Towards full simulations of high-temperature superconductors

T.A. Maier, J.B. White III, T.C. SchulthessCenter for Computational Sciences and Computer Science and Mathematics Division,

Oak Ridge National Laboratory

M. JarrellUniversity of Cincinnati

Abstract

The Cray X1 in the Center for Computational Sci-ences at Oak Ridge National Laboratory is enablingsignificant new science in the simulation of high-temperature “cuprate” superconductors. We de-scribe the method of dynamic cluster approximationwith quantum Monte Carlo, along with its computa-tional requirements. We then show the unique capa-bilities of the X1 for supporting this method, portingexperiences, performance, and the resulting new sci-entific results.

1 Introduction

Despite years of active research, the understand-ing of superconductivity in the high-temperature“cuprate” superconductors (HTSC) remains one ofthe most important outstanding problems in mate-rials science. A complete theoretical understandingof cuprate superconductors could lead to the abilityto design and synthesize room-temperature super-conductors, which would have tremendous techno-logical implications. In the superconducting stateof a material, electrons form so-called Cooper pairs,allowing them to condense into a coherent macro-scopic quantum state in which they conduct electric-ity without resistance. In conventional superconduc-tors, pairing results from an attractive interactionbetween electrons that is mediated by lattice vibra-tions (phonon-mediated pairing).

The consensus today is that the pairing mecha-nism in high-temperature superconductors is of anentirely different nature and is probably related tostrong correlations between electrons, a feature thatdistinguishes these materials from conventional su-perconductors. To address the problem theoreti-cally, one must solve the quantum many-body prob-

lem for a macroscopic number of electrons withoutbeing limited to the typical single-particle approxi-mations, such as Hartree-Fock or the local densityapproximation to density functional theory. A re-cent concurrence of new algorithmic developmentsand significant improvements in computational ca-pability has opened a clear path to solving the quan-tum many-body problem for high-temperature su-perconductors.

2 Hubbard model

The characteristic feature of all HTSC is a stronglyanisotropic layered perovskite-like crystal structurewith conducting CuO2-planes separated by insu-lating layers of other elements (see right part ofFig. 1). Superconductivity takes place within thetwo-dimensional CuO2 layers with the insulatingbarriers only providing charge carriers, usually holesto the layers and thus controlling the doping of CuO2

planes.

First-principles calculations for HTSC compoundsprovide evidence that the band which crosses theFermi surface has mainly CuO2 character (see e.g.[1] and references therein). To reduce the complexityof the problem it thus seems reasonable to restrictcalculations to a two-dimensional model with elec-trons moving in a single CuO2 layer. Justified by thestrong in-plane CuO bonds, the complexity may befurther reduced by constructing a model that treatsa whole CuO2 plaquette as a single site. The result-ing two-dimensional Hubbard model [2] is believedto capture the essential physics of HTSC [3, 4, 5]. Aschematic of its Hamiltonian,

H = −t∑

〈ij〉c†iσcσ + U

∑

i

ni↑ni↓ (1)

1

Figure 1: Crystal structure of YBa2Cu3O5 and two-dimensional Hubbard model of the hole doped CuO2

planes, with nearest neighbor hopping integral t andon-site Coulomb interaction U .

is illustrated in Fig. 1. The fermionic operator c†iσ(ciσ) creates (destroys) an electron on site i with

spin σ, and niσ = c†iσcıσ is the corresponding num-ber operator. The first term describes the hybridiza-tion between sites with amplitude t, and the secondterm the Coulomb repulsion between two electronsresiding on the same site. Because of screening, themagnitude of longer-ranged interactions is believedto be small compared to the on-site interaction.

Despite decades of intensive studies, this modelremains unsolved except in one or infinite dimen-sions. Analytical methods based on a perturba-tive approaches suffer from the large magnitude ofU , which renders these calculations at least ques-tionable. Many theorists have turned to numeri-cal approaches to close the gap between the modeldefined by its Hamiltonian and its properties. Alarge body of work has been devoted to a direct (nu-merically) exact solution of finite-size systems us-ing exact diagonalization or Quantum Monte Carlo(QMC) methods (for a review see [6]). Exact diag-onalization, however, is severely limited by the ex-ponential growth of computational effort with sys-tem size, while QMC methods suffer from what isknown as “the sign problem” at low temperatures.

Another difficulty of these methods arises from theirstrong finite-size effects, often ruling out the reliableextraction of low-energy scales, which are importantto capture the competition between different groundstates often present in correlated electron systems.

3 Dynamical Cluster Approxi-mation

Mean-field theories are defined in the thermody-namic limit and therefore do not face the finite-sizeproblems. Generally, mean-field theories divide theinfinite number of degrees of freedom into two sets.A small set of degrees of freedom is treated explic-itly, while the effects of the remaining degrees offreedom are summarized as a mean field acting onthe first set. The Dynamical Mean-Field Theory(DMFT) [7, 8] (for a review see [9]) for itinerantcorrelated systems (such as the HTSC or systemsdescribed by the model Eq. (1)) is analogous to thecoherent potential approximation for disordered sys-tems [10, 11, 12]. It retains the dynamics of localdegrees of freedom by mapping the lattice onto animpurity self-consistently embedded in a dynamicalmean-field host.

Despite its success in the description of many cor-related phenomena such as the Mott-Hubbard tran-sition, the DMFT and CPA share the critical flaw ofneglecting the effects of non-local fluctuations. Thusthe DMFT is unable to capture the effects of e.g.spin-waves in magnetic systems, localization in dis-ordered systems, or spin-liquid physics in correlatedelectron systems. Furthermore it cannot capturephase transitions to states with non-local order pa-rameters, such as the d-wave superconducting phasein the HTSC. Non-local corrections are required totreat even the initial effects of these phenomena.

Here we use the Dynamical Cluster Approxima-tion (DCA) [13, 14, 15, 16] (for a review see [17]) tostudy the properties of the Hubbard model, Eq. (1).The DCA extends the DMFT by non-local correla-tions. Instead of mapping the lattice onto a singleimpurity, the system is mapped onto a periodic clus-ter of size Nc coupled to a mean-field host represent-ing the remaining degrees of freedom (see Fig. 2). Asa result, dynamical correlations up to a range lim-ited by the cluster size are treated accurately, whilethe physics on longer length scales is described onthe mean-field level. Translational invariance of theoriginal system assures that the quantity describ-

2

Figure 2: Schematic illustration of the DCA formal-ism. The model is mapped onto a finite-size clusterself-consistently coupled to a mean-field host. Cor-relations within the cluster are treated accuratelywhile the physics on length scales beyond the clus-ter size is described on the mean-field level.

ing the mean-field host can be self-consistently de-termined from the solution of the cluster problem.The complexity of the original problem with an in-finite number of degrees of freedom is thus reducedto a self-consistent finite-size cluster problem withNc degrees of freedom. The remaining cluster prob-lem may then be solved numerically by a numberof techniques including the QMC method [16] usedhere.

4 Small Clusters

Computations with a cluster of only four sites, thesmallest cluster that can capture superconductivitywith a d-wave order parameter, on the IBM p690 atthe Center for Computational Sciences (CCS) showvery good general agreement with HTSC. Theseresults are summarized in the temperature-dopingphase-diagram shown in Fig. 3 (see also [18, 19]).At low doping, δ, the system is an antiferromag-netic insulator below the Neel temperature TN. Atfinite doping, δ ≤ 0.3, an instability is found at thecritical temperature Tc to a superconducting statedescribed by a dx2−y2-wave order parameter. Inthe normal state, low-energy spin excitations be-

0.0 0.1 0.2 0.30.00

0.02

0.04

0.06

0.08

0.10

δ

T [e

V]

TcTN

T*

d-wave superconductivity

Pseudogap

AF

Fermi-liquid

Figure 3: DCA/QMC Temperature-doping phase di-agram of the two-dimensional Hubbard model whent = 0.25 eV, U = 2 eV for a 4-site cluster. Consis-tent with experiments on HTSC, regions of antifer-romagnetism, d-wave superconductivity and pseudo-gap behavior are found.

come suppressed below the crossover temperatureT ∗. Simultaneously the electronic excitation spec-trum displays a pseudogap, i.e. a partial suppressionof low-energy spectral weight. Consistent with opti-cal experiments, computations for a four-site clustershow that the superconducting transition is accom-panied by a lowering of the electronic kinetic energy[20]. This result further shows the unconventionalcharacter of superconductivity in these systems. Itis fundamentally different from the BCS theory forconventional superconductors [21], where pairing oc-curs through a reduction of the electronic potentialenergy accompanied by a slight increase in kineticenergy.

The apparent violation of the Mermin-Wagnertheorem [22], according to which no phases with con-ventional long-range order can occur at finite tem-peratures in the two-dimensional Hubbard model, isa consequence of the small cluster size, and hencelarge mean-field character, in these simulations. Inthe case of antiferromagnetism, the Mermin-Wagnertheorem thus necessarily translates to TN = 0 for thetwo-dimensional system. Superconductivity how-ever can exist even at finite temperatures as topo-logical order below the Kosterlitz-Thouless transi-tion temperature [23]. Therefore, larger-cluster-sizestudies are needed to see if the simulations recoverthe Mermin-Wagner theorem and if superconductiv-ity survives as topological order in the infinite clustersize limit where the DCA becomes exact.

3

In the HTSC, on the other hand, a small but finitecoupling between the two-dimensional CuO2 layersinduces long-range order at finite temperatures.

5 Porting and Performance

The central quantity of the DCA code is the single-particle cluster Green function Gc, which is a matrixof size N ×N [16]. Here N = Nc ×Nl where Nl isthe number of “time-slices” in the time direction.The majority of the CPU time is spent in the in-ner loop of the QMC simulation, which updates theGreen function matrix according to the vector outerproduct

G′ = G + a ∗ bT , (2)

where a and b are two vectors of dimension N .This computation is handled by the BLAS [24] callDGER, which performs a double-precision rank-onematrix update representing O(N 2) operations. Eachiteration requires N such calls, however, resulting inO(N3) operations.

Another CPU-intensive task is the evaluation oftwo-particle correlation functions. In the QMC tech-nique this reduces to evaluating products of Greenfunctions and thus to computing matrix products.This is done by using the BLAS call CGEMM,which performs single-precision complex matrix-matrix multiplication, and one call again is O(N 3).

Porting and tuning the DCA implementation onthe Cray X1 was straightforward. The port re-quired no changes beyond the “Makefile”, andtuning involved performance profiling and adding“concurrent” directives to one file. This file con-tains a number of nested loops using indirect ad-dressing, or index arrays. The bulk of the tuningeffort was in determining which loops did and didnot iterate over repeated indices.

The Cray X1 has a number of advantages overgeneral-purpose systems in performing DCA com-putations, particularly with increasing cluster size.This advantage is demonstrated in Fig. 4, whichcompares runtimes of some early DCA runs on theX1 and the IBM p690 in the CCS, using 8 and 32processors (MSPs) on each.The figure shows run-times for production runs with a fixed value ofNc = 64 and increasing values of Nl and thus N ,where the value of N is shown. Eight X1 MSPseasily outperform thirty-two 1.3-GHz Power4 pro-cessors for the larger problem sizes.

As discussed above, the DCA implementation in-cludes two O(N3) computations built on the BLAS

Figure 4: Runtimes for a series of DCA productionruns. Each run is indicated by its value of N =NcNl. The lines connecting the data points are onlyguides to the eye.

calls CGEMM and DGER. CGEMM is a BLAS3call, which implies that it can be blocked effec-tively for cache memory, and many modern general-purpose processors can perform the operations neartheir peak. The X1 processors can also, but theyhave the added benefit of a very high peak rate aug-mented by the ability to perform single-precision op-erations at twice the rate of double-precision.

The X1 has a more significant advantage overthe prevailing cache-dependent architectures in theDGER operations. Each call depends on the resultsof the previous call, so the operations cannot be in-terleaved. DGER is a BLAS2 call, which impliesthat it does much fewer computations per memoryaccess than CGEMM, and thus is typically limitedby memory bandwidth.

We conducted separate DGER benchmarks tomeasure the advantage of the X1 in this operation,and results for the CCS Cray X1, SGI Altix (1.5 GHzItanium2), and IBM p690 (1.3 GHz Power4) are inFig. 5. The vendor-optimized DGER was used foreach system. The figure shows the performance ofDGER for a matrix of size N = 64 × 70 = 4480,which is representative of large DCA runs. Sepa-rate DGER instances were run concurrently acrossincreasing numbers of processors (MSPs), mimick-ing the processes of a Monte-Carlo simulation. TheX1 memory system is able to maintain performanceand efficiency with added processors, while the p690steadily degrades. The Altix degrades going fromone to two processors because memory bandwidth is

4

Figure 5: Per-processor performance of concurrentDGER calls using N = 4480 matrices.

shared between processor pairs. The X1 maintains8–25 times the performance and 4–10 times the effi-ciency of the other systems.

Despite the Monte-Carlo nature of the DCA algo-rithm, the X1 also has an important scalability ad-vantage over systems with weaker processors. EachDCA process has a significant fixed start-up cost,which favors splitting the Monte-Carlo iterationsacross fewer, faster processors.

Another option would be to multithread eachMonte-Carlo process, effectively using an SMP asa large single “processor”. We explore this possibil-ity in Fig. 6, which shows the performance of IBM’smultithreaded DGER on a p690, again using a ma-trix size of N = 4480. Fig. 6. The dashed lineshows the per-MSP performance of an X1 perform-ing concurrent DGER operations on 32 MSPs, thussimulating a loaded system. The solid line showsthe performance of a 32-processor p690 loaded withconcurrent DGER computations, but using differentnumbers of processors per DGER process.

The left-most point thus shows the performanceof a single processor when all 32 processors of thep690 are performing independent DGER operations,while the right-most point shows the aggregate per-formance of dedicating all 32 processors to a singleDGER. The figure indicates that dedicating a fullIBM p690 to each DGER does not match the per-formance of a single X1 MSP. No threaded versionof vendor-optimized DGER was available for the Al-tix or the X1 at the time of this test. Tests of un-tuned DGER implemented with Fortran loops andOpenMP showed little improvement on the X1 for

Figure 6: DGER performance of a fully loadedIBM p690 using different numbers of processors perthreaded process. The dotted line is the per-MSPperformance of loaded X1 nodes.

matrices of size 4480, and the Fortran/OpenMP im-plementation on the Altix was not competitive withthe single-threaded vendor-optimized DGER.

The significant performance advantage of the X1for DCA computations, as illustrated by its domi-nance in DGER performance, has allowed us to per-form simulations that are out of the reach of othersystems, all without having to resort to hybrid paral-lelization. In particular, the X1 has provided the ca-pability needed to perform DCA computation withmuch-larger cluster sizes.

6 Larger Clusters

As discussed in Sec. 4, the DCA retains a largemean-field character at small cluster sizes and con-sequently yields long-range order at finite temper-atures. Long wave-length modes which destroylong-range order at finite temperatures in two-dimensional systems are neglected. With increas-ing cluster size, however, the DCA progressively in-cludes these longer-ranged fluctuations. These areexpected to drive the Neel temperature systemati-cally to zero and thus recover the Mermin-Wagnertheorem in the infinite cluster size limit where theDCA becomes exact.

Fig. 7 displays the DCA results for the Neel tem-perature TN as a function of the inverse of the linearcluster size Lc =

√Nc. With increasing cluster size,

TN rapidly decreases and extrapolates to TN = 0

5

0.0 0.4 0.80.00

0.02

0.04

0.06

0.08

1/Lc

T NNc=1

Nc=2: Local singlet

Nc=4: RVB

Figure 7: Neel temperature at 5% doping as a func-tion of the inverse linear cluster size 1/Lc = 1/

√Nc.

in the infinite cluster size limit consistent with theMermin-Wagner theorem. The data points scatterabout a curve linear in 1/Lc, except for the “spe-cial” cluster sizes Nc = 1, 2 and 4. For Nc = 2 alocal singlet is formed on the cluster. When Nc = 4the ground state of the periodic cluster is a resonat-ing valence-bond state [25] with fluctuating singletbonds between the cluster sites. Hence antiferro-magnetic order is suppressed for these cluster sizes,and the results thus do not fall on the curve.

As discussed in Sec. 4, superconductivity may per-sist in the infinite cluster size limit as topologicalKosterlitz-Thouless order, although no conventionallong-range order is allowed.

The transition to a superconducting state withd-wave symmetry is indicated by the divergence ofthe pair-field susceptibility Pd, or equivalently bythe node of P−1

d . The DCA result for this quan-tity at 5% doping is plotted in Fig. 8 for differ-ent cluster sizes Nc. At finite cluster sizes Nc,the critical behavior found in the DCA at tempera-tures close to the transition temperature Tc definedby the node in P−1

d has to be mean-field like, i.e.P−1d ∝ |T − Tc|, since the long-ranged physics is

treated on the mean-field level. At higher tempera-tures however, where the correlation length is withinthe cluster size, the true critical behavior may beobserved in the DCA. Therefore we fit the DCA re-sults at intermediate temperatures with the functionχ = A exp(1B/(T −Tc)0.5), the critical behavior ex-pected for a Kosterlitz-Thouless transition [26]. Atlower temperatures we expect the linear dependenceto connect smoothly to this behavior.

0.0 0.1 0.2 0.30

4

8

12

T

P d-1

Nc=4 ; A=0.065, ∆A=0.006, B=0.188, ∆B=0.014, Tc=0.012, ∆Tc=0.002Nc=16 ; A=0.037, ∆A=0.002, B=0.314, ∆B=0.011, Tc=-0.04, ∆Tc=0.002Nc=18 ; A=0.054, ∆A=0.003, B=0.215, ∆B=0.013, Tc=-0.005, ∆Tc=0.002Nc=32 ; A=0.045, ∆A=0.001, B=0.262, ∆B=0.008, Tc=-0.02, ∆Tc=0.002

Fit with χ = A exp(2B/(T-Tc)1/2)

U=1.5, n=0.95

Figure 8: DCA/QMC result for the inverse d-wavepair-field susceptibility as a function of temperaturefor different cluster sizes at 5% doping when t = 0.25and U = 1.5. Superconductivity is suppressed atcluster sizes Nc > 4.

When Nc = 4, the DCA predicts a transition toa d-wave superconducting state at a finite tempera-ture Tc

1. When Nc > 4, however, the results seemto indicate the absence of a finite temperature tran-sition to a superconducting state at 5% doping.

Fig. 9 shows the results for the 15% doped system.Again, for Nc = 4 a superconducting transition isobtained at a finite temperature. In contrast to the5% doped case, the results are almost converged, i.e.independent of cluster size, when Nc > 4. This is aclear indication that at this doping correlations areshort-ranged and do not extend beyond the clustersize at the temperatures studied. Clearly, the re-sults are incompatible with superconductivity at fi-nite temperatures at 15% doping, even if the lowesttemperatures data points are extrapolated linearly.

Based on these results we infer that, despite itstendency to exhibit d-wave pairing, the purely two-dimensional Hubbard model is not enough to de-scribe high-temperature superconductivity. We con-clude that either a coupling to the third dimension,a more realistic modeling of the electronic structure,the additional inclusion of lattice degrees of free-dom or even a combination of these extensions isnecessary to stabilize superconductivity in the infi-nite cluster size limit. Work along these lines is inprogress.

1Note however that the fit function for temperatures closeto Tc changes curvature and therefore underestimates the ac-tual Tc predicted by the DCA

6

0.0 0.1 0.2 0.30

4

8

12

T

P d-1

Fit with χ = A exp(2B/(T-Tc)1/2)

U=1.5, n=0.85

Nc=4 ; A=0.093, ∆A=0.010, B=0.171, ∆B=0.015, Tc=0.009, ∆Tc=0.015Nc=16 ; A=0.041, ∆A=0.001, B=0.312, ∆B=0.007, Tc=-0.266, ∆Tc=0.002Nc=18 ; A=0.041, ∆A=0.002, B=0.313, ∆B=0.013, Tc=-0.241, ∆Tc=0.002Nc=32 ; A=0.040, ∆A=0.004, B=0.319, ∆B=0.027, Tc=-0.252, ∆Tc=0.006

Figure 9: DCA/QMC result for the inverse d-wavepair-field susceptibility as a function of tempera-ture for different cluster sizes at 15% doping whent = 0.25 and U = 1.5. The results are almostconverged for Nc > 4, where superconductivity isstrongly suppressed

7 Summary and Conclusions

The Cray X1 in the Center for Computational Sci-ences at Oak Ridge National Laboratory has en-abled significant new progress in the understandingof HTSC within a minimal microscopic model, thetwo-dimensional Hubbard model. DCA/QMC sim-ulations at small cluster size Nc = 4 show very goodgeneral agreement with HTSC, including supercon-ductivity at high temperatures. Due to the smallcluster size however, the results violate the Mermin-Wagner theorem, according to which no long-rangeorder is allowed at finite temperatures in the two-dimensional model. The significant performance ad-vantage of the X1 for the DCA/QMC computationshas provided the capability to study much largercluster sizes. Recent runs on the Cray X1 show that,with larger clusters, relevant longer-ranged fluctu-ations are captured, and the Mermin-Wagner the-orem is recovered. Furthermore, the results showthe absence of a finite-temperature superconductingtransition, and thus are incompatible with a possi-ble Kosterlitz-Thouless transition to a phase withtopological order.

These problems in the description of HTSC maybe overcome by carrying out fully three-dimensionalcalculations with an infinite set of Hubbard planescoupled along the third dimension. To eventuallyenable the design of new and optimized supercon-ductors, we further plan to parameterize the model

with ab-initio electronic-structure calculations andto generalize the method to include multiple bands.

8 Acknowledgments

This research used resources of the Center for Com-putational Sciences and was sponsored in part bythe offices of Advance Scientific Computing Re-search and Basic Energy Sciences, U.S. Depart-ment of Energy. The work was performed at OakRidge National Laboratory, which is managed byUT-Battelle, LLC under Contract No. DE-AC05-00OR22725, and where TM is a Eugene P. WignerFellow. Work at Cincinnati was supported by theNSF Grant No. DMR-0113574.

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