arXiv:0704.0313v1  [cond-mat.str-el]  3 Apr 2007
Typeset with jpsj2.cls <ver.1.2> Letter
Possibility of Gapless Spin Liquid State by One-dimensionalization
Yuta Hayashi∗ and Masao Ogata
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033
Motivated by the observation of a gapless spin liquid state in κ-(BEDT-TTF)2Cu2(CN)3, we
analyze the anisotropic triangular lattice S = 1/2 Heisenberg model with the resonating valence
bond mean-field approximation. Paying attention to the small quasi-one-dimensional anisotropy
of the material, we take an approach from one-dimensional (1D) chains coupled with frustrating
zig-zag bonds. By calculating one-particle excitation spectra changing anisotropy parameter
J ′/J from the decoupled 1D chains to the isotropic triangular lattice, we find almost gapless
excitations in the wide range from the 1D limit. This one-dimensionalization by frustration is
considered to be a candidate for the mechanism of the gapless spin liquid state.
KEYWORDS: gapless spin liquid, κ-(BEDT-TTF)2Cu2(CN)3, anisotropic triangular lattice, frustration,
one-dimensionalization
Organic conductors are one of the fascinating materi-
als which have low-dimensionality and relatively strong
electron correlations. So far, various physical states
have been observed and investigated intensively.1 Among
them, magnetism in the Mott insulating phase next to
the unconventional superconductivity has been attract-
ing considerable attention. This phase is observed in the
family of κ-(BEDT-TTF)2X, where BEDT-TTF (ET)
denotes bis(ethylenedithio)-tetrathiafulvalene and X rep-
resents a monovalent anion. Similarities to that of high-
Tc cuprates are worthy of note. Another stimulating
problem concerning magnetism is ground state proper-
ties of geometrically frustrated spin systems such as a tri-
angular lattice and a Kagomé lattice. These two intrigu-
ing issues meet in a material κ-(ET)2Cu2(CN)3, which
is a Mott insulator having a nearly isotropic triangular
lattice, and it has been in the spotlight of late.
According to 1H NMR measurements at ambient pres-
sure,2 κ-(ET)2Cu2(CN)3 shows no indication of long-
range magnetic order (LRMO) down to 32mK. This
is 4 orders of magnitude below the exchange constant
J ∼ 250K estimated from the temperature dependence
of susceptibility. Recently, a similar result has been ob-
tained by zero-field muon spin relaxation measurements,
which have observed no LRMO down to 20mK.3 These
results suggest that a quantum spin liquid state is real-
ized in the ground state. On the other hand, the static
susceptibility remains finite down to 1.9K, and spin-
lattice relaxation rate 1/T1 shows power-law temperature
dependence below 1K. These imply that almost gapless
spin excitation exists. This fact is a significant feature of
the spin liquid phase observed in this material.
Since Anderson’s proposal of a resonating valence
bond (RVB) state,5 enormous number of studies have
been made on the triangular lattice spin system. It is
now a general view that the ground state of the isotropic
triangular lattice Heisenberg model has LRMO, such as
the 120◦ structure.6–9 On the other hand, if one ne-
glects the LRMO and assumes a disordered ground state,
the mean-field theory of RVB state gives a spin-gap
∗E-mail address: yhayashi@hosi.phys.s.u-tokyo.ac.jp
Table I. Anisotropy of effective transfer integrals in κ-(ET)2X.
The definition of t and t′ are not as usual (see the text).
Anion X t′/t
Cu2(CN)3 0.94
Cu(NCS)2 1.19
Cu[N(CN)2]Br 1.33
Cu[N(CN)2]Cl 1.47
Cu(CN)[N(CN)2] 1.47
Ag(CN)2·H2O 1.67
I3 1.72
state with dx2−y2+idxy-wave symmetry, which is called
“d+id state”.10–12 This RVB state, describing an insu-
lating spin system, corresponds to a projected BCS state
at half-filling in which doubly occupied states are ex-
cluded. Thus, the existing theories show that the ground
state has LRMO in general, and if the magnetic order
is destroyed in some reason, the d+id fullgap state will
appear. If we regard the Mott insulating phase of κ-
(ET)2Cu2(CN)3 in low temperatures as an isotropic tri-
angular lattice spin system, the results of NMR and sus-
ceptibility measurements, which suggest neither LRMO
nor spin gap, cannot be explained.
In this letter, we pay attention to small anisotropy of
κ-(ET)2Cu2(CN)3 and propose a new possibility for un-
derstanding its gapless spin liquid state. As shown in Ta-
ble I, only κ-(ET)2Cu2(CN)3 has an opposite anisotropy
among the family of κ-(ET)2X studied in the past. Here,
the effective transfer integrals t and t′ are defined in-
versely to the conventional way; t = 0 corresponds to
the square lattice, and t′ = 0 the decoupled chains.
Therefore, κ-(ET)2Cu2(CN)3 has quasi-one-dimensional
(Q1D) anisotropy rather than an isotropic triangular lat-
tice. Considering that the pure 1D spin system has no
LRMO and gapless spin excitation, it is likely that this
Q1D anisotropy is concerned with the formation of the
gapless spin liquid state in κ-(ET)2Cu2(CN)3.
Based on the above consideration, we study the
Heisenberg model on an anisotropic triangular lattice,
which is equivalent to 1D chains coupled with zig-zag
http://arxiv.org/abs/0704.0313v1
2 J. Phys. Soc. Jpn. Letter Author Name
bonds as shown in Fig. 1. The Hamiltonian is given by
<i,i′>
JSi · Si′ +
<i,j>
J ′Si · Sj , (1)
where <i, i′> and <i, j> represent the summation over
intrachain and interchain nearest-neighbor pairs with an-
tiferromagnetic coupling constant J and J ′, respectively
(see Fig. 1). We investigate the anisotropy parameter
range J ′/J = 0.0-1.0, in which the model interpolates
between the decoupled chains (J ′ = 0) and the isotropic
triangular lattice (J ′ = J).
In the following, we consider a projected BCS state
defined as
∣p-BCS
, (2)
where PG is the Gutzwiller projection operator which ex-
cludes double occupancy and
is a BCS mean-field
wave function. Since it is difficult to treat the Gutzwiller
projection analytically, we apply an RVB mean-field ap-
proximation to the Hamiltonian (1) and calculate the
one-particle excitation spectra. To put it more con-
cretely, we introduce mean fields ∆ij ≡
ci↑cj↓
, ξij ≡
and obtain its excitation spectrum by
diagonalizing the mean-field Hamiltonian. This approxi-
mation is equivalent to the “Gutzwiller approximation”
which replaces the effect of the Gutzwiller projection op-
erator with the statistical weight gs as
p-BCS
∣Si ·Sj
∣p-BCS
∣Si ·Sj
. (3)
In the simplest Gutzwiller approximation, the statisti-
cal weight is given as gs = 4/(1 + δ)
2 where δ is the
density of holes,15 and in the case of half-filling (δ = 0),
gs = 4. Although double occupancy is no longer excluded
from wave functions in this approximation, it is known
in the research of high-Tc superconductivity that the
RVB mean-field (Gutzwiller) approximation gives quali-
tatively good results.
The spin operators Si ·Sj in the Hamiltonian (1) can
be rewritten by the fermion operators as
Si · Sj =
ci↑ − c†i↓ci↓
cj↑ − c†j↓cj↓
cj↑ + c
. (4)
Fig. 1. The anisotropic triangular lattice Heisenberg model with
intrachain coupling J and interchain zig-zag coupling J ′.
τ1, τ2, τ3 are lattice vectors.
By introducing the mean fields, we can rewrite the
Hamiltonian as
HMF =
ck↑+ c
+ h.c.
except for constant terms. Here, ξk and ∆k are given by
ξk ≡ −3Jξτ1cos(k · τ 1)
− 3J ′
ξτ 2cos(k · τ 2) + ξτ 3cos(k · τ 3)
, (6)
∆k ≡ 3J∆τ1cos(k · τ 1)
+ 3J ′
∆τ 2cos(k · τ 2) + ∆τ3cos(k · τ 3)
, (7)
where τ 1 = (1, 0), τ 2 = (1/2,
3/2), τ 3 = (1/2,−
as shown in Fig. 1, and
ci+τ↑
ci+τ↓
, ∆τ ≡
ci↑ci+τ↓
. (8)
On the analogy of BCS theory, we obtain self-consistent
equations at zero temperature
ξτ i = −
eik·τ i
∆τ i =
e−ik·τ i
with a quasiparticle excitation spectrum
+ |∆k|2. (10)
We determine the order parameters ∆τ i , ξτ i (i = 1, 2, 3)
by solving self-consistent equations (9) numerically, and
obtain the one-particle excitation spectrum Ek.
Firstly, we verify our method in 1D limit (J ′/J = 0).
According to the exact solution, the ground state is a
spin disordered state and the excitation spectrum is “des
Cloizeaux-Pearson mode” with S = 1.16 In the present
RVB mean-field theory, the one-particle excitation spec-
trum becomes
Ek = 3J
+ |∆τ 1 |
2 |cos kx| (11)
in the 1D limit. This clearly realizes gapless excitations
at kx = ±π/2. Note that this one-particle excitation
describes a spin singlet breaking, i.e. S = 1/2 spinon
excitation, whereas the des Cloizeaux-Pearson mode de-
scribes S = 1 spin-wave (magnon) excitation. Thus, two-
spinon excitations with kx = π/2 and kx = −π/2 form an
S = 1 magnon with kx = 0. This means that the present
gapless excitation spectrum obtained in the RVB mean-
field theory is consistent with the exact des Cloizeaux-
Pearson mode.
Nextly, we show the results of 0 ≤ J ′/J ≤ 1 case,
focusing on the following parameters
+ |∆τ 1 |
D23 ≡
+ |∆τ2 |
+ |∆τ3 |
Because of the SU(2) degeneracy at half-filling,10, 15 these
parameters are determined uniquely regardless of the de-
generate ground states. Actually, the excitation spectrum
J. Phys. Soc. Jpn. Letter Author Name 3
can be written as
= 9J2D21 cos
+ 9J ′2D223
+ cos2
Therefore, D1, D23 determine the dispersion relations
along the chains (τ 1) and between the chains (τ 2,τ 3),
respectively. Their J ′/J dependence calculated in the
system size L = 1200 (N = L2) are plotted in Fig. 2.
A notable feature is that D23 remains very small com-
pared to D1, in spite of the comparatively large J
to J ′/J ∼ 0.25. When D23 = 0 the system is a pure 1D
chain. Indeed, when J ′/J = 0, the right-hand side of the
self-consistent equations of ξτ 2 , ξτ3 , ∆τ2 , ∆τ 3 become
all equal to zero. As we show later, D23 is very small
for J ′/J . 0.25 and vanishes when J ′/J → 0. This in-
dicates that there are scarcely any correlations between
spins of different chains, and practically 1D state is real-
ized. As J ′/J approaches unity, D23 gradually increases
and becomes equal to D1.
Finally, we show in Fig. 3 the J ′/J dependence of the
one-particle excitation spectra Ek in (12). We find that
the structure of excitation spectra in 0 ≤ J ′/J . 0.25
has little difference from that of the decoupled chains
(J ′/J = 0.0). As a result, almost gapless excitations are
realized in this wide parameter range. This means that
practically 1D state is realized, which is also expected
from the behavior of D23 in Fig. 2. When J
′/J exceeds
0.25, the excitation gap gradually increases globally in
the first Brillouin zone (1BZ). However, the shape of the
whole spectrum is almost unchanged until the J ′/J be-
comes as large as about 0.6. Moreover, focusing on the
lowest energy excitations (dark areas in the contour plot
shown in Fig. 3), their locations in the 1BZ do not deviate
from those in the 1D limit (kx = ±π/2) for J ′/J . 0.8.
Additionally, when kx = ±π/2, the excitation spectrum
Ek is independent of ky, i.e., Ek = 3J
′D23. This is be-
cause the frustration of two interchain couplings (corre-
sponding to the lattice vector τ 2 and τ 3) cancel the ky
dependence. This fact is rather important, since it indi-
cates that the excited quasiparticles along the kx = ±π/2
lines feel free to move along the ky direction. This is the
same condition as in the 1D limit, except for the exis-
Fig. 2. Anisotropy dependence of D1 and D23 for L = 1200. Note
that D23 is very small compared to D1 in a wide range 0 ≤
J ′/J .0.25.
tence of a finite energy gap.
Figure 4 shows the minimum gap energy in the 1BZ
as a function of anisotropy J ′/J , changing the system
size L. We can see the almost gapless excitations in the
wide parameter range 0 ≤ J ′/J . 0.25, as is already
expected. It is quite natural that this behavior is simi-
Fig. 3. Anisotropy dependence of the one-particle excitation
spectra. Contour plots of the spectra are on the left, and sections
along ky = 0 line are on the right. The hexagons with broken
lines represent 1BZ of the triangular lattice. Up to J ′/J ∼ 0.25,
the spectra for each anisotropy are hardly distinguishable, and
the one-dimensionality strongly remains for large J ′/J .
4 J. Phys. Soc. Jpn. Letter Author Name
lar to that of D23, considering that the minimum energy
excitations are located along kx = ±π/2 for J ′/J . 0.6.
By plotting the same data for various system size, L,
in a semi-log scale (Fig. 4), we can see a discontinuous
jump for every size. We find that this critical value J ′c/J
vanishes very slowly as (lnL)−1. Thus, the discontinu-
ity is an artifact of finite-size calculation. We also find
that the minimum gap energy is finite when infinitesimal
J ′ is introduced. Actually, we can fit the J ′ dependence
as aJ ′ exp(−bJ/J ′)17 for J ′/J . 0.6 as shown in Fig.
4. Considering that the minimum gap energy is already
about 3 orders of magnitude below J at J ′/J ∼ 0.25, it
can be said that almost gapless excitation is realized in
0 ≤ J ′/J . 0.25. This result is fairly suggestive com-
pared with the previous series expansion18 and linear
spin wave19, 20 studies, all of which suggest a spin dis-
ordered state in the parameter range J ′/J . 0.25.
From the above results, we conclude that there is a
strong tendency to form a 1D-like excitation spectrum
for the triangular lattice spin system with anisotropy
0 ≤ J ′/J . 0.6. Furthermore, even if the anisotropy
is as large as 0.6 . J ′/J . 0.8, we can still expect 1D-
like behavior for quasiparticles except for the existence of
the excitation gap. Let us here discuss the relation to κ-
(ET)2Cu2(CN)3. The anisotropy of spin exchange inter-
actions in this material can be estimated from J = 4t2/U
(U being the onsite Coulomb repulsion) as J ′/J ∼ 0.89.
At this anisotropy, a rather large excitation gap exists as
shown in Fig. 4. We consider two possibilities to under-
stand the gaplessness. One is that the small gap region in
Fig. 4 expands to large values of J ′/J by some factors not
considered in the present model. For example, If long-
distance exchange interactions, quantum fluctuation or
multiple spin exchange effect14 (higher order terms of
the Heisenberg model) suppress not only LRMO but also
the spin gap, we can reproduce the gapless spin liquid
state at large J ′/J . These possibilities remain as future
problems. Another possibility is that the anisotropy J ′/J
of κ-(ET)2Cu2(CN)3 deviates from the above estimation
Fig. 4. (Color Online) Anisotropy dependence of the minimum
gap energy in the 1BZ (right axis) for L=60(diamond), 120(plus),
300(square), 600(cross) and 1200(triangle). The semi-log plots of
the same quantity are also shown (left axis). The solid line is a
fitted exponential function aJ ′ exp(−bJ/J ′), where a = 3.50 and
b = 1.61. We find that the observed critical behavior is an artifact
of finite size calculation (see the text).
due to, for example, a finite U effect.21 If it is in the
range J ′/J < 0.25, the excitation gap is sufficiently small
and the susceptibility behavior (finite at 1.9K whereas
J ∼ 250K) can be explained.
In summary, we analyzed an anisotropic triangular lat-
tice Heisenberg model using RVB mean-field approxima-
tion in order to investigate the physical origin of the gap-
less spin liquid state observed in κ-(ET)2Cu2(CN)3. We
payed attention to the Q1D anisotropy of this material,
and took an approach from the 1D limit. As a result of
calculations, we found that a practically 1D state with
almost gapless excitations is realized in the wide range of
the anisotropy parameter 0 ≤ J ′/J . 0.25. Furthermore,
one-dimensionality remained strongly even in J ′/J >
0.25 due to the geometrical frustration of interchain cou-
plings. We consider this “one-dimensionalization by frus-
tration” as a candidate for the mechanism of the gapless
spin liquid state, although the full understanding has not
yet been achieved.
This work was partly supported by a Grant-in-Aid
for Scientific Research on Priority Areas of Molecular
Conductors (No. 15073210) from the Ministry of Edu-
cation, Culture, Sports, Science and Technology, Japan,
and also by a Next Generation Supercomputing Project,
Nanoscience Program, MEXT, Japan.
1) For a review, see T.Ishiguro, K.Yamaji and G.Saito: Organic
Superconductors (Springer-Verlag, Berlin, 1998), 2nd ed.
2) Y.Shimizu, K.Miyagawa, K.Kanoda, M.Maesato and G.Saito:
Phys. Rev. Lett. 91 (2003) 107001.
3) S.Ohira, Y.Shimizu, K.Kanoda and G.Saito: J. Low Temp.
Phys. 142 (2006) 153.
4) T.Komatsu, N.Matsukawa, T.Inoue and G.Saito: J. Phys. Soc.
Jpn 65 (1996) 1340.
5) P.W.Anderson: Mater. Res. Bull. 8 (1973) 153.
6) B.Bernu, P.Lecheminant, C.Lhuillier and L.Pierre: Phys. Rev.
B 50 (1994) 10048.
7) N.Elstner, R.R.P.Singh and A.P.Young: Phys. Rev. Lett. 71
(1993) 1629.
8) P.Lecheminant, B.Bernu, C.Lhuillier and L.Pierre: Phys. Rev.
B 52 (1995) 9162.
9) L.Capriotti, A.E.Trumper and S.Sorella: Phys. Rev. Lett. 82
(1999) 3899.
10) M.Ogata: J. Phys. Soc. Jpn. 72 (2003) 1839.
11) G.Baskaran: Phys. Rev. Lett. 91 (2003) 097003.
12) T.Watanabe, H.Yokoyama, Y.Tanaka, J.Inoue and M.Ogata:
J. Phys. Soc. Jpn 73 (2004) 3404.
13) Y.Shimizu, K.Miyagawa, K.Kanoda, M.Maesato and G.Saito:
Prog. Theor. Phys. Suppl. 159 (2005) 52.
14) G.Misguich, C.Lhuillier, B.Bernu and C.Waldtmann: Phys.
Rev. B 60 (1999) 1064.
15) F.C.Zhang, C.Gros, T.M.Rice and H.Shiba: Supercond. Sci.
Technol. 1 (1988) 36.
16) J.des Cloizeaux and J.J.Pearson: Phys. Rev. 128 (1962) 2131.
17) We would like to thank T.Misawa for pointing out this possi-
bility.
18) W.Zheng, R.H.McKenzie and R.R.P.Singh: Phys. Rev. B 59
(1999) 14367.
19) J.Merino, R.H.McKenzie, J.B.Marston and C.H.Chung: J.
Phys. Condens. Matter 11 (1999) 2965.
20) A.E.Trumper: Phys. Rev. B 60 (1999) 2987.
21) H.Otsuka: Phys. Rev. B 57 (1998) 14658.

  Motivated by the observation of a gapless spin liquid state in
$\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$, we analyze the anisotropic triangular
lattice $S=1/2$ Heisenberg model with the resonating valence bond mean-field
approximation. Paying attention to the small quasi-one-dimensional anisotropy
of the material, we take an approach from one-dimensional (1D) chains coupled
with frustrating zig-zag bonds. By calculating one-particle excitation spectra
changing anisotropy parameter $J'/J$ from the decoupled 1D chains to the
isotropic triangular lattice, we find almost gapless excitations in the wide
range from the 1D limit. This one-dimensionalization by frustration is
considered to be a candidate for the mechanism of the gapless spin liquid
state.

arXiv:0704.0313v1  [cond-mat.str-el]  3 Apr 2007
Typeset with jpsj2.cls <ver.1.2> Letter
Possibility of Gapless Spin Liquid State by One-dimensionalization
Yuta Hayashi∗ and Masao Ogata
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033
Motivated by the observation of a gapless spin liquid state in κ-(BEDT-TTF)2Cu2(CN)3, we
analyze the anisotropic triangular lattice S = 1/2 Heisenberg model with the resonating valence
bond mean-field approximation. Paying attention to the small quasi-one-dimensional anisotropy
of the material, we take an approach from one-dimensional (1D) chains coupled with frustrating
zig-zag bonds. By calculating one-particle excitation spectra changing anisotropy parameter
J ′/J from the decoupled 1D chains to the isotropic triangular lattice, we find almost gapless
excitations in the wide range from the 1D limit. This one-dimensionalization by frustration is
considered to be a candidate for the mechanism of the gapless spin liquid state.
KEYWORDS: gapless spin liquid, κ-(BEDT-TTF)2Cu2(CN)3, anisotropic triangular lattice, frustration,
one-dimensionalization
Organic conductors are one of the fascinating materi-
als which have low-dimensionality and relatively strong
electron correlations. So far, various physical states
have been observed and investigated intensively.1 Among
them, magnetism in the Mott insulating phase next to
the unconventional superconductivity has been attract-
ing considerable attention. This phase is observed in the
family of κ-(BEDT-TTF)2X, where BEDT-TTF (ET)
denotes bis(ethylenedithio)-tetrathiafulvalene and X rep-
resents a monovalent anion. Similarities to that of high-
Tc cuprates are worthy of note. Another stimulating
problem concerning magnetism is ground state proper-
ties of geometrically frustrated spin systems such as a tri-
angular lattice and a Kagomé lattice. These two intrigu-
ing issues meet in a material κ-(ET)2Cu2(CN)3, which
is a Mott insulator having a nearly isotropic triangular
lattice, and it has been in the spotlight of late.
According to 1H NMR measurements at ambient pres-
sure,2 κ-(ET)2Cu2(CN)3 shows no indication of long-
range magnetic order (LRMO) down to 32mK. This
is 4 orders of magnitude below the exchange constant
J ∼ 250K estimated from the temperature dependence
of susceptibility. Recently, a similar result has been ob-
tained by zero-field muon spin relaxation measurements,
which have observed no LRMO down to 20mK.3 These
results suggest that a quantum spin liquid state is real-
ized in the ground state. On the other hand, the static
susceptibility remains finite down to 1.9K, and spin-
lattice relaxation rate 1/T1 shows power-law temperature
dependence below 1K. These imply that almost gapless
spin excitation exists. This fact is a significant feature of
the spin liquid phase observed in this material.
Since Anderson’s proposal of a resonating valence
bond (RVB) state,5 enormous number of studies have
been made on the triangular lattice spin system. It is
now a general view that the ground state of the isotropic
triangular lattice Heisenberg model has LRMO, such as
the 120◦ structure.6–9 On the other hand, if one ne-
glects the LRMO and assumes a disordered ground state,
the mean-field theory of RVB state gives a spin-gap
∗E-mail address: yhayashi@hosi.phys.s.u-tokyo.ac.jp
Table I. Anisotropy of effective transfer integrals in κ-(ET)2X.
The definition of t and t′ are not as usual (see the text).
Anion X t′/t
Cu2(CN)3 0.94
Cu(NCS)2 1.19
Cu[N(CN)2]Br 1.33
Cu[N(CN)2]Cl 1.47
Cu(CN)[N(CN)2] 1.47
Ag(CN)2·H2O 1.67
I3 1.72
state with dx2−y2+idxy-wave symmetry, which is called
“d+id state”.10–12 This RVB state, describing an insu-
lating spin system, corresponds to a projected BCS state
at half-filling in which doubly occupied states are ex-
cluded. Thus, the existing theories show that the ground
state has LRMO in general, and if the magnetic order
is destroyed in some reason, the d+id fullgap state will
appear. If we regard the Mott insulating phase of κ-
(ET)2Cu2(CN)3 in low temperatures as an isotropic tri-
angular lattice spin system, the results of NMR and sus-
ceptibility measurements, which suggest neither LRMO
nor spin gap, cannot be explained.
In this letter, we pay attention to small anisotropy of
κ-(ET)2Cu2(CN)3 and propose a new possibility for un-
derstanding its gapless spin liquid state. As shown in Ta-
ble I, only κ-(ET)2Cu2(CN)3 has an opposite anisotropy
among the family of κ-(ET)2X studied in the past. Here,
the effective transfer integrals t and t′ are defined in-
versely to the conventional way; t = 0 corresponds to
the square lattice, and t′ = 0 the decoupled chains.
Therefore, κ-(ET)2Cu2(CN)3 has quasi-one-dimensional
(Q1D) anisotropy rather than an isotropic triangular lat-
tice. Considering that the pure 1D spin system has no
LRMO and gapless spin excitation, it is likely that this
Q1D anisotropy is concerned with the formation of the
gapless spin liquid state in κ-(ET)2Cu2(CN)3.
Based on the above consideration, we study the
Heisenberg model on an anisotropic triangular lattice,
which is equivalent to 1D chains coupled with zig-zag
http://arxiv.org/abs/0704.0313v1
2 J. Phys. Soc. Jpn. Letter Author Name
bonds as shown in Fig. 1. The Hamiltonian is given by
<i,i′>
JSi · Si′ +
<i,j>
J ′Si · Sj , (1)
where <i, i′> and <i, j> represent the summation over
intrachain and interchain nearest-neighbor pairs with an-
tiferromagnetic coupling constant J and J ′, respectively
(see Fig. 1). We investigate the anisotropy parameter
range J ′/J = 0.0-1.0, in which the model interpolates
between the decoupled chains (J ′ = 0) and the isotropic
triangular lattice (J ′ = J).
In the following, we consider a projected BCS state
defined as
∣p-BCS
, (2)
where PG is the Gutzwiller projection operator which ex-
cludes double occupancy and
is a BCS mean-field
wave function. Since it is difficult to treat the Gutzwiller
projection analytically, we apply an RVB mean-field ap-
proximation to the Hamiltonian (1) and calculate the
one-particle excitation spectra. To put it more con-
cretely, we introduce mean fields ∆ij ≡
ci↑cj↓
, ξij ≡
and obtain its excitation spectrum by
diagonalizing the mean-field Hamiltonian. This approxi-
mation is equivalent to the “Gutzwiller approximation”
which replaces the effect of the Gutzwiller projection op-
erator with the statistical weight gs as
p-BCS
∣Si ·Sj
∣p-BCS
∣Si ·Sj
. (3)
In the simplest Gutzwiller approximation, the statisti-
cal weight is given as gs = 4/(1 + δ)
2 where δ is the
density of holes,15 and in the case of half-filling (δ = 0),
gs = 4. Although double occupancy is no longer excluded
from wave functions in this approximation, it is known
in the research of high-Tc superconductivity that the
RVB mean-field (Gutzwiller) approximation gives quali-
tatively good results.
The spin operators Si ·Sj in the Hamiltonian (1) can
be rewritten by the fermion operators as
Si · Sj =
ci↑ − c†i↓ci↓
cj↑ − c†j↓cj↓
cj↑ + c
. (4)
Fig. 1. The anisotropic triangular lattice Heisenberg model with
intrachain coupling J and interchain zig-zag coupling J ′.
τ1, τ2, τ3 are lattice vectors.
By introducing the mean fields, we can rewrite the
Hamiltonian as
HMF =
ck↑+ c
+ h.c.
except for constant terms. Here, ξk and ∆k are given by
ξk ≡ −3Jξτ1cos(k · τ 1)
− 3J ′
ξτ 2cos(k · τ 2) + ξτ 3cos(k · τ 3)
, (6)
∆k ≡ 3J∆τ1cos(k · τ 1)
+ 3J ′
∆τ 2cos(k · τ 2) + ∆τ3cos(k · τ 3)
, (7)
where τ 1 = (1, 0), τ 2 = (1/2,
3/2), τ 3 = (1/2,−
as shown in Fig. 1, and
ci+τ↑
ci+τ↓
, ∆τ ≡
ci↑ci+τ↓
. (8)
On the analogy of BCS theory, we obtain self-consistent
equations at zero temperature
ξτ i = −
eik·τ i
∆τ i =
e−ik·τ i
with a quasiparticle excitation spectrum
+ |∆k|2. (10)
We determine the order parameters ∆τ i , ξτ i (i = 1, 2, 3)
by solving self-consistent equations (9) numerically, and
obtain the one-particle excitation spectrum Ek.
Firstly, we verify our method in 1D limit (J ′/J = 0).
According to the exact solution, the ground state is a
spin disordered state and the excitation spectrum is “des
Cloizeaux-Pearson mode” with S = 1.16 In the present
RVB mean-field theory, the one-particle excitation spec-
trum becomes
Ek = 3J
+ |∆τ 1 |
2 |cos kx| (11)
in the 1D limit. This clearly realizes gapless excitations
at kx = ±π/2. Note that this one-particle excitation
describes a spin singlet breaking, i.e. S = 1/2 spinon
excitation, whereas the des Cloizeaux-Pearson mode de-
scribes S = 1 spin-wave (magnon) excitation. Thus, two-
spinon excitations with kx = π/2 and kx = −π/2 form an
S = 1 magnon with kx = 0. This means that the present
gapless excitation spectrum obtained in the RVB mean-
field theory is consistent with the exact des Cloizeaux-
Pearson mode.
Nextly, we show the results of 0 ≤ J ′/J ≤ 1 case,
focusing on the following parameters
+ |∆τ 1 |
D23 ≡
+ |∆τ2 |
+ |∆τ3 |
Because of the SU(2) degeneracy at half-filling,10, 15 these
parameters are determined uniquely regardless of the de-
generate ground states. Actually, the excitation spectrum
J. Phys. Soc. Jpn. Letter Author Name 3
can be written as
= 9J2D21 cos
+ 9J ′2D223
+ cos2
Therefore, D1, D23 determine the dispersion relations
along the chains (τ 1) and between the chains (τ 2,τ 3),
respectively. Their J ′/J dependence calculated in the
system size L = 1200 (N = L2) are plotted in Fig. 2.
A notable feature is that D23 remains very small com-
pared to D1, in spite of the comparatively large J
to J ′/J ∼ 0.25. When D23 = 0 the system is a pure 1D
chain. Indeed, when J ′/J = 0, the right-hand side of the
self-consistent equations of ξτ 2 , ξτ3 , ∆τ2 , ∆τ 3 become
all equal to zero. As we show later, D23 is very small
for J ′/J . 0.25 and vanishes when J ′/J → 0. This in-
dicates that there are scarcely any correlations between
spins of different chains, and practically 1D state is real-
ized. As J ′/J approaches unity, D23 gradually increases
and becomes equal to D1.
Finally, we show in Fig. 3 the J ′/J dependence of the
one-particle excitation spectra Ek in (12). We find that
the structure of excitation spectra in 0 ≤ J ′/J . 0.25
has little difference from that of the decoupled chains
(J ′/J = 0.0). As a result, almost gapless excitations are
realized in this wide parameter range. This means that
practically 1D state is realized, which is also expected
from the behavior of D23 in Fig. 2. When J
′/J exceeds
0.25, the excitation gap gradually increases globally in
the first Brillouin zone (1BZ). However, the shape of the
whole spectrum is almost unchanged until the J ′/J be-
comes as large as about 0.6. Moreover, focusing on the
lowest energy excitations (dark areas in the contour plot
shown in Fig. 3), their locations in the 1BZ do not deviate
from those in the 1D limit (kx = ±π/2) for J ′/J . 0.8.
Additionally, when kx = ±π/2, the excitation spectrum
Ek is independent of ky, i.e., Ek = 3J
′D23. This is be-
cause the frustration of two interchain couplings (corre-
sponding to the lattice vector τ 2 and τ 3) cancel the ky
dependence. This fact is rather important, since it indi-
cates that the excited quasiparticles along the kx = ±π/2
lines feel free to move along the ky direction. This is the
same condition as in the 1D limit, except for the exis-
Fig. 2. Anisotropy dependence of D1 and D23 for L = 1200. Note
that D23 is very small compared to D1 in a wide range 0 ≤
J ′/J .0.25.
tence of a finite energy gap.
Figure 4 shows the minimum gap energy in the 1BZ
as a function of anisotropy J ′/J , changing the system
size L. We can see the almost gapless excitations in the
wide parameter range 0 ≤ J ′/J . 0.25, as is already
expected. It is quite natural that this behavior is simi-
Fig. 3. Anisotropy dependence of the one-particle excitation
spectra. Contour plots of the spectra are on the left, and sections
along ky = 0 line are on the right. The hexagons with broken
lines represent 1BZ of the triangular lattice. Up to J ′/J ∼ 0.25,
the spectra for each anisotropy are hardly distinguishable, and
the one-dimensionality strongly remains for large J ′/J .
4 J. Phys. Soc. Jpn. Letter Author Name
lar to that of D23, considering that the minimum energy
excitations are located along kx = ±π/2 for J ′/J . 0.6.
By plotting the same data for various system size, L,
in a semi-log scale (Fig. 4), we can see a discontinuous
jump for every size. We find that this critical value J ′c/J
vanishes very slowly as (lnL)−1. Thus, the discontinu-
ity is an artifact of finite-size calculation. We also find
that the minimum gap energy is finite when infinitesimal
J ′ is introduced. Actually, we can fit the J ′ dependence
as aJ ′ exp(−bJ/J ′)17 for J ′/J . 0.6 as shown in Fig.
4. Considering that the minimum gap energy is already
about 3 orders of magnitude below J at J ′/J ∼ 0.25, it
can be said that almost gapless excitation is realized in
0 ≤ J ′/J . 0.25. This result is fairly suggestive com-
pared with the previous series expansion18 and linear
spin wave19, 20 studies, all of which suggest a spin dis-
ordered state in the parameter range J ′/J . 0.25.
From the above results, we conclude that there is a
strong tendency to form a 1D-like excitation spectrum
for the triangular lattice spin system with anisotropy
0 ≤ J ′/J . 0.6. Furthermore, even if the anisotropy
is as large as 0.6 . J ′/J . 0.8, we can still expect 1D-
like behavior for quasiparticles except for the existence of
the excitation gap. Let us here discuss the relation to κ-
(ET)2Cu2(CN)3. The anisotropy of spin exchange inter-
actions in this material can be estimated from J = 4t2/U
(U being the onsite Coulomb repulsion) as J ′/J ∼ 0.89.
At this anisotropy, a rather large excitation gap exists as
shown in Fig. 4. We consider two possibilities to under-
stand the gaplessness. One is that the small gap region in
Fig. 4 expands to large values of J ′/J by some factors not
considered in the present model. For example, If long-
distance exchange interactions, quantum fluctuation or
multiple spin exchange effect14 (higher order terms of
the Heisenberg model) suppress not only LRMO but also
the spin gap, we can reproduce the gapless spin liquid
state at large J ′/J . These possibilities remain as future
problems. Another possibility is that the anisotropy J ′/J
of κ-(ET)2Cu2(CN)3 deviates from the above estimation
Fig. 4. (Color Online) Anisotropy dependence of the minimum
gap energy in the 1BZ (right axis) for L=60(diamond), 120(plus),
300(square), 600(cross) and 1200(triangle). The semi-log plots of
the same quantity are also shown (left axis). The solid line is a
fitted exponential function aJ ′ exp(−bJ/J ′), where a = 3.50 and
b = 1.61. We find that the observed critical behavior is an artifact
of finite size calculation (see the text).
due to, for example, a finite U effect.21 If it is in the
range J ′/J < 0.25, the excitation gap is sufficiently small
and the susceptibility behavior (finite at 1.9K whereas
J ∼ 250K) can be explained.
In summary, we analyzed an anisotropic triangular lat-
tice Heisenberg model using RVB mean-field approxima-
tion in order to investigate the physical origin of the gap-
less spin liquid state observed in κ-(ET)2Cu2(CN)3. We
payed attention to the Q1D anisotropy of this material,
and took an approach from the 1D limit. As a result of
calculations, we found that a practically 1D state with
almost gapless excitations is realized in the wide range of
the anisotropy parameter 0 ≤ J ′/J . 0.25. Furthermore,
one-dimensionality remained strongly even in J ′/J >
0.25 due to the geometrical frustration of interchain cou-
plings. We consider this “one-dimensionalization by frus-
tration” as a candidate for the mechanism of the gapless
spin liquid state, although the full understanding has not
yet been achieved.
This work was partly supported by a Grant-in-Aid
for Scientific Research on Priority Areas of Molecular
Conductors (No. 15073210) from the Ministry of Edu-
cation, Culture, Sports, Science and Technology, Japan,
and also by a Next Generation Supercomputing Project,
Nanoscience Program, MEXT, Japan.
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