PHYSICAL REVIEW B 86, 247102 (2012)

Reply to “Comment on ‘Charge-parity symmetry observed through Friedel oscillations
in chiral charge-density waves’ ”
J. Ishioka,1 T. Fujii,2 K. Katono,3 K. Ichimura,3,4 T. Kurosawa,5 M. Oda,4,5 and S. Tanda3,4,*
1Center for Advanced Research of Energy and Materials, Hokkaido University, Sapporo 060-8628, Japan2
Department of Physics, Asahikawa Medical College, Asahikawa 078-8510, Japan3
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
4Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan
5
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan(Received 19 October 2012; published 11 December 2012)
We are responding to the Comment by J. Wezel on our paper. This study was developed from our previous work
[Ishioka et al., Phys. Rev. Lett. 105, 176401 (2010)]. In the PRL paper, HCDW was defined as a new parameterfor expressing CDW chirality for the first time. In his Comment, he claims that HCDW is ill defined. He alsoclaims that the initial phase φ of the CDW wave function is a more appropriate parameter for expressing chiral
CDW, despite our early introduction of φ to explain the experimental data described in the PRL paper. However,
we conclude that HCDW can distinguish the CDW chirality by its sign. Moreover, by considering different HCDW
signs, we had succeeded in demonstrating the difference of the spatial distributions of CDWs as shown in Fig. 4of the PRB paper [Phys. Rev. B 84, 245125 (2011)]. In our Reply, we discuss the validity of HCDW . We showthat his argument regarding the identification of the CDW with the opposite sign of q is wrong, since the logic
is inapplicable to a wave function with a nonzero φ. We also discuss the applicability of HCDW to two- or
three-dimensional CDWs in transition metal dichalcogenides.
DOI: 10.1103/PhysRevB.86.247102

PACS number(s): 71.45.Lr, 68.37.Ef, 72.80.Ga, 73.22.Gk

II. FORBIDDEN OPERATION TO
SPACE WITH CHIRALITY

I. INTRODUCTION

As we reported in our PRL study,1 in 1T-TiSe2 we
discovered chiral charge density waves (CDWs) for the firsttime by observing two-fold optical reflection and by detecting
different CDW intensities for the q vectors of three CDWs.
To explain the experimental results we introduced nonzero
relative phases between three CDW wave functions. Rightand left-handed charge distributions in real space can bereproduced by choosing relevant phase differences. Moreover,to give the CDW chirality more generality, we focused on
three-dimensional CDW q vectors and defined the CDW
chirality asHCDW ≡ q1 · (q2 × q3 ),(1)where q1 , q2 , and q3 are CDWs q vectors connecting Fermi
pockets from  to L in the first Brillouin zone of 1T-TiSe2 .
When q1 , q2 , and q3 do not exist in one plane, HCDW hasa nonzero value in the same way that cholesteric liquid
crystals have a nonzero value of n · (∇ × n)dV , where n
is a director.2 In spin systems, a similar form of scalar triple
product of spins has been proposed as a chiral order parameter.3
Since the mathematical form is the same, the introduction ofHCDW as a new measure is quite natural. As a measure forexpressing the property of the entire space, it is highly intuitive
to use a pseudoscalar.
In his Comment,4 Wezel abandoned the HCDW for the firsttime from the point of view that each q vector has invariancewhen translated by reciprocal lattice vectors so that the signof the HCDW changes arbitrarily. He insists that HCDW has nophysical meaning. However, we do not agree with this claim
because this argument is not made for the identical initial
phases of the three CDW wave functions.
1098-0121/2012/86(24)/247102(3)In the first place, no arbitrary axis inversion can preserve anentire system that does not have mirror inversion symmetry.
Let us consider a die representing systems without inversion
symmetry. As shown in Fig. 1(a), we define each axis andassume a die. When we look at the 1, 2, and 3 faces ofthe die, they are located counter clockwise [Fig. 1(a)]. If
the x axis is inverted, the arrangement of the 1, 2, and 3
faces becomes clockwise. The dice before and after axis
inversion are not identical, that is the whole system changes interms of chirality. The system with x- and y-axis inversion
is the same as the original die [Fig. 1(c)]. As shown in
Fig. 1(d), whole axis inversion changes the die’s chirality.
Therefore, as the property of the three-dimensional spaces,
arbitrary axis inversion, namely mirror operation, inverts theentire space. In the same manner, when the pseudoscalar
is defined, its sign should be changed by any mirror
operation.
In general, a scalar triple product depends on the directionsof triple q vectors. When there are three vectors, mirror
inversion must make at least one vector inversion. On theother hand, in the PRB argument,5 mirror inversion operation
corresponds to the inversion of all three q vectors, when themirror plane is perpendicular to the c∗ axis. Because threevectors are inverted by mirror inversion, the scalar triple
product of those vectors must change. When each vector
characterizes the lattice structure, the whole lattice structure
should be inverted in accordance with the vector inversion.
Therefore, HCDW reflects the chirality of the CDW structure.
√ to 6π√ .
Indeed, the HCDW value also changes from − a 26π
c 3
a2 c 3
The HCDW change corresponds to the handedness of thestates.

247102-1©2012 American Physical Society

COMMENTS

PHYSICAL REVIEW B 86, 247102 (2012)

He focuses solely on the phase difference between threeCDWs. He also insists that the sign of φ corresponds to thechirality of the ground states. We agree with the correspondence, because the change in the sign of φ corresponds to thechange in the sign of the q vector, that is,ρ  = A cos[qr + (−φ)] = A cos[−(qr + (−φ))]= A cos[−qr + φ] = ρ  .

FIG. 1. (Color online) Schematic images of dice with (a) originalstructure, (b) x axis inverted, (c) x and y axis inverted and (d) x, y,and z axis inverted. The 1, 2, and 3 faces are shown at the bottom ofeach image.

(4)

So the chiral inversion could be expressed both by the phasedifference and the q vectors. It is also possible to simulate
the same calculation in Ref. 5 by changing signs of φi .
To avoid any reader’s confusion about our previous study,1
we did not change the phases when all the q values wereinverted in PRB.5 Although φi is nonzero value, φi cannot
distinguish whether CDW is polar7 or chiral. A nonzero HCDW
clearly distinguishes tilted q vectors from in-plane q vectors.
Therefore, we think the concept of HCDW includes the sign ofthe phase φ so that HCDW is a more sophisticated concept thanφ for expressing chiral states.

III. CONCEPTUAL INCLUSION OF THE INITIAL PHASE
DIFFERENCES BY HCDW

IV. DIFFERENT APPROACHES TOWARDS
THE SAME PHENOMENA

The concept of the HCDW includes the relative phase of thewave functions. Wezel’s main claim is based on the invarianceunder the translation of each q vector. However, we think thatthe invariance of q is not demonstrated properly because heignores the phase differences of each CDW although he insists
on φ as an alternative. Within the very specific condition of φ =0 or π , the wave function with q is equal to that with −q. On theother hand, phase differences with more general values havebeen introduced to explain the experimental results as discussed in Ref. 1. The wave functions of CDW are expressed as

Regarding the paper on which Wezel’s work was based,6 werecognize it for its meaningful suggestion of the chiral state ofCDWs based on a theoretical calculation of GL free energy.6
This work strengthens our belief in the existence of chiral
CDWs because it provides multiple explanations. However, toavoid confusing the readers, he should clarify the postulate of
the theoretical analysis, which differs from that in our paperin PRL.1 Based on experimental fact, we first fixed the phasedifference between CDWs and then constructed HCDW with qvectors. In this sense, the phase difference is a postulate forexpressing chirality. On the other hand, he constructs GL freeenergy with φ as a variable and then calculates φ to minimize
the total energy.6 The calculation finally provides the nonzero
phase differences of each CDW. Although different approaches
were used, consistent results have been obtained in terms ofchirality.

ρ = A cos(qr + φ)(2)ρ  = A cos(−qr + φ),

(3)

and


where ρ is different from ρ with respect to the position of
the charge density peaks. Therefore, a total configuration
constructed with three CDW components of −qi should differ
from that with qi (i = 1,2,3). Since the formalization of thewave function4,6 given by his results in a cosine type function,
the above discussion can be applied to his discussion.
Returning to Wezel’s claim, our discussion of the PRB
paper is based on the fact mentioned above. In the numericalcalculations shown in Fig. 4 in PRB5 we first assume aset of three CDW waves and locally induced waves. Localchiral inversion was induced by the q inversion of only localcomponents, and the two different states before and after q
inversion were compared. If the invariance of q under arbitrarya∗ remains with a nonzero φ as he insists, the spatial charge
distribution of the two states must be the same. However, inthree-dimensional spaces, there is no condition satisfying this.
On the other hand, our calculation succeeded in showing thelocal chiral inversion by q inversion as the local modulationof the spatial distribution of the charge density. Therefore,in our research HCDW clearly distinguishes two states withdifferent chiralities.

V. CONCLUSION AND REMARKS

In conclusion, we think that HCDW is closely linked withthe sign of the phase differences. It is quite difficult to directly
detect phase differences. In contrast, HCDW is highly intuitive
to predict whether CDW have chirality by knowing the CDW
q vectors. It will be used as a discriminant for chiral CDWs
by testing whether HCDW can be (−,0,+). −, 0, and +corresponds to left-handed chiral CDWs, nonchiral CDWs,
and right-handed chiral CDWs, respectively.
Researchers have found magnetochiral anisotropy in theelectric conductivity of carbon nanotubes and physicallytwisted metal wires.8,9 This anisotropy depends on the structural chirality of each system. We expect a discussion tobegin on a similar effect on chiral CDWs. When severalsystems show chiral CDWs and chiral properties, we believe
HCDW is more appropriate than just φ for discussing eacheffect universally because HCDW directly reflects the structuralcharacteristics including the size of the CDW superlattice.

247102-2

COMMENTS*PHYSICAL REVIEW B 86, 247102 (2012)

Corresponding author: tanda@eng.hokudai.ac.jp
J. Ishioka, Y. H. Liu, K. Shimatake, T. Kurosawa, K. Ichimura,Y. Toda, M. Oda, and S. Tanda, Phys. Rev. Lett. 105, 176401(2010).
2
P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter
Physics (Cambridge, London, 1995).
3
R. Shankar and D. Shubashree, Phys. Rev. B 61, 12126 (2000).
4J. van Wezel, Phys. Rev. B 86, 247101 (2012).
1

5J. Ishioka, T. Fujii, K. Katono, K. Ichimura, T. Kurosawa, M. Oda,and S. Tanda, Phys. Rev. B 84, 245125 (2011).
6J. van Wezel, Europhys. Lett. 96, 67011 (2011).
7J. van Wezel, Phys. Rev. B 85, 035131 (2012).
8V. Krstić, S. Roth, M. Burghard, K. Kern, and G. L. J. A. Rikken,J. Chem. Phys. 117, 11315 (2002).
9G. L. J. A. Rikken, J. Folling, and P. Wyder, Phys. Rev. Lett. 87,236602 (2001).

247102-3