Temperature-driven transition from the Wigner Crystal to the Bond-Charge-Density
Wave in the Quasi-One-Dimensional Quarter-Filled band
R.T. Clay,1 R.P. Hardikar,1 and S. Mazumdar2
1Department of Physics and Astronomy and HPC2 Center for Computational Sciences,
Mississippi State University, Mississippi State MS 39762
2 Department of Physics, University of Arizona Tucson, AZ 85721
(Dated: August 21, 2021)
It is known that within the interacting electron model Hamiltonian for the one-dimensional 1
filled band, the singlet ground state is a Wigner crystal only if the nearest neighbor electron-electron
repulsion is larger than a critical value. We show that this critical nearest neighbor Coulomb
interaction is different for each spin subspace, with the critical value decreasing with increasing
spin. As a consequence, with the lowering of temperature, there can occur a transition from a
Wigner crystal charge-ordered state to a spin-Peierls state that is a Bond-Charge-Density Wave
with charge occupancies different from the Wigner crystal. This transition is possible because
spin excitations from the spin-Peierls state in the 1
-filled band are necessarily accompanied by
changes in site charge densities. We apply our theory to the 1
-filled band quasi-one-dimensional
organic charge-transfer solids in general and to 2:1 tetramethyltetrathiafulvalene (TMTTF) and
tetramethyltetraselenafulvalene (TMTSF) cationic salts in particular. We believe that many recent
experiments strongly indicate the Wigner crystal to Bond-Charge-Density Wave transition in several
members of the TMTTF family. We explain the occurrence of two different antiferromagnetic
phases but a single spin-Peierls state in the generic phase diagram for the 2:1 cationic solids. The
antiferromagnetic phases can have either the Wigner crystal or the Bond-Charge-Spin-Density Wave
charge occupancies. The spin-Peierls state is always a Bond-Charge-Density Wave.
PACS numbers: 71.30.+h, 71.45.Lr, 74.70.Kn
I. INTRODUCTION
Spatial broken symmetries in the quasi-one-
dimensional (quasi-1D) 1
-filled organic charge-
transfer solids (CTS) have been of strong
experimental1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 and
theoretical18,19,20,21,22,23,24,25,26 interest. The bro-
ken symmetry states include charge order (hereafter
CO, this is usually accompanied by intramolecular
distortions), intermolecular lattice distortions (hereafter
bond order wave or BOW), antiferromagnetism (AFM)
and spin-Peierls (SP) order. Multiple orderings may
compete or even coexist simultaneously. Interestingly,
these unconventional insulating states in the CTS are
often proximate to superconductivity27, the mecha-
nism of which has remained perplexing after intensive
investigations over several decades. Unconventional
behavior at or near 1
-filling has also been observed
in the quasi-two-dimensional organic CTS with higher
superconducting critical temperatures11,28,29,30,31,
sodium cobaltate32,33,34 and oxides of titanium35 and
vanadium36,37.
In spite of extensive research on 1D instabilities in the
CTS, detailed comparisons of theory and experiments re-
main difficult. Strong electron-electron (e-e) Coulomb
interactions in these systems make calculations particu-
larly challenging, and with few exceptions38,39 existing
theoretical discussions of broken symmetries in the in-
teracting 1
-filled band have been limited to the ground
state18,19,20,21,22,23,24,25,26. This leaves a number of im-
portant questions unresolved, as we point out below.
4k  1010F
AsF6 PF6SbF6 PF6
Pressure
FIG. 1: Schematic of the proposed T-vs.-P phase diagram for
(TMTCF)2X, where C=T or S, along with the charge occu-
pancies of the sites in the low T phases as determined in this
work. The P-axis reflects the extent of interchain coupling.
Open (filled) arrows indicate the ambient pressure locations
of TMTTF (TMTSF) salts.
Quasi-1D CTS undergo two distinct phase transitions
as the temperature (hereafter T) is reduced40. The 4kF
transition at higher T involves charge degrees of free-
dom (T4kF ∼ 100 K), with the semiconducting state be-
low T4kF exhibiting either a dimerized CO or a dimer-
ized BOW. Charges alternate as 0.5 + ǫ and 0.5 – ǫ
on the molecules along the stack in the dimerized CO.
The site charge occupancy in the dimerized CO state is
commonly written as · · · 1010· · · (with ‘1’ and ‘0’ denot-
ing charge-rich and charge-poor sites), and this state is
http://arxiv.org/abs/0704.1656v2
also referred to as the Wigner crystal. The 4kF BOW
has alternating intermolecular bond strengths, but site
charges are uniformly 0.5. It is generally accepted that
the dimer CO (dimer BOW) is obtained for strong (weak)
intermolecular Coulomb interactions (the intramolecular
Coulomb interaction can be large in either case, see Sec-
tion III). At T < T2kF ∼ 10 – 20 K, there occurs a sec-
ond transition in the CTS, involving spin degrees of free-
dom, to either an SP or an AFM state. Importantly,
the SP and the AFM states both can be derived from
the dimer CO or the dimer BOW. Coexistence of the SP
state with the Wigner crystal would require that the SP
state has the structure 1 = 0 = 1 · · · 0 · · · 1, with singlet
“bonds” alternating in strength between the charge-rich
sites of the Wigner crystal. Similarly, coexisting AFM
and Wigner crystal would imply site charge-spin occu-
pancies ↑ 0 ↓ 0. The occurrence of both have been sug-
gested in the literature20,26,38,39. The SP state can also
be obtained from dimerization of the dimer 4kF BOW,
in which case there occurs a spontaneous transition from
the uniform charge-density state to a coexisting bond-
charge-density wave (BCDW)18,22,25 1− 1 = 0 · · · 0. The
unit cells here are dimers with site occupancies ‘1’ and
‘0’ or ‘0’ and ‘1’, with strong and weak interunit 1–1 and
0· · · 0 bonds, respectively. The AFM state with the same
site occupancies · · · 1100· · · is referred to as the bond-
charge-spin density wave (BCSDW)21 and is denoted as
↑↓ 00.
The above characterizations of the different states and
the related bond, charge and spin patterns are largely
based on ground state calculations. The mechanism of
the transition from the 4kF state to the 2kF state lies
outside the scope of such theories. Questions that remain
unresolved as a consequence include: (i) Does the nature
of the 4kF state (charge versus bond dimerized) predeter-
mine the site charge occupancies of the 2kF state? This
is assumed in many of the published works. (ii) What de-
termines the nature of the 2kF state (AFM versus SP)?
(iii) How does one understand the occurrence of two dif-
ferent AFM phases (hereafter AFM1 and AFM2) strad-
dling a single SP phase in the proposed T vs. P (where P
is pressure) phase diagram (see Fig. 1)9 for the cationic 1
filled band quasi-1D CTS? (iv) What is the nature of the
intermediate T states between the 4kF dimerized state
and the 2kF tetramerized state? As we point out in the
next section, where we present a brief review of the ex-
perimental results, answers to these questions are crucial
for a deeper understanding of the underlying physics of
the 1
-filled band CTS.
In the present paper we report the results of our calcu-
lations of T-dependent behavior within theoretical mod-
els incorporating both e-e and electron-phonon (e-p) in-
teractions to answer precisely the above questions. One
key result of our work is as follows: in between the strong
and weak intermolecular Coulomb interaction parame-
ter regimes, there exists a third parameter regime within
which the intermolecular Coulomb interactions are in-
termediate, and within which there can occur a novel
transition from a Wigner crystal CO state to a BCDW
SP state as the T is lowered. Thus the charge or bond
ordering in the 4kF phase does not necessarily decide
the same in the 2kF state. For realistic intramolecular
Coulomb interactions (Hubbard U), we show that the
width of this intermediate parameter regime is compara-
ble to the strong and weak intersite interaction regimes.
We believe that our results are directly applicable to the
family of the cationic CTS (TMTTF)2X, where a redis-
tribution of charge upon entering the SP state from the
CO state has been observed15,16 in X=AsF6 and PF6. A
natural explanation of this redistribution emerges within
our theory. The SP state within our theory is unique and
has the BCDW charge occupancy, while the two AFM re-
gions in the phase diagram of Fig. 1 have different site
occupancies. Our theory therefore provides a simple di-
agnostic to determine the pattern of CO coexisting with
low-temperature magnetic states in the 1
-filled CTS.
In addition to the above theoretical results directly
pertaining to the quasi-1D CTS, our work gives new in-
sight to excitations from a SP ground state in a non-
half-filled band. In the case of the usual SP transition
within the 1
-filled band, the SP state is bond-dimerized
at T = 0 and has uniform bonds at T > T2kF . The site
charges are uniform at all T. This is in contrast to the
-filled band, where the SP state at T = 0 is bond and
charge-tetramerized and the T > T2kF state is dimerized
as opposed to being uniform. Furthermore, we show that
the high T phase here can be either charge- or bond-
dimerized, starting from the same low T state. This
clearly requires two different kinds of spin excitations in
the 1
-filled band. We demonstrate that spin excitations
from the SP state in the 1
-filled band can lead to two
different kinds of defects in the background BCDW.
In the next section we present a brief yet detailed sum-
mary of relevant experimental results in the quasi-1D
CTS. The scope of this summary makes the need for hav-
ing T-dependent theory clear. Following this, in Section
III we present our theoretical model along with conjec-
tures based on physical intuitive pictures. In Section IV
we substantiate these conjectures with accurate quantum
Monte Carlo (QMC) and exact diagonalization (ED) nu-
merical calculations. Finally in Section V we compare
our theoretical results and experiments, and present our
conclusions.
II. REVIEW OF EXPERIMENTAL RESULTS
Examples of both CO and BOW broken symmetry at
T < T4kF are found in the
-filled CTS. The 4kF phase
in the anionic 1:2 CTS is commonly bond-dimerized. The
most well known example is MEM(TCNQ)2, which un-
dergoes a metal-insulator transition accompanied with
bond-dimerization at 335 K41. Site charges are uniform
in this 4kF phase. The 2kF phase in the TCNQ-based
systems is universally SP and not AFM. The SP transi-
tion in MEM(TCNQ)2 occurs below T2kF = 19 K, and
low T neutron diffraction measurements of deuterated
samples41 have established that the bond tetramerization
is accompanied by 2kF CO · · · 1100· · · . X-ray
42,43 and
neutron diffraction44 experiments have confirmed a sim-
ilar low T phase in TEA(TCNQ)2. We will not discuss
these further in the present paper, as they are well de-
scribed within our previous work18,25. We will, however,
argue that the SP ground state in (DMe-DCNQI)2Ag
(as opposed to AFM) indicates · · · 1100· · · CO in this.
The cationic (TMTCF)2X, C= S and Se, exhibit more
variety, presumably because the counterions affect pack-
ing as well as site energies in the cation stack. Differ-
ences between systems with centrosymmetric and non-
centrosymmetric anions are also observed. Their overall
behavior is summarized in Fig. 1, where as is custom-
ary pressure P can also imply larger interchain coupling.
We have indicated schematically the possible locations of
different materials on the phase diagram. The most sig-
nificant aspect of the phase diagram is the occurrence of
two distinct antiferromagnetic phases, AFM1 and AFM2,
straddling a single SP phase.
Most TMTTF lie near the low P region of the phase
diagram and are insulating already at or near room tem-
perature because of charge localization, which is due to
the intrinsic dimerization along the cationic stacks1,4,11.
CO at intermediate temperatures TCO has been found
in dielectric permittivity4, NMR7, and ESR46 experi-
ments on materials near the low and intermediate P
end. Although the pattern of the CO has not been de-
termined directly, the observation of ferroelectric behav-
ior below TCO is consistent with · · · 1010· · · type CO
in this region5,23. With further lowering of T, most
(TMTTF)2X undergo transitions to the AFM1 or SP
phase (with X = Br a possible exception, see below).
X = SbF6 at low T lies in the AFM1 region
9, with a very
high TCO and relatively low Neel temperature TN = 8
K. As the schematic phase diagram indicates, pressure
suppresses both TCO and TN in this region. For P > 0.5
GPa, (TMTTF)2SbF6 undergoes a transition from the
AFM1 to the SP phase9, the details of which are not com-
pletely understood; any charge disproportionation in the
SP phase is small9. (TMTTF)2ReO4 also has a relatively
high TCO = 225 K, but the low T phase here, reached
following an anion-ordering transition is spin singlet14.
Nakamura et al. have suggested, based on NMR exper-
iments, that the CO involves the Wigner crystal state,
but the low T state is the · · · 1100· · · BCDW14. Further
along the P axis lie X = AsF6 and PF6, where TCO are re-
duced to 100 K and 65 K, respectively7. The low T phase
in both cases is now SP. Neutron scattering experiments
on (TMTTF)2PF6 have found that the lattice distortion
in the SP state is the expected 2kF BOW distortion,
but that the amplitude of the lattice distortion is much
smaller10 than that found in other organic SP materials
such as MEM(TCNQ)2. The exact pattern of the BOW
has not been determined yet. Experimental evidence ex-
ists that some form of CO persists in the magnetic phases.
For example, the splitting in vibronic modes below TCO
in (TMTTF)2PF6 and (TMTTF)2AsF6, a signature of
charge disproportionation, persists into the SP phase17,
indicating coexistence of CO and SP. At the same time,
the high T CO is in competition with the SP ground
state8, as is inferred from the different effects of pressure
on TCO and TSP : while pressure reduces TCO, it in-
creases TSP . This is in clear contrast to the effect of pres-
sure on TN in X = SbF6. Similarly, deuteration of the hy-
drogen atoms of TMTTF increases TCO but decreases
TSP . That higher TCO is accompanied by lower TSP for
centrosymmetric X (TSP= 16.4 K in X = PF6 and 11.1 K
in X = AsF6) has also been noted
48. This trend is in ob-
vious agreement with the occurrence of AFM instead of
SP state under ambient pressure in X = SbF6. Most in-
terestingly, Nakamura et al. have very recently observed
redistribution of the charges on the TMTTF molecules
in (TMTTF)2AsF6 and (TMTTF)2PF6 as these systems
enter the SP phase from CO states15,16. Charge dispro-
portionation, if any, in the SP phase is much smaller
than in the CO phase15,16, which is in apparent agree-
ment with the above observations9,14 in X = ReO4 and
SbF6.
The bulk of the (TMTTF)2X therefore lie in the
AFM1 and SP regions of Fig. 1. (TMTSF)2X, in
contrast, occupy the AFM2 region. Coexisting 2kF
CDW and spin-density wave, SDW, with the same
2kF periodicity
49,50 here is explained naturally as the
· · · 1100· · · BCSDW21,22,51. In contrast to the TMTTF
salts discussed above, charge and magnetic ordering in
(TMTTF)2Br occur almost simultaneously
46,52. X-ray
studies of lattice distortions point to similarities with
(TMTSF)2PF6
49, indicating that (TMTTF)2Br is also
a · · · 1100· · · BCSDW21. We do not discuss AFM2 re-
gion in the present paper, as this can be found in our
earlier work21,25.
III. THEORETICAL MODEL AND
CONJECTURES
The 1D Hamiltonian we investigate is written as
H = HSSH +HHol +Hee (1a)
HSSH = t
[1 + α(a
i + ai)](c
i,σci+1,σ + h.c.)
+ ~ωS
iai (1b)
HHol = g
i + bi)ni + ~ωH
ibi (1c)
Hee = U
ni,↑ni,↓ + V
nini+1 (1d)
In the above, c
i,σ creates an electron with spin σ (↑,↓) on
molecular site i, ni,σ = c
i,σci,σ is the number of electrons
with spin σ on site i, and ni =
ni,σ. U and V are the
on-site and intersite Coulomb repulsions, and a
i and b
create (dispersionless) Su-Schrieffer-Heeger (SSH)53 and
Holstein (Hol)54 phonons on the ith bond and site respec-
tively, with frequencies ωS and ωH . Because the Peierls
instability involves only phonon modes near q = π, keep-
ing single dispersionless phonon modes is sufficient for
the Peierls transitions to occur55,56. Although purely 1D
calculations cannot yield a finite temperature phase tran-
sition, as in all low dimensional theories57,58 we antici-
pate that the 3D ordering in the real system is principally
determined by the dominant 1D instability.
The above Hamiltonian includes the most important
terms necessary to describe the family of quasi-1D CTS,
but ignores nonessential terms that may be necessary for
understanding the detailed behavior of individual sys-
tems. Such nonessential terms include (i) the intrinsic
dimerization that characterizes many (TMTTF)2X, (ii)
interaction between counterions and the carriers on the
quasi-1D cations stacks, (iii) interchain Coulomb inter-
action, and (iv) interchain hopping. Inclusion of the in-
trinsic dimerization will make the Wigner crystal ground
state even less likely24, and this is the reason for exclud-
ing it. We have verified the conclusions of reference 24
from exact diagonalization calculations. The inclusion of
interactions with counterions may enhance the Wigner
crystal ordering5,23 for some (TMTTF)2X. We will dis-
cuss this point further below, and argue that it is im-
portant in the AFM1 region of the phase diagram. The
effects of intrinsic dimerization and counterion interac-
tions can be reproduced by modifying the V/|t| in our
Hamiltonian, and thus these are not included explicitly.
Rather the V in Eq. (1a) should be considered as the
effective V for the quasi-1D CTS. Interchain hopping, at
least in the TMTTF (though not in the TMTSF), are in-
deed negligible. The interchain Coulomb interaction can
promote the Wigner crystal within a rectangular lattice
framework, but for the realistic nearly triangular lattice
will cause frustration, thereby making the BCDW state
of interest here even more likely. We therefore believe
that Hamiltonian (1a) captures the essential physics of
the quasi-1D CTS.
For applications to the quasi-1D CTS we will be inter-
ested in the parameter regime18,24,25,26 |t| = 0.1 − 0.25
eV, U/|t| = 6 − 8. The exact value of V is less known,
but since the same cationic molecules of interest as well
as other related molecules (for e.g., HMTTF, HMTSF,
etc.) also form quasi-1D 1
-filled band Mott-Hubbard
semiconductors with short-range antiferromagnetic spin
correlations59,60, it must be true that V < 1
U (since for
V > 1
U the 1D 1
-filled band is a CDW61,62). Two other
known theoretical results now fix the range of V that
will be of interest. First, the ground state of the 1
-filled
band within Hamiltonian (1a) in the limit of zero e-p
coupling is the Wigner crystal · · · 1010· · · only for suffi-
ciently large V > Vc(U). Second, Vc(U → ∞) = 2|t|, and
is larger for finite U19,24,25. With the known U/|t| and
the above restrictions in place, it is now easily concluded
that (i) the Wigner crystal is obtained in the CTS for a
relatively narrow range of realistic parameters, and (ii)
even in such cases the material’s V is barely larger than
Vc(U)
24,25.
We now go beyond the above ground state theories
of spatial broken symmetries to make the following cru-
cial observation: each different spin subspace of Hamil-
tonian (1a) must have its own Vc at which the Wigner
crystal is formed. This conclusion of a spin-dependent
Vc = Vc(U, S) follows from the comparison of the fer-
romagnetic subspace with total spin S = Smax and the
S = 0 subspace. The ferromagnetic subspace is equiva-
lent to the 1
-filled spinless fermion band, and therefore
Vc(U, Smax) is independent of U and exactly 2|t|. The
increase of Vc(U, S = 0) with decreasing U
19,25 is then
clearly related to the occurrence of doubly occupied and
vacant sites at finite U in the S = 0 subspace. Since
the probability of double occupancy (for fixed U and V )
decreases monotonically with increasing S, we can in-
terpolate between the two extreme cases of S = 0 and
S = Smax to conclude Vc(U, S) > Vc(U, S+1). We prove
this explicitly from numerical calculations in the next
section.
Our conjecture regarding spin-dependent Vc(U) in turn
implies that there exist three distinct parameter regimes
for realistic U and V : (i) V ≤ V (Smax) = 2|t|, in
which case the ground state is the BCDW with 2kF CO
and the high temperature 4kF state is a BOW
18,25; (ii)
V > Vc(U, S = 0), in which case both the 4kF and
the 2kF phases have Wigner crystal CO; (iii) and the
intermediate regime 2|t| ≤ V ≤ Vc(U, S = 0). Pa-
rameter regime (i) has been discussed in our previous
work18,21,25,63. This is the case where the description of
the 1
-filled band as an effective 1
-filled band is appropri-
ate: the unit cell in the 4kF state is a dimer of two sites,
and the 2kF transition can be described as the usual SP
dimerization of the dimer lattice. We will not discuss
this parameter region further. Given the U/|t| values for
the CTS, parameter regime (ii) can have rather narrow
width. For U/|t| = 6, Vc(U, S = 0) = 3.3|t|, and there
is no value of realistic V for which the ground state is a
Wigner crystal. For U/|t| = 8, Vc(U, S = 0) = 3.0|t|, and
the widths of parameter regime (iii), 2|t| ≤ V ≤ 3|t| and
of parameter regime (ii), 3|t| ≤ V ≤ 4|t|, are comparable.
We will discuss parameter regime (ii) only briefly, as
the physics of this region has been discussed by other
authors20,26. We investigated thoroughly the intermedi-
ate parameter regime (iii), which has not been studied
previously. Within the intermediate parameter regime
we expect a novel transition from the BCDW in the 2kF
state at low T to a · · · 1010· · · CO in the 4kF phase at
high T that has not been discussed in the theoretical lit-
erature.
Our observation regarding the T-dependent behavior
of these CTS follows from the more general observation
that thermodynamic behavior depends on the free en-
ergy and the partition function. For the strong e-e inter-
actions of interest here, thermodynamics of 1D systems
at temperatures of interest is determined almost entirely
by spin excitations. Since multiplicities of spin states
increase with the total spin S, the partition function is
dominated by high (low) spin states with large (small)
multiplicities at high (low) temperatures. While at T=0
such a system must be a BCDW for V < Vc(U, S = 0), as
the temperature is raised, higher and higher spin states
begin to dominate the free energy, until V for the ma-
terial in question exceeds Vc(U, S), at which point the
charge occupancy reverts to · · · 1010· · · We demonstrate
this explicitly in the next section. A charge redistribu-
tion is expected in such a system at T2kF , as the devi-
ation from the average charge of 0.5 is much smaller in
the BCDW than in the Wigner crystal25.
The above conjecture leads to yet another novel impli-
cation. The ground state for both weak and intermediate
intersite Coulomb interaction parameters are the BCDW,
even as the 4kF phases are different in the two cases:
BOW in the former and CO in the latter. This necessarily
requires the existence of two different kinds of spin exci-
tations from the BCDW. Recall that within the standard
theory of the SP transition in the 1
-filled band57,64,65,
thermal excitations from the T = 0 ground state gener-
ates spin excitations with bond-alternation domain walls
(solitons), with the phase of bond alternation in between
the solitons being opposite to that outside the solitons.
Progressive increase in T generates more and more soli-
tons with reversed bond alternation phases, until overlaps
between the two phases of bond alternations lead ulti-
mately to the uniform state. A key difference between
the 1
-filled and 1
-filled SP states is that site charge oc-
cupancies in spin excitations in the former continue to be
uniform, while they are necessarily nonuniform in the lat-
ter case, as we show in the next section. We demonstrate
that defect centers with two distinct charge occupancies
(that we will term as type I and II), depending upon the
actual value of V , are possible in the 1
-filled band. Pre-
ponderance of one or another type of defects generates
the distinct 4kF states.
IV. RESULTS
We present in this section the results of QMC investi-
gations of Eq. (1a), for both zero and nonzero e-p cou-
plings, and of ED studies of the adiabatic (semiclassical)
limit of Eq. (1a). Using QMC techniques, we demon-
strate explicitly the spin-dependence of Vc(U), as well as
the transition from the Wigner crystal to the BCDW for
the intermediate parameter regime (iii). The ED studies
demonstrate the exotic nature of spin excitations from
the BCDW ground state. In what follows, all quantities
are expressed in units of |t| (|t|=1).
The QMC method we use is the Stochastic Series Ex-
pansion (SSE) method using the directed loop update
for the electron degrees of freedom66. For 1D fermions
with nearest-neighbor hopping SSE provides statistically
exact results with no systematic errors. While the SSE
directed-loop method is grand canonical (with fluctuat-
ing particle density), we restrict measurements to only
the 1
-filled density sector to obtain results in the canoni-
cal ensemble. Quantum phonons are treated within SSE
by directly adding the bosonic phonon creation and an-
nihilation operators in Eq. (1a) to the series expansion67.
An upper limit in the phonon spectrum must be imposed,
but can be set arbitrarily large to avoid any systematic
errors67. For the results shown below we used a cutoff of
100 SSH phonons per bond and either 30 (for g = 0.5) or
50 (for g = 0.75) Holstein phonons per site.
The observables we calculate within SSE are the
standard wave-vector dependent charge structure factor
Sρ(q), defined as,
Sρ(q) =
eiq(j−k)〈O
〉 (2)
and charge and bond-order susceptibilities χρ(q) and
χB(q), defined as,
χx(q) =
eiq(j−k)
dτ〈Oxj (τ)O
k (0)〉 (3)
In Eqs. (2) and (3) N is the number of lattice sites, O
nj,↑ + nj,↓, O
j+1,σcj,σ + h.c.), and β is the
inverse temperature in units of t.
The presence of CO or BOW can be detected by the
divergence of the 2kF or 4kF charge or bond-order sus-
ceptibility as a function of increasing system size. Strictly
speaking, in a purely 1D model these functions diverge
only at T = 0; as already explained above, we make the
reasonable assumption58 that in the presence of realis-
tic inter-chain couplings transitions involving charge or
bond-order instabilities, as determined by the dominant
susceptibility, occur at finite T.
A. Spin-dependent Vc(U)
We first present computational results within Hamil-
tonian (1a) in the absence of e-p coupling to demon-
strate that the Vc(U) at which the Wigner crystal order
is established in the lowest state of a given spin sub-
space decreases with increasing spin S. Our computa-
tional approach conserves the total z-component of spin
Sz and not the total S. Since the Lieb-Mattis theorem
E(S) < E(S+1), where E(S) is the energy of the lowest
state in the spin subspace S, applies to the 1D Hamilto-
nian (1a), and since in the absence of a magnetic field
all Sz states for a given S are degenerate, our results for
the lowest state within each different Sz must pertain to
S = Sz.
To determine Vc(U, S) we use the fact that the purely
electronic model is a Luttinger liquid (LL) for V < Vc
with correlation functions determined by a single expo-
nent Kρ (see Reference 69 for a review). The Wigner
crystal state is reached when Kρ =
. The exponent
Kρ may be calculated from the long-wavelength limit of
Sρ(q)
70. In Fig. 2(a) we have plotted our calculated Kρ
for U = 8 as a function of V for different Sz sectors. The
2.2 2.4 2.6 2.8 3 3.2
0 2 4 6 8
FIG. 2: Vc for Eq. (1a) in the limit of zero e-p interactions
(α = g = 0) as a function of Sz. Results are for a N=32
site periodic ring with U = 8. For N = 32 Sz=8 corresponds
to the fully polarized (spinless fermion) limit. (a) Luttinger
Liquid exponent Kρ as a function of V . Kρ =
determines
the boundary for the · · · 1010· · · CO phase. (b) Vc plotted vs.
temperature chosen is small enough (β = 2N) that in all
cases the results correspond to the lowest state within
a given Sz. In Fig. 2(b) we have plotted our calculated
Vc(U = 8), as obtained from Fig. 2(a), as a function of
Sz. Vc is largest for Sz = 0 and decreases with increas-
ing Sz, in agreement with the conjecture of Section III.
Importantly, the calculated Vc for Sz = 8 is close to the
correct limiting value of 2, indicating the validity of our
approach. We have not performed any finite-size scaling
in Fig. 2, which accounts for the the slight deviation from
the exact value of 2.
B. T-dependent susceptibilities
We next present the results of QMC calculations within
the full Hamiltonian (1a). To reproduce correct relative
energy scales of intra- and intermolecular phonon modes
in the CTS, we choose ωH > ωS , specifically ωH = 0.5
and ωS = 0.1 in our calculations. Small deviations from
these values do not make any significant difference. In
all cases we have chosen the electron-molecular vibration
coupling g larger than the coupling between electrons and
the SSH phonons α, thereby deliberately enhancing the
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
0 0.05 0.1 0.15 0.2 0.25
FIG. 3: (color online) QMC results for the temperature-
dependent charge susceptibilities for a N=64 site periodic
ring with U = 8, V = 2.75, α = 0.15, ωS = 0.1, g = 0.5,
and ωH = 0.5. (a) and (b) Wavevector-dependent charge
and bond-order susceptibilities. (c) 2kF and 4kF charge sus-
ceptibilities as a function of temperature. (d) 2kF and 4kF
bond-order susceptibilities as a function of temperature. If
error bars are not shown, statistical error bars are smaller
than the symbol sizes. Lines are guides to the eye.
likelihood of the Wigner crystal CO. We report results
only for intermediate and strong weak intersite Coulomb
interactions; the weak interaction regime V < 2 has been
discussed extensively in our previous work25.
1. 2 ≤ V ≤ Vc(U, S = 0)
Our results are summarized in Figs. 3–5, where we re-
port results for two different V of intermediate strength
and several different e-p couplings. Fig. 3 first shows re-
sults for relatively weak SSH coupling α. The charge sus-
ceptibility is dominated by a peak at 4kF = π (Fig. 3(a)).
Figs. 3(c) and (d) show the T-dependence of the 2kF
as well as 4kF charge and bond susceptibility. For
V < Vc(U, S = 0) the 4kF charge susceptibility does
not diverge with system size, and the purely 1D system
remains a LL with no long range CO at zero tempera-
ture. The dominance of χρ(4kF ) over χρ(2kF ) suggests,
however, that · · · 1010· · · CO will likely occur in the 3D
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
 g=0.50
 g=0.50
 g=0.75
 g=0.75
0 0.05 0.1 0.15 0.2 0.25
 g=0.50
 g=0.50
 g=0.75
 g=0.75
FIG. 4: (color online) Same as Fig. 3, but with parameters
U = 8, V = 2.25, α = 0.27, ωS = 0.1, and ωH = 0.5. In panels
(a) and (b), data are for g = 0.50 only. In panels (c) and (d),
data for both g = 0.50 and g = 0.75 are shown. Arrows
indicate temperature where χρ(2kF ) = χρ(4kF ) (solid and
broken arrows correspond to g = 0.50 and 0.75, respectively.)
system, especially if this order is further enhanced due to
interactions with the counterions9. We have plotted the
bond susceptibilities χB(2kF ) and χB(4kF ) in Fig. 3(d).
A SP transition requires that χB(2kF ) diverges as T → 0.
χB(2kF ) weaker than χB(4kF ) at low T indicates that
the SP order is absent in the present case with weak SSH
e-p coupling. This result is in agreement with earlier
result55 that the SP order is obtained only above a critical
αc (αc may be smaller for the infinite system than found
in our calculation). The most likely scenario with the
present parameters is the persistence of the · · · 1010· · ·
CO to the lowest T with no SP transition. These pa-
rameters, along with counterion interactions, could then
describe the (TMTTF)2Xmaterials with an AFM ground
state9.
In Figs. 4 and Fig. 5 we show our results for larger
SSH e-p coupling α and two different intersite Coulomb
interaction V=2.25 and 2.75. The calculations of Fig. 4
were done for two different Holstein e-p couplings g. For
both the V parameters, χρ(4kF ) dominates at high T
but χρ(2kF ) is stronger at low T in both Fig. 4(c) and
Fig. 5(c). The crossing between the two susceptibilities
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
 g=0.50
 g=0.50
0 0.05 0.1 0.15 0.2 0.25
 g=0.50
 g=0.50
FIG. 5: (color online) Same as Fig. 3, but with parameters
U = 8, V = 2.75, α = 0.27, ωS = 0.1, g = 0.5, and ωH = 0.5.
Arrow indicates temperature where χρ(2kF ) = χρ(4kF ).
is clear indication that in the intermediate parameter
regime, as T is lowered the 2kF CDW instability domi-
nates over the 4kF CO.
The rise in χρ(2kF ) at low T is accompanied by a
steep rise in χB(2kF ) in both cases (see Fig. 4(d) and
Fig. 5(d)). Importantly, unlike in Fig. 3(d), χB(2kF ) in
these cases clearly dominates over χB(4kF ) by an order
of magnitude at low temperatures. There is thus a clear
signature of the SP instability for these parameters. The
simultaneous rise in χB(2kF ) and χρ(2kF ) indicates that
the SP state is the · · · 1100· · · BCDW. Comparison of
Figs. 4 and 5 indicates that the effect of larger V is to
decrease the T where the 2kF and 4kF susceptibilities
cross. Since larger V would imply larger TCO, this result
implies that larger TCO is accompanied by lower TSP .
Our calculations are for relatively modest α < g. Larger
α (not shown) further strengthens the divergence of 2kF
susceptibilities.
The motivation for performing the calculations of
Fig. 4 with multiple Holstein couplings was to deter-
mine whether it is possible to have a Wigner crystal at
low T even for V < V (U, S = 0), by simply increas-
ing g. The argument for this would be that in strong
coupling, increasing g amounts to an effective increase
in V 71. The Holstein coupling cannot be increased arbi-
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
0 0.05 0.1 0.15 0.2 0.25
FIG. 6: (color online) Same as Fig. 3, but with parameters
U = 8, V = 3.5, α = 0.24, ωS = 0.1, g = 0.5, and ωH = 0.5.
trarily, however, as beyond a certain point, g promotes
formation of on-site bipolarons67. Importantly, the co-
operative interaction between the 2kF BOW and CDW
in the BCDW21,25 implies that in the V < V (U, S = 0)
region, larger g not only promotes the 4kF CO but also
enhances the BCDW. In our calculations in Fig. 4(b)
and (c) we we find both these effects: a weak increase in
χρ(4kF ) at intermediate T, and an even stronger increase
in the T → 0 values of χρ(2kF ) and χB(2kF ). Actually,
this result is in qualitative agreement with our obser-
vation for the ground state within the adiabatic limit
of Eq. (1a) that in the range 0 < V < V (U, S = 0),
V enhances the BCDW25. The temperature at which
χρ(2kF ) and χρ(4kF ) cross does not change significantly
with larger g. Our results for g = 0.75 for V = 2.75 (not
shown) are similar, except that the data are more noisy
now (probably because all parameters in Fig. 5 are much
too large for the larger g). We conclude therefore that in
the intermediate V region, merely increasing g does not
change the BCDW nature of the SP state. For g to have
a qualitatively different effect, V should be much closer
to V (U, S = 0) or perhaps even larger.
2. V > Vc(U, S = 0)
In principle, calculations of low temperature instabili-
ties here should be as straightforward as the weak inter-
site Coulomb interaction V < 2t regime. The 4kF CO -
AFM1 state ↑ 0 ↓ 0 would occur naturally for the case
of weak e-p coupling. Obtaining the 4kF CO-SP state
1 = 0 = 1 · · · 0 · · · 1, with realistic V < 1
U is, however,
difficult25. Previous work72, for example, finds this state
for V > 1
U . Recent T-dependent mean-field calculations
of Seo et al.39 also fail to find this state for nonzero V .
There are two reasons for this. First, the spin exchange
here involves charge-rich sites that are second neighbors,
and are hence necessarily small. Energy gained upon
alternation of this weak exchange interactions is even
smaller, and thus the tendency to this particular form
of the SP transition is weak to begin with. Second, this
region of the parameter space involves either large U (for
e.g., U = 10, for which Vc(U, S = 0) ≃ 2) or relatively
large V (for e.g. Vc(U, S = 0) ≃ 3 for U = 8). In either
case such strong Coulomb interactions make the applica-
bility of mean-field theories questionable.
Our QMC calculations do find the tendency to SP in-
stability in this parameter region. In Fig. 6 we show
QMC results for V > Vc(U = 8, S = 0). In contrast to
Figs. 4 and 5, χρ(4kF ) now dominates over χρ(2kF ) at
all T. The weaker peak at q = 2kF at low T is due to
small differences in site charge populations between the
charge-poor sites of the Wigner crystal that arises upon
bond distortion25, and that adds a small period 4 compo-
nent to the charge modulation. The bond susceptibility
χB(q) has a strong peak at 2kF and a weaker peak at 4kF ,
exactly as expected for the 4kF CO-SP state. Previous
work has shown that the difference in charge densities
between the charge-rich and charge-poor sites in the 4kF
CO-SP ground state is considerably larger than in the
BCDW25.
C. Spin excitations from the BCDW
The above susceptibility calculations indicate that in
the intermediate V region (Figs. 4 and 5) the BCDW
ground state with · · · 1100· · · CO can evolve into the
· · · 1010· · · CO as T increases. Our earlier work had
shown that for weak V , the BCDW evolves into the 4kF
BOW at high T18,25. Within the standard theory of the
SP transition57,64,65, applicable to the 1
-filled band, the
SP state evolves into the undistorted state for T > TSP .
Thus the 4kF distortions, CO and BOW, take the role of
the undistorted state in the 1
-filled band, and it appears
paradoxical that the same ground state can evolve into
two different high T states. We show here that this is inti-
mately related to the nature of the spin excitations in the
-filled BCDW. Spin excitations from the conventional
SP state leave the site charge occupancies unchanged.
We show below that not only do spin excitations from
the 1
-filled BCDW ground state lead to changes in the
site occupancies, two different kinds of site occupancies
are possible in the localized defect states that characterize
spin excited states here. We will refer to these as type
I and type II defects, and depending on which kind of
defect dominates at high T (which in turn depends on
the relative magnitudes of V and the e-p couplings), the
4kF state is either the CO or the BOW.
We will demonstrate the occurrence of type I and II
defects in spin excitations numerically. Below we present
a physical intuitive explanation of this highly unusual
behavior, based on a configuration space picture. Very
similar configuration space arguments can be found else-
where in our discussion of charge excitations from the
BCDW in the interacting 1
-filled band73.
We begin our discussion with the standard 1
-filled
band, for which the mechanism of the SP transition is
well understood57,64,65. Fig. 7(a) shows in valence bond
(VB) representation the generation of a spin triplet from
the standard 1
-filled band. Since the two phases of bond
alternation are isoenergetic, the two free spins can sepa-
rate, and the true wavefunction is dominated by VB dia-
grams as in Fig. 7(b), where the phase of the bond alter-
nation in between the two unpaired spins (spin solitons)
is opposite to that in the ground state. With increasing
temperature and increasing number of spin excitations
there occur many such regions with reversed bond al-
ternations, and overlaps between regions with different
phases of bond alternations leads ultimately to the uni-
form state.
The above picture needs to be modified for the dimer-
ized dimer BCDW state in the 1
-filled band, in which
the single site of the 1
-filled band system is replaced by
a dimer unit with site populations 1 and 0 (or 0 and 1),
and the stronger interdimer 1–1 bond (weaker interdimer
0· · · 0 bond) corresponds to the strong (weak) bond in the
standard SP case. Fig. 7(c), showing triplet generation
from the BCDW, is the 1
-filled analog of Fig. 7(a): a
singlet bond between the dimer units has been broken to
generate a localized triplet. The effective repulsion be-
tween the free spins of Fig. 7(a) is due to the absence
of binding between them, and the same is expected in
Fig. 7(c). Because the site occupancies within the dimer
units are nonuniform now, the repulsion between the
spins is reduced from changes in intradimer as well as
interdimer site occupancies: the site populations within
the neighboring units can become uniform (0.5 each), or
the site occupancies can revert to 1001 from 0110. There
is no equivalent of this step in the standard SP case.
The next steps in the separation of the resultant defects
are identical to that in Fig. 7(b), and we have shown
these possible final states in Fig. 7(d) and Fig. 7(e), for
the two different intraunit spin defect populations. For
V < Vc(U, S), defect units with site occupancies 0.5 occu-
pancies (type I defects) are expected to dominate; while
for V > Vc(U, S) site populations of 10 and 01 (type II
defects) dominate. From the qualitative discussions it is
difficult to predict whether the defects are free, in which
case they are solitons, or if they are bound, in which case
0 1 0 1.5 .5 .5 .5
1 0 0 1   0 0 1
0   0 0 1 1 0 0 1 1 0
0 1 1 0
FIG. 7: (a) and (b) S = 1 VB diagrams in the Heisenberg SP
chain. The SP bond order in between the S = 1
solitons in (b)
is opposite to that elsewhere. (c) 1
-filled band equivalent of
(a); the singlet bonds in the background BCDW are between
units containing a pair of sites but a single electron (see text).
(d) and (e) 1
-filled band equivalents of (b). Defect units with
two different charge distributions are possible. The charge
distribution will depend on V .
two of them constitute a triplet. What is more relevant
is that type I defects generate bond dimerization locally
(recall that the 4kF BOW has uniform site charge densi-
ties) while type II defects generate local site populations
1010, which will have the tendency to generate the 4kF
The process by which the BCDW is reached from the
4kF BOW or the CO as T is decreased from T2kF is ex-
actly opposite to the discussion of spin excitations from
the BCDW in Fig. 7. Now · · · 1100· · · domain walls
appear in the background bond-dimerized or charge-
dimerized state. In the case of the · · · 1010· · · 4kF state,
the driving force for the creation of such a domain wall
is the energy gained upon singlet formation (for V <
Vc(U,S)).
Fig. 7(e) suggests the appearance of localized
· · · 1010· · · regions in excitations from the · · · 1100· · ·
BCDW SP state for 2|t| ≤ V ≤ Vc(U, S = 0). We have
verified this with exact spin-dependent calculations for N
= 16 and 20 periodic rings with adiabatic e-p couplings,
HASSH = t
[1− α(ui − ui+1)](c
i,σci+1,σ + h.c.)
(ui − ui+1)
2 (4a)
HAHol = g
vini +
v2i (4b)
and Hee as in Eq. (1d). In the above ui is the displace-
ment from equilibrium of a molecular unit and vi is a
molecular mode. Note that due to the small sizes of
the systems we are able to solve exactly, the e-p cou-
plings in Eq. (4a) and Eq. (4b) cannot be directly com-
pared to those in Eq. (1a). In the present case, we
take the smallest e-p couplings necessary to generate the
BCDW ground state self-consistently within the adia-
batic Hamiltonian22,23,25. We then determine the bond
distortions, site charge occupancies and spin-spin corre-
4 8 12 16 20
-0.04
-0.04
FIG. 8: (a) Self-consistent site charges (vertical bars) and
spin-spin correlations (points, lines are to guide the eye) in
the N=20 periodic ring with U = 8, V = 2.75, α2/KS = 1.5,
g2/KH = 0.32 for Sz = 2. (b) Same parameters, except
Sz = 3. Black and white circles at the bottoms of the panels
denote charge-rich and charge-poor sites, respectively, while
gray circles denote sites with population nearly exactly 0.5.
The arrows indicate locations of the defect units with nonzero
spin densities.
lations self-consistently with the same parameters for ex-
cited states with Sz > 0.
In Fig. 8(a) we have plotted the deviations of the site
charge populations from average density of 0.5 for the
lowest Sz=2 state for N = 20, for U = 8, V = 2.75 and e-p
couplings as given in the figure caption. Deviations of the
charge occupancies from the perfect · · · 0110· · · sequence
identify the defect centers, as seen from Fig. 7(c) - (e).
In the following we refer to dimer units composed of sites
i and j as [i,j]. Based on the charge occupancies in the
Fig, we identify units [1,2], [7,8], [13,14] and [14,15] as the
defect centers. Furthermore, based on site populations of
nearly exactly 0.5, we identify defects on units [1,2] and
[7,8] as type I; the populations on units [13,14] and [14,15]
identify them type II. Type I defects appear to be free
and soliton like, while type II defects appear to be bound
into a triplet state, but both could be finite size effects.
Fig. 7(d) and (e) suggest that the spin-spin z-
component correlations, 〈Szi S
j 〉, are large and positive
only between pairs of sites which belong to the defect cen-
ters, as all other spins are singlet-coupled. For our char-
acterization of units [1,2], [7,8], [13,14] and [14,15] to be
correct therefore, 〈Szi S
j 〉 must be large and positive if
sites i and j both belong to this set of sites, and small
(close to zero) when either i or j does not belong to
this set (as all sites that do not belong to the set are
singlet-bonded). We have superimposed the calculated
z-component spin-spin correlations between site 2 and
all sites j = 1 – 20. The spin-spin correlations are in
complete agreement with our characterization of units
[1,2], [7,8], [13,14] and [14,15] as defect centers with free
spins. The singlet 1–1 bonds between nearest neighbor
charge-rich sites in Fig. 7(c) - (e) require that spin-spin
couplings between such pairs of sites are large and neg-
ative, while spin-spin correlations between one member
of the pair any other site is small. We have verified such
strong spin-singlet bonds between sites 4 and 5, 10 and
11, and 18 and 19, respectively (not shown). Thus mul-
tiple considerations lead to the identification of the same
sites as defect centers, and to their characterization as
types I and II.
We report similar calculations in Fig. 8(b) for Sz = 3.
Based on charge occupancies, four out of the six defect
units with unpaired spins in the Sz = 3 state are type II;
these occupy units [3,4], [9,10], [13,14] and [19,20]. Type
I defects occur on units [1,2] and [11,12]. As indicated in
the figure, spin-spin correlations are again large between
site 2 and all other sites that belong to this set, while they
are again close to zero when the second site is not a defect
site. As in the previous case, we have verified singlet spin
couplings between nearest neighbor pairs of charge-rich
sites. There occur then exact correspondences between
charge densities and spin-spin correlations, exactly as for
Sz = 2.
The susceptibility calculations in Fig. 4 and Fig. 5 are
consistent with the microscopic calculations of spin de-
fects presented above. As 4kF defects are added to the
2kF background, the 2kF susceptibility peak is expected
to broaden and shift towards higher q. This is exactly
what is seen in the charge susceptibility, Fig. 4(a) and
Fig. 5(a) as T is increased. A similar broadening and
shift is seen in the bond order susceptibility as well.
V. DISCUSSIONS AND CONCLUSIONS
In summary, the SP state in the 1
-filled band CTS is
unique for a wide range of realistic Coulomb interactions.
Even when the 4kF state is the Wigner crystal, the SP
state can be the · · · 1100· · · BOW. For U = 8, for exam-
ple, the transition found here will occur for 2 < V < 3.
This novel T-dependent transition from the Wigner crys-
tal to the BCDW is a consequence of the spin-dependence
of Vc. Only for V > 3 here can the SP phase be the
tetramerized Wigner crystal 1 = 0 = 1 · · · 0 · · · 1 (note,
however, that V ≤ 4 for U = 8). We have ignored the
intrinsic dimerization along the 1D stacks in our reported
results, but this increases Vc(U, S = 0) even further, and
makes the Wigner crystal that much more unlikely24.
Although even larger U (U = 10, for example) reduces
Vc(U, S = 0), we believe that the Coulomb interactions
in the (TMTTF)2X lie in the intermediate range.
A Wigner crystal to BCDW transition would explain
most of the experimental surprises discussed in Section
II. The discovery of the charge redistribution upon en-
tering the SP phase15,16 in X = AsF6 and PF6 is prob-
ably the most dramatic illustration of this. Had the SP
state maintained the same charge modulation pattern
as the Wigner crystal that exist above T2kF , the dif-
ference in charge densities between the charge-rich and
the charge-poor sites would have changed very slightly25.
The dominant effect of the SP transition leading to
1 = 0 = 1 · · · 0 · · · 1 is only on the charge-poor sites,
which are now inequivalent (note that the charge-rich
sites remain equivalent). The difference in charge den-
sity of the charge-rich sites and the average of the charge
density of the charge-poor sites thus remains the same25.
In contrast, the difference in charge densities between
the charge-rich and the charge-poor sites in the BCDW
is considerably smaller than in the Wigner crystal25,
and we believe that the experimentally observed smaller
charge density difference in the SP phase simply reflects
its BCDW character.
The experiments of references 15,16 should not be
taken in isolation: we ascribe the competition between
the CO and the SP states in X = PF6 and AsF6, as
reflected in the different pressure-dependences of these
states8, to their having different site charge occupan-
cies. The observation that the charge density difference
in X = SbF6 decreases considerably upon entering the SP
phase from the AFM1 phase9 can also be understood if
the AFM1 and the SP phases are assigned to be Wigner
crystal and the BCDW, respectively. The correlation be-
tween larger TCO and smaller TSP
48 is expected. Larger
TCO implies larger effective V , which would lower TSP .
This is confirmed from comparing Figs. 4 and 5: the
temperature at which χρ(2kF ) begins to dominate over
χρ(4kF ) is considerably larger in Fig. 4 (smaller V ) than
in Fig. 5 (larger V ). The isotope effect, strong enhance-
ment of the TCO (from 69 K to 90 K) with deutera-
tion of the methyl groups in X = PF6, and concomi-
tant decrease in TSP
6,47 are explained along the same
line. Deuteration decreases ωH in Eq. (1a), which has
the same effect as increasing V . Thus from several dif-
ferent considerations we come to the conclusion that the
transition from the Wigner crystal to the BCDW that we
have found here theoretically for intermediate V/|t| does
actually occur in (TMTTF)2X that undergo SP transi-
tion. This should not be surprising. Given that the 1:2
anionic CTS lie in the “weak” V/|t| regime, TMTTF with
only slightly smaller |t| (but presumably very similar V ,
since intrastack intermolecular distances are comparable
in the two families) lies in the “intermediate” as opposed
to “strong” V/|t| regime.
Within our theory, the two different antiferromagnetic
regions that straddle the SP phase in Fig. 1, AFM1 and
AFM2, have different charge occupancies. The Wigner
crystal character of the AFM1 region is revealed from the
similar behavior of TN and TCO in (TMTTF)2SbF6 un-
der pressure9, indicating the absence of the competition
of the type that exists between CO and SP, in agree-
ment with our assignment. The occurrence of a Wigner
crystal AFM1 instead of SP does not necessarily imply
a larger V/|t| in the SbF6. A more likely reason is that
the interaction with the counterions is strong here, and
this interaction together with V pins the electrons on al-
ternate sites. (TMTSF)2X, and possibly (TMTTF)2Br,
belong to the AFM2 region. The observation that the
CDW and the spin-density wave in the TMTSF have the
same periodicities1 had led to the conclusion that the
charge occupancy here is · · · 1100· · · 21. This conclusion
remains unchanged. Finally, we comment that the ob-
servation of Wigner crystal CO3 in (DI-DCNQI)2Ag is
not against our theory, as the low T phase here is an-
tiferromagnetic and not SP. We predict the SP system
(DMe-DCNQI)2Ag to have the · · · 1100· · · charge order-
In summary, it appears that the key concept of spin-
dependent Vc within Eq. (1a) can resolve most of the
mysteries associated with the temperature dependence
of the broken symmetries in the 1
-filled band CTS. One
interesting feature of our work involves demonstration
of spin excitations from the BCDW state that necessar-
ily lead to local changes in site charges. Even for weak
Coulomb interactions, the 1
-filled band has the BCDW
character18. The effects of magnetic field on 1
-filled band
CDWs is of strong recent interest74,75. We are currently
investigating the consequences of the Zeeman interaction
on the mixed spin-charge excitations of the BCDW.
VI. ACKNOWLEDGMENTS
We acknowledge illuminating discussions with S.E.
Brown. This work was supported by the Department
of Energy grant DE-FG02-06ER46315 and the American
Chemical Society Petroleum Research Fund.
1 J. P. Pouget and S. Ravy, J. Physique I 6, 1501 (1996).
2 B. Dumoulin, C. Bourbonnais, S. Ravy, J. P. Pouget, and
C. Coulon, Phys. Rev. Lett. 76, 1360 (1996).
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  It is known that within the interacting electron model Hamiltonian for the
one-dimensional 1/4-filled band, the singlet ground state is a Wigner crystal
only if the nearest neighbor electron-electron repulsion is larger than a
critical value. We show that this critical nearest neighbor Coulomb interaction
is different for each spin subspace, with the critical value decreasing with
increasing spin. As a consequence, with the lowering of temperature, there can
occur a transition from a Wigner crystal charge-ordered state to a spin-Peierls
state that is a Bond-Charge-Density Wave with charge occupancies different from
the Wigner crystal. This transition is possible because spin excitations from
the spin-Peierls state in the 1/4-filled band are necessarily accompanied by
changes in site charge densities. We apply our theory to the 1/4-filled band
quasi-one-dimensional organic charge-transfer solids in general and to 2:1
tetramethyltetrathiafulvalene (TMTTF) and tetramethyltetraselenafulvalene
(TMTSF) cationic salts in particular. We believe that many recent experiments
strongly indicate the Wigner crystal to Bond-Charge-Density Wave transition in
several members of the TMTTF family. We explain the occurrence of two different
antiferromagnetic phases but a single spin-Peierls state in the generic phase
diagram for the 2:1 cationic solids. The antiferromagnetic phases can have
either the Wigner crystal or the Bond-Charge-Spin-Density Wave charge
occupancies. The spin-Peierls state is always a Bond-Charge-Density Wave.

Temperature-driven transition from the Wigner Crystal to the Bond-Charge-Density
Wave in the Quasi-One-Dimensional Quarter-Filled band
R.T. Clay,1 R.P. Hardikar,1 and S. Mazumdar2
1Department of Physics and Astronomy and HPC2 Center for Computational Sciences,
Mississippi State University, Mississippi State MS 39762
2 Department of Physics, University of Arizona Tucson, AZ 85721
(Dated: August 21, 2021)
It is known that within the interacting electron model Hamiltonian for the one-dimensional 1
filled band, the singlet ground state is a Wigner crystal only if the nearest neighbor electron-electron
repulsion is larger than a critical value. We show that this critical nearest neighbor Coulomb
interaction is different for each spin subspace, with the critical value decreasing with increasing
spin. As a consequence, with the lowering of temperature, there can occur a transition from a
Wigner crystal charge-ordered state to a spin-Peierls state that is a Bond-Charge-Density Wave
with charge occupancies different from the Wigner crystal. This transition is possible because
spin excitations from the spin-Peierls state in the 1
-filled band are necessarily accompanied by
changes in site charge densities. We apply our theory to the 1
-filled band quasi-one-dimensional
organic charge-transfer solids in general and to 2:1 tetramethyltetrathiafulvalene (TMTTF) and
tetramethyltetraselenafulvalene (TMTSF) cationic salts in particular. We believe that many recent
experiments strongly indicate the Wigner crystal to Bond-Charge-Density Wave transition in several
members of the TMTTF family. We explain the occurrence of two different antiferromagnetic
phases but a single spin-Peierls state in the generic phase diagram for the 2:1 cationic solids. The
antiferromagnetic phases can have either the Wigner crystal or the Bond-Charge-Spin-Density Wave
charge occupancies. The spin-Peierls state is always a Bond-Charge-Density Wave.
PACS numbers: 71.30.+h, 71.45.Lr, 74.70.Kn
I. INTRODUCTION
Spatial broken symmetries in the quasi-one-
dimensional (quasi-1D) 1
-filled organic charge-
transfer solids (CTS) have been of strong
experimental1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 and
theoretical18,19,20,21,22,23,24,25,26 interest. The bro-
ken symmetry states include charge order (hereafter
CO, this is usually accompanied by intramolecular
distortions), intermolecular lattice distortions (hereafter
bond order wave or BOW), antiferromagnetism (AFM)
and spin-Peierls (SP) order. Multiple orderings may
compete or even coexist simultaneously. Interestingly,
these unconventional insulating states in the CTS are
often proximate to superconductivity27, the mecha-
nism of which has remained perplexing after intensive
investigations over several decades. Unconventional
behavior at or near 1
-filling has also been observed
in the quasi-two-dimensional organic CTS with higher
superconducting critical temperatures11,28,29,30,31,
sodium cobaltate32,33,34 and oxides of titanium35 and
vanadium36,37.
In spite of extensive research on 1D instabilities in the
CTS, detailed comparisons of theory and experiments re-
main difficult. Strong electron-electron (e-e) Coulomb
interactions in these systems make calculations particu-
larly challenging, and with few exceptions38,39 existing
theoretical discussions of broken symmetries in the in-
teracting 1
-filled band have been limited to the ground
state18,19,20,21,22,23,24,25,26. This leaves a number of im-
portant questions unresolved, as we point out below.
4k  1010F
AsF6 PF6SbF6 PF6
Pressure
FIG. 1: Schematic of the proposed T-vs.-P phase diagram for
(TMTCF)2X, where C=T or S, along with the charge occu-
pancies of the sites in the low T phases as determined in this
work. The P-axis reflects the extent of interchain coupling.
Open (filled) arrows indicate the ambient pressure locations
of TMTTF (TMTSF) salts.
Quasi-1D CTS undergo two distinct phase transitions
as the temperature (hereafter T) is reduced40. The 4kF
transition at higher T involves charge degrees of free-
dom (T4kF ∼ 100 K), with the semiconducting state be-
low T4kF exhibiting either a dimerized CO or a dimer-
ized BOW. Charges alternate as 0.5 + ǫ and 0.5 – ǫ
on the molecules along the stack in the dimerized CO.
The site charge occupancy in the dimerized CO state is
commonly written as · · · 1010· · · (with ‘1’ and ‘0’ denot-
ing charge-rich and charge-poor sites), and this state is
http://arxiv.org/abs/0704.1656v2
also referred to as the Wigner crystal. The 4kF BOW
has alternating intermolecular bond strengths, but site
charges are uniformly 0.5. It is generally accepted that
the dimer CO (dimer BOW) is obtained for strong (weak)
intermolecular Coulomb interactions (the intramolecular
Coulomb interaction can be large in either case, see Sec-
tion III). At T < T2kF ∼ 10 – 20 K, there occurs a sec-
ond transition in the CTS, involving spin degrees of free-
dom, to either an SP or an AFM state. Importantly,
the SP and the AFM states both can be derived from
the dimer CO or the dimer BOW. Coexistence of the SP
state with the Wigner crystal would require that the SP
state has the structure 1 = 0 = 1 · · · 0 · · · 1, with singlet
“bonds” alternating in strength between the charge-rich
sites of the Wigner crystal. Similarly, coexisting AFM
and Wigner crystal would imply site charge-spin occu-
pancies ↑ 0 ↓ 0. The occurrence of both have been sug-
gested in the literature20,26,38,39. The SP state can also
be obtained from dimerization of the dimer 4kF BOW,
in which case there occurs a spontaneous transition from
the uniform charge-density state to a coexisting bond-
charge-density wave (BCDW)18,22,25 1− 1 = 0 · · · 0. The
unit cells here are dimers with site occupancies ‘1’ and
‘0’ or ‘0’ and ‘1’, with strong and weak interunit 1–1 and
0· · · 0 bonds, respectively. The AFM state with the same
site occupancies · · · 1100· · · is referred to as the bond-
charge-spin density wave (BCSDW)21 and is denoted as
↑↓ 00.
The above characterizations of the different states and
the related bond, charge and spin patterns are largely
based on ground state calculations. The mechanism of
the transition from the 4kF state to the 2kF state lies
outside the scope of such theories. Questions that remain
unresolved as a consequence include: (i) Does the nature
of the 4kF state (charge versus bond dimerized) predeter-
mine the site charge occupancies of the 2kF state? This
is assumed in many of the published works. (ii) What de-
termines the nature of the 2kF state (AFM versus SP)?
(iii) How does one understand the occurrence of two dif-
ferent AFM phases (hereafter AFM1 and AFM2) strad-
dling a single SP phase in the proposed T vs. P (where P
is pressure) phase diagram (see Fig. 1)9 for the cationic 1
filled band quasi-1D CTS? (iv) What is the nature of the
intermediate T states between the 4kF dimerized state
and the 2kF tetramerized state? As we point out in the
next section, where we present a brief review of the ex-
perimental results, answers to these questions are crucial
for a deeper understanding of the underlying physics of
the 1
-filled band CTS.
In the present paper we report the results of our calcu-
lations of T-dependent behavior within theoretical mod-
els incorporating both e-e and electron-phonon (e-p) in-
teractions to answer precisely the above questions. One
key result of our work is as follows: in between the strong
and weak intermolecular Coulomb interaction parame-
ter regimes, there exists a third parameter regime within
which the intermolecular Coulomb interactions are in-
termediate, and within which there can occur a novel
transition from a Wigner crystal CO state to a BCDW
SP state as the T is lowered. Thus the charge or bond
ordering in the 4kF phase does not necessarily decide
the same in the 2kF state. For realistic intramolecular
Coulomb interactions (Hubbard U), we show that the
width of this intermediate parameter regime is compara-
ble to the strong and weak intersite interaction regimes.
We believe that our results are directly applicable to the
family of the cationic CTS (TMTTF)2X, where a redis-
tribution of charge upon entering the SP state from the
CO state has been observed15,16 in X=AsF6 and PF6. A
natural explanation of this redistribution emerges within
our theory. The SP state within our theory is unique and
has the BCDW charge occupancy, while the two AFM re-
gions in the phase diagram of Fig. 1 have different site
occupancies. Our theory therefore provides a simple di-
agnostic to determine the pattern of CO coexisting with
low-temperature magnetic states in the 1
-filled CTS.
In addition to the above theoretical results directly
pertaining to the quasi-1D CTS, our work gives new in-
sight to excitations from a SP ground state in a non-
half-filled band. In the case of the usual SP transition
within the 1
-filled band, the SP state is bond-dimerized
at T = 0 and has uniform bonds at T > T2kF . The site
charges are uniform at all T. This is in contrast to the
-filled band, where the SP state at T = 0 is bond and
charge-tetramerized and the T > T2kF state is dimerized
as opposed to being uniform. Furthermore, we show that
the high T phase here can be either charge- or bond-
dimerized, starting from the same low T state. This
clearly requires two different kinds of spin excitations in
the 1
-filled band. We demonstrate that spin excitations
from the SP state in the 1
-filled band can lead to two
different kinds of defects in the background BCDW.
In the next section we present a brief yet detailed sum-
mary of relevant experimental results in the quasi-1D
CTS. The scope of this summary makes the need for hav-
ing T-dependent theory clear. Following this, in Section
III we present our theoretical model along with conjec-
tures based on physical intuitive pictures. In Section IV
we substantiate these conjectures with accurate quantum
Monte Carlo (QMC) and exact diagonalization (ED) nu-
merical calculations. Finally in Section V we compare
our theoretical results and experiments, and present our
conclusions.
II. REVIEW OF EXPERIMENTAL RESULTS
Examples of both CO and BOW broken symmetry at
T < T4kF are found in the
-filled CTS. The 4kF phase
in the anionic 1:2 CTS is commonly bond-dimerized. The
most well known example is MEM(TCNQ)2, which un-
dergoes a metal-insulator transition accompanied with
bond-dimerization at 335 K41. Site charges are uniform
in this 4kF phase. The 2kF phase in the TCNQ-based
systems is universally SP and not AFM. The SP transi-
tion in MEM(TCNQ)2 occurs below T2kF = 19 K, and
low T neutron diffraction measurements of deuterated
samples41 have established that the bond tetramerization
is accompanied by 2kF CO · · · 1100· · · . X-ray
42,43 and
neutron diffraction44 experiments have confirmed a sim-
ilar low T phase in TEA(TCNQ)2. We will not discuss
these further in the present paper, as they are well de-
scribed within our previous work18,25. We will, however,
argue that the SP ground state in (DMe-DCNQI)2Ag
(as opposed to AFM) indicates · · · 1100· · · CO in this.
The cationic (TMTCF)2X, C= S and Se, exhibit more
variety, presumably because the counterions affect pack-
ing as well as site energies in the cation stack. Differ-
ences between systems with centrosymmetric and non-
centrosymmetric anions are also observed. Their overall
behavior is summarized in Fig. 1, where as is custom-
ary pressure P can also imply larger interchain coupling.
We have indicated schematically the possible locations of
different materials on the phase diagram. The most sig-
nificant aspect of the phase diagram is the occurrence of
two distinct antiferromagnetic phases, AFM1 and AFM2,
straddling a single SP phase.
Most TMTTF lie near the low P region of the phase
diagram and are insulating already at or near room tem-
perature because of charge localization, which is due to
the intrinsic dimerization along the cationic stacks1,4,11.
CO at intermediate temperatures TCO has been found
in dielectric permittivity4, NMR7, and ESR46 experi-
ments on materials near the low and intermediate P
end. Although the pattern of the CO has not been de-
termined directly, the observation of ferroelectric behav-
ior below TCO is consistent with · · · 1010· · · type CO
in this region5,23. With further lowering of T, most
(TMTTF)2X undergo transitions to the AFM1 or SP
phase (with X = Br a possible exception, see below).
X = SbF6 at low T lies in the AFM1 region
9, with a very
high TCO and relatively low Neel temperature TN = 8
K. As the schematic phase diagram indicates, pressure
suppresses both TCO and TN in this region. For P > 0.5
GPa, (TMTTF)2SbF6 undergoes a transition from the
AFM1 to the SP phase9, the details of which are not com-
pletely understood; any charge disproportionation in the
SP phase is small9. (TMTTF)2ReO4 also has a relatively
high TCO = 225 K, but the low T phase here, reached
following an anion-ordering transition is spin singlet14.
Nakamura et al. have suggested, based on NMR exper-
iments, that the CO involves the Wigner crystal state,
but the low T state is the · · · 1100· · · BCDW14. Further
along the P axis lie X = AsF6 and PF6, where TCO are re-
duced to 100 K and 65 K, respectively7. The low T phase
in both cases is now SP. Neutron scattering experiments
on (TMTTF)2PF6 have found that the lattice distortion
in the SP state is the expected 2kF BOW distortion,
but that the amplitude of the lattice distortion is much
smaller10 than that found in other organic SP materials
such as MEM(TCNQ)2. The exact pattern of the BOW
has not been determined yet. Experimental evidence ex-
ists that some form of CO persists in the magnetic phases.
For example, the splitting in vibronic modes below TCO
in (TMTTF)2PF6 and (TMTTF)2AsF6, a signature of
charge disproportionation, persists into the SP phase17,
indicating coexistence of CO and SP. At the same time,
the high T CO is in competition with the SP ground
state8, as is inferred from the different effects of pressure
on TCO and TSP : while pressure reduces TCO, it in-
creases TSP . This is in clear contrast to the effect of pres-
sure on TN in X = SbF6. Similarly, deuteration of the hy-
drogen atoms of TMTTF increases TCO but decreases
TSP . That higher TCO is accompanied by lower TSP for
centrosymmetric X (TSP= 16.4 K in X = PF6 and 11.1 K
in X = AsF6) has also been noted
48. This trend is in ob-
vious agreement with the occurrence of AFM instead of
SP state under ambient pressure in X = SbF6. Most in-
terestingly, Nakamura et al. have very recently observed
redistribution of the charges on the TMTTF molecules
in (TMTTF)2AsF6 and (TMTTF)2PF6 as these systems
enter the SP phase from CO states15,16. Charge dispro-
portionation, if any, in the SP phase is much smaller
than in the CO phase15,16, which is in apparent agree-
ment with the above observations9,14 in X = ReO4 and
SbF6.
The bulk of the (TMTTF)2X therefore lie in the
AFM1 and SP regions of Fig. 1. (TMTSF)2X, in
contrast, occupy the AFM2 region. Coexisting 2kF
CDW and spin-density wave, SDW, with the same
2kF periodicity
49,50 here is explained naturally as the
· · · 1100· · · BCSDW21,22,51. In contrast to the TMTTF
salts discussed above, charge and magnetic ordering in
(TMTTF)2Br occur almost simultaneously
46,52. X-ray
studies of lattice distortions point to similarities with
(TMTSF)2PF6
49, indicating that (TMTTF)2Br is also
a · · · 1100· · · BCSDW21. We do not discuss AFM2 re-
gion in the present paper, as this can be found in our
earlier work21,25.
III. THEORETICAL MODEL AND
CONJECTURES
The 1D Hamiltonian we investigate is written as
H = HSSH +HHol +Hee (1a)
HSSH = t
[1 + α(a
i + ai)](c
i,σci+1,σ + h.c.)
+ ~ωS
iai (1b)
HHol = g
i + bi)ni + ~ωH
ibi (1c)
Hee = U
ni,↑ni,↓ + V
nini+1 (1d)
In the above, c
i,σ creates an electron with spin σ (↑,↓) on
molecular site i, ni,σ = c
i,σci,σ is the number of electrons
with spin σ on site i, and ni =
ni,σ. U and V are the
on-site and intersite Coulomb repulsions, and a
i and b
create (dispersionless) Su-Schrieffer-Heeger (SSH)53 and
Holstein (Hol)54 phonons on the ith bond and site respec-
tively, with frequencies ωS and ωH . Because the Peierls
instability involves only phonon modes near q = π, keep-
ing single dispersionless phonon modes is sufficient for
the Peierls transitions to occur55,56. Although purely 1D
calculations cannot yield a finite temperature phase tran-
sition, as in all low dimensional theories57,58 we antici-
pate that the 3D ordering in the real system is principally
determined by the dominant 1D instability.
The above Hamiltonian includes the most important
terms necessary to describe the family of quasi-1D CTS,
but ignores nonessential terms that may be necessary for
understanding the detailed behavior of individual sys-
tems. Such nonessential terms include (i) the intrinsic
dimerization that characterizes many (TMTTF)2X, (ii)
interaction between counterions and the carriers on the
quasi-1D cations stacks, (iii) interchain Coulomb inter-
action, and (iv) interchain hopping. Inclusion of the in-
trinsic dimerization will make the Wigner crystal ground
state even less likely24, and this is the reason for exclud-
ing it. We have verified the conclusions of reference 24
from exact diagonalization calculations. The inclusion of
interactions with counterions may enhance the Wigner
crystal ordering5,23 for some (TMTTF)2X. We will dis-
cuss this point further below, and argue that it is im-
portant in the AFM1 region of the phase diagram. The
effects of intrinsic dimerization and counterion interac-
tions can be reproduced by modifying the V/|t| in our
Hamiltonian, and thus these are not included explicitly.
Rather the V in Eq. (1a) should be considered as the
effective V for the quasi-1D CTS. Interchain hopping, at
least in the TMTTF (though not in the TMTSF), are in-
deed negligible. The interchain Coulomb interaction can
promote the Wigner crystal within a rectangular lattice
framework, but for the realistic nearly triangular lattice
will cause frustration, thereby making the BCDW state
of interest here even more likely. We therefore believe
that Hamiltonian (1a) captures the essential physics of
the quasi-1D CTS.
For applications to the quasi-1D CTS we will be inter-
ested in the parameter regime18,24,25,26 |t| = 0.1 − 0.25
eV, U/|t| = 6 − 8. The exact value of V is less known,
but since the same cationic molecules of interest as well
as other related molecules (for e.g., HMTTF, HMTSF,
etc.) also form quasi-1D 1
-filled band Mott-Hubbard
semiconductors with short-range antiferromagnetic spin
correlations59,60, it must be true that V < 1
U (since for
V > 1
U the 1D 1
-filled band is a CDW61,62). Two other
known theoretical results now fix the range of V that
will be of interest. First, the ground state of the 1
-filled
band within Hamiltonian (1a) in the limit of zero e-p
coupling is the Wigner crystal · · · 1010· · · only for suffi-
ciently large V > Vc(U). Second, Vc(U → ∞) = 2|t|, and
is larger for finite U19,24,25. With the known U/|t| and
the above restrictions in place, it is now easily concluded
that (i) the Wigner crystal is obtained in the CTS for a
relatively narrow range of realistic parameters, and (ii)
even in such cases the material’s V is barely larger than
Vc(U)
24,25.
We now go beyond the above ground state theories
of spatial broken symmetries to make the following cru-
cial observation: each different spin subspace of Hamil-
tonian (1a) must have its own Vc at which the Wigner
crystal is formed. This conclusion of a spin-dependent
Vc = Vc(U, S) follows from the comparison of the fer-
romagnetic subspace with total spin S = Smax and the
S = 0 subspace. The ferromagnetic subspace is equiva-
lent to the 1
-filled spinless fermion band, and therefore
Vc(U, Smax) is independent of U and exactly 2|t|. The
increase of Vc(U, S = 0) with decreasing U
19,25 is then
clearly related to the occurrence of doubly occupied and
vacant sites at finite U in the S = 0 subspace. Since
the probability of double occupancy (for fixed U and V )
decreases monotonically with increasing S, we can in-
terpolate between the two extreme cases of S = 0 and
S = Smax to conclude Vc(U, S) > Vc(U, S+1). We prove
this explicitly from numerical calculations in the next
section.
Our conjecture regarding spin-dependent Vc(U) in turn
implies that there exist three distinct parameter regimes
for realistic U and V : (i) V ≤ V (Smax) = 2|t|, in
which case the ground state is the BCDW with 2kF CO
and the high temperature 4kF state is a BOW
18,25; (ii)
V > Vc(U, S = 0), in which case both the 4kF and
the 2kF phases have Wigner crystal CO; (iii) and the
intermediate regime 2|t| ≤ V ≤ Vc(U, S = 0). Pa-
rameter regime (i) has been discussed in our previous
work18,21,25,63. This is the case where the description of
the 1
-filled band as an effective 1
-filled band is appropri-
ate: the unit cell in the 4kF state is a dimer of two sites,
and the 2kF transition can be described as the usual SP
dimerization of the dimer lattice. We will not discuss
this parameter region further. Given the U/|t| values for
the CTS, parameter regime (ii) can have rather narrow
width. For U/|t| = 6, Vc(U, S = 0) = 3.3|t|, and there
is no value of realistic V for which the ground state is a
Wigner crystal. For U/|t| = 8, Vc(U, S = 0) = 3.0|t|, and
the widths of parameter regime (iii), 2|t| ≤ V ≤ 3|t| and
of parameter regime (ii), 3|t| ≤ V ≤ 4|t|, are comparable.
We will discuss parameter regime (ii) only briefly, as
the physics of this region has been discussed by other
authors20,26. We investigated thoroughly the intermedi-
ate parameter regime (iii), which has not been studied
previously. Within the intermediate parameter regime
we expect a novel transition from the BCDW in the 2kF
state at low T to a · · · 1010· · · CO in the 4kF phase at
high T that has not been discussed in the theoretical lit-
erature.
Our observation regarding the T-dependent behavior
of these CTS follows from the more general observation
that thermodynamic behavior depends on the free en-
ergy and the partition function. For the strong e-e inter-
actions of interest here, thermodynamics of 1D systems
at temperatures of interest is determined almost entirely
by spin excitations. Since multiplicities of spin states
increase with the total spin S, the partition function is
dominated by high (low) spin states with large (small)
multiplicities at high (low) temperatures. While at T=0
such a system must be a BCDW for V < Vc(U, S = 0), as
the temperature is raised, higher and higher spin states
begin to dominate the free energy, until V for the ma-
terial in question exceeds Vc(U, S), at which point the
charge occupancy reverts to · · · 1010· · · We demonstrate
this explicitly in the next section. A charge redistribu-
tion is expected in such a system at T2kF , as the devi-
ation from the average charge of 0.5 is much smaller in
the BCDW than in the Wigner crystal25.
The above conjecture leads to yet another novel impli-
cation. The ground state for both weak and intermediate
intersite Coulomb interaction parameters are the BCDW,
even as the 4kF phases are different in the two cases:
BOW in the former and CO in the latter. This necessarily
requires the existence of two different kinds of spin exci-
tations from the BCDW. Recall that within the standard
theory of the SP transition in the 1
-filled band57,64,65,
thermal excitations from the T = 0 ground state gener-
ates spin excitations with bond-alternation domain walls
(solitons), with the phase of bond alternation in between
the solitons being opposite to that outside the solitons.
Progressive increase in T generates more and more soli-
tons with reversed bond alternation phases, until overlaps
between the two phases of bond alternations lead ulti-
mately to the uniform state. A key difference between
the 1
-filled and 1
-filled SP states is that site charge oc-
cupancies in spin excitations in the former continue to be
uniform, while they are necessarily nonuniform in the lat-
ter case, as we show in the next section. We demonstrate
that defect centers with two distinct charge occupancies
(that we will term as type I and II), depending upon the
actual value of V , are possible in the 1
-filled band. Pre-
ponderance of one or another type of defects generates
the distinct 4kF states.
IV. RESULTS
We present in this section the results of QMC investi-
gations of Eq. (1a), for both zero and nonzero e-p cou-
plings, and of ED studies of the adiabatic (semiclassical)
limit of Eq. (1a). Using QMC techniques, we demon-
strate explicitly the spin-dependence of Vc(U), as well as
the transition from the Wigner crystal to the BCDW for
the intermediate parameter regime (iii). The ED studies
demonstrate the exotic nature of spin excitations from
the BCDW ground state. In what follows, all quantities
are expressed in units of |t| (|t|=1).
The QMC method we use is the Stochastic Series Ex-
pansion (SSE) method using the directed loop update
for the electron degrees of freedom66. For 1D fermions
with nearest-neighbor hopping SSE provides statistically
exact results with no systematic errors. While the SSE
directed-loop method is grand canonical (with fluctuat-
ing particle density), we restrict measurements to only
the 1
-filled density sector to obtain results in the canoni-
cal ensemble. Quantum phonons are treated within SSE
by directly adding the bosonic phonon creation and an-
nihilation operators in Eq. (1a) to the series expansion67.
An upper limit in the phonon spectrum must be imposed,
but can be set arbitrarily large to avoid any systematic
errors67. For the results shown below we used a cutoff of
100 SSH phonons per bond and either 30 (for g = 0.5) or
50 (for g = 0.75) Holstein phonons per site.
The observables we calculate within SSE are the
standard wave-vector dependent charge structure factor
Sρ(q), defined as,
Sρ(q) =
eiq(j−k)〈O
〉 (2)
and charge and bond-order susceptibilities χρ(q) and
χB(q), defined as,
χx(q) =
eiq(j−k)
dτ〈Oxj (τ)O
k (0)〉 (3)
In Eqs. (2) and (3) N is the number of lattice sites, O
nj,↑ + nj,↓, O
j+1,σcj,σ + h.c.), and β is the
inverse temperature in units of t.
The presence of CO or BOW can be detected by the
divergence of the 2kF or 4kF charge or bond-order sus-
ceptibility as a function of increasing system size. Strictly
speaking, in a purely 1D model these functions diverge
only at T = 0; as already explained above, we make the
reasonable assumption58 that in the presence of realis-
tic inter-chain couplings transitions involving charge or
bond-order instabilities, as determined by the dominant
susceptibility, occur at finite T.
A. Spin-dependent Vc(U)
We first present computational results within Hamil-
tonian (1a) in the absence of e-p coupling to demon-
strate that the Vc(U) at which the Wigner crystal order
is established in the lowest state of a given spin sub-
space decreases with increasing spin S. Our computa-
tional approach conserves the total z-component of spin
Sz and not the total S. Since the Lieb-Mattis theorem
E(S) < E(S+1), where E(S) is the energy of the lowest
state in the spin subspace S, applies to the 1D Hamilto-
nian (1a), and since in the absence of a magnetic field
all Sz states for a given S are degenerate, our results for
the lowest state within each different Sz must pertain to
S = Sz.
To determine Vc(U, S) we use the fact that the purely
electronic model is a Luttinger liquid (LL) for V < Vc
with correlation functions determined by a single expo-
nent Kρ (see Reference 69 for a review). The Wigner
crystal state is reached when Kρ =
. The exponent
Kρ may be calculated from the long-wavelength limit of
Sρ(q)
70. In Fig. 2(a) we have plotted our calculated Kρ
for U = 8 as a function of V for different Sz sectors. The
2.2 2.4 2.6 2.8 3 3.2
0 2 4 6 8
FIG. 2: Vc for Eq. (1a) in the limit of zero e-p interactions
(α = g = 0) as a function of Sz. Results are for a N=32
site periodic ring with U = 8. For N = 32 Sz=8 corresponds
to the fully polarized (spinless fermion) limit. (a) Luttinger
Liquid exponent Kρ as a function of V . Kρ =
determines
the boundary for the · · · 1010· · · CO phase. (b) Vc plotted vs.
temperature chosen is small enough (β = 2N) that in all
cases the results correspond to the lowest state within
a given Sz. In Fig. 2(b) we have plotted our calculated
Vc(U = 8), as obtained from Fig. 2(a), as a function of
Sz. Vc is largest for Sz = 0 and decreases with increas-
ing Sz, in agreement with the conjecture of Section III.
Importantly, the calculated Vc for Sz = 8 is close to the
correct limiting value of 2, indicating the validity of our
approach. We have not performed any finite-size scaling
in Fig. 2, which accounts for the the slight deviation from
the exact value of 2.
B. T-dependent susceptibilities
We next present the results of QMC calculations within
the full Hamiltonian (1a). To reproduce correct relative
energy scales of intra- and intermolecular phonon modes
in the CTS, we choose ωH > ωS , specifically ωH = 0.5
and ωS = 0.1 in our calculations. Small deviations from
these values do not make any significant difference. In
all cases we have chosen the electron-molecular vibration
coupling g larger than the coupling between electrons and
the SSH phonons α, thereby deliberately enhancing the
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
0 0.05 0.1 0.15 0.2 0.25
FIG. 3: (color online) QMC results for the temperature-
dependent charge susceptibilities for a N=64 site periodic
ring with U = 8, V = 2.75, α = 0.15, ωS = 0.1, g = 0.5,
and ωH = 0.5. (a) and (b) Wavevector-dependent charge
and bond-order susceptibilities. (c) 2kF and 4kF charge sus-
ceptibilities as a function of temperature. (d) 2kF and 4kF
bond-order susceptibilities as a function of temperature. If
error bars are not shown, statistical error bars are smaller
than the symbol sizes. Lines are guides to the eye.
likelihood of the Wigner crystal CO. We report results
only for intermediate and strong weak intersite Coulomb
interactions; the weak interaction regime V < 2 has been
discussed extensively in our previous work25.
1. 2 ≤ V ≤ Vc(U, S = 0)
Our results are summarized in Figs. 3–5, where we re-
port results for two different V of intermediate strength
and several different e-p couplings. Fig. 3 first shows re-
sults for relatively weak SSH coupling α. The charge sus-
ceptibility is dominated by a peak at 4kF = π (Fig. 3(a)).
Figs. 3(c) and (d) show the T-dependence of the 2kF
as well as 4kF charge and bond susceptibility. For
V < Vc(U, S = 0) the 4kF charge susceptibility does
not diverge with system size, and the purely 1D system
remains a LL with no long range CO at zero tempera-
ture. The dominance of χρ(4kF ) over χρ(2kF ) suggests,
however, that · · · 1010· · · CO will likely occur in the 3D
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
 g=0.50
 g=0.50
 g=0.75
 g=0.75
0 0.05 0.1 0.15 0.2 0.25
 g=0.50
 g=0.50
 g=0.75
 g=0.75
FIG. 4: (color online) Same as Fig. 3, but with parameters
U = 8, V = 2.25, α = 0.27, ωS = 0.1, and ωH = 0.5. In panels
(a) and (b), data are for g = 0.50 only. In panels (c) and (d),
data for both g = 0.50 and g = 0.75 are shown. Arrows
indicate temperature where χρ(2kF ) = χρ(4kF ) (solid and
broken arrows correspond to g = 0.50 and 0.75, respectively.)
system, especially if this order is further enhanced due to
interactions with the counterions9. We have plotted the
bond susceptibilities χB(2kF ) and χB(4kF ) in Fig. 3(d).
A SP transition requires that χB(2kF ) diverges as T → 0.
χB(2kF ) weaker than χB(4kF ) at low T indicates that
the SP order is absent in the present case with weak SSH
e-p coupling. This result is in agreement with earlier
result55 that the SP order is obtained only above a critical
αc (αc may be smaller for the infinite system than found
in our calculation). The most likely scenario with the
present parameters is the persistence of the · · · 1010· · ·
CO to the lowest T with no SP transition. These pa-
rameters, along with counterion interactions, could then
describe the (TMTTF)2Xmaterials with an AFM ground
state9.
In Figs. 4 and Fig. 5 we show our results for larger
SSH e-p coupling α and two different intersite Coulomb
interaction V=2.25 and 2.75. The calculations of Fig. 4
were done for two different Holstein e-p couplings g. For
both the V parameters, χρ(4kF ) dominates at high T
but χρ(2kF ) is stronger at low T in both Fig. 4(c) and
Fig. 5(c). The crossing between the two susceptibilities
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
 g=0.50
 g=0.50
0 0.05 0.1 0.15 0.2 0.25
 g=0.50
 g=0.50
FIG. 5: (color online) Same as Fig. 3, but with parameters
U = 8, V = 2.75, α = 0.27, ωS = 0.1, g = 0.5, and ωH = 0.5.
Arrow indicates temperature where χρ(2kF ) = χρ(4kF ).
is clear indication that in the intermediate parameter
regime, as T is lowered the 2kF CDW instability domi-
nates over the 4kF CO.
The rise in χρ(2kF ) at low T is accompanied by a
steep rise in χB(2kF ) in both cases (see Fig. 4(d) and
Fig. 5(d)). Importantly, unlike in Fig. 3(d), χB(2kF ) in
these cases clearly dominates over χB(4kF ) by an order
of magnitude at low temperatures. There is thus a clear
signature of the SP instability for these parameters. The
simultaneous rise in χB(2kF ) and χρ(2kF ) indicates that
the SP state is the · · · 1100· · · BCDW. Comparison of
Figs. 4 and 5 indicates that the effect of larger V is to
decrease the T where the 2kF and 4kF susceptibilities
cross. Since larger V would imply larger TCO, this result
implies that larger TCO is accompanied by lower TSP .
Our calculations are for relatively modest α < g. Larger
α (not shown) further strengthens the divergence of 2kF
susceptibilities.
The motivation for performing the calculations of
Fig. 4 with multiple Holstein couplings was to deter-
mine whether it is possible to have a Wigner crystal at
low T even for V < V (U, S = 0), by simply increas-
ing g. The argument for this would be that in strong
coupling, increasing g amounts to an effective increase
in V 71. The Holstein coupling cannot be increased arbi-
T=0.25t
T=0.125t
T=0.042t
0 0.2 0.4 0.6 0.8 1
T=0.25t
T=0.125t
T=0.042t
0 0.05 0.1 0.15 0.2 0.25
FIG. 6: (color online) Same as Fig. 3, but with parameters
U = 8, V = 3.5, α = 0.24, ωS = 0.1, g = 0.5, and ωH = 0.5.
trarily, however, as beyond a certain point, g promotes
formation of on-site bipolarons67. Importantly, the co-
operative interaction between the 2kF BOW and CDW
in the BCDW21,25 implies that in the V < V (U, S = 0)
region, larger g not only promotes the 4kF CO but also
enhances the BCDW. In our calculations in Fig. 4(b)
and (c) we we find both these effects: a weak increase in
χρ(4kF ) at intermediate T, and an even stronger increase
in the T → 0 values of χρ(2kF ) and χB(2kF ). Actually,
this result is in qualitative agreement with our obser-
vation for the ground state within the adiabatic limit
of Eq. (1a) that in the range 0 < V < V (U, S = 0),
V enhances the BCDW25. The temperature at which
χρ(2kF ) and χρ(4kF ) cross does not change significantly
with larger g. Our results for g = 0.75 for V = 2.75 (not
shown) are similar, except that the data are more noisy
now (probably because all parameters in Fig. 5 are much
too large for the larger g). We conclude therefore that in
the intermediate V region, merely increasing g does not
change the BCDW nature of the SP state. For g to have
a qualitatively different effect, V should be much closer
to V (U, S = 0) or perhaps even larger.
2. V > Vc(U, S = 0)
In principle, calculations of low temperature instabili-
ties here should be as straightforward as the weak inter-
site Coulomb interaction V < 2t regime. The 4kF CO -
AFM1 state ↑ 0 ↓ 0 would occur naturally for the case
of weak e-p coupling. Obtaining the 4kF CO-SP state
1 = 0 = 1 · · · 0 · · · 1, with realistic V < 1
U is, however,
difficult25. Previous work72, for example, finds this state
for V > 1
U . Recent T-dependent mean-field calculations
of Seo et al.39 also fail to find this state for nonzero V .
There are two reasons for this. First, the spin exchange
here involves charge-rich sites that are second neighbors,
and are hence necessarily small. Energy gained upon
alternation of this weak exchange interactions is even
smaller, and thus the tendency to this particular form
of the SP transition is weak to begin with. Second, this
region of the parameter space involves either large U (for
e.g., U = 10, for which Vc(U, S = 0) ≃ 2) or relatively
large V (for e.g. Vc(U, S = 0) ≃ 3 for U = 8). In either
case such strong Coulomb interactions make the applica-
bility of mean-field theories questionable.
Our QMC calculations do find the tendency to SP in-
stability in this parameter region. In Fig. 6 we show
QMC results for V > Vc(U = 8, S = 0). In contrast to
Figs. 4 and 5, χρ(4kF ) now dominates over χρ(2kF ) at
all T. The weaker peak at q = 2kF at low T is due to
small differences in site charge populations between the
charge-poor sites of the Wigner crystal that arises upon
bond distortion25, and that adds a small period 4 compo-
nent to the charge modulation. The bond susceptibility
χB(q) has a strong peak at 2kF and a weaker peak at 4kF ,
exactly as expected for the 4kF CO-SP state. Previous
work has shown that the difference in charge densities
between the charge-rich and charge-poor sites in the 4kF
CO-SP ground state is considerably larger than in the
BCDW25.
C. Spin excitations from the BCDW
The above susceptibility calculations indicate that in
the intermediate V region (Figs. 4 and 5) the BCDW
ground state with · · · 1100· · · CO can evolve into the
· · · 1010· · · CO as T increases. Our earlier work had
shown that for weak V , the BCDW evolves into the 4kF
BOW at high T18,25. Within the standard theory of the
SP transition57,64,65, applicable to the 1
-filled band, the
SP state evolves into the undistorted state for T > TSP .
Thus the 4kF distortions, CO and BOW, take the role of
the undistorted state in the 1
-filled band, and it appears
paradoxical that the same ground state can evolve into
two different high T states. We show here that this is inti-
mately related to the nature of the spin excitations in the
-filled BCDW. Spin excitations from the conventional
SP state leave the site charge occupancies unchanged.
We show below that not only do spin excitations from
the 1
-filled BCDW ground state lead to changes in the
site occupancies, two different kinds of site occupancies
are possible in the localized defect states that characterize
spin excited states here. We will refer to these as type
I and type II defects, and depending on which kind of
defect dominates at high T (which in turn depends on
the relative magnitudes of V and the e-p couplings), the
4kF state is either the CO or the BOW.
We will demonstrate the occurrence of type I and II
defects in spin excitations numerically. Below we present
a physical intuitive explanation of this highly unusual
behavior, based on a configuration space picture. Very
similar configuration space arguments can be found else-
where in our discussion of charge excitations from the
BCDW in the interacting 1
-filled band73.
We begin our discussion with the standard 1
-filled
band, for which the mechanism of the SP transition is
well understood57,64,65. Fig. 7(a) shows in valence bond
(VB) representation the generation of a spin triplet from
the standard 1
-filled band. Since the two phases of bond
alternation are isoenergetic, the two free spins can sepa-
rate, and the true wavefunction is dominated by VB dia-
grams as in Fig. 7(b), where the phase of the bond alter-
nation in between the two unpaired spins (spin solitons)
is opposite to that in the ground state. With increasing
temperature and increasing number of spin excitations
there occur many such regions with reversed bond al-
ternations, and overlaps between regions with different
phases of bond alternations leads ultimately to the uni-
form state.
The above picture needs to be modified for the dimer-
ized dimer BCDW state in the 1
-filled band, in which
the single site of the 1
-filled band system is replaced by
a dimer unit with site populations 1 and 0 (or 0 and 1),
and the stronger interdimer 1–1 bond (weaker interdimer
0· · · 0 bond) corresponds to the strong (weak) bond in the
standard SP case. Fig. 7(c), showing triplet generation
from the BCDW, is the 1
-filled analog of Fig. 7(a): a
singlet bond between the dimer units has been broken to
generate a localized triplet. The effective repulsion be-
tween the free spins of Fig. 7(a) is due to the absence
of binding between them, and the same is expected in
Fig. 7(c). Because the site occupancies within the dimer
units are nonuniform now, the repulsion between the
spins is reduced from changes in intradimer as well as
interdimer site occupancies: the site populations within
the neighboring units can become uniform (0.5 each), or
the site occupancies can revert to 1001 from 0110. There
is no equivalent of this step in the standard SP case.
The next steps in the separation of the resultant defects
are identical to that in Fig. 7(b), and we have shown
these possible final states in Fig. 7(d) and Fig. 7(e), for
the two different intraunit spin defect populations. For
V < Vc(U, S), defect units with site occupancies 0.5 occu-
pancies (type I defects) are expected to dominate; while
for V > Vc(U, S) site populations of 10 and 01 (type II
defects) dominate. From the qualitative discussions it is
difficult to predict whether the defects are free, in which
case they are solitons, or if they are bound, in which case
0 1 0 1.5 .5 .5 .5
1 0 0 1   0 0 1
0   0 0 1 1 0 0 1 1 0
0 1 1 0
FIG. 7: (a) and (b) S = 1 VB diagrams in the Heisenberg SP
chain. The SP bond order in between the S = 1
solitons in (b)
is opposite to that elsewhere. (c) 1
-filled band equivalent of
(a); the singlet bonds in the background BCDW are between
units containing a pair of sites but a single electron (see text).
(d) and (e) 1
-filled band equivalents of (b). Defect units with
two different charge distributions are possible. The charge
distribution will depend on V .
two of them constitute a triplet. What is more relevant
is that type I defects generate bond dimerization locally
(recall that the 4kF BOW has uniform site charge densi-
ties) while type II defects generate local site populations
1010, which will have the tendency to generate the 4kF
The process by which the BCDW is reached from the
4kF BOW or the CO as T is decreased from T2kF is ex-
actly opposite to the discussion of spin excitations from
the BCDW in Fig. 7. Now · · · 1100· · · domain walls
appear in the background bond-dimerized or charge-
dimerized state. In the case of the · · · 1010· · · 4kF state,
the driving force for the creation of such a domain wall
is the energy gained upon singlet formation (for V <
Vc(U,S)).
Fig. 7(e) suggests the appearance of localized
· · · 1010· · · regions in excitations from the · · · 1100· · ·
BCDW SP state for 2|t| ≤ V ≤ Vc(U, S = 0). We have
verified this with exact spin-dependent calculations for N
= 16 and 20 periodic rings with adiabatic e-p couplings,
HASSH = t
[1− α(ui − ui+1)](c
i,σci+1,σ + h.c.)
(ui − ui+1)
2 (4a)
HAHol = g
vini +
v2i (4b)
and Hee as in Eq. (1d). In the above ui is the displace-
ment from equilibrium of a molecular unit and vi is a
molecular mode. Note that due to the small sizes of
the systems we are able to solve exactly, the e-p cou-
plings in Eq. (4a) and Eq. (4b) cannot be directly com-
pared to those in Eq. (1a). In the present case, we
take the smallest e-p couplings necessary to generate the
BCDW ground state self-consistently within the adia-
batic Hamiltonian22,23,25. We then determine the bond
distortions, site charge occupancies and spin-spin corre-
4 8 12 16 20
-0.04
-0.04
FIG. 8: (a) Self-consistent site charges (vertical bars) and
spin-spin correlations (points, lines are to guide the eye) in
the N=20 periodic ring with U = 8, V = 2.75, α2/KS = 1.5,
g2/KH = 0.32 for Sz = 2. (b) Same parameters, except
Sz = 3. Black and white circles at the bottoms of the panels
denote charge-rich and charge-poor sites, respectively, while
gray circles denote sites with population nearly exactly 0.5.
The arrows indicate locations of the defect units with nonzero
spin densities.
lations self-consistently with the same parameters for ex-
cited states with Sz > 0.
In Fig. 8(a) we have plotted the deviations of the site
charge populations from average density of 0.5 for the
lowest Sz=2 state for N = 20, for U = 8, V = 2.75 and e-p
couplings as given in the figure caption. Deviations of the
charge occupancies from the perfect · · · 0110· · · sequence
identify the defect centers, as seen from Fig. 7(c) - (e).
In the following we refer to dimer units composed of sites
i and j as [i,j]. Based on the charge occupancies in the
Fig, we identify units [1,2], [7,8], [13,14] and [14,15] as the
defect centers. Furthermore, based on site populations of
nearly exactly 0.5, we identify defects on units [1,2] and
[7,8] as type I; the populations on units [13,14] and [14,15]
identify them type II. Type I defects appear to be free
and soliton like, while type II defects appear to be bound
into a triplet state, but both could be finite size effects.
Fig. 7(d) and (e) suggest that the spin-spin z-
component correlations, 〈Szi S
j 〉, are large and positive
only between pairs of sites which belong to the defect cen-
ters, as all other spins are singlet-coupled. For our char-
acterization of units [1,2], [7,8], [13,14] and [14,15] to be
correct therefore, 〈Szi S
j 〉 must be large and positive if
sites i and j both belong to this set of sites, and small
(close to zero) when either i or j does not belong to
this set (as all sites that do not belong to the set are
singlet-bonded). We have superimposed the calculated
z-component spin-spin correlations between site 2 and
all sites j = 1 – 20. The spin-spin correlations are in
complete agreement with our characterization of units
[1,2], [7,8], [13,14] and [14,15] as defect centers with free
spins. The singlet 1–1 bonds between nearest neighbor
charge-rich sites in Fig. 7(c) - (e) require that spin-spin
couplings between such pairs of sites are large and neg-
ative, while spin-spin correlations between one member
of the pair any other site is small. We have verified such
strong spin-singlet bonds between sites 4 and 5, 10 and
11, and 18 and 19, respectively (not shown). Thus mul-
tiple considerations lead to the identification of the same
sites as defect centers, and to their characterization as
types I and II.
We report similar calculations in Fig. 8(b) for Sz = 3.
Based on charge occupancies, four out of the six defect
units with unpaired spins in the Sz = 3 state are type II;
these occupy units [3,4], [9,10], [13,14] and [19,20]. Type
I defects occur on units [1,2] and [11,12]. As indicated in
the figure, spin-spin correlations are again large between
site 2 and all other sites that belong to this set, while they
are again close to zero when the second site is not a defect
site. As in the previous case, we have verified singlet spin
couplings between nearest neighbor pairs of charge-rich
sites. There occur then exact correspondences between
charge densities and spin-spin correlations, exactly as for
Sz = 2.
The susceptibility calculations in Fig. 4 and Fig. 5 are
consistent with the microscopic calculations of spin de-
fects presented above. As 4kF defects are added to the
2kF background, the 2kF susceptibility peak is expected
to broaden and shift towards higher q. This is exactly
what is seen in the charge susceptibility, Fig. 4(a) and
Fig. 5(a) as T is increased. A similar broadening and
shift is seen in the bond order susceptibility as well.
V. DISCUSSIONS AND CONCLUSIONS
In summary, the SP state in the 1
-filled band CTS is
unique for a wide range of realistic Coulomb interactions.
Even when the 4kF state is the Wigner crystal, the SP
state can be the · · · 1100· · · BOW. For U = 8, for exam-
ple, the transition found here will occur for 2 < V < 3.
This novel T-dependent transition from the Wigner crys-
tal to the BCDW is a consequence of the spin-dependence
of Vc. Only for V > 3 here can the SP phase be the
tetramerized Wigner crystal 1 = 0 = 1 · · · 0 · · · 1 (note,
however, that V ≤ 4 for U = 8). We have ignored the
intrinsic dimerization along the 1D stacks in our reported
results, but this increases Vc(U, S = 0) even further, and
makes the Wigner crystal that much more unlikely24.
Although even larger U (U = 10, for example) reduces
Vc(U, S = 0), we believe that the Coulomb interactions
in the (TMTTF)2X lie in the intermediate range.
A Wigner crystal to BCDW transition would explain
most of the experimental surprises discussed in Section
II. The discovery of the charge redistribution upon en-
tering the SP phase15,16 in X = AsF6 and PF6 is prob-
ably the most dramatic illustration of this. Had the SP
state maintained the same charge modulation pattern
as the Wigner crystal that exist above T2kF , the dif-
ference in charge densities between the charge-rich and
the charge-poor sites would have changed very slightly25.
The dominant effect of the SP transition leading to
1 = 0 = 1 · · · 0 · · · 1 is only on the charge-poor sites,
which are now inequivalent (note that the charge-rich
sites remain equivalent). The difference in charge den-
sity of the charge-rich sites and the average of the charge
density of the charge-poor sites thus remains the same25.
In contrast, the difference in charge densities between
the charge-rich and the charge-poor sites in the BCDW
is considerably smaller than in the Wigner crystal25,
and we believe that the experimentally observed smaller
charge density difference in the SP phase simply reflects
its BCDW character.
The experiments of references 15,16 should not be
taken in isolation: we ascribe the competition between
the CO and the SP states in X = PF6 and AsF6, as
reflected in the different pressure-dependences of these
states8, to their having different site charge occupan-
cies. The observation that the charge density difference
in X = SbF6 decreases considerably upon entering the SP
phase from the AFM1 phase9 can also be understood if
the AFM1 and the SP phases are assigned to be Wigner
crystal and the BCDW, respectively. The correlation be-
tween larger TCO and smaller TSP
48 is expected. Larger
TCO implies larger effective V , which would lower TSP .
This is confirmed from comparing Figs. 4 and 5: the
temperature at which χρ(2kF ) begins to dominate over
χρ(4kF ) is considerably larger in Fig. 4 (smaller V ) than
in Fig. 5 (larger V ). The isotope effect, strong enhance-
ment of the TCO (from 69 K to 90 K) with deutera-
tion of the methyl groups in X = PF6, and concomi-
tant decrease in TSP
6,47 are explained along the same
line. Deuteration decreases ωH in Eq. (1a), which has
the same effect as increasing V . Thus from several dif-
ferent considerations we come to the conclusion that the
transition from the Wigner crystal to the BCDW that we
have found here theoretically for intermediate V/|t| does
actually occur in (TMTTF)2X that undergo SP transi-
tion. This should not be surprising. Given that the 1:2
anionic CTS lie in the “weak” V/|t| regime, TMTTF with
only slightly smaller |t| (but presumably very similar V ,
since intrastack intermolecular distances are comparable
in the two families) lies in the “intermediate” as opposed
to “strong” V/|t| regime.
Within our theory, the two different antiferromagnetic
regions that straddle the SP phase in Fig. 1, AFM1 and
AFM2, have different charge occupancies. The Wigner
crystal character of the AFM1 region is revealed from the
similar behavior of TN and TCO in (TMTTF)2SbF6 un-
der pressure9, indicating the absence of the competition
of the type that exists between CO and SP, in agree-
ment with our assignment. The occurrence of a Wigner
crystal AFM1 instead of SP does not necessarily imply
a larger V/|t| in the SbF6. A more likely reason is that
the interaction with the counterions is strong here, and
this interaction together with V pins the electrons on al-
ternate sites. (TMTSF)2X, and possibly (TMTTF)2Br,
belong to the AFM2 region. The observation that the
CDW and the spin-density wave in the TMTSF have the
same periodicities1 had led to the conclusion that the
charge occupancy here is · · · 1100· · · 21. This conclusion
remains unchanged. Finally, we comment that the ob-
servation of Wigner crystal CO3 in (DI-DCNQI)2Ag is
not against our theory, as the low T phase here is an-
tiferromagnetic and not SP. We predict the SP system
(DMe-DCNQI)2Ag to have the · · · 1100· · · charge order-
In summary, it appears that the key concept of spin-
dependent Vc within Eq. (1a) can resolve most of the
mysteries associated with the temperature dependence
of the broken symmetries in the 1
-filled band CTS. One
interesting feature of our work involves demonstration
of spin excitations from the BCDW state that necessar-
ily lead to local changes in site charges. Even for weak
Coulomb interactions, the 1
-filled band has the BCDW
character18. The effects of magnetic field on 1
-filled band
CDWs is of strong recent interest74,75. We are currently
investigating the consequences of the Zeeman interaction
on the mixed spin-charge excitations of the BCDW.
VI. ACKNOWLEDGMENTS
We acknowledge illuminating discussions with S.E.
Brown. This work was supported by the Department
of Energy grant DE-FG02-06ER46315 and the American
Chemical Society Petroleum Research Fund.
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