Spectral weight function for the half-filled Hubbard model: a singular value decomposition approach

10 Pages·0.16 MB·English

by  C. E. Creffield

Spectral weight function for the half-filled Hubbard model: a singular value
decomposition approach
C.E. Creffield, E.G. Klepfish, E.R. Pike and Sarben Sarkar
Department of Physics, King’s College London, Strand, London WC2R 2LS, UK
The singular value decomposition technique is used to reconstruct the electronic spectral weight
function for a half-filled Hubbardmodel with on-site repulsion U =4t from QuantumMonte Carlo
data. A two-band structure for the single-particle excitation spectrum is found to persist as the
latticesizeexceedsthespin-spincorrelationlength. Theobservedbandsareflatinthevicinityofthe
(0,π),(π,0)pointsintheBrillouinzone,inaccordancewithexperimentaldataforhigh-temperature
superconducting compounds.
5 PACS numbers: 74.20.-z, 74.20.Mn, 74.25.Dw
9
9
1
The study of the normal properties of the high- SWF,ledtheauthorsofRef.[8]toconcludethatthepseu-
n
a temperature superconducting compounds and the at- dogap in the single-particle spectrum observed in small
J temptstoconstructamicroscopictheoryofsuperconduc- clusters (up to 82) is a lattice-size artifact which disap-

5 tivity in these materialshas beenin largepartdedicated pearsasthesizeofthesimulatedsystemincreases. They
2 totheinvestigationoftheFermisurfacestructureandits suggestedthereforethatforon-siterepulsionweakerthan

dependence onthe strong electroniccorrelationsandan- the Hubbard bandwidth (8t) the antiferromagnetic cor-
1
tiferromagnetic order. Angle-resolved photoemission ex- relationlength would be comparable to the size of a suf-
v
9 periments(ARPES)haverecentlyprovidedacomprehen- ficiently small lattice and would effectively create long-
1 sivemappingoftheFermisurfaceinsomeofthecuprates range antiferromagnetic order. Based on this conclusion
1 (YBa Cu O and Bi Sr CaCu O )[1],[2],[3]. These the U = 4t coupling (where U is the on-site Coulomb
2 4 2 2 2 2 8+x
1
results are particularly relevant to the high-temperature repulsionand t is the nearest-neighbourhopping param-
0
superconductivity scenario in which the phenomenon is eter) was regarded as belonging to the weak-coupling
5
9 related to the presence of van Hove singularities in the regime, where the band structure is essentially similar
/ density of states close to the Fermi surface [1]. Quan- to that of noninteracting electrons for all temperatures
t
a tum Monte Carlosimulationshaverecently beenused to above zero, in accordance with the Mermin-Wagner the-
m
comparetheseexperimentalresultswithnumericalcalcu- orem. Following this argument, a gap originating from
- lations in models of strongly correlated electrons [4],[5]. spin-density waves(SDW) will be both temperature and
d
An essential part of these numerical results was the ex- finite-sizesensitive. Thelow-temperaturesimulations,as
n
o tractionofthespectralweightfunction(SWF)forsingle- presentedin[10]forβ =12showindeedastablegapeven
c particleexcitationsfromtheQuantumMonteCarlodata for relatively large lattice size. Such temperature depen-
: calculated for imaginary time. dence of the gap would be expected to correspond to a
v
i The SWF is obtained as a solution of the following strong temperature dependence of the spin-spin correla-
X
inverse problem [7]: tions. However,as the numerical results of Refs.[11] and
r [12]show,thespin-spincorrelationsatU =4tand42 lat-
a G(~k,τ)= ∞ dω e−ωτ A(~k,ω); (0<τ ≤β) (1) ticefallrapidlywithin2latticespacingswithonlyminor
Z−∞ 1+e−ωβ dependence on the temperature in the rangeβ =4−12.
The SWF presented in Ref.[8] shows a clear two-peak
where G(~k,τ) is the Matsubara Green’s function for lat- structure for this value of U andlattice linear dimension
tice momentum ~k and imaginary-time separation τ, ob- larger than the correlation length. Hence the possibility
tainedinthefinite-temperatureMonteCarlosimulations thatthe bandstructureisdue totheshort-rangeantifer-
at temperature 1/β, and A(~k,ω) is the corresponding romagnetic order (as was suggested in Ref.[3]) demands
SWF. an accurate mapping of the on-site repulsion regimes for
This inverse problem admits an infinite class of solu- coupling U <8t.
tions for A(~k,ω) which will satisfy Eq.(1) within small AdetailedstudyoftheSWFasasolutionofaninverse
perturbationsofthel.h.s.,originatingfromthestatistical problemrequires a quantitive criterionfor the resolution
noise on the simulation data. In recent works the one- limits of the reconstruction technique. Without such
electronSWFforthesingle-band2DHubbardmodelwas a criterion an existing gap might be overlooked in the
obtained from Monte Carlo data using a maximum en- reconstruction procedure. Moreover, the single-particle
tropy approach[8],[9],[13]. The finite-size effects evident spectrum derived from the distribution of the function
in the data itself, as well as in the reconstruction of the A(~k,ω) along the Brillouin zone is based on the identifi-
1
cation of the peaks of this function. Thus the resolution The solutionof Eq.(1)expanded inthe functions {u }
k
of the two-peak structure, even if there is no clear evi- is given by:
denceofthevanishingofthisfunctionbetweenthepeaks,
is important. nτ 1
A(ω)= <G,v >u (ω) (6)
k k
In our work we use a method based on the singular α
X k
k=1
value decomposition (SVD) approachwidely used in the
fieldofinverseproblems[14]. Thistechniqueallowsusto where <,> denotes the inner product in the data space.
definetheresolutionlimitsquantitatively,thusproviding Thesmallestsingularvalueswillgreatlyamplifythecon-
an upper bound on the size of the gap with respect to tribution of the statistical noise to the expansion coeffi-
the lattice size. cients in Eq.(6). We therefore introduce a cut-off into
The work presented is an investigation performed on the expansiontoexclude termswith αk smallerthanthe
α1
lattice sizes rangingbetween42 and122. The simulation statistical error in G(τ) [15].
temperature was chosen as β = 5 and on-site repulsion The“resolutionratio”dependsonthenumberofnodes
asU =4t. Toavoidthesignproblemthemodelwassim- in the function with the highest index kmax included
ulated at half-filling. However, the SVD treatment can in the truncated expansion [16]. Here prior information
be extended to finite doping, as well as to other Green’s about the support of the solution is essential. Since we
functions. assume that the A(ω) is localised within a certain range
Eq.(1) can be rewritten in operator form as follows ofω,weconfinethesingularfunctionstothisrange. This
[14]: is done by multiplying the kernel in Eq.(1) by a pro-
filefunctionwhichisapproximately1withinthesupport
G(τ )=(KA)(τ ). (2)
i i andsmoothlyvanishesoutsideit[16]. Therateofthede-
K is the integral operator of the r.h.s. of (1), with the creaseofthesingularvaluesforthelimitedsupportgrows
variable τ in the kernel discretised, acting on the func- astheprofilefunctiongetstighter,therefore,the trunca-
tion A. This operator defines a transformation from the tion in (6) will have a smaller kmax. However, since the
L2 space of the SWF to the data space of which G(τ ) is support of the singular functions is limited, their nodes
i
avector. Thistransformationmayberegardedasarect- will be more closely spaced, thus giving better resolu-
angular matrix, K, one of whose dimensions is infinite tion. To investigate the resolution limits, artificial data
and the other equal to the dimension of the data space, was generated by substituting an A(ω) consisting of two
namely to the number of data points n . A conjugate Gaussian peaks into Eq.(1). The inverse problem was
τ
operator K∗ acts in the data space and is determined by then solved for A(ω) with the expansion (6) truncated
the following equality of inner products: at α1 exceeding a hypothetical noise level. Similar to
αk
the results of Ref.[16]we found that for a fixed noise the
nτ
∗ resolution improves as the range of support decreases.
W (KA)(τ )G(τ )= dωA(ω)(K G)(ω), (3)
X τiτj i j Z The reconstructionof the SWF for~k at the vicinity of
i,j=1
~k =(π,π)isgiveninFig.1. Itcanbeseenthatthereare
with a metric tensor W which canaccount for the corre- 2 2
negative side-lobes. Just as a positive δ−function, im-
lations in the data points. A common choice is to define
aged over a limited Fourier bandwidth, turns into Airy
this metric as the inverse of the covariance matrix [14].
rings which have negative side-lobes, we must expect
∗
The operator KK is represented by a symmetric square
a localised SWF, reconstructed over a limited singular
matrixwithapositivesetofn eigenvalues{α2}andcor-
τ i function bandwidth, to have similar features. A regu-
responding set of orthonormal eigenvectors {v }. These
i larizationprocedure which would restrictthe solutionto
∗
vectorsformabasisinthedataspace. TheoperatorK K
positive values only may also reduce the resolutionof its
acting in the space to which the solution A(ω) belongs reconstruction[15]. Wepresentheretheresultsofthe82-
hasasetofnτ orthonormaleigenfunctions {ui}withthe 122 lattices, since the existence of the gap in the smaller
same eigenvalues. If the set {α }nτ is arranged in de-
i i=1 lattices is beyond controversy. Limiting the support of
scending order each vector vk and function uk will have A(~k,ω) to the range [−8,8], allowed us to identify a gap
k−1 nodes. These two sets {u } and {v } and the set
i i unseen in an infinite-support reconstruction. According
{α }nτ form a singular system satisfying the equations:
i i=1 to our statistical noise we truncated the expansion of
A(~k,ω) with the ratio α1 not exceeding 100 (for a 1%
K∗uk =αkvk (4) statistical error as estimαakted for 4000 Monte Carlo mea-
K v =α u .
k k k surements) which allowed us to use 9 singular functions
The matrix representation of the operator KK∗ is ob- intheunlimited-supportcaseand7inthereconstruction
tained[14] from the kernel of Eq.(1) as: with the finite profile.
Since the statisticalnoise determines the leveloftrun-
(KK∗) = W ∞ dωe−ω(τk+τn). (5) cation of the singular-function expansion it is clear that
τmτn X τmτkZ−∞ (1+eωβ)2 ataparticulartruncationlevel,notallthe dataincluded
τk
2
inthe n points areofidenticalrelevance. Inaccordance and the same moments for the density of states. We
τ
with the application of sampling theory in other inverse present in Table 1 the values of these moments for the
problems we identify as essential the nodes of the v . density of states, showing excellent agreement with the
kmax
ThisisageneralisationoftheNyquistrateinFourierthe- analytic values 1,0,8 for the zeroth, first and second mo-
ory. For example, in the inversion of the Laplace trans- ments respectively [8].
form [17] the optimal sampling was found to be expo-
lattice µ µ µ
nential which corresponds to the nodes of v . Appli- 0 1 2
kmax 82 1.00 0.00 7.99
cation of this sampling technique to the inverse problem
(1) is performed as follows: we estimate k from the 102 1.00 0.00 7.79
max
knowledge of the noise and the spectrum of the singular 122 1.00 0.00 7.77
values. Then the data is evaluated at points nearest to Table 1. The first three moments of the density of
the nodes of vkmax and the SVD reconstruction is done states versus lattice size.
forthesepoints only. Thisprocedureleadsto anobvious
Fig.3 shows the single-particle spectrum derived from
advantage from the computational point of view, since
identifying E(~k) as the location of a peak of A(~k,ω),
the evaluation of the Matsubara Green’s functions is re-
duced to 1 of the total points. Moreover, by separating where the values of the momentum~k scan over the Bril-
5
the data points by a large spacing we reduce the corre- louinzone. Inthecalculationofthespectrumweobserve
lations within the data and the metric onthe data space a significant difference between the 82 and the 122 re-
can be taken as Euclidean, which also substantially re- sults. Both lattices exhibit a clear two-bandstructure of
ducesthecalculationaleffort. Forthe82latticeweevalu- the single-particle spectrum. However, while the former
atedtheSWFusing7datapointsonlyandcomparedthe can be fitted well with the mean-field SDW formula:
results with that obtained from using 40 equally spaced
points. For the latter case we accounted for the corre- E(~k)MF = (ǫ(~k)2+∆2) (10)
q
lations using the covariance matrix as the metric in the
dataspace. Theresultsshowremarkableagreement,thus whereǫ(~k)=−2t(cosk +cosk )and∆=0.64t,the122
x y
fully justifying the sampling approach in this problem. lattice shows a bigger gap at the corners of the nonin-
In Fig.2 we presentthe density of states calculatedfor teracting Fermi surface than at the point (π,π). Indeed
2 2
the three biggest lattice sizes as the solution N(ω) of: the122latticesimulationgivesasmallergapatthispoint
∞ e−ωτ than the 82 and the 102 (see Fig.1). However,at the the
G˜(0,τ)= dω N(ω) (7) points(π,0)thegapremainspersistentlyclearforallthe
Z−∞ 1+e−ωβ lattices. This result can be interpreted as suggesting a
with G˜(0,τ) being the zero-space-separation Matsubara gap due to an anisotropic pairing channel [2] (although
Green’s function. In this reconstruction the range of the not necessarily of dx2−y2 symmetry). Higher resolution
calculations, based on data with better statistics will be
support for N(ω) is [-12,12].
requiredto examine this possibility. We also observethe
The gap decreases as the lattice size increases, in cor-
feature of a flat band near ~k = (π,0) similar to the re-
respondence with the results of Veki´c andWhite [8], but
it is still clear even in the 122 lattice. In fact the dif- sultsofDagottoet al.[4] andthe experimentaldata from
ARPES [2]. This flatness may lead to a van Hove-like
ference in the size and depth of the gap is very small as
the lattice size increases from 82 to 122, while the lin- singularity in the density of states.
The SVD method brings to this inversion problem a
ear dimension of the lattice is considerably larger than
more rigorous and controlled approach than the maxi-
the SDW correlation length. The existence of two clear
peaksinthedensityofstates(aswellasinA(~k ,ω)) mum entropy technique. Contrary to previous findings
Fermi
the resultsofourcalculationssuggestthatU =4tis ina
suggests a two-band structure of the spectrum.
regime in which the spectrum of quasiparticle (damped)
The reliability of our results has been tested against
excitations has a two-band structure, in spite of the on-
thefollowingcriteria: thesuccessfulreconstructionofthe
Monte-Carlodata whenthe solution ofA(~k,ω)is substi- siterepulsionbeing sufficientlyweaktoallowsubstantial
double occupancy. There is some support for the rel-
tutedinther.h.s. ofEq.(1);thesuccessfulreconstruction
evance of U = 4t regime to high T superconductivity
of the same data by: c
from a recent work by Beenen and Edwards [18].
kmax WethankJ.Jefferson,A.Bratkovsky,D.Edwardsand
G= <G,v >v ; (8)
k k P. Kornilovitch for stimulating discussions. This work
X
k was suported by SERC grant GR/J18675 and our gen-
and, finally, by checking the first three moments of the eral development of SVD techniques by the US Army
SWF: Research Office, agreement no. DAAL03-92-G-0142.
µ = ωnA(~k,ω)dω (n=0,1,2) (9)
n Z
3
References Figure captions
Fig.1a: A(π/2,π/2,ω) for a 82 lattice. Solid line – SVD
1. K. Gofronet al., Phys. Rev. Lett. 73, 3302 (1994)
solutionwithinfinitesupport,dottedline–SWFsupport
is limited in the range [-8,8].
2. D.S. Dessau et al., Phys. Rev. Lett. 71, 2781
Fig.1b: A(2π/5,3π/5,ω) for a 102 lattice. Solid line –
(1993);
SVD solution with infinite support, dotted line – SWF
D.M.Kingetal.,Phys. Rev. Lett. 73,3298(1994).
support is limited in the range [-8,8].
3. P. Aebi et al., Phys. Rev. Lett. 72, 2757 (1994) Fig.1c: A(π/2,π/2,ω)fora 122 lattice. Solidline –SVD
solutionwithinfinitesupport,dottedline–SWFsupport
4. E. Dagotto, A. Nazarenko and M. Boninsegni,
is limited in the range [-7,7].
Phys. Rev. Lett. 73, 728 (1994).
Fig.2: Density of states as a function of ω.
Fig.3a: 82 lattice. Spectrum of single-particle excita-
5. S. Haas, A. Moreo and E. Dagotto, unpublished.
tions. Lower band – stars; upper band – circles. Dot-
6. A. Moreo, S. Haas, A. Sandvik and E. Dagotto, ted line – ǫ(~k) = −2t(cosk + cosk ); solid line –
x y
unpublished. E(~k) =± ǫ(~k)2+∆2. Points in the Brillouin zone:
MF
q
7. H.-B. Schu¨ttler and D.J. Scalapino, Phys. Rev. Γ – (0,0); M¯ – (π,π); M – (π,π); X – (π,0).
2 2
B34, 4744 (1986). Fig.3b: Same as (a) for a 122 lattice.
8. S.R. White, Phys. Rev. B44, 4670 (1991);
M. Veki´c and S.R. White, Phys. Rev. B47, 1160
(1993).
9. N. Bulut, D.J. Scalapino and S.R. White, Phys.
Rev. Lett. 73, 748 (1994).
10. S.R. White, Phys. Rev. B46, 5678 (1992).
11. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
12. G. Fano, F. Ortolani and A. Parola, Phys. Rev.
B46, 1048 (1992)
13. N. Bulut, D.J. Scalapino and S.R. White, Phys.
Rev. Lett. 72, 705 (1994).
14. M.Bertero,C.DeMolandE.R.Pike,InverseProb-
lems 1, 301 (1985);
M. Bertero and E.R. Pike, Handbook of Statistics
10,Eds. N.K.BoseandC.R.Rao,ElsevierScience
Publishers (1993).
15. M. Bertero, P. Branzi, E.R. Pike and L. Rebolia,
Proc. R. Soc. London A 415, 257 (1988);
M.Bertero,C.DeMolandE.R.Pike,InverseProb-
lems 4, 573 (1988).
16. M. Bertero, P. Boccacci and E.R. Pike, Proc. R.
Soc. London A 383, 15 (1982); A 393, 51 (1984);
M.Bertero,P.BranziandE.R.Pike,Proc. R. Soc.
London A 398, 23 (1985).
17. M. Bertero and E.R. Pike, Inverse Problems 7, 1
(1991); 21 (1991).
18. J. Beenen and D.M. Edwards, to be published in
Jour. of Low Temp. Phys. .
4
Fig. 1a
0.40
0.30
)
ω
,
2
/
π 0.20
,
2
/
π
(
A
0.10
0.00
-0.10
-8.0 -4.0 0.0 4.0 8.0
ω
Fig. 1b
0.40
0.30
)
ω
,
5
/
π
3 0.20
,
5
/
π
2
(
A
0.10
0.00
-0.10
-8.0 -4.0 0.0 4.0 8.0
ω
Fig. 1c
0.30
)
ω
,
2
/
π
,
2
/
π
(
A
0.10
-0.10
-8.0 -4.0 0.0 4.0 8.0
ω
Fig.2
0.20
2
8
2
10
2
12
0.15
)
ω
0.10
(
N
0.05
0.00
-8.0 -4.0 0.0 4.0 8.0
ω
Fig.3a
5.0
3.0
1.0
)
p
(
E
-1.0
-3.0
-5.0
−
Γ Γ
M M X
Fig. 3b
4.0
2.0
)
p 0.0
(
E
-2.0
-4.0
−
Γ Γ
M M X