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Computer Physics Communications 169 (2005) 394–399

www.elsevier.com/locate/cp

Quantum simulations of realistic systems by auxiliary fields

Shiwei Zhang∗, Henry Krakauer, Wissam A. Al-Saidi, Malliga Suewattana

Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA

Available online 12 April 2005

Abstract

To treat interacting quantum systems, it is often crucial to have accurate calculations beyond the mean-field levebody simulations based on field-theoretical approaches are a promising tool for this purpose and are applied in sefields of physics, in closely related forms. An major difficulty is the sign or phase problem, which causes the Montvariance to increase exponentially with system size. We address this issue in the context of auxiliary-field simulations oelectronic systems in condensed matter physics. We show how to use importance sampling of the complex fieldsthe phase problem. An approximate approach is formulated with a trial determinant to constrain the paths in field scompletely eliminate the growth of the noise. Forab initio electronic structure calculations, this gives a many-body approathe form of a “coherent” superposition of mean-field calculations, allowing direct incorporation of state-of-the-art techfrom the latter (non-local pseudopotentials; high quality basis sets, etc.). In our test calculations, single Slater detefrom density functional theory or Hartree–Fock calculations were used as trial wave functions, with no additional optimThe calculated dissociation energies of various molecules and the cohesive energy of bulk Si are in excellent agreeexperiment and are comparable to or better than the best existing theoretical results. 2005 Elsevier B.V. All rights reserved.

PACS:71.15.-m; 02.70.Ss; 31.25.-v

Keywords:Sign/phase problem; Quantum Monte Carlo; Constrained-path; Electronic structure;Ab initio many-body calculations;Auxiliary-fields; Functional integrals; Slater determinants

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1. Introduction

Quantum simulations based on field-theoreticalproaches are a promising general tool for studyinteracting many-body systems. They are used in cdensed matter physics, nuclear physics, high-en

* Corresponding author.E-mail address:shiwei@physics.wm.edu(S. Zhang).

0010-4655/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.cpc.2005.03.087

physics, and quantum chemistry. These methods[1,2]allow essentially exact calculations of ground-stand finite-temperature equilibrium properties of intacting many fermion systems. The required CPU tiscales in principle as a power law with system siand the methods have been applied to study a vaof problems.

Perhaps the largest obstacle for further, more uversal applications of these methods is the sign

.

S. Zhang et al. / Computer Physics Communications 169 (2005) 394–399 395

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phase problem for fermionic (and sometimes ebosonic[3]) systems, which results in an exponetially increasing noise and therefore breakdown ofmethod. In a recent paper[4], we addressed this issuusing the example of auxiliary-field simulations of ralistic electronic systems in condensed matter physWe showed the general principles of how to use imptance sampling of the complex determinants to conthe phase problem. An approximate approach wasmulated with a trial determinant to constrain the pain field space and eliminate the growth of the noiThe method was tested on atoms and molecules,in bulk Si using fcc supercells consisting of up toatoms (216 valence electrons). The results were incellent agreement with experiments and comparato the best existing theoretical results. Here we ga more general review of the method, and presentliminary results from some current applications.

2. Formalism

2.1. Ground-state projection

The Hamiltonian for any many-fermion systewith two-body interactions can be written in any onparticle basis in the general form

H = H1 + H2

(1)=M∑i,j

Tij c†i cj + 1

2

M∑i,j,k,l

Vijklc†i c

†j ckcl,

whereM is the size of the chosen one-particle baandc

†i andci are the corresponding creation and a

nihilation operators. Both the one-body (Tij ) and two-body matrix elements (Vijkl) are known.

To obtain the ground state|ΨG〉 of H , QMC meth-

ods use the imaginary time evolution operator e−τH

acting on a trial wave function|ΨT 〉: limn→∞(e−τH )n ·|ΨT 〉 ∝ |ΨG〉. |ΨT 〉 must not be orthogonal to|ΨG〉,and we will assume that it is of the form of a singSlater determinant or a linear combination of Sladeterminants. The time stepτ is chosen to be smaenough so thatH1 and H2 in the propagator can baccurately separated with the Trotter decompositio

The operation of e−τH1 on a Slater determinantstraightforward to calculate, and it simply yields aother determinant. The ground-state projection wo

therefore turn into the propagation of a single Sladeterminant if it were possible to write the two-bo

propagator e−τH2 as the exponential of a one-body oerator as well. This is realized in mean-field methoIn the Hartree–Fock approximationH2 is replaced byone-body operators times expectations with respethe current determinant, schematically:

(2)c†i c

†j ckcl → c

†i cl〈c†

j ck〉 − c†i ck〈c†

j cl〉.In the local density approximation (LDA) in densifunctional theory (DFT)[5], H2 is replaced by the density operator in real-space times a functional oflocal density. In both these cases, an iterative produre can be used to project out the solution to theproximate Hamiltonians, in the form of a single Sladeterminant.

2.2. Functional integral with auxiliary fields

It turns out that the two-body propagator e−τH2 canbe expressed, exactly, as a linear combination of obody evolution operators. Any two-body operator cbe written as a quadratic form of one-body operato

(3)H2 = −1

2

∑α

λαv2α,

whereλα is a real number andvα is a one-body operator. The Hubbard–Stratonovich (HS) transformtion [6] then allows us to write

(4)e12τλv2 = 1√

2π

∞∫−∞

e− 12σ2

e√

τσ√

λv dσ.

Introducing vector representationsσ ≡ σ1, σ2, . . .andv = √λ1v1,

√λ2v2, . . ., we obtain

(5)e−τH =∫

P(σ)B(σ)dσ,

whereP(σ) is the normal distribution in Eq.(4) and

(6)B(σ) ≡ e−τH1/2e√

τσ ·ve−τH1/2

is in the desired one-body form.The imaginary-time propagation thus requires ev

uating the multidimensional integral in Eq.(5) overtime slicesn and the corresponding auxiliary fieldMonte Carlo (MC) techniques are the only wayevaluate such integrals efficiently. We use a rand

396 S. Zhang et al. / Computer Physics Communications 169 (2005) 394–399

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walk approach[7]. In each step, a walker|φ〉, whichis a single Slater determinant, is propagated to aposition|φ′〉: |φ′(σ )〉 = B(σ)|φ〉, whereσ is a randomvariable sampled fromP(σ). After a sufficient numbeof steps (iterations), the ensemble of random walkis a MC representation of the ground-state wave fution: |ΨG〉 .= ∑

φ′ |φ′〉. The explicit form of the Slatedeterminants and some of the relevant propertiesgiven in Refs.[7,8].

Although the formalism of our method is indepedent of details ofvα and the transformation, it is important to emphasize that the quadratic form forH2 isnot unique. The number of terms in the sum, whichin general ofO(M2), depends on the particular breaup, as does the actual form ofvα . (In the applicationsin Section5, the number of fields isO(M).) Differentchoices lead to different HS decompositions, whichturn can affect the quality of the results (e.g., seveof the sign or phase problem).

3. Sign/phase problem

In condensed matter physics, auxiliary-field QMmethods have mostly been applied to lattice modwhere the form of the interaction is simple. The Hubard model is the best known example, whereinteraction in Eq.(1) is on-site:U

∑i ni,↑ni,↓. The

auxiliary-fields can be chosen as integers in this cresulting in one Ising-like field for each lattice sitEven in such cases, a sign problem generally occThe exceptions are systems (e.g., the half-filledpulsive Hubbard model, or the attractive model)which symmetry makes the integrand (the fermionterminant) in the path integral non-negative. A claof algorithms have been developed, based on funmental properties of the path integrals, to controlsign problem when the auxiliary-fields are real, bofor the ground state[7,9] and for finite-T , grand-canonical-ensemble simulations[10]. These methodeliminate the sign in the integrano by a constraintthe fermion determinant. They are approximate,have been shown to be quite accurate.

In general λα cannot be made all positive iEq. (3) [11]. (Although this is in principle possiblby an overall shift to the potential[2] or by introduc-ing many more auxiliary fields, they both cause lafluctuations[4].) The one-body vector operatorv is

Fig. 1. Illustration of the phase problem and constraints to ctrol it. The total valence energy (in Ry) of an fcc Si primitive c(2 atoms) is shown as a function of projection timeβ = nτ , withτ = 0.05 Ry−1. All except the square-dotted line were from 10 0walkers. Simple generalization of the constraint that worked welreal determinants leads to poor results. The new method givesrate results (note the agreement with the solid free projection cuwhich is exact, until the latter becomes too noisy atβ ∼ 1.5).

therefore complex. As the projection proceeds, thebitals in the random walkers will become complex.a result, the statistical fluctuations in the MC represtation of |ΨG〉 increase exponentially with projectiotime β ≡ nτ . This is the phase problem referredearlier. It is of the same origin as the sign problem toccurs whenB(σ) is real. The phase problem is mosevere, however, because for each|φ〉, instead of a+|φ〉 and−|φ〉 symmetry[7], there is now an infiniteseteiθ |φ〉 (θ ∈ [0,2π)) from which the random walkcannot distinguish. At largeβ, the phase of each|φ〉becomes random, and the MC representation of|ΨG〉becomes dominated by noise. InFig. 1, the curves la-beled “free projection” illustrate the phase problem

The phase problem here is unique becauseonly do the determinants acquire overall phases,the internal structures of their orbitals become coplex. The real-space analogy would be to have wers whose coordinates become complex. This mastraightforward generalization of real-space or reAF approaches ineffective. For example, similar toconstrained path approximation[7] we could imposethe condition Re〈ΨT |φ〉 > 0. Or, in the spirit of thefixed-phase approximation in real space[12] we couldproject the walker by including a factor cos(θ) inthe weight, whereθ is the phase of〈Ψ |φ′〉/〈Ψ |φ〉.

T T

S. Zhang et al. / Computer Physics Communications 169 (2005) 394–399 397

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They give similar results and are inaccurate[13]. Theformer is shown inFig. 1(“simple constraint”).

4. New method

4.1. “Importance sampling” transformation

To formulate a new method that can better serate the overall phase from the determinant, we fiborrow from the idea of importance sampling[14],although our choice of the so-called importance fution, 〈ΨT |φ〉, is actuallycomplex. We modify Eq. (5)to obtain the following new propagator for|φ〉:

(7)

∫ ⟨ΨT |φ′(σ − σ )

⟩P(σ − σ )B(σ − σ )

1

〈ΨT |φ〉 dσ,

where|φ′〉 is related to|φ〉 by: |φ′(σ − σ )〉 = B(σ −σ )|φ〉, and we have included a constant shift[15] σ

in the integral in Eq.(5), which does not affect thequality. Eq.(7) can be re-written as

(8)∫

P(σ)W(σ,φ)B(σ − σ )dσ,

where

(9)W(σ,φ) ≡ 〈ΨT |B(σ − σ )|φ〉〈ΨT |φ〉 eσ ·σ− σ ·σ

2 .

The new propagator in Eq.(8) defines a new random walk. In each step the walker|φ〉 is propa-gated to|φ′〉 by B(σ − σ ), whereσ is again sampledfrom P(σ). W(σ,φ) is a c-number which can be acounted for by having every walker carry an overweight factor and updating them according to:wφ′ =W(σ,φ)wφ . Formally the MC representation of|ΨG〉is now:

(10)|ΨG〉 .=∑φ′

wφ′|φ′〉

〈ΨT |φ′〉 .

For any choice of the shiftσ , the new random walkis an exact procedure to realize the imaginary tipropagation, in the sense of Eq.(10). The optimalchoice ofσ is the one that minimizes the fluctuatioof W(σ,φ) with respect toσ . ToO(

√τ) it is

(11)σ = −√τ

〈ΨT |v|φ〉〈ΨT |φ〉 .

With this choice the leadingσ -dependent term inWis reduced toO(τ ) and, by expandingB(σ − σ ) inEq.(9), we can manipulateW into:

W(σ,φ).= exp

[−τ

〈ΨT |H |φ〉〈ΨT |φ〉

]

(12)≡ exp[−τEL(φ)

].

The local energyEL and the shiftσ are both inde-pendent of any overall phase factor of|φ〉. The weightof the walker in the new random walk is determinby EL. In the limit of an exact|ΨT 〉, EL is a real con-stant, and the weight of each walker remains real.so-called mixed estimate for the energy is phasele

(13)EG = 〈ΨT |H |ΨG〉〈ΨT |ΨG〉

.=∑

φ′ wφ′EL(φ′)∑φ′ wφ′

.

With a general|ΨT 〉 which is not exact, a natural approximation is to replaceEL in Eqs.(12) and (13)byits real part, Re(EL). We have thus obtained a phasless formalism for the random walk, with real and poitive weights in Eqs.(10) and (13).

4.2. Projection

Despite this, an additional constraint is still rquired. To illustrate the problem we consider the ovlap 〈ΨT |φ′〉 during the random walk. Let us denothe phase of〈ΨT |φ′(σ − σ )〉/〈ΨT |φ〉 by θ , whichis in general non-zero (of order−σ Imag(σ )). Thismeans that, the walkers will undergo a random win the complex plane defined by〈ΨT |φ′〉. At largeβ

they will therefore populate the complex plane symetrically, independent of their initial positions. Ituseful to contrast the situation with the special casa real v. For any v the shift σ diverges as a walkeapproaches the origin in the complex plane, i.e.〈ΨT |φ′〉 → 0. The effect of the divergence is to mothe walker away from the origin. With areal v, θ = 0and the random walkers move only on the real aIf they are initialized to have positive overlaps wi|ΨT 〉, σ will ensure that the overlaps remain positithroughout the random walk, much like in fixed-nodiffusion Monte Carlo (DMC) in real space. Thusthis case the phaseless formalism reduces to thestrained path Monte Carlo method of Ref.[7], and italone is sufficient to control the sign problem. Focomplexv, however, the random walk is “rotational

398 S. Zhang et al. / Computer Physics Communications 169 (2005) 394–399

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h

tedrends

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invariant” in the complex plane, and the divergenceσ is not enough to prevent the build-up of a finite desity at the origin. Near the origin the local energyEL

diverges, which causes diverging fluctuations inweights of walkers. To address this we make anditional approximation. We project the random wato “one-dimension” and multiply the weight of eacwalker in each step by max0,cos(θ).

We have tested several alternative schemesprojection. (These are different from the “simpconstraint” inFig. 1, because they are appliedafterthe transformation in Section4.1.) One that seemeto work equally well was to project withexp−[Imag(σ )]2/2, which is the same as cos(θ)

in the limit of small θ . Another was to imposeRe〈ΨT |φ′〉 > 0, which gave similar results, but witsomewhat larger variance.

We note that the ground-state energy compuwith Eq. (13) is not variational, because of the natuof these approximations. The systematic error depeon |ΨT 〉, vanishing when|ΨT 〉 is exact.

5. Application in electronic structure

We employ a normalized plane-wave basis usperiodic boundary conditions withN electrons in asimulation cell of volumeΩ : |ψ〉 = 1√

Ωexp(iG · r),

where G is a reciprocal lattice vector. As in stadard plane-wave DFT calculations, our basis csists of plane-waves with|G|2/2 Ecut, where theparameterEcut is a cutoff energy. Norm-conservinLDA Kleinman–Bylander (KB) non-local pseudoptentials[16] are used to replace “pseudo-ions” suchSi4+. The one-body kinetic energy, local and non-lopseudopotential terms in the Hamiltonian are given

K = 1

2

∑i

G2i c

†i ci ,

(14)Vei,L =∑i =j

VL(Gi − Gj )c†i cj ,

Vei,NL =∑i,j

VNL(Gi ,Gj )c†i cj ,

with Gi andGj must both be within the plane-wavcut-off. The two-body terms in the Hamiltonian canmanipulated into the desired form of sum of squar

Fig. 2. Relative errors in atomization energies of di-atomic mocules in comparison with experiment. Our QMC results weretained using single determinant trial wave functions from LDStandard diffusion Monte Carlo (DMC) results[18] used optimizedJastrows times single- or multi-determinants as trial wave functiResults from the corresponding LDA calculations are also sho(DFT with a generalized gradient approximation (GGA) wouldbetter than LDA, typically between 10–15% higher than expments. Our results on O2 were obtained with GGA.)

The number of HS fields is given by the numbof uniqueQ-vectors, ofO(M). With the use of fasFourier Transforms (FFTs), the computational coseach MC step (advancing all fields simultaneously)a walker scales roughly asN2M lnM .

As Fig. 2 illustrates, applications to date on molcules have yielded very good accuracy compareexperiments. We have also studied bulk silicon wup to 54-atom (216 electrons) in a supercell, obtaing a cohesive energy of 4.59(3) eV after finite-scorrection[4], compared to 4.62(8) eV from experment. Particularly worth noting is that all our resuwere obtained with trial wave functions that are sgle determinants formed by orbitals from DFT LDcalculations (using ABINIT[17]), with no additionalparameters or optimization.

Without an exact solution to the sign/phase prlem, reducing the reliance on trial wave functionsclearly of key importance to increasing the predtive power of QMC. For continuum electronic systemsuch as our test cases above, fixed-node DMC haten been the most accurate theoretical method[19]. Itis encouraging that the new method gave comparresults to DMC, needing only simple mean-field sotions as the trial wave functions. Another advantagthe present method is in the straightforward and eximplementation of non-local pseudopotentials. In stdard DMC calculations a locality approximation[20]

S. Zhang et al. / Computer Physics Communications 169 (2005) 394–399 399

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6. Concluding remarks

We have presented a general framework. Vous possibilities exist for further improvement of tmethod, including improved|ΨT 〉, the use of alter-native one-particle basis, and/or the use of differHS transformations. The method provides an apprmate way to control the phase problem in all AF-baQMC methods, while retaining many of their advatages. We have shown that the method gave accuresults for systems from an atom to a large supercusing a simple trial wave function. We believe the geeral idea of the method is applicable to the differflavors of QMC based on field-theoretical approachincluding at finite-temperatures.

In condensed matter physics, it is hoped thatpresent method will allow quantum many-body siulations for a much wider variety of systems. Thincludesab initio simulations of real materials in thframework that we have demonstrated. It also inclulattice models with more realistic Hamiltonians, whemore complicated forms of interactions can be trea

Acknowledgements

We thank E.J. Walter, W. Purwanto, and H. Kwfor help and collaborations. This work is supportby NSF (DMR-9734041), ONR (N000149710049 aN000140110365), and the Research Corporation.acknowledge computing support by the CenterPiezoelectrics by Design, the NCSA at UIUC, andComputational Science Cluster at William and Mar

References

[1] R. Blankenbecler, D.J. Scalapino, R.L. Sugar, Monte Cacalculations of coupled boson–fermion systems. I, Phys. RD 24 (1981) 2278.

[2] G. Sugiyama, S.E. Koonin, Auxiliary field Monte-Carlo foquantum many-body ground states, Ann. Phys. (NY) 1(1986) 1.

[3] W. Purwanto, S. Zhang, Quantum Monte Carlo method forground state of many-boson systems, Phys. Rev. E (2004press, physics/0403146.

[4] S. Zhang, H. Krakauer, Quantum Monte Carlo method usphase-free random walks with Slater determinants, Phys.Lett. 90 (2003) 136401.

[5] W. Kohn, Nobel lecture: Electronic structure of matter—wafunctions and density functional, Rev. Mod. Phys. 71 (191253, and references therein.

[6] J. Hubbard, Calculations of partition functions, Phys. RLett. 3 (1959) 77.

[7] S. Zhang, J. Carlson, J.E. Gubernatis, Constrained path MCarlo method for fermion ground states, Phys. Rev. B(1997) 7464.

[8] S. Zhang, in: D. Sénéchal, A.-M. Tremblay, C. Bourbonn(Eds.), Theoretical Methods for Strongly Correlated ElectroCRM Series in Mathematical Physics, Springer, New Yo2003, pp. 39–74.

[9] S. Zhang, in: M.P. Nightingale, C.J. Umrigar (Eds.), QuantMonte Carlo Methods in Physics and Chemistry, Kluwer Ademic Publishers, 1999, cond-mat/9909090.

[10] S. Zhang, Finite-temperature calculations for systems wfermions, Phys. Rev. Lett. 83 (1999) 2777.

[11] P.L. Silvestrelli, S. Baroni, R. Car, Auxiliary-field quantuMonte-Carlo calculations for systems with long-range repsive interactions, Phys. Rev. Lett. 71 (1993) 1148.

[12] G. Ortiz, D.M. Ceperley, R.M. Martin, New stochastic methfor systems with broken time-reversal symmetry: 2d fermioin a magnetic field, Phys. Rev. Lett. 71 (1993) 2777.

[13] M.T. Wilson, B.L. Gyorffy, A constrained path auxiliary-fielquantum Monte Carlo method for the homogeneous elecgas, J. Phys. Condens. Matter 7 (1995) 371.

[14] M.H. Kalos, D. Levesque, L. Verlet, Helium at zero tempeture with hard-sphere and other forces, Phys. Rev. A 9 (19257.

[15] N. Rom, D.M. Charutz, D. Neuhauser, Shifted-contoauxiliary-field Monte Carlo: circumventing the sign difficultfor electronic-structure calculations, Chem. Phys. Lett.(1997) 382.

[16] A.M. Rappe, K.M. Rabe, E. Kaxiras, J.D. Joannopoulos, Omized pseudopotentials, Phys. Rev. B 41 (1990) 1227.

[17] X. Gonze, et al., First-principles computation of material prerties: the ABINIT software project, Comput. Mat. Sci. 2(2002) 478.

[18] J.C. Grossman, Benchmark quantum Monte Carlo calctions, J. Chem. Phys. 117 (2002) 1434.

[19] W.M.C. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, QuanMonte Carlo simulations of solids, Rev. Mod. Phys. 73 (2033, and references therein.

[20] L. Mitas, E.L. Shirley, D.M. Ceperley, Nonlocal pseudoptentials and diffusion Monte Carlo, J. Chem. Phys. 95 (193467.