Breakdown of the Meissner effectat the exceptional pointin the non-Hermitian two-band BCS model
Novel Quantum States in Condensed Matter2022
たいらA
東⼤⽣研 平良 敬信AU-Tokyo IISA
Takanobu TairaA
Based on: 2211.11422

Room: Y208
taira904@iis.u-tokyo.ac.jp

Freeg Hermitian BCS"$BathGKSL

Non-Hermitian effective theory#$Two-component complex Ginzburg-Landau
Extended London equation

Breakdown of the Meissner effectbeyond the critical temp while gaplessat the exceptional point while gapfulHSIntroduction / Result

2/25[Kato, Perturbation theory for linear operators(1980)][Heiss, Sannino, J. Phys A, (1990).]'%=#(#(&'/"""'=1/*± = ± ' − (−'
23(()
0 0
Exceptional point%#$ =0 0"0 1%%$ =0 0
Diabolic point!
./(()

Introduction Interesting/ Result Physics at EP[Goldzak, et. al, PRL, (2018)][Kozii, Fu, ArXiv, (2017)]

3/25

Introduction Interesting/ Result Physics at EP[Hanai, et al., PRL (2019)]

4/25[Dembowski, et al., PRL, (2003)]

Experiment in microwave cavity

Introduction Interesting/ Result Physics at EP[Fring, Taira, EPJP, (2022)] [Fring, Taira, PLB, (2022)]

Particle physics

Breakdown of Higgs
mechanism while
finite vacuum

Complex two
Breakdown ofcomponent GinzburgMeissner effectLandau type model Task 2
while gapfull
Task 1
Task 3
Microscopic theory inat exceptional point
open system

5/25

Task 1

Freeg Hermitian BCS"$BathGKSL

Non-Hermitian effective theory#$

Two-component complex Ginzburg-Landau
Extended London equation

Breakdown of the Meissner effectTask 3beyond the critical temp while gaplessat the exceptional point while gapful

HSTask 2

Task 1g Hermitian BCS

7/25"Non-Hermitian effective theory#$Task 1

8/25

Effective theory of ultra-cold atom
[Yamamoto, et al., PRL123, (2019)][Dürr, et al., PARA 79 (2009)][Liu, et al., ArXiv:2209.10427, (2022)][Takasu, et al,. PTEP, (2020)]

Task 1

9/25)
4Tot 8& , 8&):Tot [<& , <& ]Coherent statepath integral[Yamamoto, et al., PRL, (2019)]

Non-Hermitian
Mean field theory))@@" : Auxiliary fields in MFT
Δ* ≠ Δ* , Δ" ≠ ΔB)
A@:[<,<Δ,Δ]&&&&@*
BΔ=0)
A@& ]:[<& , <& Δ& , Δ

Task 1)
A@& ]:[<& , <& Δ& , Δ

10/25

@& ]
:eff [Δ& , Δ)
Integrate out{<& , <& }Want to expand@&trace log in Δ& , Δ

B:eff
=0
@&
BΔ
Gap equations

Task 1

11/25

Gap equations|Δ+ |@, Δ, )*⁄"(ΔDropping J H 0
from gap eq
Δ+ ≡ Δ* − #HΔ"
Δ, ≡ Δ" + BΔ*--.

/H= ,B=g
Hermitian
BCS theory"$NonHermitiantheory

Task 1

12/25

Two-component complex Ginzburg-Landau type modelTask 1

Free

g Hermitian BCS"$BathGKSL

Non-Hermitian effective theory#$Two-component complex Ginzburg-Landau
Extended London equation

Breakdown of the Meissner effectTask 3beyond the critical temp while gaplessat the exceptional point while gapful

HSTask 2

Task 2

14/25

Minimal coupling∇→ ∇ + /X
B:233
∇×
Y=0
BX& 454!

London equation

Task 2

15/25

B:233
Inverse of London∇×Y=0∇×X ≡ [penetration depth
BX& 454!
Extended London equation=0Gapless=0Gapless ?
No!!!

#!
!! ≡ % 1& ("& + *%⃗$",%⃗&)! ⁄&*,!
=.
2'tanh 0/00Task 2

16/25

If and only if

Meissner effect breaks at this temp\ ≡ \" ⁄\*Gapful!!

The breakdown of Meissner effect

17/25

Gap parameters Δ* and Δ"

Exceptional temperature

Finite gap parameter !!

gHermitian
BCS theory"$Non-Hermitiantheory

H#!=$Gapful region
Gapless region

The breakdown of Meissner effect

18/25

Gap parameters Δ* and Δ"

Exceptional temperature+5*⁄*"6T%$^789:

Finite gap parameter !!

gHermitianBCS theory"$Non-Hermitiantheory

H#!=$Gapful region
Gapless region

Task 1Freeg Hermitian BCS"$BathGKSL

Non-Hermitian effective theory#$Two-component complex Ginzburg-Landau
Extended London equation

Breakdown of the Meissner effectTask 3beyond the critical temp while gaplessat the exceptional point while gapfulHSTask 2Task 3)@*
Δ* = Δ)@"Δ" ≠ Δ
Δ* ≡ _* + # `* , Δ)* ≡ _* − # `*Action is U(1) symmetric !
Tayler expand
around vacuum20/25Task 3)
Δ* ≡ _* + # `* , Δ* ≡ _* − # `*Action is U(1) symmetric !

21/25

Exceptional point iswhen this is nondiagonalisable

Mass matrix

Task 322/25

Mass matrix
Exceptional point of
this mass matrix

Extended London equation

Task 1

Freeg Hermitian BCS"$BathGKSL

Non-Hermitian effective theory#$

Two-component complex Ginzburg-Landau
Extended London equation

Breakdown of the Meissner effectTask 3beyond the critical temp while gaplessat the exceptional point while gapfulHSTask 2

Conclusion

24/25• We have shown that the Meissner effect breakdown
at the exceptional point.
• However, this was observed for a particular example.
• The exceptional temperature is sensitive to smallchange of H. But why?
• Experimental realisation?

Thank you

25/25

Thank you for listening

Room: Y208
taira904@iis.u-tokyo.ac.jp

Motivation

26/25[Kato1980Book],Exceptional[Heiss1990JPhys]point
c '* , '" , * ≔ det(4 '* , '" − *f)
4 '* , '" b& = *& b&c '* , '" , * = 0,>? @" ,@# ,BCB= 0,Diabolic point
Exceptional point{'* , '" } = {'*#$ , '"#$ }'* = '%$
Interesting
Physics*& = *; ,a< ≠a=*& = *; ,a< = a=e.g. (edges of) Fermi arce.g. Dirac cone

Task 328/25

Mass matrix
Exceptional point of
this mass matrix

g* ≡ h* − 1ij + h" B "g" ≡ h" − 1i#$k ≡ −#Hh* + Bh" + #H 1ij

London equation

Task 329/25

Exceptional point iswhen this is nondiagonalisable

Δ* ≡ _* + # `* , Δ)* ≡ _* − # `*Action is U(1) symmetric !

Mass matrix

GKSL30/25GKSL

31/25

GKSL)
4: XD l32/25

=FmE + HD)
XD |l⟩