02 0.03 0.04 DCA,φ(0)
1/N
c
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 4: (Color online) Imaginary part of the self-energy T
for K = (π,0) and ω = πT at half-filling (cid:104)n(cid:105) = 1,U =
0
8t,T = 0.15t versus the inverse cluster size 1/N obtained
c
with DCA and φ(0) (blue) and φ(2)(red) coarse-graining and FIG.5: TheaverageQMCsignversustemperatureforU/t=
with determinantal QMC calculations of a finite size lattice 8and(cid:104)n(cid:105)=0.9forthe16Acluster. TheuseoftheDCA+and
(black). With increasing cluster size, the results obtained the φ(2) patching lead to a significantly larger average sign.
fromthestandardφ(0) andtheinterlacedφ(2) coarse-graining
converge to the same large cluster DQMC result.
Here we study the effect of the coarse-graining on the
sign problem. Fig. 5 shows the temperature dependence
5
of the average QMC sign for the 16A cluster with
t(cid:48) = 0, U = 8t and (cid:104)n(cid:105) = 0.9 for both DCA and
0.03
DCA+ calculations with φ(0) and φ(2) coarse-graining.
At low temperatures, one sees that the QMC sign falls
rapidly to zero, and as noted, the DCA+algorithm gives
0.02
an improved sign relative to the DCA algorithm. As one c
T
sees, a significant further improvement is achieved with
theinterlacedφ(2) coarse-graining. Whencombinedwith
0.01
theDCA+algorithm,ithasasignificantlylargeraverage DCA+,φ(0)
sign at low temperatures than the DCA algorithm with
DCA+,φ(2)
standard φ(0) patching. For example, at T = 0.2t, the
0.00
sign in the DCA/φ(0) calculation has fallen to 0.5, while 0 10 20 30 40 50
the sign in the DCA+ /φ(2) calculation remains almost N
c
1. As a consequence, one sees that the DCA+ /φ(2)
combination enables calculations with a sizeable sign at
much lower temperatures than those that are accessible FIG. 6: (Color online) d-wave superconducting transition
with just the DCA or the standard coarse-graining. temperature T versus cluster size from DCA+ calculations
c
Finally, we illustrate the benefits of the improved sign withtheinterlacedφ(2) andstandardφ(0) coarse-grainingfor
problem by calculating the superconducting transition U = 4t, t(cid:48) = 0 and (cid:104)n(cid:105) = 0.9. The improved sign problem
temperature T as a function of cluster size N . To of the φ(2) coarse-graining algorithm enables calculations on
c c
much larger clusters, for which the results converge to the
calculate T , we determine the eigenvalues λ and
c α asymptotic large cluster limit.
eigenvectors φ (k) of the Bethe-Salpeter equation13
α
T (cid:88)
− Γpp(k,k(cid:48))G(k(cid:48))G(−k(cid:48))φα(k(cid:48))=λαφα(k), (7) have longer-range than those taken into account by the
N
k φ(0) coarse-graining in the smaller clusters. This is
particularly useful for the study of phase transitions
where k = (k,iω ) and Γpp(k,k(cid:48)) is the irreducible
n where the critical behavior is determined by the long-
particle-particlevertexonthebulklattice. IntheDCA+, rangecorrelations. Oneseesthattheφ(2) coarse-graining
just as the self-energy in Eq. (3), the lattice vertex
gives results with smooth cluster size dependence in the
Γpp(k,k(cid:48)) is determined from inverting the equation N ≥ 32 region, which is not accessible by the standard
c
φ(0) coarse-graining. As previously discussed in Ref.14,
N2 (cid:88)
Γpcp(K,K(cid:48))= Nc2 φK(k)Γpp(k,k(cid:48))φK(cid:48)(k(cid:48)) (8) iKtoissteprloitszs-ibTlheoutolesfistsctahleingrecsuurltvse ainndthdiesterremgiinone awTithfoar
K,K(cid:48) c
the exact infinite cluster size limit.
as described in Ref.14. At T , the leading eigenvalue
c
crosses1andonefindsthatthecorrespondingeigenvector
has d symmetry14. IV. SUMMARY AND CONCLUSIONS
x2−y2
In Fig. 6, we plot T (N ) for U = 4t, t(cid:48) = 0 and
c c
(cid:104)n(cid:105) = 0.9 from DCA+ calculations with the standard To conclude, we have introduced and studied a new
φ(0) and the new interlaced φ(2) coarse-graining. In form of an interlaced coarse-graining for the DCA
the region with N ≤ 32 the trends are clearly and DCA+ algorithms to map the bulk lattice to an
c
different: The φ(0) coarse-graining gives a T that effective cluster problem and compared it with the
c
decreases with N and thus apparently overestimates standardcoarse-graining. Thisinterlacedcoarse-graining
c
the pairing correlations in small clusters, while the averages over a more extended region in momentum
φ(2) coarse-graining gives an increasing T (N ). This space and thus gives a more localized self-energy with
c c
increase can again be traced to the stronger locality of weaker k-dependence. Non-local correlations are thus
the φ(2) coarse-graining. The d-wave pairing strength potentially underestimated in small clusters and, in the
arises from a pairing interaction Γpp(k,k(cid:48)) that increases absence of the QMC sign problem, the standard coarse-
with momentum transfer k − k(cid:48)13. In small clusters, graining converges faster to the exact infinite cluster
the φ(2) coarse-graining underestimates this momentum size result. However, the interlaced coarse-graining
dependence and thus T . With increasing cluster size, generally gives more controlled results with weaker
c
this underestimation is reduced, and T increases with cluster shape and smoother cluster size dependence that
c
N . For N = 32, both approaches give similar T . converge with the results from the standard coarse-
c c c
While the sign problem of the standard coarse-graining graining with increasing cluster size. Most importantly,
prevents calculations for N > 32, the interlaced coarse- we find that the interlaced coarse-graining significantly
c
grainingallowssimulationsofsignificantlylargerclusters. reduces the sign problem of the underlying QMC solver,
As one sees from Fig. 2, the φ(2) coarse-graining takes therebyenablingcalculationswithlargerclustersizes,for
into account correlations in these larger clusters that which longer-ranged correlations are taken into account
6
and the underestimation of shorter-ranged correlations recursion 0 1 2 3 k
is not an issue. The new coarse-graining is thus no. triangles 6 24 96 384 6×4k
particularly well suited for large cluster studies of phase
no. stripes 1 2 4 8 2k
transitions where the critical behavior is determined by
max. no. periods 0 1 2 4 2k−1
the long-range correlations. As an example, we have
shownthattheinterlacedcoarse-grainingincombination
TABLEI:Detailsofthek-meshrefinementforthe16Bcluster
with the DCA+ algorithm enables calculations of the
uptothreerecursionsteps. Theinitialnumberoftrianglesfor
superconducting T on cluster sizes, for which the
c thehexagonalshapedpatchissix. Eachrecursionstepdivides
results converge to the asymptotic Kosterlitz-Thouless
thetrianglesintofoursimilar,smallertriangles. Thenumber
scaling curve. Thus, while care should be taken in of stripes is doubled each time. The maximum number of
interpreting results on small clusters, the new coarse- periods is the number of stripes divided by two.
graining introduced in this paper gives access to much
largerclustersizesandthuscanenableafinitesizescaling
analysis to recover the exact infinite cluster size result. p are then obtained by reflecting those triangles across
this facet for which
(cid:18) (cid:19)
Acknowledgments 2πp
sin k ·n <0. (A.1)
n·n cm
We would like to thank Richard Scalettar for useful
comments. Part of this research was conducted at For the 16B cluster this is illustrated in Fig. 7 for one
the Center for Nanophase Materials Sciences, which period (blue-yellow) and three periods (blue-green).
is a DOE Office of Science User Facility. An award First note that by construction stripes always get
of computer time was provided by the Innovative and reflected as a whole. As its name indicates, p is just
NovelComputationalImpactonTheoryandExperiment the number of periods of the sine along the normal
(INCITE) program. This research used resources of the vector n and as such twice the numbers of reflected
Oak Ridge Leadership Computing Facility, which is a stripes. Therefore, the total number of stripes in the
DOE Office of Science User Facility supported under initial triangle must be a multiple of the number of
Contract DE-AC05-00OR22725. periods, p, and at least 2p. Consequently, the number
of periods determines the minimum number of recursion
steps required.
Appendix: Setup of interlaced coarse-graining
patches
Inthissectionwedescribethealgorithmthatgenerates
the interlaced coarse-graining patches for the two
dimensional case. We start from the traditional coarse-
graining patches defined as the Brillouin zones of the
superlattice. We call the edges and corners of these
patches facets and simplexes, respectively. First, each
initial patch is divided into triangles by connecting the
corners of the patch with its center, i.e. the cluster
momentum K. For example, the square shaped initial
patches of the 16A cluster are split into four triangles
while the hexagonal shaped initial patches of the 16B
cluster are divided into six triangles. Each of these
triangles has a unique facet as one edge. The vector
perpendiculartothefacetthatconnectsthecentertothe
facetisdefinedasthenormalvectornofthetriangle. We
simplex
then recursively subdivide each triangle into four similar facet
triangles by connecting the midpoints of each side. By
construction, this k-mesh refinement at the same time
divides the initial triangle into stripes of equal width FIG. 7: (Color online) Construction of the patches for
parallel to its facet (see Fig. 7). The details of the first the 16B cluster. Successive recursion steps of the k-mesh
three recursion steps are listed in Tab. I for the 16B refinementareshown: 0(plainblue),1(blue-yellow),2(blue-
cluster. purple) and 3 (blue-green). Stripes are reflected with the
For each of the smallest triangles we compute the maximum number of periods possible: 0, 1, 2 and 4. The
centerofmassk andprojectitontothenormalvector sine,thatdetermineswhetherastripeisreflected,issketched
cm
n of its initial triangle. The interlaced patches of period for p=1 and p=4 in recursion step 1 and 3, respectively.
7
One advantage of this new approach of generating integrate over. The traditional coarse-graining is a
the coarse-graining patches is its recursive nature. The special case and corresponds to zero recursion steps.
larger the recursion depth, the more the patches become Last but not least, this new coarse-graining approach
interlaced. But at the same time the only geometric can easily be generalized to three dimensions, in which
structures occurring are triangles, which are easy to triangles become tetrahedrons.
1 M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, H. Krishnamurthy, Physical Review B 61, 12739 (2000).
T. Pruschke, and H. R. Krishnamurthy, Physical Review 9 E. Gull, P. Werner, O. Parcollet, and M. Troyer,
B 58, R7475 (1998). Europhysics Letters 82, 57003 (2008).
2 T.Maier,M.Jarrell,T.Pruschke, andM.Hettler,Reviews 10 E. Gull, P. Staar, S. Fuchs, P. Nukala, M. Summers,
of Modern Physics 77, 1027 (2005). T.Pruschke,T.Schulthess, andT.Maier,PhysicalReview
3 M. Jarrell, T. Maier, C. Huscroft, and S. Moukouri, B 83, 075122 (2011).
Physical Review B 64, 195130 (2001). 11 R. Blankenbecler, D. Scalapino, and R. Sugar, Physical
4 S. Okamoto, A. J. Millis, H. Monien, and A. Fuhrmann, Review D 24, 2278 (1981).
Physical Review B 68, 195121 (2003). 12 E.Y.Loh,J.E.Gubernatis,R.T.Scalettar,S.R.White,
5 P.Staar,T.Maier, andT.C.Schulthess,PhysicalReview D. J. Scalapino, and R. L. Sugar, Physical Review B 41,
B 88, 115101 (2013). 9301 (1990).
6 M. Troyer and U. Wiese, Physical Review Letters 94, 13 T. A. Maier, M. S. Jarrell, and D. J. Scalapino, Physical
170201 (2005). Review Letters 96, 047005 (2006).
7 E. Gull, M. Ferrero, O. Parcollet, A. Georges, and 14 P.Staar,T.Maier, andT.C.Schulthess,PhysicalReview
A. Millis, Physical Review B 82, 155101 (2010). B 89, 195133 (2014).
8 M. Hettler, M. Mukherjee, M. Jarrell, and
