Permanent Loop Current
in Strongly Correlated Electron Systems
based on fRG
fRG in condensedmatter physics

Rina Tazai (YITP, Kyoto university)outline

1. Introduction: fRG in condensed matter physics

2. Recent study of “non-local phase transition” based on fRG
ex. permanent loop current①coupled-chain Hubbrad model②kagome superconductor (2019~)
RT et al., Phys. Rev. B 103, L161112 (2021).
RT et al., Sci. Adv. 8, eabl4108 (2022).

3. summary

Itineracy and locality of electron

1Hubbard model
electrons in metalhopping(itineracy)

itineracy

Coulomb repulsion（locality）

locality

duality:spin
ion: iintermediate region of itineracy and locality（U≈t）
・magnetic order・charge order・superconductivity

fRG is powerful inintermediate region.

ion j

ion i’

fRG in Hubbard model・g-ology fRG → 1D・parquet fRG (Shultz) → 2D
・N-patch fRG (Metzner)・fRG for orbital-order・fRG for loop current/bond-order

fRG in Hubbard model

2

Fourier transformation

p1

p3

p2

p4

→ band= initial 4-point vertex
cos(k) band dispersion
logarithmic energy cutoffδabsence of UV divergenceby band-width-π0k・・・πl=0,1,2,3・・・
δ: small value-π0kπfRG in Hubbard model3

low energy effective model

full energy band

integrate out
from Hilbert space
only low energy
excitation
integrate out
from Hilbert space-π0kπ-π0k

low energy effective interaction

fRG in Hubbard model

W. Metzner, et al., Rev. Mod. Phys. 84, 299 (2012).
T. Enss, PhD thesis (2005).

low energy effective interaction

fRG differential equation
Wick-ordered fRG

Cooperon shellPeierls
low energy

2-loop6-point vertex

45fRG in Hubbard model
neglected・・・
Cooper2-loopPeierls

48-patches

Parquet diagrams are automatically generated.
4848

6-point: 484 -channels

advantage of fRG✓

advantage✓ Higher-order scatterings of U
are automatically considered.

6× disadvantage✓ Physical meaning is unclear.
ex. Which diagram/process
are important ?

✓ Vertex-corrections areautomatically considered.

Both fRG ＆ diagrammatic
calculation are needed.

beyond RPA/mean-fieldfRG

DWequation

diagrammatic calculation with vertex corrections (later)outline

1. Introduction: fRG in condensed matter physics

2. Recent study of “non-local phase transition” based on fRG
ex. permanent loop current①coupled-chain Hubbrad model②kagome superconductor (2019~)
RT et al., Phys. Rev. B 103, L161112 (2021).
RT et al., Sci. Adv. 8, eabl4108 (2022).

3. summary

7

non-local phase transitions
recent discovery
Based on fRG,
“new types of phase transitions” have been discovered!
= non-local orderion 1enhancedion 2

suppressedion 3= bond order

order parameter of non-local order-a+a
ion 1current

ion 2

a: real #current
２= loop current= spontaneous symmetry breaking of hopping+ia+ia

local VS non-local order

8local ordernon-local orderex. spin order, charge orderex. loop current order

order parameter<Si>≠０i: lattice index

1

2

3

static current
order parameterσ: spinsame site

different site

loop current by Haldane
Haldane’s loop current

9Phys. Rev. Lett. 61 2015 (1988)→ Novel Prize in 2016

effective Aharonov-Bohm (AB) phase

Origin of this imaginary
hopping was unknown.

cf. AB phase in magnetic-field

Hermite system

space inversion→ evenoddpure imaginary hopping= effective AB phase* time reversal symmetry is broken.
* odd parity

A: vector potentialrecent discover: various loop current
Cuprates

experiment

Varma’s loop
current

10

experimentexperiment

IridatekagomeSr2IrO42-leg ladder
Sr14Cu24O41

C. M. Varma,
PRB 55,14554 (1997).
PRB 99, 224516 (2019).
D Bounoua et al.,Comm Phys 3, 123 (2020).

C. Mielke et al., arXiv:2106.13443.
H. Murayama et al.,Phys. Rev. X 11, 011021 (2021). C. Guo et al., arXiv:2203.09593

I. Affleck et al.,
PRB 37, 3774 (1988).

open problem

Emergence of loop currents could not be explained by RPA/ mean-field theory.
Can we explain by fRG? = our motivation

classification of non-local ordernon-local order

11

Fourier transformation
σWe call fq(k) “form-factor”.

field theoryfσq(k) = symmetry breaking of self energy

spin
↑↔↓

S=+1i ↔ j pbond＝+1

TRS

T=+1

S=-1

pbond＝-1

T=-1

pbond＝+1

T=-1

S＊Pbond＊T=1

pbond＝-1

T=+1

Various non-local order are
explained by “non-local fq(k)”.
RT et al., arXiv:2205.02280 (2022)
to be published in Phys. Rev. B

study of non-local order by fRG

12fRG + “optimized non-local form factor”
pp

ph

ph

effective interaction(with vertex correction)
on-shellenergy cutoff

low energy

We optimize the form-factor fq(k).

3-point vertex= form-factor fq(k)Susceptibility
higher-order many-body effects are considered.

a=coefficientoutline

1. Introduction: fRG in condensed matter physics

2. Recent study of “non-local phase transition” based on fRG
ex. permanent loop current①coupled-chain Hubbrad model②kagome superconductor (2019~)
RT et al., Phys. Rev. B 103, L161112 (2021).
RT et al., Sci. Adv. 8, eabl4108 (2022).

3. summarycoupled chain Hubbard model

13

interaction

kinetic
chain3organic metals etc.,

chain2alattice constant2I,j: site-index: intra-chain hopping

fix

chain1: inter-chain hopping

V: Coulomb repulsion
spin frustration
spin→Spin order is suppressed.

fixcoupled chain Hubbard model

14

interaction

kinetic
chain3organic metals etc.,

chain2
alattice constant2I,j: site-index: intra-chain hopping

fix

chain1
V: inter-chain hopping
48-patches in k-space: Coulomb repulsion
Brillouin zone

absence of UVdivergenceblue: constant energy line

2 Fermi surface

fix

15
RT et al., Phys. Rev. B 103, L161112 (2021)

coupled chain Hubbard model→ SU(2)-symmetry

2
kinetic

interactionkq

We calculate generalized susceptibility by fRG.

qk+qgeneralized susceptibility

n=1with non-local form factorcoefficient a is optimized.
n=7

fkq is optimized
to maximize χc(s)(q).

16
RT et al., Phys. Rev. B 103, L161112 (2021)

coupled chain Hubbard model→ SU(2)-symmetry

2interaction

kinetickq

We calculate generalized susceptibility by fRG.

qk+qgeneralized susceptibility

T=0quantum fluctuation

εFT=∞
thermal fluctuation

our interest = T<<εF
Quantum fluctuation is important !

n=1

coefficient a is optimized.
n=7

fkq is optimized
to maximize χc(s)(q).

17
RT et al., Phys. Rev. B 103, L161112 (2021)

coupled chain Hubbard model
2interaction

kinetic

RG flow ofparticle-hole susceptibility= spin,charge, loop current, etc.,with 49-basis

susceptibility

Vloop current emerges!

Λ0
RG flowcoupled chain Hubbard model

quantum susceptibility

V18
RT et al., Phys. Rev. B 103, L161112 (2021)-loop current emerges!
half-filling

loop current

RG flow* Development of loop current fluctuation is absence in RPA

19phase diagram by fRG
antiferro magnet
J. Kishine etal., JPSJ 67,2590 (1998)

Loop current appears.

1D region

loop current

Loop current→ new!

superconductivityt⊥spin frustration
RT et al., Phys. Rev. B 103, L161112 (2021)

We discover loop current, which was overlooked for years.

20

origin of loop current
Maki-Thompson (MT) vertex correction
spin fluctuation

DW equation=qδ∑kksymmetry broken
component of self-energyk’+ mean-fieldk’+qk+qqGreen’s functionδ∑k’
MT vertex correction

new mechanism of loop current
revealed by fRG ＆DW equation= MT-vertex correction
by spin fluctuationinduce loop current.

=spin fluctuation
cause loop current.

RT et al., Phys. Rev. B 103, L161112 (2021)
RT et al., Sci. Adv. 8, eabl4108 (2022).

electronspinfluctuation

holetime : toutline

1. Introduction: fRG in condensed matter physics

2. Recent study of “non-local phase transition” based on fRG
ex. permanent loop current①coupled-chain Hubbrad model②kagome superconductor (2019~)
RT et al., Phys. Rev. B 103, L161112 (2021).
RT et al., Sci. Adv. 8, eabl4108 (2022).

3. summary

21

New superconductor: AV3Sb5 (2019~)
2019～Kagome network of Vanadium-ion

bond orderSuperconductivity (SC)aa: lattice constantbond order (David-star)= Hopping-integral isspontaneously enhanced.
→ Translational symmetry is broken.

origin of bond order

R. Tazai et al., Sci. Adv. 8, eabl4108 (2022).

22

We reveal that bond order is induced by AL vertex correction.
= interference of spin-fluctuations

spin fluctuations
bond order (David-star)

Aslamasov-Larkin vertex correction
interference mechanism

bond orderspin fluctuation 1
spin fluctuation 2

κ-(BEDT-TTF)2X

bond-order by fRG
RT et al., Phys. Rev. Research 3, 022014 (2021).

23

New superconductor: AV3Sb5 (2019~)
2019～

Kagome network of Vanadium-ion

Charge Density Wave

loop current
by experiment
Superconductivity (SC)(2020～）
aa: lattice constant
T=0

Superconductivity

loop current

bond order90[K]no orderT=∞
TU (1) broken

time reversal
symmetry broken

translation symmetry broken

24loop current in kagome metal
T=0Superconductivity

loop current

bond order90[K]no orderT=∞
TU (1) broken

time reversal
symmetry broken

translation symmetry broken

25

loop current in kagome metal
T=0Superconductivity

loop current phasebond order90[K]no orderT=∞
TU (1) broken

time reversal
symmetry broken

translation symmetry broken

Origin of loop current was unknown.

We reveal “Maki-Thompson vertex
corrections induce loop current”.
R. Tazai et al., arXiv:2207.08068 (2022).

new mechanism !

26
R. Tazai et al., Sci. Adv. 8, eabl4108 (2022).

loop current in kagome metal

We reveal a new origin of “loop current” by Maki-Thompson vertex correction.
Kagome metalelectronbond orderfluctuation

hole
coupled-chain model

difference

MT-vertex
correction

electronspin
fluctuation
holeSummary27

We discover the microscopic origin of new phase transition by using fRG.
ex. loop current“Beyond mean-field vertex correction
is necessary“
new phase transition(ex. loop current)
formfactor
DWequation

fRG

Maki-Thompson VC
cause loop current inKagome (2021～)
・static, without joule heating
space-dependent

・By optimizing the form-factor in fRG,
new phase transition has been revealed.

・By comparing fRG ＆ DW equation (=diagrammatic),origin of phase transition is clearly understood.