Comprehensive ab initio investigation of the phase diagram of quasi-one-dimensional molecular solids Kazuyoshi Yoshimi1 Takahiro Misawa12 Takao Tsumuraya34 and Hitoshi Seo56

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Comprehensive ab initio investigation of the phase diagram of quasi-one-dimensional

molecular solids

Kazuyoshi Yoshimi1,†, Takahiro Misawa1,2,†, Takao Tsumuraya3,4, and Hitoshi Seo5,6

1Institute for Solid State Physics, University of Tokyo,

5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

2Beijing Academy of Quantum Information Sciences, Haidian District, Beijing 100193, China

3Priority Organization for Innovation and Excellence,

Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan

4Magnesium Research Center, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan

5Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

6Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan and

†The authors contributed to this work equally.

(Dated: September 6, 2023)

An ab initio investigation of the family of molecular compounds TM2Xis conducted, where

TM is either TMTSF or TMTTF and Xtakes centrosymmetric monovalent anions. By deriving

the extended Hubbard-type Hamiltonians from ﬁrst-principles band calculations and evaluating not

only the intermolecular transfer integrals but also the Coulomb parameters, we discuss their material

dependence in the uniﬁed phase diagram. Furthermore, we apply the many-variable variational

Monte Carlo method to accurately determine the symmetry-breaking phase transitions, and show

the development of the charge and spin orderings. We show that the material-dependent parameter

can be taken as the correlation eﬀect, represented by the value of the screened on-site Coulomb

interaction Urelative to the intrachain transfer integrals, for the comprehensive understanding of

the spin and charge ordering in this system.

Introduction—. Molecular solids serve as textbook ma-

terials to study many-body physics in condensed mat-

ter, despite their complex crystal structures with many

atoms in the unit cell, owing to the success in modeling

the low-energy electronic properties based on the frontier

molecular orbitals [1,2]. A general challenge is to un-

derstand multiple symmetry-broken phases observed in

diﬀerent systems in a systematic way, where the charge,

spin, and lattice degrees of freedom are coupled to each

other, based on the eﬀective Hamiltonians and identify-

ing the factors governing them [3].

The family of compounds constituted by TMTSF (=

tetramethyltetraselenafulvalene) or TMTTF (= tetram-

ethyltetrathiafulvalene) molecules, which we call together

TM here, forming 2:1 salts with monovalent anions X−1,

TM2X, has been studied for decades as a typical example

showing such a rich variety of phases [2–5]. TMTSF2X

historically outstands as the ﬁrst organic system under-

going superconductivity [6]. Unconventional supercon-

ductivity with an anisotropic gap function is realized,

whose ‘glue’ for the Cooper pair formation is the strong

spin ﬂuctuation, supported by the pressure-temperature

phase diagram with the superconducting phase neigh-

boring an antiferromagnetic (AFM) phase driven by the

nesting of the Fermi surface [7–11].

Later, the isostructural TMTTF counterparts became

realized as strongly-correlated electron systems, owing to

the atomic substitution resulting in smaller bandwidths

than in the TMTSF compounds [5,12,13]. In particular,

at high temperatures of the order of 100 K, TMTTF2X

hosts a competition between a Mott insulating state, the

so-called dimer-Mott (DM) insulator, and a charge or-

dering (CO) state with electron-rich and poor molecules

spontaneously arranging to avoid the Coulomb repulsion:

a common framework for the understanding of correlated

quarter-ﬁlled systems [14–16]. In the low-temperature re-

gion (∼10K), on the other hand, where the magnetic in-

teractions start to develop between the localized spins in

the strongly correlated regime, the AFM state competes

with the spin-Peierls instability accompanying lattice dis-

tortion owing to the quasi-one-dimensionality [17–19].

The variation among the symmetry-breaking states,

in which both the charge and spin degrees of freedom

are involved, depending on the applied pressure to the

compounds and on the choice of the combination of TM

and X(chemical pressure), has been extensively dis-

cussed [20–24]. Nevertheless, there is room toward a

comprehensive understanding of them. There are dif-

ferent views on how the choice of Xand the amount

of applied pressure aﬀect the microscopic parameters,

and the impact of the diﬀerence between TMTSF and

TMTTF is still obscure. Conventionally, the two sys-

tems are drawn in a uniﬁed pressure-temperature phase

diagram [4,5,22,23], as shown in Fig. 1(a). Although

there are diﬀerent versions in the literature, here we sim-

ply draw the ‘ambient pressure’ positions of the represen-

tative compounds in an equally spaced manner, together

with the reported transition temperatures [13,27–30].

From the early days, in the TMTSF salts, it is dis-

cussed that the applied pressure increases the dimension-

ality [6]. On the other hand, the degree of dimeriza-

tion along the one-dimensional chain direction was also

pointed out to be an important factor, especially for the

TMTTF salts [31]. And more recently, to understand the

arXiv:2210.13726v2 [cond-mat.str-el] 4 Sep 2023

1 2

(b) (c)

(d)

ta1

ta2

tq1

tq2

Energy [eV]

T [K]

NbF6SbF6PF6Br PF6

AsF6NbF6AsF6

(a)

100

150

200

250

FCO

Metal

TMTTF TMTSF

AFM AFM

DMI

300

-3

-2

-1

FIG. 1. (a) Schematic phase diagram of (TM)2X. DMI, FCO, AFM, and SP indicate dimer-mott insulating, ferroelectric charge

ordered, antiferromagnetic, and spin-Peierls phases, respectively. Symbols are the ambient-pressure transition temperatures

for each material [25]. We note that, since the reported AFM transition temperatures for the TMTSF salts are the same [25],

the order is determined by the volume of the unit cell. (b) Band structure of TMTTF2PF6. The thin solid lines are the DFT

results, while the bold lines are obtained using the MLWFs, overlapping the valence bands. We set the Fermi energy to zero

(the dotted line). (c) Drawings of the two MLWFs for TMTTF2PF6. (d) Deﬁnition of the transfer integrals. The TMTTF

molecules and MLWFs are drawn by using VESTA [26].

complicated interplay between the phases toward the left-

hand end of the phase diagram, it was proposed that the

CO instability can be the controlling factor [22]; this sug-

gests the essential role of electron correlation. Whether

we can choose a parameter that varies monotonically as

we go along the compounds listed in the horizontal axis

is the question we ask here.

In this work, to investigate these issues, we apply

ﬁrst-principles density functional theory (DFT) calcu-

lations [32,33] to a number of members of the TM2X

family with centrosymmetric X; we numerically derive

not only the transfer integrals between the molecules but

also the correlation parameters. We then use these pa-

rameters as inputs to a highly-accurate numerical solver,

i.e., the many-variable variational Monte Carlo (mVMC)

method [34–36], for the eﬀective extended Hubbard-type

Hamiltonian and seek possible long-range ordering in the

charge and spin degree of freedoms. As a result, we

can directly compare with the experimentally observed

phases and comprehensively extract the determining fac-

tor at a quantitative level.

Ab initio derivation of microscopic parameters—

.The DFT calculations are performed using Quantum

Espresso (version 6.6) [37] for the experimental crys-

tal structures [13]. We employ norm-conserving pseu-

dopotentials based on the Vanderbilt formalism with

plane-wave basis sets [38,39]. The exchange-correlation

functional used in this study is the generalized gradient

approximation by Perdew, Burke, and Ernzerhof [40],

which is commonly used for accurately describing elec-

tronic states of molecular compounds [41]. The cutoﬀ

energies for plane waves and charge densities are 70 and

280 Ry, respectively. A 7 ×7×3 uniform k-point mesh

was used with a Gaussian smearing method during self-

consistent loops.

For deriving the interaction parameters, we use

the constrained random phase approximation (cRPA)

method [42,43]. In fact, previous studies applying the

cRPA method to molecular solids show good agreement

between the results for the derived eﬀective Hamiltoni-

ans and the experiments [44–49]. On the basis of the

obtained DFT electronic states, we construct maximally

localized Wannier function(MLWF)s and derive the pa-

rameters using RESPACK [50]. In the calculations, the

energy cutoﬀ for the dielectric function was set to be 3

Ry.

Figures 1(b) and (c) show the band structure and the

MLWFs, respectively, of TMTTF2PF6as an example.

The two bands crossing the Fermi energy are constituted

from the bonding and anti-bonding combination of the

highest occupied molecular orbitals situated on the two

TMTTF molecules in the unit cell shown in Fig. 1(b).

Using these MLWFs, we evaluated the transfer integrals

and the density-density interactions, i.e. the on-site and

oﬀ-site Coulomb interactions (the exchange interactions

are negligibly small so we omit them in the following).

The obtained microscopic parameters are listed in Ta-

ble II in the supplemental materials [25]. We adopt the

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Comprehensiveabinitioinvestigationofthephasediagramofquasi-one-dimensionalmolecularsolidsKazuyoshiYoshimi1,†,TakahiroMisawa1,2,†,TakaoTsumuraya3,4,andHitoshiSeo5,61InstituteforSolidStatePhysics,UniversityofTokyo,5-1-5Kashiwanoha,Kashiwa,Chiba277-8581,Japan2BeijingAcademyofQuantumInformationSciences,...

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Comprehensive ab initio investigation of the phase diagram of quasi-one-dimensional molecular solids Kazuyoshi Yoshimi1 Takahiro Misawa12 Takao Tsumuraya34 and Hitoshi Seo56

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