Charge-loop current order and Z₃ nematicity mediated by bond order fluctuations in kagome metals

1.

2.

3.

4.

5.

6.

7.

Methods

Self-energy due to BO fluctuations

To understand the BO+cLC phase diagram and the energy scale of

these orders accurately, we have to include the self-energy that

describes the quasiparticle properties. We calculate the on-site selfenergy due to BO fluctuations as

Σm ðϵn Þ =

T X

G 00 0 ðk + q,ϵn + ωl Þ

N k,q,m00 ,m000 m m

ð8Þ

8.

9.

10.

× Bmm,m00 m0 ðk,qÞ,

00

Bmm,m00 m0 ðk,qÞ = g qm m ðkÞg qm m ðkÞ  yvð1 + vχ g ðqÞÞ,

11.

ð9Þ

which is shown in Fig. 2d. Then, the Green function is given as

^ ÞÞ1 . The effect of thermal fluctuations

 Σðϵ

GðkÞ

= ðiϵn + μ  hðkÞ

described by the self-energy is essential to reproduce the

T-dependence of various physical quantities. Here, y = 1/2 when H

int

is given in Eq. (2). In the present numerical study, we calculate

χ g ðqÞ = χ 0g ðqÞ=ð1  vχ 0g ðqÞÞ and Σm(ϵn) in Eq. (8) self-consistently.

12.

13.

Kernel function of the DW equation

The kernel function due to BO fluctuations in Eq. (7) is given as

14.

0 0

lm

I llq ,mm ðk,pÞ =  g m

pk ðkÞyvð1 + vχ g ðk  pÞÞg kp ðp + qÞ

+ g llq ðkÞvg mm

q ðpÞ ,

ð10Þ

15.

16.

which is expressed in Fig. 3a and Supplementary Fig. 4a. The first term,

the MT term, is important when αBO ≲ 1, and its first term is the Fock

term. The second term, the Hartree term, vanishes when the eigenfunction ^f q ðkÞ is orthogonal to the BO form factor g^ q ðkÞ, like the cLC

Nature Communications | (2023)14:7845

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Acknowledgements

We are grateful to S. Onari, A. Ogawa, Y. Matsuda, T. Shibauchi, K.

Hashimoto, and T. Asaba for fruitful discussions. This study has been

supported by Grants-in-Aid for Scientific Research from MEXT of Japan

(JP20K03858, JP20K22328, JP22K14003), and by the Quantum Liquid

Crystal No. JP19H05825 KAKENHI on Innovative Areas from JSPS of Japan.

Author contributions

R.T. executed the calculations in discussion with Y.Y and H.K., and R.T.

and H.K. wrote the paper.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information The online version contains

supplementary material available at

https://doi.org/10.1038/s41467-023-42952-6.

Correspondence and requests for materials should be addressed to

Rina Tazai.

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