PHYSICAL REVIEW B 109, L081407 (2024)
Letter

Conformal aspect of charge density waves: Theory and experiment
Keiji Nakatsugawa ,1,2,* Tatsuhiko N. Ikeda,3 Takeshi Toshima ,4 and Satoshi Tanda
1

1,5,*

Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan
2
Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan
3
RIKEN Center for Quantum Computing, Wako, Saitama 351-0198, Japan
4
Department of General Education, National Institute of Technology, Toyama College, Toyama 939-8630, Japan
5
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan(Received 13 March 2023; revised 24 January 2024; accepted 24 January 2024; published 27 February 2024)
The variety of two-dimensional (2D) discommensurate charge density wave (DC-CDW) phases has been
explained by a phenomenological CDW free-energy theory, where the wave vector of different DC-CDW
phases corresponds to different local minima of a multivalley free-energy landscapes. Here, we discover thatexperimental 2D DC-CDW structures in transition-metal dichalcogenides can be understood elegantly by aconformal transformation with complex numbers. Specifically, we represent a 2D CDW wave vector Q by acomplex number Q and find that by minimizing the CDW free energy with respect to Q, different wave vectors√√
Q and Q are connected by a conformal transformation. Consequently, n × n (n = 7, 9, 13) honeycomb and(quasi-)stripe DC-CDWs can be understood using simple conformal groups, which link DC-CDW wave vectors(multivalley landscape) to the incommensurate and commensurate wave vectors.
DOI: 10.1103/PhysRevB.109.L081407

Introduction. Charge density waves (CDWs) are electronic crystals that are described by so-called macroscopic
wave functions such as superconductors, superfluids, fractional quantum Hall liquids, etc. [1–3]. Transition-metal
dichalcogenides (MX 2 ) are layered compounds that induce
typical two-dimensional (2D) CDW phases with wave vectors Q(i) (i = 1, 2, 3) satisfying the triple-Q condition Q(1) +Q(2) + Q(3) = 0. Such CDW phases include the incommensurate (IC) phase, commensurate (C) phase, and CDW phaseswith discommensurate (DC) domain walls, such as the nearlycommensurate (NC), stripe [4], and triclinic (T) phases [5,6].
Moreover, other CDW phases are continually being discovered [7–9].
A phenomenological Ginzburg-Landau theory of CDW
phases in MX 2 was introduced by McMillan [10] to explain
the IC to C phase transition and 1D stripe discommensuration.
However, McMillan heuristically included discommensurations, and hence could not correctly explain known CDW
phases [11]. Nakanishi and Shiba extended this free energy
theoryto√explain the appearance of the NC√√ phase√ in 1T -TaS2( 13 × 13 structure) and 2H-TaSe2 ( 9 × 9 structure)using CDW harmonics, where the domain walls and harmonic
wave vectors are all obtained from the fundamental wave
vectors Q(i) [12,13]. In general, the search for CDW freeenergy local minima is a difficult task. Some of the present
authors revisited the Nakanishi-Shiba-McMillan models formonolayer 1T -TaS2 and 2H-TaSe2 and discovered the multivalley free-energy landscape, where free energy local minimacorrespond to different DC-CDW phases with domain walls(NC, stripe, and T phases) [14]. The existence of these CDW*

These authors contributed equally to this work.

2469-9950/2024/109(8)/L081407(9)phases in monolayer MX 2 (pure 2D system) is confirmed bothexperimentally [15] and theoretically [14].
Despite the development of CDW free-energy theory,
why such different DC-CDW phases can appear in modern
quantum crystals is still an open question. The reason for
the appearance of various CDW phases is not simply because
there are various magnitudes of nesting vectors Q. For nesting
vectors that happen to be close to the C condition, there
seems to be a law that locks in CDW superlattice to the MX 2
lattice. Within the framework of the Nakanishi-ShibaMcMillan theory, we reveal that this law is given by aconformal transformation between DC wave vectors. Here, aconformal transformation refers to a composition of translation, rotation, dilation, and inversion [16].
In this paper, we discover that experimental 2D DC-CDW
structures in MX 2 can be explained by a conformal transformation with complex numbers. Specifically, we represent a
2D CDW wave vector Q by a complex number Q and minimize the phenomenological CDW free energy with respect toQ, which leads to a multivalley free-energy landscape where
different wave vectors Q and Q are connected√ by a√conformal transformation. Consequently, we explain n × n (n =7, 9, 13) honeycomb and (quasi)stripe DC-CDWs using simple conformal groups, which link DC-CDW wave
to√ vectors√
the IC and C wave vectors. Consequently, n × n (n =7, 9, 13) honeycomb and (quasi)stripe DC-CDWs can be explained beautifully by a discrete conformal transformation
with Eisenstein integers Z[ω]. The CDW phases such as NC,
T, and stripes, which seem to be completely unrelated at firstglance, actually have a common relation linking their Q to
QIC and QC . Then, we apply our conformal method to explainthe√ origin√ of an experimental NC phase in TaSe2 thin-film( 7 × 7 structure).

L081407-1©2024 American Physical Society

NAKATSUGAWA, IKEDA, TOSHIMA, AND TANDAPHYSICAL REVIEW B 109, L081407 (2024)

FIG. 1. (a) C-CDW superlattice vectors can be represented by Eisenstein integers (this figure shows theand ν = 1 ). (b) Domain wall wave vectors q (i) are given as deviation from QC(i) . (c) CDW harmonics k (i) .

The commensurate structure and CDW free energy. First,
we review the C structure and CDW free energy theory in
MX 2 . Consider a triangular lattice with√ primitive lattice vectors a1 = (1, 0)a and a2 = 21 (−1, 3)a where a = a1  =a2 ,  ·  represents the norm of a vector. The CDW superlattice vector AC of the C phase is defined as [1]AC = (μ + ν)a1 + νa2 ,(1)

where μ and ν are positive integers that√ on mate√ depend
rials [e.g., μ = 3 and ν = 1 for the 13 × 13 structure
in 1T -TaS2 as shown in Fig. 1(a)]. Reciprocal lattice vectors b1 and b2 which √satisfy ai · b j = 2π δi j (i,√ j = 1, 2) are
given by b1 = (1, 1/ 3)/a and b2 = (0, 2/ 3)/a. ai andbi are not parallel but are separated by 30◦ . However, forthe description of CDW it would be convenient to have a
set of reciprocal lattice vectors Gi which are parallel to ai .
From the threefold rotational symmetry of the system, we
require that G1 and G2 are separated by 120◦ and we also
define a3 = −a1 − a2 , G3 = −G1 − G2 . The two sets of reciprocal lattice vectors are related by b1 = 2/3(G1 − G3 ) andb2 = 2/3(G2 − G3 ). Gi (i = 1, 2, 3) are normalized by thecondition ai · Gi = 2π . For i = j, ai and G j are mutually
separated by 120◦ , which implies ai · G j = 2π cos(2π /3) =−π . Consequently, we have the condition ai · G j = 2π δi j −π (1 − δi j ) = π (3δi j − 1). Then, the wave vector which satisfies AC · QC = 2π is defined as
QC =(μ + ν)G1 + νG2,
AC 2 /a2(2)
where AC /a = μ2 + μν + ν 2 . There are actually threeequivalent superlattice vectors AC(1) , AC(2) , AC(3) , which are rotated from AC by 0◦ , 120◦ , and 240◦ , respectively, and threeequivalent wave vectors QC(i) , which are defined likewise(Fig. 1).
We also consider general CDW phases in equilibrium.
MX 2 exhibits three types of isotropic CDW phases, where
each Q(i) is separated by 120◦ , namely, the IC, NC, and C
phases. There are also the anisotropic stripe phase [4,17](i)
are usually determined by Fermi
and the T phase [5,6]. QIC
surface nesting or van Hove singularity [2,3,18–21], which is
independent of the underlying lattice. On the other hand, QC(i)are determined from lattice symmetry and satisfy Eq. (2). DC
phases between the C phase and the IC phase have domain

√

13 ×

√
13 structure with μ = 3

walls (discommensurations) that originate from commensurability with the underlying base lattice [10,12,13,22].
(i)and QC(i) are known. The following freeSuppose that QIC
energy, which was originally proposed by McMillan [10],describes the IC-C phase transition2
F = d r a(r)α(r)2 − b(r)α(r)3 + c(r)α(r)4+ d (r)3|ψi (r)ψi+1 (r)|2 +

i=1

3



ψi (r)∗ ei (−i∇ )ψi (r) .

i=1

(3)
Here, r is the 2D spatial coordinate. The complex orderparameters ψi (r) are related to the charge density ρ(r) =ρ0 (r)[1 + α(r)], where α(r) = Re[ψ1 (r) + ψ2 (r) + ψ3 (r)].
ρ0 (r) is the charge density in the normal phase. Each coefficient has periodicity of the MX 2 lattice and is written asa(r) = a0 + 2a1 3i=1 cos(Gi · r), etc. The third-order termreflects the triple-Q condition Q(1) + Q(2) + Q(3) = 0. The
constant terms a0 , b0 , c0 , d0 are sufficient to discuss the IC
phase, but the Umklapp terms such as c1 are necessary to
discuss the C√phase.
determine√For instance, b1 and c1 √√ the stability of the 9 × 9 C phase [13] and the 7 × 7 C phase(Supplemental√ Note 2 [23]), c1 determines the sta√ Material
bility of the 13 × 13 C phase [12], and so on. Moreover,(i) 2ei (Q) = sQ − QIC /Gi 2 , where s is a constant, has been(i)introduced to determine QIC. [Other forms of ei (Q) have also
been proposed, which may be more appropriate in some experiments. See Supplemental Material Note 1 for consistency
with other forms of ei (Q(i) ) [23]]. Surprisingly, it was shownthat the free energy Eq. (3) can also describe the NC phase in
1T -TaS2 using the Nakanishi-Shiba expansion [12,13,24]

 (i)
(i)
ψi (r) =(4)
lmn exp iQlmn · r ,l,m,n0l·m·n=0(i)where l, m, n are integers; Qlmn
= Q(i) + lk (i) + mk (i+1) +
(i+2)
(i)
(i)
(i+1)
, k = μq − νq
are CDW harmonics; andnkq(i) = Q(i) − QC(i) are domain-wall wave vectors. The
free energy has been solved numerically by imposing acutoff N  l, m, n  0. N = 0 represents the charge density
modulation by the fundamental wave Q(i) . The free-energy

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PHYSICAL REVIEW B 109, L081407 (2024)

minimum for the NC phase appears as higher order harmonics
N = 1, 2, 3, . . . are included.
Analytically, Eq. (3) can be integrated using Eq. (4) (seeSupplemental Material Note 2 for a sample calculation, which
is analogous to Refs. [12,13,23,24]). After integration, we
obtain
F=

3



(i)∗
lmn

(i)
(i)
lmn ei (Qlmn ) + · · · ,where the terms in (· · · ) are independent of Q. The freeenergy is minimized by the condition3
i=1 l,m,n0 ∂

∂F

(i)
lmn +

d
(i)

lmn

3


q (i) = Q(i) − QC(i) ,(5)

i=1 l,m,n0l·m·n=0

dF =where we used Eqs. (9) and ω∗ = ω−1 = ω2 = −1 − ω. For
brevity, we set G1 = 1 (i.e., a = 2π ) in the following, which
implies |Gi | = 1 for i = 1, 2, 3.
We also write general CDW wave vectors Q(i) =(i)(Qx , Qy(i) ) using complex wave numbers Q(i) = Qx(i) + iQy(i) .
q(i) and k (i) are also written using complex numbers (Fig. 1)(i)Using complex numbers the gradient ∇ Q
is replaced by(i)∗∂/∂Q , so the CDW wave vectors for local minima between
the C and IC phases are given as solutions of
3


∂F
=
|
(i)∗
∂Q
j=1 l,m,n0∇Q(i) F · dQ(i) = 0.

i=1l·m·n=0

∂F

= 0,

(7)

∇ Q(i) F = 0.

(8)

∂

(i)
lmn

Equation (7), which contains all coefficients a(r), b(r), . . . ,ei (Q) in the free energy, leads to an infinite set of nonlinear
differential equations which cannot be solved analytically.
(i)
Numerical calculation of lmn
with fixed Q(i) has been donein our previous study [14], where we unveiled the existenceof the multivalley free-energy landscape in 1T -TaS2 and2H-TaSe2 . On the other hand, we show in the next section thatEq. (8) can be solved analytically, where the solution is a listof possible Q vectors of DC-CDW phases.
Conformal aspects of CDW phases. In previous studies of
CDW free energy, the free-energy local minima are obtainedby calculating F for all values of Q(i) . However, we observethat local minima can also be obtained by differentiating F
with respect to Q(i) . This major simplification is achieved byrepresenting CDW vectors using complex numbers.
√
The correspondence (1, 0) ↔ 1 and (0, 1) ↔ i = −1
maps a√1 and a2 to a and aω, respectively, where ω = e2πi/3 =− 21 + 23 i. Then, the triangular lattice can be mapped to theset of Eisenstein integers Z[ω] with elements z = m + nω(m, n ∈ Z) [25]. Consequently, AC(i) are equivalent to the
Eisenstein integers [Fig. 1(a)]:AC = [(μ + ν) + νω]a,AC(i) = AC ωi−1 .

2πG1QC = ∗ =,ACμ − νω

QC(i) = QC ωi−1 ,

∂Q

(11)

where
(i)
(i)∗
Q∗ − QIC,ei (Q) = s Q − QIC

(12)

(i)
Qlmn
= Q(i) + lk (i) + mk (i+1) + nk (i+2) .

(13)

We solve Eq. (11) for three cases: (i) the isotropic NC phases,(ii) the T phases, and (iii) the stripe phases.
a. The NC phases. In the case of isotropic NC phases,
we have the following simplification: (i) The wave vectors
are separated by 120◦ , which implies Q(i+1) = ωQ(i) . Consequently, (ii) the summation in Eq. (4) is a summation overEisenstein integers zlmn = l + mω + nω2 (note the identity
ω2 + ω + 1 = 0) and (iii) the CDW harmonics have the simple expression k (i) = ω (i−1) (Q/QC − 1). Then, Eq. (13) with
Eqs. (10) imply
Q(i)

(i)
=
Qlmn

QC(1) + zlmn − zlmn ωi−1 .

QC(1)Using ∂Q/∂Q∗ = 0, we obtain
3


∂F
=
s
(i)∗
∂Q
j=1 l,m,n0( j) 2lmn

l·m·n=0××(9)

Gi are mapped to Gi = 2π a−1 ω1−i and AC · QC = 2π corresponds to A∗C QC = 2π where the asterisk denotes complex
conjugation (this correspondence can be formalized, for instance, using 2D Clifford algebra [26]). Consequently, QC(i) areequivalent to( j)( j) 2 ∂e j (Qlmn )
|
= 0,
lmn
(i)∗

l·m·n=0(6)(i)
Regarding different lmn
and Q(i) as independent degrees of(i)freedom and assuming that lmn
does not explicitly depend on(i)
Q , then minimization of F requires us to solve the following
differential equations:k (i) = μq (i) − νq (i+1) .

Q( j)
QC(1)
ωi− j
QC(1)∗(Q(1)


+ zlmn ) − ω

j−1

(1)
QIC
+ zlmn∗QC(1)∗ + zlmn

= 0.

(14)

We assume that components with different j values vanish(i)
individually such that the wave vector of the NC phase QNC
isgiven by(10)

L081407-3

(i)
QNC

(i)
QIC
+ zi

QC

QC(i) + zi(i)≡
= f QIC
(i)

,

(15)

NAKATSUGAWA, IKEDA, TOSHIMA, AND TANDAPHYSICAL REVIEW B 109, L081407 (2024)

FIG. 2. (a) CDW local minima of 1T -TaS2 are shown for N = 3 harmonics in units with G1 = 1 (see also Supplemental Material Video
2 for larger N [23]). Different CDW free energy local minima correspond to distinct Eisenstein integers zlmn = l + mω + nω2 . The green dot
represents the C phase and the black dots [circles for (d) and (e)] represent local minima. (b) Each local minima is obtained from QIC by theaction of Eq. (17). (c) Successive CDW phase transitions can be explained by a composition of Eq. (17). The local minima in the free-energy
landscape for (d) 1T -TaS2 and (e) 2H -TaSe2 (obtained as in our previous work [14]) agrees with the local minima of Eq. (17). We used
experimental values of wave vectors in (f).

where we defined the complex number(i) 2 i−1∗zlmn QC(1)∗ + zlmn
l,m,n0lmn ωl·m·n=0
zi =
.

(i) 2
∗
QC(1)∗ + zlmn
l,m,n0lmn

implies that the wave vectors of NC phases corresponding to
the free-energy local minima with the label lmn are given by
the following solution:(16)(1)
QNC,lmn

l·m·n=0

Equation (15) shows that the wave vector of an NC phase canbe obtained from the IC and C wave vectors with a complex
number zi . However, the form of zi is complicated and gives(i)little intuition, and we still need to solve for lmn
to obtain the(i)numerical value of QNC . So, we investigate its meaning withthe following approximation. In our previous work, we reported the free-energy landscape where different free-energy
local minima correspond to distinct l, m, n values (Ref. [17]).
(i)
Numerical results show that the amplitudes lmn
are localizednear free energy local minima (Supplemental Material Fig. 3[23]). Then, we have the approximation zi ≈ ωi−1 zlmn , which

QC(1)

(1)
≡
= flmn QIC(1)
QIC+ zlmnQC(1) + zlmn

.

(17)

Equation (17) means that if QIC and QC are known, then
CDW phases between IC and C are described by distinct
Eisenstein integers zlmn (Supplemental Material Video 1[23]). Figure 2 shows the free energy local minima. Figure √2(a) shows√ the CDW local minima for μ = 3 and ν =1 ( 13 × 13 C structure) with N = 3 harmonics (seealso Supplemental Material Video 2 for larger N [23]).
We emphasize that the position of each local minima isdetermined from Eq. (8) which contains only the term

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PHYSICAL REVIEW B 109, L081407 (2024)

proportional to ei . In other words, we find that the ei termalone can explain the structure of the free energy landscapewith a good approximation, without the need to solve for the(i)amplitudes lmn.
Mathematically, Eq. (15), which includes Eq. (17) as an
approximation, is a conformal transformation. The set oftransformations f have a one-to-one correspondence witha commutative subgroup of the projective linear group
PSL(2, Z[ω]). This group property implies that transition
between different local minima (different CDW phases) is
obtained by a composition of f and f −1 . Three successive local minima lie on a circle whose center and radius
can be calculated as in Supplemental Material Note 3 [23]:These can explain successive CDW phase transitions, suchas the IC → NC → C phase transition in 1T -TaS2 with decreasing temperature [14]. Figure 2(b) shows flmn (QIC ) for|zlmn |  3. This type of transformation can explain how each
local minima is obtained from the IC phase. Figure 2(c)shows flmn ( fl  m n (QIC )) for |zlmn |, |zl  m n |  1. This transformation can explain successive CDW phase transitions. The
free energy landscape in Fig. 2(d) 1T -TaS2 and Fig. 2(e)2H-TaSe2 are obtained as in our previous work [14] (except
for 2H-TaSe2 , we used T − TIC = −0.1 to show local minima
more clearly). We used experimental values of wave vectors

ti,lmn =in Fig. 2(f). The free-energy calculation agrees with the local
minima of Eq. (17).
b. The T and stripe phases. This conformal method also applies to anisotropic stripe and T phases as follows. Anisotropic
CDW phases have domain walls that are tilted from each otherby an angle that is different from 120◦ . This angle can beobtained with good accuracy using real-spacing imaging suchas scanning tunneling microscope, so we use it as an input
parameter.
Suppose that the domain wall wave vectors q (i) of the Tphase are related by q (1) = w1 q (1) , q (2) = w2 q (1) , and q (3) =w3 q (1) . For isotropic phases, we have w2 = ω and w3 = ω2 .
Here, w1 = 1 is introduced for convenience such that theharmonic wave vectors are given byk (i) = (μwi − νwi+1 )q (1)

(18)

(i)
and Qlmn
can be written as

(i)
=
Qlmn

Q(i)
QC(i)

QC(i) + ti,lmn − ti,lmn ,(19)

where we defined(lwi + mwi+1 + nwi+2 )μ − (lwi+1 + mwi+2 + nwi )ν i−1ω .
wi (μ − νω)

For isotropic phases, we obtain ti,lmn = zlmn ωi−1 as expected. Consequently, Eq. (11) can be written as

3
( j)


w ∗j 1∂F( j) 2 Q
( j)
( j)
Q
−
Q
QC( j)∗ + t ∗j,lmn = 0.
=
s
+
t
+
t
j,lmn
j,lmn
C
IC
lmn
∗ ( j)∗( j)∂Q(i)∗
w
QC
i QC
j=1 l,m,n0

(20)

(21)

l·m·n=0

We assume that components with different j values vanish
individually such that the wave vector of the T phase QT(i) is
given by
QT(i)
QC(i)

=

(i)
QIC
+ ti

QC(i) + ti

,

(2)
(3)
and Qlmn
can be written as
Qlmn

(2)
=
Qlmn

(22)

where we defined the complex numberti =

l,m,n0l·m·n=0

(3)
Qlmn
=

(i) 2
(i)∗
∗
lmn ti,lmn QC + ti,lmn



(i) 2
∗
QC(i)∗ + ti,lmn
lmn

l,m,n0l·m·n=0

.

QC(i)

=

(i)
QIC
+ ti,lmn

QC(i) + ti,lmn

.

QC(2)
Q(3)
QC(3)

QC(2) + s2,lmn − s2,lmn ,QC(3) + s3,lmn − s3,lmn ,(23)
where we defined(i)are localized nearIf we further assume that different lmn
different local minima, then each local minima gives the Tphase with the following wave vectors:(i)QT,lmnQ(2)(24)

Next, the stripe CDW phase has domain-wall wave vectors(1)q (1) = 0 and q (2) = −q (3) . In this case, we have lmn= 0 sos2,lmn =(l − m)μ − (n − l )ν,μ − νω

(25)

s3,lmn =(l − n)μ − (n − m)ν
,μ − νω

(26)

which can also be obtained from Eq. (20) with w1 → 0
and w3 = −w2 . Consequently, Eq. (11) can be written as

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NAKATSUGAWA, IKEDA, TOSHIMA, AND TANDAPHYSICAL REVIEW B 109, L081407 (2024)(i = 2, 3)
 
∂F
= −s
(i)∗
∂Q
j=1,2 l,m,n0( j) 2lmn

l·m·n=0

×
×

Q

( j)



QC( j) + s j,lmn
QC( j)1
QC( j)∗

−

( j)
QIC
+ s j,lmn

QC( j)∗ + s j,lmn = 0,(27)

= −1. We assume that components
where we have used ∂Q∂Q(3)with different j values vanish individually such that the wave
vector of the stripe phase QS(i) is given by
(2)

QS(i)

(i)
QIC
+ si

QC

QC(i) + si

=
(i)

,

(28)

where we defined the complex number

(i) 2
(i)∗
∗
l,m,n0lmn si,lmn QC + si,lmn
l·m·n=0si = 
.
(i) 2
(i)∗
∗
l,m,n0lmn (QC + si,lmn )(29)

l·m·n=0(i)
are localized nearIf we further assume that different lmn
different local minima, then each local minima give the stripe
phase with the following wave vectors:(i)QS,lmn

QC(i)

=

(i)
QIC
+ si,lm

QC(i) + si,lm

.

(30)

It should be emphasized that the free energy local minima,
which correspond to the NC, T, and stripe phases, are all
obtained from Eq. (11). In the approximation where lmn are
localized near the local minima, a good approximation of the
DC wave vectors was obtained from Eq. (8) only. This result is
independent of the stability of each minima. Therefore, fromthis analysis, we reveal the different roles of stability andposition in the CDW free-energy structure. From this pointof view, isotropic and anisotropic DC-CDW phases can beclearly understood in a unified way.
Consistency with experiments. We have shown that thewave vectors of DC-CDW phases can be described by conformal transformations. Which of the local minima become
stable depends on many factors such as temperature andsample thickness, but our method gives a list of possible Q
vectors of new CDW phases. Since we have already shown theconsistency with known CDWs, we test our theoretical results
using experimental data. TaSe2 thin films with thickness of
10 nm order have been synthesized by the dechalcogenide
method. The details of this method are given in Refs. [8,27].
The CDW structure was observed by transmission electron
microscope (TED). The crystal structure is shown in Figs. 3(a)and 3(b). Their TED patterns (obtained at room temperature)are √shown√in Figs. 3(d) and 3(e). Figure 3(d) is identified as
the 7 × 7 C phase as shown in Fig. 3(c) [8]. In addition to(1)|/|G1 | = 0.341(2) and φ =the C phase, a NC phase with |QNC◦
25.0(2) was observed in two different samples (Fig. 3(e) [27]and Supplemental Material Fig. 2) [23]. The fundamental NC(i)wave vectors QNC
are determined as in Supplemental Material
Fig. 2 [23], where we made sure that the strongest satellite

FIG. 3. (a) Structure of TaSe2 thin film. Red spheres represent
Ta atoms√and yellow
spheres represent Se atoms. (b) Top view of (a).
√(c) The 7 × 7 commensurate structure involving seven atoms, acentral Ta atom (green) √surrounded√ by six Ta atoms (red), in the unit
cell. TED pattern of (d) 7 × 7 C phase [8] and (e) NC phase [27]in TaSe2 thin film. (f) The CDW free energy landscape for TaSe2 thinfilm is calculated as a function of a general wave vector Q = (Qx , Qy )with N = 3. See Supplemental Material Note 5 [23] for free-energy
parameters. The C phase is shown with a green dot. Experimental
value of the NC phase is shown with a red dot. An enlarged image
with local minima from Eq. (17) is shown in Supplemental Material
Fig. 4 [23]. (g) Radius of circles shared by N = 1 harmonics as afunction of commensurate angle φC .

peaks are included. Although it looks like the threefold rotational symmetry is broken, this is due to the inclination of thesample during the TEM measurement which affects the intensity of the CDW satellite peaks. To explain the appearance of
this experimental NC phase, we first use Eq. (17) to see which
local minimum it corresponds to, then we calculate the CDW
free energy using Eq. (3).
The C and NC phases in TaSe2 thin films are obtainedat room temperature, so QIC with a much higher transition
temperature could not be measured. However, we can obtainits approximate value based on the experimental NC phase. In
1T -TaS2 and 2H-TaSe2 , QC and QIC differ by no more than a

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PHYSICAL REVIEW B 109, L081407 (2024)

few percent [1,12,13]: We assume that this is also true for our
TaSe2 thin film.
Different MX 2 polytypes can have different nesting mechanisms. The IC phase in 2H-TaSe2 can be formed by saddle
point (van Hove singularity) nesting [19,20]. In thinner systems, the attractive force caused by the van Hove singularity
could be stronger than the force caused by Fermi surface
nesting. Saddle point nesting can occur both in the -M direction which is parallel to G1 , or in the -K ( -K  ) which istilted 30◦ (−30◦ ) from G1 . From Ref. [19], one can estimate|QIC | = 0.53 in the -M direction, but this value is too large.
On the other hand, in the -K or -K  direction, one canestimate |QIC | = 0.31 [19,20]. In fact, using Eq. (17), we
observe that the local minimum f031 (QIC ) corresponds to theexperimental NC phase if we set |QIC | = 0.35 ∼ 0.38 and
φ = −30◦ (Supplemental Material Video 3 [23]). The −30◦
angle implies that the IC phase in this sample is formed by
van Hove singularity rather than Fermi-surface nesting.
In the above, we found that the experimental QNC value
is included in the list of possible Qs if |QIC | = 0.35 ∼ 0.38
and φ = −30◦ . We confirm this result by calculating the freeenergy Eq. (3) numerically with the temperature dependenceof the form a0 /2 = T − TIC where TIC is the IC-CDW transition temperature. The result of numerical calculation for
TaSe2 thin film is shown in Fig. 3. Note the multivalley
free-energy landscape which coincides with the local minima obtained by the conformal method. If Eq. (17) was notavailable, one would need performing numerical calculations
repeatedly by changing the values of QIC and other freeenergy parameters. Even if a result is obtained that fits wellwith QNC , there is no guarantee whether this result is uniqueor whether it was obtained by chance. On the other hand,
Eq. (17) allows us to determine the local minimum that fits
QNC using only the values of QIC and QC . Then, if necessary, one can refine the free-energy parameters such that thisminimum becomes stable. Therefore, we conclude that our
method is ideal for analyzing CDW phases with almost no
assumptions.
Conclusion and discussion. In this paper, we investigatedthe conformality of CDW in MX 2 compounds. We showedthat DC-CDW phases can be understood using conformaltransformations. This method forms a basis to describe successive IC-DC-C phase transitions. If a CDW did not interact
with the underlying base lattice, then Q(i) may take arbitrary
continuous values, so the discreteness of this transformationis due to the commensurability of CDW with the underlying
base lattice. We also applied our analysis to experimental√√
results of a NC phase in TaSe2 thin-film with a 7 × 7 C
structure.
Why could we describe DC-CDW phases using a conformal transformation? Our reason is as follows. CDW has
the aspect of a quantum crystal with a macroscopic wave
function. The best way to describe a quantum crystal seems
to be a method that keeps local angles and produces constanttension (crystal properties) and allows local deformations(wave property). It has been found that the deformation of 2D
aperiodic lattices is governed by conformal transformations
that satisfy these requirements [28,29]. Regarding CDWs as
deformable superlattice crystals, it is reasonable that they canalso be described by conformal transformations. Recall thata conformal transformation refers to a composition of translation, rotation, dilation, and inversion [16]. The first threetransformations can describe the change of CDW domain
structure, and the last transformation connects the real-spacestructure to the momentum space. For nesting vectors that
happen to be close to the C condition, there is a conformal
law that locks in a CDW superlattice to the MX 2 lattice. On
the other hand, the ratio between the IC-CDW wave vector
and the MX 2 lattice constant is an irrational number, whilethat of the C-CDW wave vector is an integer. Therefore, aconformal description of CDW and the IC-DC-C CDW phasetransitions should be described by a conformal function thatconnects continuous numbers (IC) and discrete numbers (C).
So, a conformal method with complex numbers, which naturally satisfies these requirements, is ideal for studying CDW
phases. In what follows, we give some implications of ourresults.
First, note that QC is a stationary point of the transformation (17), i.e., flmn (QC ) = 1 (Supplemental Material Video
2 [23]). Therefore, we can regard QIC as a source, QC as adrain, and QNC as crossing points of the transformation. From
this point of view, discommensuration may be understoodas deviation from fixed points. This picture is a natural one
because QIC values usually determine QC .
Second, in this paper, we described DC-CDW phases using the discrete conformal group PSL(2,Z[ω]), but a unified
description of FQHL phases was also made using the discrete
conformal group PSL(2,Z) [30–35]. These systems are bothincompressible with a gapped spectrum. Therefore, we surmise that the field K in the group PSL(2,K) is discrete forincompressible systems, and K is continuous for compressible
systems. These are summarized in Table I.
Third, local minima with the same value of |zlmn | lie on acircle (Supplemental Material Note 2 [23]). Figures 2(d), 2(e)and 3(f) show the case with |zlmn | = 1. Figure 3(g) shows theradius R of these circles as a function of the C angle φC . Using
experimental values of Q we find that R increases with√ φC .
The√ difference between |QC | and√ |QIC√| in 1T -TaS2 ( 13 ×
13 structure) and 2H-TaSe2 ( 9 × 9 structure) are both2 − 3%, but the local minima are spread more widely (i.e.,
has larger R) in 1T -TaS2 . On the other hand, NC is reported in
1T -TaS2 but not in 2H-TaSe2 . Moreover, our TaSe2 thin film
has large R, which allows |QC | and |QIC | to differ by morethan 5%. Therefore, we surmise that, in addition to the normof Q, the phase of Q must be considered to obtain a stable NC
phase.
Fourth, we consider the temperature dependence of the
IC-DC-C phase transition. The amplitudes lmn change withtemperature, which can be confirmed by calculating Eq. (7)numerically. So the wave vectors given by Eq. (15), Eq. (22),and Eq. (28) also change with temperature. If the amplitudes lmn change smoothly with temperature, then Q will
also change smoothly. The detail of transition depends onthe choice of ei (Q). But, in any case, we confirm that the
distribution of free-energy local minima barely changes (seeSupplemental Material Fig. 1 [23]).
Fifth, we have shown that the variety of CDW domain
structures can be understood using only the IC and C wave
vectors. Therefore, our method is both simple and powerful, making it an ideal tool for analyzing CDW domain

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TABLE I. Comparison of 2D electron systems and their representative groups. The incompressibility of a system is associated with thepresence of a band gap. We surmise that field K in group PSL(2,K) is discrete for incompressible systems, and K is continuous for compressible
systems.
2D system

Compressibility/gap type

MaterialGroup

IC1 ↔ IC2Compressible? [21]MX 2

PSL(2, C)IC ↔ NC
NC1 ↔ NC2
IC ↔ C

Incompressible, [31] Comme. gap

1T-TaS2
TaSe2 thin-film
2H-TaSe2 VSe2

PSL(2, Z[ω])QHL

Incompressible, magnetoroton gap

GaAs

PSL(2, Z)walls. For example, we√can show
√ that the DC structure in
Cu-doped TiSe2 (with 4 × 4 C structure) [36] can beunderstood as the T phase and stripe phase. On the other
hand, the McMillan-Nakanishi-Shiba model can give only
free-energy results that are averaged over the system. Therefore, some of the results reported in Ref. [36], such as thedisappearance of domain wall by removal of impurity, arenot covered by our results. On the other hand, our theory
can predict what kind of domain-wall structure (wave vectors)
can appear.
Finally, possible future studies are as follows. Our conformal method, which keeps angles and produces constanttension (crystal properties) and allows local deformations(wave property), is ideal for describing CDWs as quantum
crystals and would be applicable to quantum liquid crystals and Moiré solids. We expect that our formalism is also
applicable to other superlattice systems [37,38]: 3 He A/Binterface [39] and other triple-Q systems like skyrmion lattices. Although we have focused on a triangular lattice, similar
results may be obtained in square lattice mutatis-mutandis
using Gauss integers Z[i] with elements z = a + ib, a, b ∈ Z
to describe CDWs in CuO, FeTe, and charge order.
Acknowledgments. The authors thank the Supercomputer
Center, the Institute for Solid State Physics, and the Universityof Tokyo for use of the facilities. T.N.I. was supported by JSPS
KAKENHI Grants No. JP18K13495 and No. 21K13852.

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