Rotationally invariant slave-boson and density matrix embedding theory: A unified framework and a comparative study on the 1D and 2D Hubbard Model
Tsung-Han Lee, Thomas Ayral, Yong-Xin Yao, Nicola Lanata, Gabriel Kotliar
RRotationally invariant slave-boson and density matrix embedding theory:A unified framework and a comparative study on the 1D and 2D Hubbard Model
Tsung-Han Lee, Thomas Ayral,
1, 2
Yong-Xin Yao, Nicola Lanata, and Gabriel Kotliar
1, 5 Physics and Astronomy Department, Rutgers University, Piscataway, New Jersey 08854, USA Atos Quantum Lab, Les Clayes-sous-Bois, France Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Aarhus University, 8000, Aarhus C, Denmark. Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973, USA
We present detailed benchmark ground-state calculations of the one- and two-dimensional Hub-bard model utilizing the cluster extensions of the rotationally invariant slave-boson (RISB) mean-field theory and the density matrix embedding theory (DMET). Our analysis shows that the overallaccuracy and the performance of these two methods are very similar. Furthermore, we proposea unified computational framework that allows us to implement both of these techniques on thesame footing. This provides us with a new line of interpretation and paves the ways for developingsystematically new generalizations of these complementary approaches.
I. INTRODUCTION
Strongly correlated electron systems are still a mostchallenging problem in condensed-matter physics. In thisarea, quantum embedding approaches have proven to beinvaluable tools for studying their electronic structure. Inparticular, dynamical mean-field theory (DMFT) , den-sity matrix embedding theory (DMET) and their respec-tive cluster extensions have been successfully applied tomany interacting model Hamiltonians as well as to realmaterials . The common basic idea underlying theseschemes is to map the fully interacting lattice to a self-consistently determined impurity problem, for which afragment of the original lattice, termed cluster, is treatedas a correlated impurity coupled to a self-consistentlydetermined non-interacting bath. The accuracy can besystematically improved by increasing the reference clus-ter size towards the thermodynamic limit (TL) and thesize of the Hilbert space representing the non-interactingbath.Another important theoretical method widely usedfor studying strongly correlated electron systems is therotationally-invariant slave-boson theory (RISB) ,which is equivalent to the multi-orbital Gutzwiller ap-proximation at the mean-field level and generallyprovides predictions almost as accurate as DMFT (especially for the ground-state properties) while beingmuch less computationally demanding. Even if the foun-dation of the RISB mean-field theory is based on seem-ingly distinct ideas, it turns out that also this frameworkcan be viewed as a quantum-embedding theory. In fact,it has been recently shown that the RISB equationscan be cast, similarly to DMET, in terms of ground-state calculations of auxiliary impurity systems named“ embedding Hamiltonians ”, whose non-interacting bathis determined self-consistently based on the variationalprinciple. Subsequently, it has been also shown thatDMET can be formally recovered from the RISB equa-tion derived in Ref. 19 by setting to unity the variational parameters encoding the mass renormalization weights.RISB and DMET are especially essential for the sit-uations where the computational cost of DMFT be-comes prohibitively large due to the exponentially grow-ing Hilbert space and/or the sign problem in quantumMonte Carlo. This usually happens for the 5 f systems,where the crystal-field effects, spin-orbit-coupling inter-action and lattice relaxation have to be taken into ac-count simultaneously, and for the large-scale cluster sim-ulations of the Hubbard model. Many challenging prob-lems, such as the equations of state of elemental actinidesand the phase diagram of the high T c superconductors,rely on such approximations to gain a qualitative or evenquantitative understanding . Hence, it is of impor-tant interest to characterize the respective accuracy andperformance of these two approaches.Here we perform comparative RISB and DMET bench-mark calculations on the 1D and 2D Hubbard modelagainst the available exact solution and the DMET valuesextrapolated to the TL. Our numerical results indi-cate that the accuracy and the performance of these twomethods are very similar for all the quantities studied, e.g. , the total energy and local observables. Small differ-ences between the two methods are found only for smallcluster sizes, where RISB provides slightly more accuratepredictions for the local observables (such as occupancy,double occupancy and local moments) as well as for themetal-insulator transition in the 2D Hubbard model.Finally, we derive an alternative numerical imple-mentation of DMET featuring a modified RISB algo-rithm with mass renormalization weights set to unity ,which provides us with a new line of interpretationand paves the way for developing new generalizationsand synergistic combination of these approaches ( e.g. ,to systems at finite temperature and/or with inter-site electron-electron interactions or electron-phononinteractions ). This implementation makes italso possible to pattern an interface between densityfunctional theory (DFT) and DMET after previous a r X i v : . [ c ond - m a t . s t r- e l ] D ec DFT+RISB and DFT+DMFT works .The paper is organized as follows: The Hubbard modelis introduced in Sec. II. The RISB and DMET formalismand algorithmic structure are outlined in Sec. III. InSection IV are presented our benchmark simulation ofthe Hubbard model in 1D and 2D. Finally, Sec. V isdevoted to concluding remarks.
II. MODEL
Let us consider the 1D and 2D Hubbard model withthe nearest neighbor hopping, H = t (cid:88) σ, (cid:104) i,j (cid:105) c † iσ c jσ + (cid:88) i U n i ↑ n i ↓ , (1)where t is the hopping amplitude, i and j are the indicesfor the lattice sites, and the σ is the spin label, and U is the local Coulomb interaction. c ( † ) iσ is the annihilation(creation) operator for the electron at site i and spin σ .The cluster extensions of RISB and DMET are bothimplemented by tiling the original lattice with clustersof increasing size . Thus, the degrees of freedom of thesingle-band Hubbard model belonging to each cluster aretreated as a single impurity, i.e. , as if they were elemen-tary (orbital) degrees of freedom of a multi-orbital Hub-bard Hamiltonian represented as follows: H = (cid:88) (cid:104) ij (cid:105) ,α,β ˜ t αβij c † iα c jβ + (cid:88) i H loc [ { c iα , c † iα } ] , (2)where the indices i, j = 1 , ..., N /N c denote the enlargedunit cell, N is the total number of atoms and N c isthe number of atoms within each cluster and the labels α, β = 1 , ..., N c indicate the cluster spin and atom de-grees of freedom.In order to utilize the RISB and DMET theory, it isuseful to define the inter-cluster hopping matrix as fol-lows: ˜ t αβij = (cid:110) t αβij if i (cid:54) = j . (3)The terms corresponding to the intra-cluster hoppingparameters t iα,iβ are included within the operator H loc [ { c iα , c † iα } ], along with the chemical potential andthe local Coulomb interaction.In our calculations, the translational invariance is ex-ploited only partially, i.e. , we represent the hopping ma-trix defined as: ˜ ε αβ k = (cid:88) i e − i k · r i ˜ t αβi , (4) where the momentum k belongs to the reduced Brillouinzone (RBZ) of the enlarged unit cell containing the clus-ter. The resulting Hamiltonian in the momentum spaceis represented as follows: H = (cid:88) k ∈ RBZ ,α,β ˜ ε αβ k c † k α c k β + (cid:88) ˜ i H loc [ { c iα , c † iα } ] , (5)where H loc [ { c iα, c † iα } ] contains all the local one- and two-body terms. III. METHODS
As shown in Refs. 2, 27, and 30, the RISB and DMETground-state solution of the Hubbard Hamiltonian [Eq.(5)] is obtained by solving recursively two auxiliary sys-tems: (i) a non-interacting system termed “ effective-medium ” or “ quasiparticle Hamiltonian ” and (ii) an in-teracting embedding impurity problem called “ embeddingHamiltonian .”The structure of the effective-medium Hamiltonian isthe following: H eff = (cid:88) k ∈ RBZ (cid:104) R aα ˜ ε αβ k R † βb + λ ab (cid:105) f † k a f k b , (6)where ˜ ε k was defined in Eq. (4), R and λ are 2 N c × N c complex matrices (the factor 2 arises from the spin de-grees of freedom) and λ is Hermitian. As we are goingto show in Sec. III A, in RISB both R and λ are deter-mined self-consistently and their converged entries areconnected to the self-energy Σ( ω ) as follows: Σ( ω ) = − ω − R † RR † R + 1 R λ R † . (7)On the other hand, in DMET only the entries of λ (called u in the DMET literature) can vary while R = , i.e. ,the self-energy consist exclusively of the part representingthe on-site energy shifts: Σ( ω ) = λ, (8)see Sec. III A.The embedding Hamiltonian describes a multi-orbitaldimer molecule containing a correlated impurity c ( † ) α anda non-correlated bath f ( † ) a . It reads: H emb = H loc (cid:2) { c † α , c α } (cid:3) + (cid:88) αa (cid:0) D aα c † α f a + H.c. (cid:1) + (cid:88) ab λ cab f b f † a , (9)where H loc is defined in Eq. (2), D and λ c are 2 N c × N c complex matrices and λ c is Hermitian. The entries ofboth matrices are determined self-consistently ,see Secs. III A and III B. After convergence, the reduceddensity matrix of the impurity degrees of freedom (whichis formally obtained by tracing out the bath degrees offreedom) provides the local reduced density matrix of theoriginal physical system. In other words, the expectationvalue of any local operator ˆ O (cid:2) { c † α , c α } (cid:3) , such as the dou-ble occupancy or the local stagger magnetic moment, canbe calculated from the ground state wavefunction | Φ (cid:105) of H emb as follows: (cid:104) O (cid:105) = (cid:104) Φ | ˆ O (cid:2) { c † α , c α } (cid:3) | Φ (cid:105) . (10) A. Rotationally invariant slave-boson mean-fieldtheory
The RISB theory is, in principle, an exact reformu-lation of the Hubbard system constructed by introduc-ing auxiliary “ slave ” bosons coupled to “ quasiparticle ”fermionic degrees of freedom.
As shown in Ref. 27,the RISB mean-field theory is entirely encoded in thefollowing Lagrange function: L [ | Φ (cid:105) , R, λ, ∆ p ; E c , D , λ c ] = − β N c N (cid:88) k ∈ RBZ (cid:88) iω n Tr log (cid:2) iω n − R aα ˜ ε αβ k R † βb − λ ab (cid:3) e iω n + + (cid:88) i Tr (cid:104) E c ( (cid:104) Φ | Φ (cid:105) −
1) + (cid:104) Φ | H emb | Φ (cid:105) (cid:105) − (cid:88) iab (cid:0) λ ab + λ cab (cid:1) ∆ pab − (cid:88) icaα (cid:0) D aα R cα + c.c (cid:1)(cid:2) ∆ p (1 − ∆ p ) (cid:3) / ca , (11)where: R and λ are the renormalization coefficients of thequasiparticle Hamiltonian introduced in Eq. (6), H emb , D and λ c are the parameters of the embedding Hamiltonianintroduced in Eq. (9), | Φ (cid:105) is the ground state wavefunc-tion of H emb , E c is a Lagrange multiplier enforcing thenormalization of | Φ (cid:105) and ∆ p is the local density matrix of H eff (see Eq. (12)).The self-consistency conditions determining the pa-rameters of H emb and H eff , see Eqs. (6) and (9), are ob-tained by extremizing the mean-field Lagrange functionwith respect to | Φ (cid:105) , R, λ, ∆ p , E c , D , and λ c , whichleads to the following equations:∆ pab = N c N (cid:88) k ∈ RBZ (cid:2) f T ( R ˜ ε k R † + λ ) (cid:3) ba , (12) (cid:2) ∆ p (1 − ∆ p ) (cid:3) / ac D cα = N c N (cid:88) k ∈ RBZ (cid:2) ˜ ε k R † f T ( R ˜ ε k R † + λ ) (cid:3) αa , (13) (cid:88) cbα ∂∂d ps (cid:2) ∆ p (1 − ∆ p ) (cid:3) cb (cid:2) D (cid:3) bα (cid:2) R (cid:3) cα + c.c + (cid:2) l + l c (cid:3) s = 0 , (14) H emb | Φ (cid:105) = E c | Φ (cid:105) , (15) (cid:104) F (1) (cid:105) ab ≡ (cid:104) Φ | f b f † a | Φ (cid:105) − ∆ pab = 0 , (16) (cid:104) F (2) (cid:105) αa ≡ (cid:104) Φ | c † α f a | Φ (cid:105) − R cα (cid:2) ∆ p (1 − ∆ p ) (cid:3) ca = 0 . (17)where the symbol f T stands for the Fermi function of asingle-particle matrix at temperature T and we utilizedthe following matrix parameterizations:∆ p = (cid:88) s d ps t h s , (18) λ c = (cid:88) s l cs h s , (19) λ = (cid:88) s l s h s , (20) R = (cid:88) s r s h s , (21)where the set of matrices h s are an orthonormal basisof the space of Hermitian matrices (with respect to thecanonical trace inner product). The parameters d ps , l cs and l s are real, while r s is complex. The RISB saddle-point equations can be solved as follows: Embedding Mapping E ff ective Medium Embedding Hamiltonian Hartree-Focknon-interacting density matrix interacting density matrixExact-diagonalizationMinimizing di ff . inmatrix elementsDMET: R= Figure 1. Schematic representation of the RISB and DMETalgorithm. The black boxes denote the extra constraints forthe DMET algorithm.
1. Starting with an initial guess of R and λ , compute∆ p from Eq. (12).2. From ∆ p , calculate D from Eq. (13).3. With D and ∆ p , compute λ c from Eq. (14).4. From D and λ c , construct H emb from Eq. (9) andcalculate its ground state | Φ (cid:105) .5. From | Φ (cid:105) and ∆ p , calculate Eqs. (16) and (17) and utilize quasi-Newton methods to estimate the new R and λ .6. The convergence is achieved if Eqs. (16) and (17)are satisfied. Otherwise, continue the root search-ing with the new R and λ .This structure is summarized schematically in Fig. 1.Note that the Lagrange function [Eq. 11] evaluated forthe converged parameters reduces to: E = (cid:88) k ∈ RBZ (cid:88) ab (cid:2) R ˜ ε k R † f T ( R ˜ ε k R † + λ ) (cid:3) ab + (cid:88) i (cid:104) Φ | H i,loc (cid:2) c † iα , c iα (cid:3) | Φ (cid:105) , (22)which is the total energy of the system. It can bestraightforwardly verified that, as long as Eqs. (12)-(17)are satisfied, the total energy can be equivalently ex-pressed also as follows: E = (cid:88) i (cid:104) Φ | (cid:88) αa ( D αa c † α f a ) + H i,loc [ { c † α c α } ] | Φ (cid:105) . (23) B. Density matrix embedding theory
The self-consistency conditions determining the pa-rameters of H emb and H eff in DMET can be formulatedas follows: ∆ pab = N c N (cid:88) k ∈ RBZ (cid:2) f T (˜ ε k + λ ) (cid:3) ba , (24) (cid:2) ∆ p (1 − ∆ p ) (cid:3) / ac D cα = N c N (cid:88) k ∈ RBZ (cid:2) ˜ ε k f T (˜ ε k + λ ) (cid:3) αa , (25) (cid:88) cb ∂∂d ps (cid:2) ∆ p (1 − ∆ p ) (cid:3) cb (cid:2) D (cid:3) bc + c.c + (cid:2) l + l c (cid:3) s = 0 , (26) H emb | Φ (cid:105) = E c | Φ (cid:105) , (27) (cid:104) F (1) (cid:105) ab ≡ (cid:104) Φ | f b f † a | Φ (cid:105) − ∆ pab , (28) (cid:104) F (2) (cid:105) αa ≡ (cid:104) Φ | c † α f a | Φ (cid:105) − (cid:2) ∆ p (1 − ∆ p ) (cid:3) αa , (29) (cid:104) F (3) (cid:105) αβ ≡ (cid:104) Φ | c † α c β | Φ (cid:105) − ∆ pαβ , (30) λ min := argmin λ (cid:0) (cid:107)F (1) (cid:107) F + (cid:107)F (2) (cid:107) F + (cid:107)F (3) (cid:107) F (cid:1) , (31)where the symbol (cid:107) ... (cid:107) F in Eq. 31 indicates the Frobe-nius norm. Note that Eqs. (24)-(29) are equivalent toEqs. (12)-(17) with R = and the constraint Eq. (30)was originally considered also in the Gutzwiller approxi-mation (equivalent to RISB), but later was found to be unnecessary .The DMET equations can be solved as follows, seeFig. 1:1. Starting with an initial guess of λ , calculate ∆ p using Eq. (24). E / t (a) DMET U = 1 tU = 4 tU = 8 t n E / t (b) RISB U = 1 tU = 4 tU = 8 t BANc=1Nc=2Nc=4
Figure 2. Energy
E/t for (a) DMET and (b) RISB as afunction of occupancy n in the 1D Hubbard model with thenearest neighbor hopping at U = 1 t, t, t for cluster size N c = 1 , ,
4, indicated by the blue solid, green dashed, andred dotted lines, respectively. The solid black lines denote theresults from BA.
2. Compute D and λ c from Eq. (25) and Eq. (26)and construct the H emb .3. Compute the ground state | Φ (cid:105) and the correspond-ing single-particle density matrix, i.e. : (cid:104) Φ | f b f † a | Φ (cid:105) , (cid:104) Φ | c † α f a | Φ (cid:105) and (cid:104) Φ | c † α c β | Φ (cid:105) .4. From (cid:104) Φ | f b f † a | Φ (cid:105) , (cid:104) Φ | c † α f a | Φ (cid:105) and (cid:104) Φ | c † α c β | Φ (cid:105) , de-termine the entries of λ min that minimize Eq. 31 (note that such a minimum is generally larger thanzero in interacting systems ).5. Iterate until λ min is converged.A quasi-Newton method is usually utilized to acceleratethe convergence of DMET iteration. Once convergenceis reached, the DMET total energy is computed from Eq.(23). IV. RESULTS
Here, we benchmark RISB and DMET with clustersizes N c = 1 , , , n (a) DMET U = 1 tU = 4 tU = 8 t µ/t n (b) RISB U = 1 tU = 4 tU = 8 t BANc=1Nc=2Nc=4
Figure 3. Occupancy n for (a) DMET and (b) RISB as afunction of chemical potential µ in the 1D Hubbard modelwith the nearest neighbor hopping at U = 1 t, t, t for clustersize N c = 1 , ,
4, indicated by the blue solid, green dashed,and red dotted lines, respectively. The solid black lines denotethe results from BA.
A. 1D Hubbard model
In Fig. 2 the DMET and RISB behaviors of the ener-gies as a function of the occupation n for U = 1 t, t, t with N c = 1 , , solutions. Overall, the DMETand RISB approximations to the total energies are verysimilar for all cluster sizes, and both techniques repro-duce the BA results with less than 2% error alreadyfor N c = 4. The only difference observed is that theDMET energies are slightly more accurate at half-filling,while the RISB energies are more accurate away fromhalf-filling.In Figure 3 are shown the behaviors of the DMET andRISB occupancies n as a function of the chemical poten-tial µ for U = 1 t, t, t with N c = 1 , ,
4, in com-parison with the BA. The Mott insulating phase is char-acterized by a constant n with compressibility dndµ = 0.At the Mott insulator-metal transition point µ c the com-pressibility dndµ diverges . In the metallic phase, n de-creases monotonically by decreasing µ . We observe thatboth DMET and RISB capture the correct behavior for N c ≥
2. Moreover, RISB yields more accurate n and µ c at N c = 2. However, at N c = 4 both DMET and RISBpredicts very precise occupancy and µ c with less than 5%error. › n ↑ n ↓ fi (a) DMET0 2 4 6 8 10 U/t › n ↑ n ↓ fi (b) RISB BANc=1Nc=2Nc=4
Figure 4. Double occupancy (cid:104) n ↑ n ↓ (cid:105) for (a) DMET and (b)RISB as a function of interaction U in the half-filled 1DHubbard with the nearest neighbor hopping for cluster size N c = 1 , ,
4, indicated by the blue solid, green dashed, andred dotted lines, respectively. The solid black lines denote theresults from BA.
In Fig. 4 are shown the behaviors of the DMET andRISB double occupancies (cid:104) n ↑ n ↓ (cid:105) with N c = 1 , ,
4, incomparison with the BA. At N c = 1 the DMET solutionsare always metallic for every U ; consequently, the doubleoccupancy deviates from the BA results at large U . Onthe other hand in RISB, the double occupancy vanishesat the critical point U c ∼ t , i.e. , the charge fluctuationsare not captured in the Mott phase . For N c = 2 bothmethods predict behaviors of (cid:104) n ↑ n ↓ (cid:105) that closely followthe BA values, although RISB is slightly more accurate.At N c = 4, both methods are very accurate with lessthan 7% error compared to BA.We also analyze the convergence of the energy as afunction of cluster size at filling n = 1 and n = 0 .
75 with U = 4 t and U = 8 t for DMET and RISB as shown in Fig.5. DMET gives a better estimation for the ground-stateenergy at half-filling, while RISB yields more accurateenergies at n = 0 .
75. However, as the cluster size grows,both methods converge to the BA value rapidly. Ourresults are consistent with the data extracted from Ref.13, where an antiferromagnetic ground state was assumed(in 1D the ground state is non-magnetic). E / t BA(a) U=4t n=10.350.300.250.200.150.100.05 E / t BA(b) U=8t n=10.780.800.820.84 E / t BA(c) U=4t n=0.750.2 0.4 0.6 0.8 1.0 /Nc E / t BA(d) U=8t n=0.75
DMET PMRISB PMZheng et. al AFM
Figure 5. Energy
E/t as a function of inverse cluster size 1 /N c in the 1D Hubbard model with the nearest neighbor hoppingfor (a) U = 4 t and n = 1, (b) U = 8 t and n = 1, (c) U = 4 t and n = 0 .
75, and (d) U = 8 t and n = 0 .
75. The blue circlescorrespond to the DMET values in our simulation. The redsquares are our RISB results. The green triangles are the datafrom Zheng et al. with antiferromagnetic order . The blacksolid lines are the results from BA. B. 2D Hubbard model (a) (b)(c) (d)
Figure 6. Clusters with sizes (a) N c = 1, (b) N c = 2, (c) N c = 4, and (d) N c = 6, used in our simulation. The redarrows indicate the lattice vectors. The blue lines delimit theunit cells. Figure 7. Energy
E/t for (a) DMET and (b) RISB as a func-tion of interaction U in the half-filled 2D Hubbard model ona square lattice with the nearest neighbor hopping at clustersize, N c = 1 , ,
4, indicated by the blue, green, and red line,respectively. The solid, dashed, and dotted lines representthe PM metal, PM insulator, and AFM solutions, respec-tively. The critical interaction U c is indicated by the verticleline. The black solid circles indicate the results in the TLfrom Ref. 13 and 15. The grey arrow indicates the U c fromCellular-DMFT with N c = 4 in Ref. 43. The inset of (a)shows the magnified plot around U c . Here we investigate the behaviors of the RISB andDMET solutions of the 2D Hubbard model on a squarelattice with cluster sizes N c = 1 , , ,
6, see Fig. 6.These geometries are chosen so that the antiferromag-netic (AFM) ground state can be reproduced for N c ≥ E as a function of the Hubbard in-teraction U at half-filling n = 1 in the PM metal, PMinsulating and AFM insulating phase, with cluster sizes N c = 1 , , N c = 1, DMET does not capture the Mott metal-insulator transition (MIT), i.e. , it predicts a metallic so-lution for every value of U. On the other hand, RISBpredicts a MIT at U c = 12 . t , where the total energyvanishes . For N c ≥
2, both methods capture a MIT,as indicated by the crossing of the PM metal and PMinsulator energies. Moreover, the energies of the AFM n (a) DMET PM8 6 4 2 0 2 4 6 8 µ/t › n ↑ n ↓ fi (b) RISB PM Nc=2Nc=4
Figure 8. Occupancy n as a function of chemical potential µ in the PM phase of the 2D Hubbard model on a square latticewith the nearest neighbor hopping at U = 12 t for cluster sizes N c = 2 and 4, indicated by the green dashed and red dottedline, respectively. solutions are lower than the PM solutions, consistentlywith previous studies .It is also interesting to see how U c varies with the clus-ter size. We observe that in DMET U c is almost inde-pendent of the cluster size, e.g. , U c = 8 . t for N c = 2and U c = 9 . t for N c = 4. On the other hand, in RISB U c decreases from 12 . t for N c = 1 to 6 . t for N c = 4(which is very close to the CDMFT value U c = 6 . t forthe same cluster size ).Figure 8 shows the DMET and RISB occupancy n asa function of chemical potential µ at U = 12 t with N c =2 ,
4. We observe that in DMET the difference in theoccupancy and the µ c between N c = 2 and N c = 4 islarge, while in RISB, the discrepancy between the twocluster sizes is small (less than 3% error). We concludethat RISB provides a slightly better description of thePM solutions.The ground-state energy predicted from DMET andRISB are shown in Tabs. I and II for n = 1 AFM phaseand n = 0 . U and N c . Our numerical values are compared to the DMETresults at N c = 4 and in the TL in Refs. 15 and 13, whichare also shown as black solid dots in Fig. 7 at n = 1.We observe that at half-filling n = 1 DMET gives over-all more accurate predictions to the ground-state energiesin the AFM phase compared to the TL energies (seeTab. I and Fig. 7). However, the discrepancy betweenthe two methods is already small at N c = 4 (less than 3%error). Away from half-filling ( n = 0 . N c = 2 N c = 4 N c = 6 N c = 4 Ref. 13 TL Ref. 15Method DMET RISB DMET RISB DMET RISB DMET DMET U/t = 2 -1.1804 -1.1673 -1.1790 -1.1693 -1.1790 -1.1704 -1.179 -1.1764
U/t = 4 -0.8681 -0.8428 -0.8654 -0.8459 -0.8658 -0.8472 -0.863 -0.8604
U/t = 6 -0.6541 -0.6306 -0.6545 -0.6362 -0.6553 -0.6376 -0.652 -0.6562
U/t = 8 -0.5115 -0.4942 -0.5155 -0.5023 -0.5157 -0.5100 - -0.5234
U/t = 12 -0.3497 -0.3400 -0.3566 -0.3487 -0.3563 -0.3565 - -0.3685Table I. Energy
E/t for DMET and RISB in the AFM phase of the 2D Hubbard model at half-filled n = 1 with the nearestneighbor hopping for N c = 2 , , U = 2 t, t, t, t, t . The values in the last two columns are the soltions at N c = 4and in the TL extracted from Ref. 13 and 15. N c = 2 N c = 4 N c = 6 TL Ref. 15Method DMET RISB DMET RISB DMET RISB DMET U/t = 2 -1.312 -1.300 -1.309 -1.302 -1.310 -1.302 -1.306
U/t = 4 -1.129 -1.083 -1.122 -1.086 -1.120 -1.091 -1.108
U/t = 6 -1.015 -0.927 -1.002 -0.938 -1.002 -0.942 -0.977
U/t = 8 -0.950 -0.823 -0.932 -0.838 -0.923 -0.846 -0.880Table II. Energy
E/t for DMET and RISB in the PM phase of the 2D Hubbard model at n = 0 . N c = 2 , , U = 2 t, t, t, t . The values in the last two columns are the solutions at N c = 4 and in the TLextracted from Ref. 15. energies predicted by RISB and DMET are equally accu-rate compared to the energies in the TL . Our DMETresults are consistent with previous studies .The double occupancies (cid:104) n ↑ n ↓ (cid:105) at n = 1 in the AFMphase with different N c and U are shown in Tab. III.DMET yields slightly more precise double occupancy at N c = 2 for smaller U compared to the TL results . How-ever, for N c = 4, both methods obtained very accuratedouble occupancy close to the TL (less than 3% error).In Tab. IV we present the prediction of the AFM mag-netic moment m for both methods with different clustersizes N c and U . Overall, we found the DMET and RISBmagnetic moment are very similar, with RISB slightlycloser to the TL . V. CONCLUSIONS
We have performed comparative benchmark calcula-tions of RISB and DMET on the 1D and 2D (squarelattice) Hubbard model with cluster sizes ranging from N c = 1 to 6. We found that the overall performances ofthe two methods are very similar. Small differences areobserved only for small cluster sizes, where RISB gen-erally predicts slightly more accurate Mott MIT criticalpoints, magnetic moments, occupancies and double occu-pancies. The DMET ground-state energy is usually moreaccurate around half-filling, while the RISB ground-stateenergy is more precise away from half-filling.Furthermore, we proposed an alternative implementa-tion of DMET featuring a modified RISB algorithm witha unity mass renormalization matrix. This formalismpaves the ways for many generalizations. For example,the DFT+RISB derived in Ref. 27 can now be read-ily transposed to DFT+DMET. The non-equilibrium ex-tensions of both methods are also available . A systematic way of improving the accuracy of RISBwithout breaking translational symmetry has been re-cently proposed by introducing auxiliary “ ghost ” de-grees of freedom , and similar ideas have been ap-plied also within the DMET framework . Otherpossible directions may be to generalize DMET tofinite-temperature or extending RISB to systemswith electron-phonon interactions or inter-site electron-electron interactions . VI. ACKNOWLEDGEMENTS
T.-H. L. thanks G. Booth and Q. Chen for useful dis-cussions on the DMET algorithm. Y. Y. thanks for thesupports from BNL CMS center. T.-H. L, T. A., and G.K. were supported by the Department of Energy underGrant No. DE-FG02-99ER45761. N. L. was supportedby the VILLUM FONDEN via the Centre of Excellencefor Dirac Materials (Grant No. 11744). This work usedthe Extreme Science and Engineering Discovery Environ-ment (XSEDE) funded by NSF under Grants No. TG-DMR170121. N c = 2 N c = 4 N c = 6 TL Ref. 15Method DMET RISB DMET RISB DMET RISB DMET U/t = 2 0.1937 0.1942 0.1934 0.1953 0.1935 0.1950 0.1913
U/t = 4 0.1281 0.1314 0.1274 0.1300 0.1277 0.1300 0.1261
U/t = 6 0.0819 0.0841 0.0815 0.0829 0.0816 0.0830 0.0810
U/t = 8 0.0538 0.0548 0.0538 0.0542 0.0539 0.0541 0.0540
U/t = 12 0.0268 0.0269 0.0272 0.0270 0.0272 0.0270 0.0278Table III. Double occupancy (cid:104) n ↑ n ↓ (cid:105) for DMET and RISB in the AFM phase of the half-filled 2D Hubbard model with thenearest neighbor hopping for N c = 2 , , U = 2 t, t, t, t, t . The values in the last column are the solutions in theTL extracted from Ref. 15. N c = 2 N c = 4 N c = 6 N c = 4 Ref. 13 TL Ref. 13Method DMET RISB DMET RISB DMET RISB DMET DMET U/t = 2 0.161 0.158 0.155 0.147 0.151 0.143 0.152 0.115
U/t = 4 0.304 0.293 0.298 0.289 0.296 0.288 0.299 0.226
U/t = 6 0.382 0.376 0.368 0.368 0.367 0.365 0.372 0.275Table IV. Staggered magnetic moment m for DMET and RISB in the AFM phase of the half-filled 2D Hubbard model with thenearest neighbor hopping for N c = 2 , , U = 2 t, t, t . The values in the last two columns are the solutions at N c = 4and in the TL extracted from Ref. 13. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Reviews of Modern Physics , 13 (1996). G. Knizia and G. K.-L. Chan, Physical Review Letters , 186404 (2012). G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. ,865 (2006). T. A. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler,Reviews of Modern Physics , 1027 (2005). M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pr-uschke, and H. R. Krishnamurthy, Physical Review B ,R7475 (1998). M. H. Hettler, M. Mukherjee, M. Jarrell, and H. R. Kr-ishnamurthy, Physical Review B , 12739 (1999). A. I. Lichtenstein and M. I. Katsnelson, Physical ReviewB , R9283 (2000). G. Kotliar, S. Savrasov, G. P´alsson, and G. Biroli, Phys-ical Review Letters , 186401 (2001). G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin,A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N.Rubtsov, and K. Held, Rev. Mod. Phys. , 025003 (2018). G. Knizia and G. K.-L. Chan, Journal of Chemical Theoryand Computation , 1428 (2013). S. Wouters, C. A. Jim´enez-Hoyos, Q. Sun, and G. K.Chan, Journal of Chemical Theory and Computation ,2706 (2016). B.-X. Zheng and G. K.-L. Chan, Physical Review B ,035126 (2016). B.-X. Zheng, J. S. Kretchmer, H. Shi, S. Zhang, and G. K.-L. Chan, Physical Review B , 045103 (2017). B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers,M.-P. Qin, R. M. Noack, H. Shi, S. R. White,S. Zhang, and G. K.-L. Chan, Science , 1155 (2017),http://science.sciencemag.org/content/358/6367/1155.full.pdf. J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bu-lik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero,T. M. Henderson, C. A. Jim´enez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. V. Prokof’ev, M. Qin, G. E. Scuseria,H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn, S. R.White, S. Zhang, B.-X. Zheng, Z. Zhu, and E. Gull (Si-mons Collaboration on the Many-Electron Problem), Phys.Rev. X , 041041 (2015). M. Motta, D. M. Ceperley, G. K.-L. Chan, J. A. Gomez,E. Gull, S. Guo, C. A. Jim´enez-Hoyos, T. N. Lan, J. Li,F. Ma, A. J. Millis, N. V. Prokof’ev, U. Ray, G. E. Scuse-ria, S. Sorella, E. M. Stoudenmire, Q. Sun, I. S. Tupitsyn,S. R. White, D. Zgid, and S. Zhang (Simons Collaborationon the Many-Electron Problem), Phys. Rev. X , 031059(2017). R. Fr´esard and P. W¨olfle, International Journal of ModernPhysics B , 685 (1992). F. Lechermann, A. Georges, G. Kotliar, and O. Parcollet,Physical Review B , 155102 (2007). N. Lanat`a, Y. Yao, X. Deng, V. Dobrosavljevi´c, andG. Kotliar, Physical Review Letters , 126401 (2017). G. Kotliar and A. E. Ruckenstein, Physical Review Letters , 1362 (1986). J. B¨unemann and F. Gebhard, Physical Review B ,193104 (2007). N. Lanat`a, P. Barone, and M. Fabrizio, Phys. Rev. B ,155127 (2008). A. Isidori and M. Capone, Physical Review B , 115120(2009). M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet,G. Kotliar, and A. Georges, Europhysics Letters , 57009(2008). M. Ferrero, P. Cornaglia, L. De Leo, O. Parcollet,G. Kotliar, and A. Georges, Physical Review B , 064501(2009). I. I. Mazin, H. O. Jeschke, F. Lechermann, H. Lee, M. Fink,R. Thomale, and R. Valent´ı, Nature communications ,4261 (2014). N. Lanat`a, Y. X. Yao, C. Z. Wang, K. M. Ho, andG. Kotliar, Physical Review X , 11008 (2015). C. Piefke and F. Lechermann, Phys. Rev. B , 125154(2018). M. Behrmann and F. Lechermann, Physical Review B ,075110 (2015). T. Ayral, T.-H. Lee, and G. Kotliar, Phys. Rev. B ,235139 (2017). N. Lanat`a, X. Deng, and G. Kotliar, Phys. Rev. B ,081108 (2015). W.-S. Wang, X.-M. He, D. Wang, Q.-H. Wang, Z. D. Wang,and F. C. Zhang, Phys. Rev. B , 125105 (2010). M. Sandri, M. Capone, and M. Fabrizio, Phys. Rev. B ,205108 (2013). B. Sandhoefer and G. K.-L. Chan, Phys. Rev. B , 085115(2016). T. E. Reinhard, U. Mordovina, C. Hubig, J. S. Kretchmer,U. Schollwck, H. Appel, and A. A. Sentef, Michael Rubio,arXiv:1811.00048. N. Lanat`a, T.-H. Lee, Y.-X. Yao, and V. Dobrosavljevi´c,Phys. Rev. B , 195126 (2017). M. Fabrizio, Phys. Rev. B , 165110 (2007). B.-X. Zheng, arXiv:1803.10259 . P. Pulay, Chemical Physics Letters , 393 (1980). E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. , 1445 (1968). M. Capone, M. Civelli, S. S. Kancharla, C. Castellani, andG. Kotliar, Phys. Rev. B , 195105 (2004). W. F. Brinkman and T. M. Rice, Phys. Rev. B , 4302(1970). H. Park, K. Haule, and G. Kotliar, Phys. Rev. Lett. ,186403 (2008). M. Schir´o and M. Fabrizio, Physical Review Letters ,076401 (2010). M. Schir´o and M. Fabrizio, Physical Review B , 165105(2011). G. Mazza and A. Georges, Physical Review B , 064515(2017). J. S. Kretchmer and G. K.-L. Chan, The Jour-nal of Chemical Physics , 054108 (2018),https://doi.org/10.1063/1.5012766. E. Fertitta and G. H. Booth, Phys. Rev. B98