Importance of anisotropy in the spin-liquid candidate Me3EtSb[Pd(dmit)2]2

A. C. Jacko, Luca F. Tocchio, Harald O. Jeschke, and Roser ValentıInstitut fur Theoretische Physik, Goethe-Universitat Frankfurt,Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany

Organic charge transfer salts based on the molecule Pd(dmit)2 display strong electronic correla-tions and geometrical frustration, leading to spin liquid, valence bond solid, and superconductingstates, amongst other interesting phases. The low energy electronic degrees of freedom of thesematerials are often described by a single band model; a triangular lattice with a molecular or-bital representing a Pd(dmit)2 dimer on each site. We use ab initio electronic structure calcula-tions to construct and parametrize low energy effective model Hamiltonians for a class of Me4−n

EtnX[Pd(dmit)2]2 (X=As, P, N, Sb) salts and investigate how best to model these systems byusing variational Monte Carlo (VMC) simulations. Our findings suggest that the prevailing modelof these systems as a t − t′ triangular lattice is incomplete, and that a fully anisotropic triangularlattice (FATL) description produces importantly different results, including a significant lowering ofthe critical U of the spin-liquid phase.

Introduction.- The Me4−nEtnX[Pd(dmit)2]2 family oforganic crystals is known for its many interesting elec-tronic and magnetic phases; these materials can havesuperconducting, Mott insulating, spin-liquid, valencebond solid and spin density wave orders, determinedby the cation as well as by temperature and appliedpressure.1–4 The reason for this rich variety of phasesis the competition between frustration effects and elec-tronic correlations. As such, these materials are the focusof very active research. Here we will investigate the sig-nificance of anisotropy and correlations in these materialsby parameterizing and solving model Hamiltonians. Wewill abbreviate Me4−nEtnX[Pd(dmit)2]2 as X-n, follow-ing Ref. 5.

These materials share a generic crystal structure, il-lustrated in Fig. 1, and are found in a variety of orderedphases. As-0 enters an anti-ferromagnetic (AFM) phasebelow 35 K; Sb-1 has a spin-liquid ground state, andspin susceptibility measurements imply that it has anexchange interaction 220 K ≤ J ≤ 250 K; Sb-2 has nolow temperature AFM transition (unlike P-2 and mostof the X-0s); N-0 is the only X-0 material with a super-

FIG. 1: Structure of the spin-liquid candidateEtMe3Sb[Pd(dmit)2]2 (Sb-1). The Pd(dmit)2 moleculesdimerise, and the dimers form 2D triangular lattice layers,interspersed with cation layers (in this case, EtMe3Sb).

conducting transition (6.2 K at 6.5 kbar), while at lowerpressures (away from the SC phase) it exhibits a spin-density wave transition.1–4 Developing a unified descrip-tion of this wide variety of phases is an ongoing challenge.

Here we calculate the electronic structure and useWannier orbitals for the frontier bands to parameterizemodel Hubbard Hamiltonians. By solving these Hamilto-nians with variational Monte Carlo (VMC), we show thatthere are important qualitative differences between treat-ing the system as a t − t′ triangular lattice and consid-ering it as a fully anisotropic triangular lattice (FATL).A discussion on the role of full anisotropy within theHeisenberg model may be found in Ref. 6.

Methods.- We perform density functional theory cal-culations of the electronic structure and then constructlocalized Wannier orbitals for the frontier bands. Fromthese we parameterize tight-binding model Hamiltonians,both for the t − t′ and for the FATL (ta − tb − tc). Wethen solve these models with a VMC approach. The ap-proach used here allows for the description of metallic andspin-liquid states with the same variational wavefunction,which is compared to a variational wavefunction that de-scribes the magnetic spiral ordered insulating state.

Electronic Structure.- The electronic structure cal-culations presented here were performed in an all-electron full-potential local orbital basis using the FPLOpackage.7 The densities were converged on a (6×6×6) kmesh using a generalized gradient approximation (GGA)functional.8

To move from density functional theory calculations tomodel Hamiltonians, it is convenient to construct Wan-nier orbitals to represent the frontier bands of the sys-tem. In principle the Wannier orbitals are simply Fouriertransforms of the Bloch wavefunctions; however in thisprocedure there are still many degrees of freedom. Herethey are constrained by the requirements that the Wan-nier orbitals be represented by Kohn-Sham orbitals in anarrow energy window; by projecting onto the FPLO ba-sis orbitals ensures that they will form a good basis for atight-binding model.9 With these nearly optimally local-ized Wannier orbitals we compute real-space overlaps to

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obtain tight-binding parameters. This method has sev-eral advantages over band fitting, as has been discussedpreviously in the case of molecular organic crystals.10

Effective Modelling.- To model these systems we con-sider the Hubbard Hamiltonian. As we will show later,a half-filled single orbital per site Hubbard model is sug-gested by the electronic structure, so it is this kind ofmodel we will focus on. The Hamiltonian is defined by:

H = −∑i,j,σ

tijc†i,σcj,σ + U

∑i

ni,↑ni,↓, (1)

where c†i,σ(ci,σ) creates (destroys) an electron with spin

σ on site i, ni,σ = c†i,σci,σ is the electronic density, tijis the hopping amplitude and U is the on-site Coulombrepulsion. Here we calculate a VMC solution includingbackflow correlations, which allows us to provide an ac-curate description of a system with hundreds of latticesites.11

A major issue in the Hubbard model on the anisotropictriangular lattice is the possibility of stabilizing a spin-liquid phase (as seen in the experimental data). Inaddition, for generic values of the hopping parametersmagnetic states with incommensurate order may be ex-pected and indeed can be obtained within mean-field ap-proaches, like for instance the Hartree-Fock (HF) approx-imation.12,13

Here, we approach this problem by implementing cor-related variational wave functions which describe bothspin-liquid states and magnetic states with generic in-commensurate order. In this way, we are able to treatincommensurate spiral order and non-magnetic states atthe same level of theory, and therefore have a sensiblecomparison of their energies vs U .14

Spiral Magnetic States.- We start with the magneticstates obtained at the HF level, with the only constraintfor the spin order to be coplanar in the x − y plane.Our magnetic HF solutions display spiral order, which (in2D) may be parametrized by two pitch angles θ and θ′,where θ (θ′) is the angle between nearest-neighbor spinsalong the hopping direction tb (tc). Since we use finiteclusters, only certain commensurate angles are allowed.For a lattice size L = l × l, the allowed values are θ =2πn/l and θ′ = 2πn′/l, with n and n′ integers. We testedvarious lattice sizes ranging from 12× 12 to 20× 20, anddetermined the best pair of pitch angles for each latticesize.

Our VMC magnetic states are then constructed byapplying correlation terms on top of the HF spi-ral states. We employ a spin-spin Jastrow fac-tor to correctly describe fluctuations orthogonal tothe plane where the magnetic order lies, i.e., Js =exp[ 12

∑i,j ui,jS

zi S

zj ].15 A further density-density Jastrow

factor Jc = exp[ 12∑i,j vi,jninj ] (that includes the on-site

Gutzwiller term vi,i) is considered to adjust electron cor-relations. All the ui,j ’s and the vi,j ’s are optimized foreach independent distance |i− j|. The correlated state isthen given by |ΨSP〉 = JsJc|SP〉.

Non-Magnetic States.- In order to describe a non-magnetic state, we construct an uncorrelated wave func-tion given by the ground state |BCS〉 of a superconduct-ing BCS Hamiltonian:16,17

HBCS =∑i,j,σ

tijc†i,σcj,σ − µ

∑i,σ

c†i,σci,σ

+∑i,j

∆ijc†i,↑c†j,↓ + h.c.,

(2)

where both the variational hopping amplitudes tij , thepairing fields ∆ij , and the chemical potential µ are vari-ational parameters to be independently optimized. Forthe majority of the results reported here we constrain allof these variational parameters to be real.

The correlated state |ΨBCS〉 = Jc|BCS〉 allows us todescribe a non-magnetic Mott insulator for a sufficientlysingular Jastrow factor vq ∼ 1/q2 (vq being the Fouriertransform of vi,j),

18 while a metallic state can be ob-tained whenever vq ∼ 1/q.

A size-consistent and efficient way to further improvethe correlated states |ΨBCS〉 and |ΨSP〉 is based on back-flow correlations. In this approach, each orbital that de-fines the unprojected states |BCS〉 and |SP〉 is taken todepend upon the many-body configuration, in order toincorporate virtual hopping processes.11 All results pre-sented here are obtained by fully incorporating the back-flow corrections and optimizing individually every varia-tional parameter in the mean-field BCS equation, and inthe Jastrow factors Jc and Js, as well as in the backflowcorrections.

Determination of an appropriate model.- The spin-liquid candidate Sb-1 has been the subject of much recentstudy. We will focus on this material in discussing theprocess of determining the appropriate model Hamilto-nian for this class of systems. To move towards the goal ofunderstanding the phase diagram of Sb-1, and the originof the various ordered phases, one needs a sensible choiceof a model Hamiltonian with reliable parameters. In con-structing some minimal model, one must also be awareof what the model neglects, and what effect that has onthe physics it predicts. Here we analyze the electronicstructure to determine an appropriate minimal model,and then use Wannier orbitals to parameterize it.

Fig. 2 shows the band structure and the density ofstates for Sb-1 in a wide energy window around theFermi energy. From this figure, we can see that there areseveral pairs of bands separated from the others; thesebands are the bands identified as arising from bonding (-b) and anti-bonding (-ab) hybrids of Pd(dmit)2 highestoccupied molecular orbitals (HOMO) and lowest unoccu-pied molecular orbitals (LUMO) (highlighted in green).The pair crossing the Fermi energy are HOMO-ab bands,while the pairs on either side are the LUMO-b and -abbands. The fourth highlighted pair at the top of the bulkvalence bands is the HOMO-b pair. The origin of thesebands is illustrated in Fig. 3. The bands come in pairsbecause for each dimer orbital in one plane, there is an

3

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2

Γ X XY YYZ XYZ Z Γ

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FIG. 2: Band structure and density of states of Sb-1 in awide energy window around the Fermi energy. The four pairsof bands highlighted in green are those identified as being ofPd(dmit)2 HOMO or LUMO origin. The pair that crossesthe Fermi energy is of HOMO-ab origin. These are the bandsused in a minimal one-orbital model of the system. The directenergy gap between the HOMO-ab bands and the others ismore than 0.1 eV, although the indirect gap is smaller.

FIG. 3: Schematic of the hybridisation of the orbital of thetwo Pd(dmit)2 in each dimer, and the resultant crossing ofenergy levels leading to the unusually ordered frontier bandsof Pd(dmit)2 complexes.

identical one in the other plane, related to the first by atranslation and a rotation about the tb direction (the yaxis). These pairs of bands are well separated from eachother, with typical direct band gaps on the order of 100’sof meV, well above kBT for most experiments on thesesystems.

Modeling the Frontier Bands.- As Fig. 2 shows, theHOMO-ab bands are well separated from the otherbands, and so they form a good basis for a low energy ef-fective model Hamiltonian. In Fig. 4 we show one of theWannier orbitals for these bands, clearly showing theirHOMO-ab character (the other Wannier orbital is thesame but in the other layer of Pd(dmit)2 molecules). Ta-ble I shows the three t’s in the first-nearest neighbourshell (see Fig. 4) computed from the HOMO-ab Wannierorbitals of several X-n systems (note that including moreneighbours explicitly has no effect on these parameters,unlike in a band-fitting computation). These results are

FIG. 4: Wannier orbital for the HOMO-ab bands of Sb-1 (left)and the 2D tight-binding lattice generated from this Wannierorbital (right). The anti-bonding HOMO character of theorbital is clearly visible. The widths of the various cylindersin the lattice are linearly proportional to the magnitude ofthe corresponding t (see Table I).

X-n µ tb ta tc Ref.

N-0 34.8 44.3 48.6 38.9 22*

As-0 28.6 44.5 55.6 32.6 22*

P-1 29.3 39.8 48.4 46.4 23

Sb-1 32.3 46.9 56.5 39.8 24

Sb-2 32.4 35.2 45.5 44.1 25*

TABLE I: Comparison of the sets of one orbital model param-eters for several X-n systems. All energies are given in meV.Starred references did not include H coordinates so they wereinserted manually. All structures used here were obtained atroom temperature, except for Sb-1 which was obtained at 4K.

consistent with those found in past work.19–21

With the t’s obtained from the Wannier orbitals, wecan now explore the Hubbard model for Sb-1 with VMC.We find that the spiral magnetic state |SP〉 has optimalpitch angles θ = 7π/9, θ′ = 3π/9, that are commensurateto an 18× 18 lattice size. The BCS wavefunction has fi-nite pairing fields for U/ta > 6.75 (i.e. when the systemis insulating) and they are highly anisotropic; with thelargest component along the ta direction, the tb compo-nent approximately half as large and with opposite sign,and the tc component nearly zero. Fig. 5 shows the opti-mized energies of these two wavefunctions as a functionof U/ta; |ΨSP〉 is favorable for 6.75 < U/ta < 11, and|ΨBCS〉 is favorable outside of this region.

The charge gap, G, can be calculated from the staticstructure factor, N(q), by assuming that the low mo-mentum excitations are collective modes.11,26 With thisapproximation, one finds that G ∝ limq→0

q2

N(q) , where

N(q) = 〈n−qnq〉 and nq = 1/√L∑r,σ e

iqrnr,σ. As such,

the metallic phase is characterized byN(q)/q → const. asq → 0, implying a vanishing gap at q = 0, and the insulat-ing phase by N(q)/q → 0 as q → 0, implying a finite gap.Fig. 5 shows N(q)/q versus q for the |ΨBCS〉 and |ΨSP〉states at U/ta = 6.4 and U/ta = 7.1, either side of thepoint the |ΨSP〉 state becomes favorable. While the spi-ral magnetic |ΨSP〉 state is insulating on both sides of thetransition, the |ΨBCS〉 state changes from metallic to in-

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-0.84

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E/J

U/ta

ΨBCS

ΨSP

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0 0.25 0.5 0.75 1 1.25

N(q)/|q|

|q|/π

FIG. 5: (Color online) Upper panel: Variational energiesper site for the FATL model with the parameters for Sb-1,by using the non-magnetic state (red squares), |ΨBCS〉, andthe magnetic spiral state (blue circles), |ΨSP〉, as a functionof U/ta and in units of J = 4t2a/U . Both wavefunctions werecomputed on a L = 324 lattice, for which the optimal pitchangles (θ = 7π/9, θ′ = 3π/9) are commensurate. Lower panel:N(q)/q at U/ta = 6.4 by using |ΨBCS〉 (empty triangles) and|ΨSP〉 (empty diamonds) and at U/ta = 7.1 by using |ΨBCS〉(full triangles) and |ΨSP〉 (full diamonds). Data show themetal to insulator transition in |ΨBCS〉, while the magneticstate is always insulating. Plots are presented along the lineconnecting the point Q = (π, π/

√3) to the point Γ = (0, 0)

in reciprocal space.

sulating. Thus, we clearly see this is the metal-insulatortransition (MIT). By examining N(q)/q, we confirm thatthe |ΨBCS〉 state remains insulating all the way above theMIT, in particular in the region U/ta > 11, where it be-comes favorable.

We note that an existing calculation of the interactionparameters using the constrained random-phase approx-imation (cRPA) finds U/ta ∼ 11 for Sb-1, in good agree-ment with our results for the location of the spin-liquidregion.20

t − t′ model VS FATL.- Pd(dmit)2 systems are oftenrepresented by a t− t′ model, despite this symmetry notbeing found in the various t estimates, including our es-timate in Table I.19–21 Here we compare model resultsusing the t − t′ approximation with a fully anisotropictriangular lattice (FATL) model, focusing our discussionon the spin-liquid candidate Sb-1. If we average the twolarger t’s of Sb-1, assuming t = (ta + tb)/2 and t′ = tc,we find that the equivalent t − t′ model has t′/t = 0.77.

FIG. 6: Phase diagram versus U/tmax, where tmax is thebiggest hopping parameter in the model, for the isotropic,t−t′, and FATL models for Sb-1; showing the metallic (green),spiral ordered magnetic insulator (orange) and spin liquid(cyan) phases. The spin-liquid phase is strongly enhanced byincluding the full anisotropy of the system, while the metal-insulator transition is only slightly changed.

This model has been previously studied with variationalMonte Carlo,14 where for this value of t′/t the critical Ufor the spin-liquid transition is located at U/t ∼ 22, whilefor the FATL this value is strongly reduced to U/ta ∼ 11.On the other hand, the metal-insulator transition is onlyslightly affected, raising from Uc/t ∼ 6 for the t − t′

model to Uc/ta ∼ 6.75 for the FATL. These results areillustrated in the phase diagram in Fig. 6.

We would like to point out that if we repeat our calcu-lation for the P-1 system (which is almost truly t− t′) wefind that in both the FATL and in the t − t′ model, thetransition to spin liquid occurs at U/tmax ∼ 13, (wheretmax is the biggest hopping parameter). Thus the enhanc-ment of the spin-liquid phase is driven by anisotropy.

Interestingly, if we allow the pairing fields to be com-plex, we find that finite imaginary components lower theenergy of the |ΨBCS〉 state slightly. This imaginary com-ponent appears for U/ta > 11, the same point where the|ΨBCS〉 state becomes favorable again and it seems to bedue to the anisotropy of the system; indeed by consid-ering the less anisotropic P-1 system, we find a smallerimaginary component, while it is not seen in t−t′ models.

Conclusions.- We have shown that the high anisotropyin Pd(dmit)2 materials can have important effects on thephysics. Relative to a t−t′, using a FATL model for Sb-1increases the critical U for the metal-insulator transitiononly by ∼ 10%, while it halves the critical U for the spin-liquid transition. With this reduction, existing parame-terisations of Sb-1 place it in the spin-liquid regime. Inaddition, the spin-liquid phase develops a complex pair-ing function as it becomes favorable for U/ta > 11, some-thing not seen in the t− t′ model.

We would like to acknowledge useful discussions withD. Cocks, C. Gros, and H. Seo. We acknowledge thesupport of the German Science Foundation through the

5

grant SFB/TR49.

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