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King’s Research Portal DOI: 10.1103/PhysRevB.99.205156 10.1103/PhysRevB.99.205156 Document Version Peer reviewed version Link to publication record in King's Research Portal Citation for published version (APA): Sheridan, E., Weber, C. R., Plekhanov, E., & Rhodes, C. (2019). Continuous-time quantum Monte Carlo solver for dynamical mean field theory in the compact Legendre representation. Physical Review B (Condensed Matter and Materials Physics), 99(20), [205156]. https://doi.org/10.1103/PhysRevB.99.205156, https://doi.org/10.1103/PhysRevB.99.205156 Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. •Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. •You may not further distribute the material or use it for any profit-making activity or commercial gain •You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 11. Oct. 2020

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Page 1: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

King’s Research Portal

DOI:10.1103/PhysRevB.99.20515610.1103/PhysRevB.99.205156

Document VersionPeer reviewed version

Link to publication record in King's Research Portal

Citation for published version (APA):Sheridan, E., Weber, C. R., Plekhanov, E., & Rhodes, C. (2019). Continuous-time quantum Monte Carlo solverfor dynamical mean field theory in the compact Legendre representation. Physical Review B (Condensed Matterand Materials Physics), 99(20), [205156]. https://doi.org/10.1103/PhysRevB.99.205156,https://doi.org/10.1103/PhysRevB.99.205156

Citing this paperPlease note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this maydiffer from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination,volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you areagain advised to check the publisher's website for any subsequent corrections.

General rightsCopyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyrightowners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights.

•Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research.•You may not further distribute the material or use it for any profit-making activity or commercial gain•You may freely distribute the URL identifying the publication in the Research Portal

Take down policyIf you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access tothe work immediately and investigate your claim.

Download date: 11. Oct. 2020

Page 2: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

A novel continuous time quantum Monte Carlo solver for dynamical mean field theoryin the compact Legendre representation

Evan Sheridan,1 Cedric Weber,1 Evgeny Plekhanov,1 and Christopher Rhodes∗2

1Theory and Simulation of Condensed Matter, Department of Physics,King’s College London, The Strand, London WC2R 2LS, UK.

2Theory and Simulation of Condensed Matter, Department of Physics,King’s College London, The Strand, London WC2R 2LS,UK. & AWE, Aldermaston, Reading, RG7 4PR, UK.

Dynamical mean-field theory (DMFT) is one of the most widely-used methods to treat accuratelyelectron correlation effects in ab-initio real material calculations. Many modern large-scale imple-mentations of DMFT in electronic structure codes involve solving a quantum impurity model with aContinuous-Time Quantum Monte Carlo (CT-QMC) solver [1–4]. The main advantage of CT-QMCis that, unlike standard quantum Monte Carlo approaches, it is able to generate the local Green’sfunctions G(τ) of the correlated system on an arbitrarily fine imaginary time τ grid, and is free ofany systematic errors. In this work, we extend a hybrid QMC solver proposed by Khatami et al. [5]and Rost et al. [6] to a multi-orbital context. This has the advantage of enabling impurity solverQMC calculations to scale linearly with inverse temperature, β, and permit its application to d andf band materials. In addition, we present a novel Green’s function processing scheme which gen-erates accurate quasi-continuous imaginary time solutions of the impurity problem which overcomeerrors inherent to standard QMC approaches. This solver and processing scheme are incorporatedinto a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations forSrVO3 properties are presented. The computational efficiency of this method is also demonstrated.

I. INTRODUCTION

Density functional theory (DFT), in both its local den-sity (LDA) and generalised gradient (GGA) approxima-tions, is a highly effective method for the calculation ofquantitatively accurate ab-initio ground-state propertiesof a wide range of real materials [8]. However, for somematerials DFT’s outstanding predictive capabilities be-come significantly less reliable. Such materials are foundtypically amongst the transition metal oxides, as wellas in lanthanide and actinide-based compounds. Thisis problematic because of the potential application ofthese materials in the fields of quantum computing, datastorage and high temperature superconductivity. Thesematerials characteristically have narrow bandwidths (W∼ 2− 3 eV), so the correlations induced by the Coulombinteraction between valence electrons (U ∼ 4− 5 eV) arestrong, implying that i.e. U/W & 1. When bandwidthsare broader the ratio of electron correlation to bandwidthis reduced (i.e. U/W . 1), so the effects of electron cor-relation are weaker. In the latter case, the typical ap-proximations involved at the DFT level work well, oftenwith a quantitative precision that sustains comparisonwith experiment.

A single unified framework is sought in which the ef-fects of electron correlation can be addressed for real ma-terial calculations over a wide range of temperatures andcorrelation strengths U/W . In recent years, a combina-tion of DFT and dynamical mean-field theory (DMFT)has proved to be an effective way of interpolating be-tween the itinerant and strongly localised limits of theelectronic behaviour. The DFT+DMFT approach has

been well established, and allows extending the capabil-ities of DFT-based approaches into calculations whereelectron correlation effects are significant. DFT+DMFThas evolved into a powerful method for dealing with cor-relation effects, and is being frequently used for material-specific applications. In addition to DFT+DMFT, theGW+DMFT approach is continuing to mature into an ef-fective way of undertaking material-specific calculationsfor correlated systems [9].

The critical component of a DFT+DMFT calculationis the so-called “impurity solver”, which represents thequantum many-body physics interactions of the corre-lated electrons in the material. It is essential that thesolver contains an accurate representation of the physicsof the interacting electrons, and can efficiently computesolutions in fast and stable way over a wide range of pa-rameters (U , W and T ). A variety of solvers are availablefor use in DMFT calculations, and they can be selected tomatch the computational resources available to completea calculation in a time-efficient way. These solvers can bebased on quasi-analytical methods (e.g. Hubbard I, Iter-ated Perturbation Theory (IPT), Non-Crossing Approx-imation (NCA) or fluctuation exchange approximation(FLEX)) or numerical methods (e.g. Numerical Renor-malisation Group (NRG), Exact Diagonalisation (ED),QMC) [10]. Of all these methods it is the quantum MonteCarlo methods which have proved to be particularly pop-ular. This is because QMC methods are conceptuallystraightforward and can return stable results (to statis-tical accuracy) over a wide range of parameter values.

A QMC technique known as continuous-time QMC(CT-QMC) [1–4] has emerged lately as a popular solver

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for many DMFT applications. The main advantage ofCT-QMC is that it overcomes one of the most signifi-cant limitations of conventional QMC methods - namely,the systematic errors which arise due the Trotter-Suzukidecomposition and the concomitant discretisation of theimaginary time interval. Errors of this kind can gener-ate substantial bias in the results of DMFT calculations.This is because they can significantly shift the convergedfixed point of the DMFT iteration [6]. However, a disad-vantage of most of the readily available CT-QMC solversis that the computational effort scales cubically with in-verse temperature β = 1/kBT , and this severely limitstheir access to low temperature phases. Moreover, be-cause the order of perturbation of CT-(HYB)QMC scaleswith the product of kinetic energy and β, it is costly toapply to systems far from the localisation-delocalisationtransition, especially at low temperatures. Although ourmethod is applicable over a large range of parameters, weconcede that in the limit of very large Coulomb repulsiona technique such as CT-HYB may be more effective. Formulti-orbital or cluster DMFT applications, the inversetemperature scaling limitation becomes even more sig-nificant. A key computational bottleneck in calculationsof fully charge self-consistent DFT+DMFT properties ofreal materials is the requirement to generate accurate im-purity model solutions on a fast time-scale with medium-scale computational resources. For such challenging ap-plications QMC methods remain a competitive techniquefor solving quantum impurity problems, and our inten-tion here is to introduce a complementary QMC tech-nique that has advantage for some applications. Also,when calculating a material equation of state, numerousDFT+DMFT total energy calculations are needed overa range of temperatures and pressures [12]. To performthese calculations to a level comparable with experimen-tal data demands accurate and computationally efficientimpurity solvers [13, 14].

In this paper we present a QMC solver which scaleslinearly in inverse temperature [5, 6] extended to a multi-orbital context, and supplement the solver with a robustLagrange/Chebyshev-based Green’s function interpola-tion technique that facilitates controlled extrapolationsof the impurity Green’s functions on multiple discre-tised imaginary time grids to a solution that is quasi-continuous in imaginary time. Because the solver self-energy is very sensitively dependent upon this Green’sfunction it is imperative that the impurity model solutionis an accurate one. In this way we are able to calculateaccurate quasi-continuous time quantum impurity modelsolutions for multi-orbital systems faster than CT-QMCsolvers, particularly at temperatures those solvers findchallenging to access.

The paper is organised as follows. In Section II weintroduce the QMC solver and describe how it is inte-grated into a DFT+DMFT scheme for real material cal-culations using the CASTEP plane-wave DFT code [7].

In Section III a new technique designed to generate ac-curate quasi-CT Green’s functions using fits of multiplelow-resolution QMC solutions is presented. Some of theparticular issues of fitting Green’s functions from realmaterials are discussed. In Section IV, to test the tech-nique fully, we undertake a full DFT+DMFT calculationon strontium vanadium oxide SrVO3 by integrating oursolver with both the CASTEP DFT code and the quasi-CT Green’s function scheme. Where appropriate, calcu-lations are base-lined against equivalent quasi-CT QMCcalculations using the TRIQS solver that is embedded inCASTEP [7] [15]. All energies and temperatures are ineV.

II. QMC IMPURITY SOLVER AND DFT+DMFTCALCULATIONS

1. The QMC algorithm

The purpose of a quantum impurity solver in a DMFTcalculation is to adequately capture the physics of spin-dependent electron-electron interactions, using either anumerical model Hamiltonian representation or a quasi-analytic approach [10]. The initial conditions for themodel are derived from a material-specific DFT calcu-lation, and then the solver solution is processed and fedback to the DFT code in an iterative cycle. The key toefficient DFT+DMFT calculations of real material prop-erties is finding a fast, accurate and stable solution to theimpurity model (without a fermionic sign problem [16])over a wide range of model parameters (U , J and W )and inverse temperatures, β.

Most early DFT+DMFT calculations employed aHirsch-Fye (HF) solution of the single impurity Ander-son model to represent electron correlation effects [10].The appeal of this solver is that it is QMC-based methodwhich can be applied over a broad parameter range andband fillings without being compromised by fermionicsign problems. However, this method contains a sys-tematic error inherent in the Trotter decomposition andimaginary time discretisation. Also, its inverse temper-ature scaling goes as β3. More recently HF-QMC hasbeen complemented by the CT-QMC method (and vari-ants thereof) discussed above. But, as noted there, it tooscales as β3. Another popular impurity solver is basedon exact diagonalisation (ED) of the impurity Hamilto-nian [17, 18]. The advantage of that method is that itgives an exact Green’s function solution and scales lin-early in β. However, it relies on a discretisation of theWeiss field (bath Green’s function), and the computa-tional load scales exponentially with the number of bathsites which can be limiting for applications beyond ap-plication to single site, single band, impurity models.

Khatami et al. [5] and later Rost et al. [6] introduceda novel Hamiltonian-based QMC scheme that combined

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the virtues of ED with a standard QMC-type solution,resulting in an impurity solver that scales cubically inthe number of discretised bath sites and linearly in in-verse temperature. Moreover, it has the same fermionsign performance as HF-QMC. Rost et al. [6] demon-strated the advantages of using this type of solver inDMFT studies of the single impurity model on a Bethelattice [10]. The solver still exhibits the systematic Trot-ter errors characteristic of QMC methods, but, as weshow below, these can be treated by the application ofa novel Green’s function interpolation and extrapolationtechnique which generates a quasi-CT solution.

The formulation of the QMC algorithm for single bandphysics is described in detail in [5] and [6]. In what fol-lows we will be extending the method in a multi-bandcontext. For the general case of electrons on a single im-purity site which interact both on and between 2M or-bital/spin pairs (where 1 < m ≤ 2M), the Hamiltonianis given by

H = H0LDA +

∑m

Ummnm↑nm↓ −∑mσ

µnmσ

+∑

m<m′σ

Umm′nmσnm′−σ

+∑

m<m′σ

(Umm′ − Jmm′)nmσnm′σ (1)

where Umm = U , Jmm′ = J , and Umm′ = U−2J for m 6=m′. Also, nmσ = c†mσcmσ where c

(†)mσ are the annihilation

(creation) operators for electrons in orbital m with spinσ. In this formulation there are Hubbard-like interactionsbetween opposite-spin electrons in the same orbitals andneighbouring orbitals, and Ising-like Hund’s coupling be-tween same-spin electrons on neighbouring orbitals. Thefull Slater-Kanamori Hamiltonian for multi-orbital inter-actions is rotationally invariant in spin-space [19] andalso includes “spin-flip” and “pair-hopping” terms, whichare likewise determined by ≈ Jmm′ . These interactionscannot be straightforwardly factorised into products ofquadratic operators nmσ, which is essential in any auxil-iary field QMC-based implementation. Various schemeshave been proposed to include these additional termsinto QMC formulations [19–24], but this is not whollystraightforward to do without introducing additional signproblems, especially at low temperatures. To obviatesuch sign problems we do not include spin-flip and pair-hopping terms in this work, and this is a current lim-itation of this method. However, future work willaddress this deficiency. ED and CT-QMC approachescan handle the full Hamiltonian, and this must be bornein mind when comparing results generated by differentmulti-orbital impurity solvers.

Our solver uses a multi-orbital generalisation of thestandard determinantal QMC procedure. For a multi-orbital interaction matrix

∑mm′ Umm′nmnm′ the usual

Hubbard-Stratonovich transformation can be applied,

i.e.

e−Umm′∆τ(nm−1/2)(nm′−1/2) =1

2

∑smm′=±1

eλmm′smm′ (nm − nm′) (2)

where λmm′ = acosh[e( 12 ∆τUmm′ )] and Umm′ is one of U ,

(U − 2J) or (U − 3J), depending on which orbital/spinpair is being considered. There is an auxiliary spin fieldss for each inter and intra orbital interaction.

Following conventional QMC procedure, at each imag-inary time slice each of the M(2M − 1) auxiliary spinfields is flipped sequentially and then tested for accep-tance or rejection using the ratio R = RmRm′ where

Rm = 1 + {(1−Gm)γm}Rm′ = 1 + {(1−Gm′)γm′} (3)

with γm = e2λmm′smm′−1 and γm′ = e−2λmm′smm′−1 . Ifthe spin flip is accepted the Green’s function is updatedaccording to the standard BSS [25] prescription

Gj,k,m = Gj,k,m − (δj,1 −Gj,1,m)γmG1,k,m/Rm

Gj,k,m′ = Gj,k,m′ − (δj,1 −Gj,1,m′)γm′G1,k,m′/Rm′ . (4)

If the spin flip is rejected the Green’s functions remainthe same. The condition for half-filling in this Hamilto-nian is given by µ = (U+(M−1)(U−2J)+(M−1)(U−3J))/2.

In the exact diagonalisation method [17] the infinitelattice surrounding the impurity site is approximated bya discretised bath of finite size Nb, and the identical pro-cedure is followed in this QMC solver. The first step isto parameterise the Weiss field G0 for each orbital andspin in terms of a finite number of bath parameters byapproximating the Weiss Green’s function in terms of anon-interacting Anderson impurity model as follows:

G−1And,m(iωn) = iωn + µ− εimpm −

k=Nb∑k=1

Vm,kV∗m,k

iωn − εm,k(5)

where k is the index for the bath level, m is the indexfor each orbital/spin. Essentially, this entails minimisingthe difference between the Weiss field and equation (5)using a cost function, typically

χ2[εk, Vk] =

n=nc∑n=0

An|GAnd(iωn : {εk, Vk})− G0(iωn)|2.

(6)In practice it has been found advisable to weight the costfunction towards smaller imaginary frequencies by usingpre-factor An ≈ 1/iω2

n. This avoids squandering fittingcost on the asymptotic regions of the Green’s function,and has been found particularly apposite when attempt-ing accurate fits to real material Green’s functions. A

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robust multi-dimensional fitting routine was used to de-termine the set of εk and Vk for each orbital, which aresubsequently used to parameterise the QMC solver. Toavoid falling into local minima, multiple fits over a widerange of initial conditions are performed and the best-fitis then selected.

Once a set of εk, Vk has been found to fit the Andersonimpurity Green’s function the QMC part of the calcula-tion proceeds as a standard Monte Carlo simulation for amulti-orbital context, where the imaginary time interval0 ≤ τ ≤ β is discretised into L time-slices (β = L∆τ).A sufficient number of Monte Carlo steps are run on aparallelised processor configuration, and the imaginarytime impurity Green’s functions are calculated for eachorbital and spin using the procedure of White et al. [26],and Loh and Gubernatis [27].

2. DFT+DMFT implementation with CASTEP

In our implementation of DFT+DMFT the plane-waveDFT code CASTEP was used [7]. It is a widely avail-able DFT code that can be readily used for calculat-ing ab-initio material properties. Previous work withCASTEP has demonstrated the integration of a HubbardI solver [15] and an ED-based solver [18] for material-specific calculations, and what follows builds on that.The multi-orbital solver described above was integratedwith CASTEP to better represent the electron correla-tion effects. Many of the issues associated with inte-grating an impurity solver with a DFT code are genericand a variety of methods have been used. Here wegive a description of our CASTEP implementation ofDFT+DMFT, and this can be compared to other ap-proaches.

A CASTEP calculation generates an Linear Combi-nation of Atomic Orbital (LCAO) basis set using eithernorm-conserving or ultra-soft pseudopotentials. Thesetwo cases can be dealt with on an equal footing by defin-ing an “overlap matrix”, S. The basis set functionsgenerated from norm-conserving pseudopotentials are or-thogonal by construction, so the overlap matrix is theidentity matrix. In the case of ultra-soft pseudopoten-tials these states are overlapping with a matrix:

〈χm′R′ |S|χmR〉 = δm′,mδR,R′ . (7)

This implies that the Kohn-Sham (KS) equations trans-form from a standard eigenvalue problem into a gener-alised one

HKSk |Ψk,ν〉 = Ek,νS|Ψk,ν〉 (8)

where Ψk,ν are the KS eigenstates.

3. Projectors

The standard DMFT technique is defined in terms ofcompletely localised electronic states (e.g. as in Hubbardmodel). On the other hand, in a wide class of ab-initiocodes, including CASTEP, the electrons are described interms of completely delocalised plane wave states. Con-sequently, a key feature of the DFT+DMFT calculationis the selection of a correlated sub-space of orbitals, andthe means to bridge the DFT Bloch space basis and thebasis of the correlated sub-space. To do this we definethe orthonormal projectors PL,ν(k), where

PL,ν(k) = 〈χL|S|Ψk,ν〉. (9)

We now have two distinct spaces - i) the KS Bloch spaceindexed by k and ν, and ii) the localised (correlated)subspace indexed by L.

To go from χL to Ψk,ν i.e. “up-folding” , we use

|ak,ν〉 =∑L

P ∗ν,L(k)|bL〉. (10)

Conversely, to go from Ψk,ν to χL i.e. “down-folding” ,we use

|bL〉 =∑k,ν

PL,ν(k)|ak,ν〉, (11)

where ak,ν is a vector defined in Bloch space and bL is avector defined in the space of correlated orbitals.

The projector matrix satisfies the following condition:

∑k,ν

PL,ν(k)P ∗ν,L′(k) = δL,L′ . (12)

In this scheme an up-folding operation followed by adown-folding operation is equivalent to an identity oper-ation. Both the up-folding and down-folding operationsare each used once in each DMFT iteration cycle, as wenow discuss.

4. The DMFT cycle

To begin the DMFT iteration for SrVO3 we useCASTEP to calculate the Bloch Green’s function

GBν,ν′(k, iωn) = {(iωn + µ− εk,ν)δν,ν′ − ΣBν,ν′(k, iωn)}−1

(13)which is the Fourier transform F.T.[〈r|G|r〉], where

G(r, iωn) = (iωn + µ+1

2∇2 − νKS(r)− ΣB(r, iωn))−1.

(14)(An adjustment to the chemical potential µ is necessaryat this point in the cycle to ensure correct level of elec-tron occupancy - see Appendix A.) We now consider a

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correlated atom at location R. The basis functions in thecorrelated sub-space are denoted by m. The local Green’sfunction for the correlated sites is now obtained from theBloch Green’s function by down-folding and then aver-aging over the Brillouin zone, as follows:

Glocm,m′(iωn) =1

Nk

∑ν,ν′,k

Pm,ν(k)GBν,ν′(k, iωn)P ∗ν′,m′(k).

(15)We now make the identification of Gloc with Gimp - soin the DMFT impurity Dyson equation we calculate theWeiss field

[G0(iωn)]−1m,m′ = Σimpm,m′(iωn) + [Gimp(iωn)]−1

m,m′ . (16)

On the first iteration we make a guess for the self-energy(typically Σimp = 0). We then parameterise the Weissfield [G−1

0 ]m,m′ using equation (5) for G−1And.

It is now possible to run the multi-orbital QMC solver(described above) to obtain a set of impurity Green’sfunctions (GQMC(τ)). Following the interpolation andextrapolation procedure (described in Section III), theFourier Transformed Green’s functions are used to calcu-late a revised impurity self-energy, i.e.

Σimpm,m′ = [G0(iωn)]−1m,m′ − [GQMC(iωn)]−1

m,m′ . (17)

At this point it is necessary to make allowance for the“double counting” term, V DCm,m′ (see Appendix B), andthen up-fold the impurity self-energy back to Bloch space, i.e.

ΣBν,ν′(k, iωn) =∑m,m′

P ∗ν,m(k)(Σimpm,m′(iωn)−V DCm,m′)Pm′,ν′(k).

(18)The self-energy is purely local when expressed in the setof correlated orbitals, but it acquires momentum depen-dence when up-folded to the Bloch basis set. The up-folded self-energy is returned to equation (13), and theiteration sequence is continued until an acceptable levelof convergence is reached in the self-energy (or chemicalpotential).

System properties can also be calculated during theDMFT iteration cycle. Of particular value are the Bloch-level occupation matrix:

Ne =1

Nk

∑ν,k

∑n

GBν,ν′(k, iωn)eiωn0+

, (19)

the spectral density

Aν,ν′(k, ω) = − 1

πImGBν,ν′(k, ω), (20)

and the density of states (DOS)

D(ω) =1

Nk

∑ν,k

Aν,ν′(k, ω). (21)

To calculate the spectral density and DOS, the analyticcontinuation from imaginary (iωn) to real (ω) frequencyis made using the Pade approximation [28].

As well as our implementation here, which uses theBSS method to update the Green’s function, a version ofthe multi-orbital solver code has also been implementedwhich uses the Nukala et al. [29] fast updating method.The results are identical to BSS, but the Nukala methodwill be better suited to multi-orbital cluster DMFT ap-plications.

III. A NOVEL QUASI-CONTINUOUSIMAGINARY TIME SOLUTION

The conventional QMC method calculates imaginarytime Green’s functions G(τ) on a grid of (L) regularlyspaced τ points over the interval 0 ≤ τ ≤ β(= L∆τ).In DMFT calculations this Green’s function is Fouriertransformed and then used to calculate the self-energyof the impurity problem. However, any attempt to cal-culate the frequency-dependent self-energy Σ(iωn) usingthe raw QMC Green’s functions immediately generatestwo problems, both related to the fact that the Green’sfunction is calculated on a discrete imaginary time grid.Firstly, the Trotter decomposition implies that the result-ing self-energy will be in error - to first order ∼ U∆τ2.Secondly, attempting to Fourier transform the Green’sfunction GQMC(τ) as it stands will introduce aliasing er-rors in the transformed function GQMC(iωn). Aliasingarises from the fact that the Fourier transform of theimaginary time Green’s function is a periodic functionin imaginary frequency with poles at multiples of theNyquist frequency (2πL/β). This leads to weight beingfolded back into the transform from frequencies abovethe Nyquist frequency, and generates incorrect asymp-totic behaviour of GQMC(iωn).

In this section we present a novel multi-scale Green’sfunction processing technique that simultaneously ad-dresses these two issues. The method interpolates a set ofcomparatively coarse time-scale QMC impurity Green’sfunctions on to a fine-scale τ grid (to eliminate alias-ing problems), and extrapolates them to a generate aquasi-continuous imaginary time solution for the multi-orbital impurity problem (to eliminate systematic Trot-ter errors). This method generates Green’s functions andself-energies with the correct behaviour across all imag-inary frequencies that can be seamlessly integrated intothe DMFT iteration cycle.

Splining procedures (with and without quasi-CT ex-trapolation) have been used previously to process imag-inary time Green’s functions so they can be used inDMFT calculations [6]. In real material calculations theimaginary time Green’s function can substantially changeits magnitude over the space of few imaginary time in-tervals, ∆τ . For strong electron interactions and low

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temperature, this behaviour can be particularly acute.Because of this care must be taken when interpolat-ing and extrapolating Green’s functions so as not toinadvertently incorporate spurious processing patholo-gies into calculated Green’s functions. Moreover, QMCmethods generate noisy data which can cause additionalchallenges for splining schemes, so here we describe anew Green’s function splining and extrapolation schemewhich is robust and can fit noisy data before generatinga quasi-continuous time solution to the DMFT Hamil-tonian. As we will show subsequently, it is straightfor-ward to integrate this scheme into a full material-specificDFT+DMFT calculation.

1. Polynomial basis method

One way to reduce the systematic error introduced bythe Trotter decomposition is to simply reduce ∆τ by in-creasing the number of imaginary time steps L. How-ever, the price for an increasingly accurate representationof G(τ) is a significantly more computationally intensiveQMC calculation. For example, the ubiquitous Hirsch-Fye solver scales as ∼ L3.

The objective of our new Green’s function processingprotocol is to demonstrate that by using a novel polyno-mial basis interpolation for GQMC(τ), and an extrapola-tion scheme, we can perform QMC calculations on mul-tiple, relatively coarse, imaginary time grids to generatea quasi-continuous imaginary time solution.

We expand G(τ) in an arbitrary orthogonal polynomial

basis P(k)i [x(τ)] (e.g. Legendre, Chebyshev, etc) where i

is the polynomial order, k is the polynomial species andx(τ) = 2τ

β − 1 is the transformation from [−1,+1] to

[0, β]. The expansion is:

G(k)(τ) =∑i≥0

P(k)i [x(τ)]G

(k)i . (22)

To isolate the basis coefficients we apply the orthogonal-ity constraints obeyed by the polynomials, i.e.,∫ β

0

dτG(k)(τ)P(k)i′ [x(τ)]W (x(τ)) =∫ β

0

dτP(k)i [x(τ)]P

(k)i′ [x(τ)]W (x(τ))G

(k)i . (23)

The general orthogonality condition obeyed by the family

of polynomials P(k)i [x(τ)] is∫ β

0

dτP(k)i [x(τ)]P

(k)i′ [x(τ)]W (x(τ)) = W (i)δi,i′ , (24)

and so the basis coefficients can be calculated as

G(k)i =

1

W (τ)

∫ β

0

dτG(k)(τ)P(k)i [x(τ)]W (x(τ)). (25)

Here we restrict our analysis only to the Legendre poly-nomials, where W (τ) = 1 and W (i) = 1

2i+1 .Calculating Gi, allows us to express G(τ) on an arbi-

trarily fine imaginary time grid.It is imperative to first obtain a reliable representa-

tion of G(τ) in the Legendre basis. To achieve this, it ispossible to formulate a controlled fitting procedure thatuses the Legendre basis coefficients gl as parameters tobe adjusted to the raw QMC data.

2. Green’s function fitting procedure

Expressing G(τ) in the Legendre basis allows a fittingprocedure to be formulated, which amounts to the min-imisation of the function

min{gl}[GQMC(τ)−G{gl}FIT (τ)], (26)

where the fitted (model) function G{gl}FIT (τ) is

parametrised by the basis coefficients gl, i.e.

G{gl}FIT (τ) =

Nl∑l≥0

√2l + 1

βPl(x(τ))gl. (27)

It is straightforward to find G{Gl}FIT (τ) using the conjugate

gradient method, and since Nl (the number of Legendrecoefficients) is generally quite modest. In our case be-cause Nl ≈ 20, this procedure is exceptionally compu-tationally efficient. Moreover, by shifting the paradigmof dealing with a statistical problem, i.e. Monte Carlo,to that of an optimisation one, allows the advantage ofincluding a-priori information. This strategy becomesparticularly attractive when dealing with unrefined QMCdata on exceedingly coarse imaginary time grids. In thisrespect, the Legendre basis proves to be an extremelyuseful tool for two outstanding reasons; the first is itssimple relationship to the moments of the GQMC(iωn),and the second is due to the convergence properties of glin the Kernel Polynomial Method (KPM) [30]. The finalsignificant constraint that can be imposed on the param-eters gl in an a-priori fashion is the convexity of G(τ).We now discuss how incorporating this additional infor-mation on GFIT (τ) can reliably improve its accuracy.

In many-body calculations it is essential that theGreen’s function solutions have the correct high fre-quency tail. Often the high frequency tail, which hasa 1/iωn behaviour, is fitted onto the low frequency re-sult. However, in the Legendre basis there is an exactrelationship between the moments of the Green’s func-tion and the Legendre coefficients [31]. By introducinga set of Lagrange parameter penalty terms into equation(26) information on the tail can be included into the fit,thereby eliminating the need for an ad-hoc fit of the tail.The term added to equation (26) to ensure the correct

Page 8: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

7

high frequency tail asymptotic behaviour is

λ1

c1 +∑

l≥0,even

2√

2l + 1

βgl

+ λ2

c2 − ∑l≥0,odd

2√

2l + 1

β2gll(l + 1)

(28)

where λ1 is the Lagrange parameter controlling thefirst moment c1, and λ2 is the Lagrange parameter con-trolling the second moment c2. The inclusion of the twoLagrange penalty terms relies on knowing the values forc1 and c2 for G(iωn) before the fit is performed. For-tunately, for c1 it is known that in the high frequencylimit of GQMC(iωn), it behaves as 1/(iωn), and thereforec1 = 1. The equivalent considerations for c2 are some-what more involved, but it can be shown that

c2 = µ− ε+ Σ′(∞), (29)

where µ is the chemical potential, ε is the impurity leveland Σ′(∞) is the high frequency real asymptotic self-energy of an isolated impurity. Σ′(∞) is attainable bysolving the Anderson Impurity Model with a finite set ofbath orbitals; for simplicity, this can be achieved usingan ED-solver [18], or Hubbard I solver [10].

It is an unavoidable fact that the truncation of a func-tion in any polynomial basis that is, in principle, infinitecan lead to convergence issues. In the case where thereare non-differentiable points or singularities it is espe-cially problematic and can lead to the well establishedGibbs oscillations. The severity of the oscillations nearthese ill-defined points can be damped by the introduc-tion of a kernel kl in equation (27) such that gl → klgl.The process of truncating this series and modifying thebasis coefficients amounts convolving G(τ) with a kernelkl. Since G(τ) is continuously differentiable (except atits boundaries) it is possible to pick a kernel that willguarantee this behaviour, and in the process filter outany spurious noise. In this work, we concern ourselvesonly with the kernels of Dirichlet and Jackson.

3. Green’s function extrapolation for ∆τ2 scaling

Representing G(τ) in a polynomial basis is the firststage in the two-step quasi-continuous method by gen-erating an accurate parameterisation of the raw QMCGreen’s functions on an imaginary time grid of arbitraryresolution. The second step is the systematic removal ofthe Trotter error by engineering a well defined extrap-olation procedure of the Legendre basis coefficients ondifferent Green’s functions Gλ(τ), where λ is the imagi-nary time grid index, related to ∆τλ = β/Nλ, with Nλbeing the number of imaginary time points.

The procedure begins with defining a measure of erroron each grid and time point, i.e. Fλ(τi) such that,

Fλ(τi) =

Np∑l=1

[gλl − Tl]Pl[x(τi)] = α(τi)∆τ2λ , (30)

where Tl are Legendre basis coefficients of quasi-continuous Green’s function GQC-QMC(τ), absent of thesystematic Trotter error. The α(τi) are the scaling coef-ficients of each imaginary time point.

The motivation for defining such an object is that we

would like to find the set of coefficients {gλQC

l } such thatFλQC(τi) = 0. To find this set of coefficients we definethe following minimisation problem:

minTl,α(τi)

[Fλ(τi)− α(τi)∆τ

2], (31)

over the completely continuous target parameters Tland scaling coefficients α(τi). It is possible to simplifythis procedure by introducing the matrix Aλ(τi) suchthat,

Aλ(τi) =

Np∑l=1

[gλl − Tl]∆τ2

i

= α(τi), (32)

and noticing that across each grid λ that Aλ(τi) re-mains unchanged. As a result, it is possible to map theminimisation problem of equation (31) to that of a sim-pler one,

minTl

∣∣∣∣ Aλ(τi)

Aλ−1(τi)− 1

∣∣∣∣ , (33)

that is simply a minimisation with respect to the targetparameters Tl and not the scaling coefficients α(τi).

Of critical importance in generating a controlled es-timate of the high frequency tails of GQC-QMC(iωn),and thus also ΣQC-QMC(iωn), is the implementationof the additional constraint on the second moment ofGQC-QMC(iωn), as discussed earlier. In principle, it isonly sensible to add this constraint here, at the level ofthe Green’s function without any systematic error, sinceGQC-QMC(iωn) truly represents a physically relevant ob-ject while the Gλ(iωn) are inherently non-physical.

IV. APPLICATION TO SRVO3

1. DFT+DMFT for SrVO3

In this section we integrate the three main compo-nents of our DFT+DMFT scheme: the CASTEP DFT

Page 9: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

8

code, the multi-orbital QMC-based solver, and the quasi-continuous imaginary time Green’s function processingprotocol.

We apply this scheme to a full iterated DFT+DMFTcalculation of SrVO3 properties, and, where relevant,show comparison calculations made using the TRIQSCT-HYB solver.

The lack of any structural or magnetic phase transi-tion behaviour over a broad temperature range makesthe metallic transition metal oxide SrVO3 an ideal testmaterial against which to benchmark our CASTEP first-principles DFT+DMFT technique. There have been pre-vious experimental and theoretical investigations of thismaterial (using a variety of flavours of DFT+DMFT)against which results using our DFT+DMFT scheme canbe compared [32–35]. These studies highlight the neces-sity to explicitly account for electron correlation whencalculating material properties, beyond those given bybasic one-particle LDA. SrVO3 has a perovskite structurewith completely occupied oxygen 2p bands, and partiallyoccupied vanadium 3d bands.

2. Calculation of SrVO3 Green’s functions andself-energies

A basic electronic structure calculation for SrVO3 wascarried out in CASTEP. SrVO3 has a perovskite unit cell(space group of crystal = 221: Pm-3m, -P 4 2 3) withlattice parameters a = b = c = 3.8421A [35], giving aunit cell volume V = 56.72A3. We have used a 20 x 20x 20 Monkhorst-Pack k-point grid, with 550 irreduciblek-points. The pseudopotentials for all three elements Sr,V and O were taken from the C17 CASTEP set, and theplane-wave basis cut-off was automatically determinedto be 653.07 eV. The calculations were performed at atemperature T = 0.1 eV (β = 10 eV−1).

Figure 1 shows the LDA band structure and DoS cal-culation for SrVO3 obtained from CASTEP. The DoS forO(2p) and V (3d) orbitals are shown, along with the to-tal DoS. There is an isolated set of three partially oc-cupied bands around the Fermi level, which originatemainly from the triply degenerate vanadium t2g orbitals,(dxy, dxz, dyz). The contribution from V (eg) and O(2p)orbitals is minimal in the vicinity of εf .

Particularising our QMC solver to the three vanadiumt2g − d orbitals, M = 3, gives µ = 5U/2 − 5J , andM(2M − 1) = 15 auxiliary fields are required for themulti-orbital quantum Monte Carlo simulation.

Previous studies with ED-like solvers have used < 5bath levels to parameterise the Weiss field [36] [6] at in-verse temperatures of β ≈ 10. In our calculation we setthe number of bath levels Nb = 5. The QMC solver canin fact deal with a large number of bath orbitals, and testcalculations with Nb as great as 9 showed no significantchange to the accuracy of the impurity Green’s function

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8

DO

S

ω (eV)

O-pV-dTot

-8

-6

-4

-2

0

2

4

6

8

R Γ X M Γ

Energ

y (

eV

)

FIG. 1. (Upper panel) The corresponding metallic DoSfor perovskite SrVO3 from CASTEP LDA.. (Lower panel)CASTEP LDA bandstructure for perovskite SrVO3. The iso-lated set of three partially occupied bands around the FermiLevel (blue line) are formed from the almost triply degeneratevanadium t2g orbitals, (dxy, dxz, dyz).

at this temperature. It should be noted that the com-putational load of the QMC simulation scales as ∝ N3

b β[5]. Furthermore the fitting error of the parametrisedWeiss field can be systematically controlled and in thiscalculation was of the order of 10−6.

The results of the impurity QMC Green’s function areshown in Figure 2a for the first iteration of the DMFTcalculation for SrVO3 across the interval 0 ≤ τ ≤ β.Two QMC Green’s functions are shown for one of thet2g−d orbitals for imaginary time discretisations L = 50and L = 80, along with fits to these functions using themethod described in Section III. The parameters cho-sen for this calculation are U = 4, J = 0 and β = 10,and match those used in other DFT+DMFT studies ofSrVO3. That said techniques such as constrained-RPAhave been successfully used to give ab-initio estimates ofmodel parameters for this kind of application [37]. Theeffect of the Trotter error is evident, and in Figure 2b adetailed portion of the Green’s function is shown, alongwith the Green’s function that results from our integratedfitting and extrapolation procedure. In this example a setof grids L = 50, 60, 70, 80 were used. However identical

Page 10: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

9

results are also seen when using either L = 50, 60, 70 orL = 60, 70, 80, for instance. Additionally, a minimum ofNl = 22 Legendre polynomials are needed to correctly ex-press G(τ) across the interval, but especially so near theboundaries where its derivative is larger. For compari-son, the equivalent TRIQS CT-QMC result is also shown,and the agreement between the two results can be read-ily observed. This demonstrates that the QMC solvercombined with our interpolation-extrapolation methodis capable of generating quasi-continuous time solutionsto the impurity problem.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

G(τ

)

τ

L=50L=50 fit

L=80L=80 fit

CT-QMCExtrapolated

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

1 1.5 2 2.5 3 3.5 4

G(τ

)

τ

L=50L=50 fit

L=80L=80 fit

CT-QMCExtrapolated

FIG. 2. BSS-QMC impurity Green’s functions for a selectedt2g SrVO3 orbital at U = 4, J = 0, T = 0.1 on successive dis-crete imaginary time grids for the first DMFT iteration. Theextrapolation of the fitted functions, which use Nl = 22 Leg-endre polynomials, to ∆τ → 0 is shown. Also included is theCT-QMC result, which is free of Trotter error. The extrap-olation procedure is using the data obtained at respectivelyL = 50, 60, 70, 80.

To test the method further, an additional calculationwas performed for U = 8, J = 0.65 and β = 10. Fig-ure 3a shows the Green’s functions and their fits for thesame time discretisations as before, illustrating the en-

hanced Trotter error for this larger U value. In Figure3b a comparison with the extrapolated Green’s functionwith a QMC calculation for L = 200 is shown. It isexpected that the quasi-continuous solution will be veryclose to the very fine-scale (but computationally demand-ing) QMC calculation, and this is indeed the case.

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

G(τ

L=50L=50 fit

L=80L=80 fitL=200

Extrapolated

-0.08

-0.075

-0.07

-0.065

-0.06

-0.055

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

3 3.5 4 4.5 5 5.5 6 6.5 7

G(τ

)

τ

L=50L=50 fit

L=80L=80 fitL=200

Extrapolated

FIG. 3. BSS-QMC impurity Green’s functions for a selectedt2g SrVO3 orbital at U = 8, J = 0.65, T = 0.1.

The next step is the calculation of the self-energy,which involves Fourier transforming the Green’s functionfrom τ to iωn i.e.

Σ(iωn) = G−10 (iωn)−G−1

QMC(iωn) (34)

where G−1QMC is the inverse of the calculated quasi-

continuous time Green’s function. Figure 4 shows theself-energy result for the first DMFT iteration for a setof different L values at U = 8, J = 0.65, β = 10 . The ex-trapolated result, for both the real (Figure 4a) and imag-inary (Figure 4b) parts, is a close match to the fine-scaleQMC calculation. Clearly, discretisation of the imagi-nary time interval has a significant effect of self-energies.

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10

12

12.1

12.2

12.3

12.4

12.5

12.6

12.7

12.8

12.9

13

0 10 20 30 40 50 60

Σ' (iωn)(eV)

iωn (eV)

L=50L=60L=70L=80

L=200Extrapolated

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 10 20 30 40 50 60

Σ''

(iωn)(eV)

iωn (eV)

L=50L=60L=70L=80

L=200Extrapolated

FIG. 4. The DMFT self-energy at U = 8, J = 0.65 andT = 0.1 in the limit of ∆τ → 0 for the first DMFT iteration.The excellent agreement of the extrapolated data in both realand imaginary parts with the L = 200 result is a consequenceof the systematic nesting of the coarse-grid set of discreteBSS-QMC results, their fitting and subsequent extrapolation.

In Figure 5 the self-energy calculations are now takento full DMFT convergence. It is not computationally fea-sible to perform a fully converged self-consistent calcula-tion using an imaginary time discretisation of L = 200.As justified in Figure 4, by extrapolating a finite set ofcoarse discrete time grids it is, however, possible to repli-cate the continuous result with this method. Therefore,it is possible to converge the calculation using a subset ofsuccessively coarse grids that achieve the same accuracyas that of the more expensive fine grid calculation. Wesee that after three iterations the DMFT converges, asSrVO3 is relatively weakly correlated.

The QMC solver and its associated quasi-continuoustime protocol can be straightforwardly applied to ever-larger values of electron correlation. In Figure 6 calcu-lations are shown for U = 12, J = 0.65, β = 10. Forthis large U value the Trotter error on the Green’s func-

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0 10 20 30 40 50 60

Σ'' (iωn)(eV)

iωn (eV)

Iteration 1Iteration 3

Converged

12

12.1

12.2

12.3

12.4

12.5

12.6

12.7

12.8

0 10 20 30 40 50 60

Σ' (iωn)(eV)

iωn (eV)

Iteration 1Iteration 3

Converged

FIG. 5. The self-energy during a self consistent DMFTcycle of a selected t2g SrVO3 orbital at U = 8, J = 0.65and T = 0.1. SrVO3 is weakly correlated with a small quasi-particle weight, and it converges after five iterations.

tion calculation is non-negligible. For one of the orbitals,Figure 6a shows the raw QMC Green’s function (withinterpolations), along with an extrapolation to quasi-continuous time. Figure 6b shows that at this significantlevel of correlation the Trotter error is dramatically en-hanced in contrast to smaller values of U . As a result,it is necessary to extrapolate with a finer set of discretegrids, i.e. L= 80, 90, 100, while also increasing the size ofthe Legendre basis to Nl = 28 polynomials. In doing so,it is possible to recover the correct form of the extrapola-tion scaling and therefore remove the non-trivial Trottererror induced by such strong interactions.

We conclude this section with some observations on thecomparative computational efficiency of the TRIQS CT-QMC solver and the quasi-CT BSS-QMC solver methoddescribed above, using U = 4, J = 0 at β = 10.

Figure 7a compares the BSS-QMC calculation methodrun for 20 core hours to the 100 core hour CT-QMC(TRIQS) result. A factor five times speed-up gen-

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11

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

1 1.5 2 2.5 3

G(τ

)

τ

L=80L=80 fitL=100

L=100 fitL=200

Extrapolated

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

G(τ

)

τ

L=80L=80 fitL=100

L=100 fitL=200

Extrapolated

FIG. 6. Upper panel: (6a) BSS-QMC impurity Green’s func-tion in the strongly interacting limit of U = 12, J = 0.65 andT = 0.1. Lower Panel: (6b) The relative separation of thesuccessive grids is indicative of the severe systematic Trottererror present in BSS-QMC calculations for large values of U .Using the quasi-continuous orthogonal polynomial method,by extrapolating respectively the L = 80, 90, 100 grids withNl = 28 Legendre polynomials, illustrates how it can beremedied by comparing with the approximately continuousL = 200 data.

erates a smooth Green’s function of comparable accuracyto the TRIQS result. Figure 7b shows that the very smallresidual difference between these two results is resolvedby using a BSS-QMC calculation of 50 core hours.

The BSS-QMC calculations shown here use imaginarytime grids of L = 50, 60, 70 and 80 time slices, withNl = 22 Legendre fitting polynomials. To make a com-parison, the CT-QMC was run to achieve sufficient statis-tics to be of comparable accuracy to the BSS-QMC re-sult: these calculations used 104 imaginary time pointswith a binning of 200, for 106 update steps, for ∼ 100core hours.

In Figure 8 we run an even faster BSS-QMC calculation

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

G(τ

)

τ

CT-QMC (100 core hours)20 core hours

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0 0.5 1 1.5 2

G(τ

) τ

CT-QMC (100 core hours)20 core hours50 core hours

FIG. 7. Upper panel: (7a) Comparison of CT-QMC andBSS-QMC impurity Green’s functions over the full imaginarytime range. Lower Panel: (7b) Detail of the Green’s functionshowing the comparison of a 50 core hour calculation with theCT-QMC result.

using a grid of L = 40, 50, 60 and 70 imaginary timeslices, with Nl = 12 Legendre fitting polynomials, whichtakes 8 core hours. This represents more than an order-of-magnitude speed-up compared with TRIQS to achievea comparable result.

The computational advantage of the solver describedin this paper will be further enhanced at lower tempera-tures due to the advantageous temperature scaling of themethod over the CT-QMC approach.

V. CONCLUSIONS

In recent years the DFT+DMFT framework hasemerged as a powerful and effective procedure for under-taking ab-initio calculations of the properties of real ma-terials when electron correlation effects are a significantinfluence. The method builds on the well-established ca-pabilities of DFT, whilst seeking to enhance the way theelectron correlation problem is addressed by utilising aquantum many-body physics approach.

A big step towards demonstrating the full potentialof the DFT+DMFT scheme would be to attempt fullycharge self-consistent calculations for many more mate-

Page 13: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

12

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10

G(τ

)

τ

CT-QMC (100 core hours)8 core hours

FIG. 8. A calculation comparable with TRIQS can beachieved an order of magnitude faster using the BSS-QMCsolver.

rials. However, to do this requires computationally effi-cient and accurate quantum impurity solvers that workover a very broad range of parameters. The work pre-sented in this paper presents one possible implementationof a QMC-based solver that addresses this need. Addi-tionally, applications such as the calculation of materialequations of state, where many repeated DFT+DMFTcalculations are needed, also require very fast and accu-rate QMC solvers. The solver we have presented herehas the advantage of scaling linearly in inverse temper-ature, and moreover provides a quasi-continuous imagi-nary time solution. We have integrated the solver withthe popular DFT code CASTEP, thus opening the wayto fully charge self-consistent calculations on real mate-rials. Though the validation of this approach has beendone here against SrVO3 it will be of particular valuewhen used in the more challenging context of modellingthe properties of f -band materials, and particularly forcalculating materials’ equations of state where the avail-ability of an accurate and computationally efficient solverwill be essential.

ACKNOWLEDGEMENTS

The authors thank the EPSRC for support throughgrant EP/M011038/1, and AWE for support through itsFuture Technologies fund. This work used the ARCHERUK National Supercomputing Service, for which accesswas obtained via the UKCP consortium, and funded byEPSRC grant EP/P022561/1.

APPENDICES

Appendix A: Updating the chemical potential

During each cycle in the DMFT iteration it is neces-sary to adjust the chemical potential in the Bloch Green’sfunction (equation (13)) in order to maintain the overallcorrect number of electrons. Essentially, at each itera-tion a value of µ is sought that ensures the constancy ofthe overall electron occupancy, equation (19). A searchalgorithm employing the Brent method [38] is used to fixµ.

Appendix B: Double-counting correction

CASTEP calculations already include a partial repre-sentation of electron correlation effects through the DFTexchange-correlation potential. In order not to count thecontribution due to electron correlation twice the firststep is to subtract the effect of the DFT potential throughthe “double-counting” approximation.

Double counting is not unambiguously resolved inDFT+DMFT, and there is no incontestable way to per-form this correction. In CASTEP there is a choice ofthree double counting corrections: i) Fully localised limit(FLL); ii) Around mean-field limit, and iii) Held’s correc-tion [10] [39] [40],[15]. A selection can be made accordingto the modelling context. Taking each in turn:

i) FLL: In this approximation it is assumed that anorbital occupation is either 0 or 1 i.e. empty or full.Denoting Nσ =

∑m nmσ, Ntot =

∑σ Nσ and taking an

average value for U and J as follows

Uavg = U =1

(2l + 1)2

∑m,m′

Umm′

and

Uavg − Javg = U − J =1

2l(l + 1)

∑m,m′

(Umm′ − Jmm′),

the double counting is found to be

EDC =1

2UNtot(Ntot − 1)− 1

2J∑σ

Nσ(Nσ − 1).

Differentiating with respect to Nσ gives

V DCσ = U

(Ntot −

1

2

)− J

(Nσ −

1

2

).

This approximation is better suited to insulating sys-tems.

ii) AMF: Here it is assumed that the average occupa-tion of an orbital (nmσ) is independent of m, so that

nmσ = nσ ≡Nσ

2l + 1

Page 14: King s Research Portal - King's College London · into a full DFT+DMFT calculation using the CASTEP DFT code [7]. Benchmark calculations for SrVO 3 properties are presented. The computational

13

where Nσ is the total occupation of the impurity site(with spin σ) and l orbitals. The double-counting energyis given by

EDC = Un↑n↓ +2l

2l + 1

U − J2

(n2↑ + n2

↓)

and the potential by

VDC = U

(Ntot −

nσ2l + 1

)− J

(nσ −

nσ2l + 1

).

This approximation is better suited to metallic systems.iii) Held’s correction: In this approximation an average

Coulomb repulsion Uav is introduced as follows

Uav =U + (l − 1)(U − 2J) + (l − 1)(U − 3J)

2l − 1

where l is the degeneracy of the shell. The double count-ing and associated potential as given as

EDC =UavNtot(Ntot − 1)

2

and

V DCσ = Uav

(Ntot −

1

2

).

[email protected]

[1] A. N. Rubtsov, V. V. Savkin and A. I. Lichten-stein, Continuous-time quantum Monte Carlo method forfermions. Physical Review B 72, 035122 (2005).

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