Adiabatic vs Sudden Flux Insertion andNonlinear Electric Conduction
Talk 2 by Masaki Oshikawa (ISSP, UTokyo)

Condensed Matter Physics
in All the Cities 2020
26 June 2020@Zoom
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This presentation ﬁle is based on what was used in the actual talk at#CMPCity2020, but slightly revised and modiﬁed

Talk 1 (Last week, Thursday 18 June)
Applications of Adiabatic Flux Insertion!toQuantum Many-Body Systems:
A Pedagogical Introduction
M. O. and T. Senthil, PRL 96, 060601 (2006)
Talk 2 (Today, Friday 26 June)
Adiabatic vs Sudden Flux Insertion andNonlinear Electric Conduction
M. O. PRL 84, 1535 (2000) / PRL 90, 236401; 90 109901 (E) (2003)
Haruki Watanabe and M.O., arXiv:2003.10390
Haruki Watanabe,Yankang Liu, and M. O., arXiv:2004.04561
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Adiabatic Flux Insertion(i) Increase Aharonov-Bohm ﬂux Φ adiabatically from|Ψ0 ! → |Ψ′0 !
0 to Φ0(=2̟)
Hamiltonian for the ﬁnal state isΦdifferent from the original one, but
we can(ii) eliminate the unit ﬂux quantum by
the large gauge transformationUx H(Φ = 2π)Ux −1 = H(Φ = 0)Ux = exp|Ψ0 ! → |Ψ′0 ! → Ux |Ψ′0 !
3!

2πi "xn!rLx!
r#Many Particles on Periodic Lattice
For example, consider a many-particle system on thesquare lattice of Lx × Ly with periodic boundary conditions
assume particle number conservation (U(1) symmetry)Φ

assume that the system is gapped, and consider theadiabatic insertion of unit ﬂux quantum through the “hole”
M. O. 2000
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Translation InvarianceTranslation invariance⇒ Momentum Conservation!
P! = −i∇

Lattice model / periodic potential
Tx = eiPxdiscrete (lattice) translation:
Let us now consider the adiabatic ﬂux insertion
Φ0 tAx =T Lx

Hamiltonian is always translation invariant
momentum is exactly conserved!

⇒initial state:Px(0)⇒ ﬁnal state:
5Px(0)

Which Momentum?
What is conserved exactly is the“canonical momentum” which is NOT gauge-invariant!
!
P!canonical = −i∇! −A!
P!kinetic = −i∇kinetic momentum = covariant derivative(gauge invariant)
After the insertion of the unit ﬂux quantum,
the system is equivalent to zero ﬂux but in the differentgauge!
We must eliminate the vector potential
by the large gauge transformation
6Large Gauge Transformation
Initial Groundstate |Ψ0 !
iPx(0)|Ψ′0 ! = Fx |Ψ0 !

Final State|Ψ0 !

′iPx(0)Tx |Ψ0 ! = e|Ψ′0 !

groundstate of H(0)groundstate of H(2π)

Tx |Ψ0 ! = e

Large gauge transformationmust be a groundstate of H(0)|Ψ̃′0 ! ≡ Ux |Ψ′0 !
Ux = exp!

2πi "xn!rLx!
r#i(′Tx |Ψ̃0 ! = eUx−1Tx Ux = Tx exp!
(0)2πPx + Lxr"r n"7!

2πi "n!rLx) |Ψ̃′ !
0!
r#Momentum Shift!
2πPx(0) → Px(0) +n!rLx!
rtotal number of particles(conserved)

We are usually interested in the thermodynamic limit
for a ﬁxed particle density (particle # / unit cell) νpSuppose ν =and choose Ly to be a coprime with qqp
2π∆Px =Lx Ly ν = 2πLyLxq

Lattice momentum is deﬁned modulo 2̟momentum shifted if q ≠ 1 (fractional ﬁlling)
The ﬁnal state is different from the initial ground state
ground-state degeneracy!

⇒8

“Lieb-Schultz-Mattis Theorem”
General constraint on the spectrum ofquantum many-body Hamiltonian on a periodic lattice
Periodic (translation invariant) lattice
U(1) symmetry⇒ unit cell⇒ conserved particle numberν : number of particle per unit cell (ﬁlling fraction)
ν = p/q⇒“ingappability”- system is gaplessOR- gapped with q-fold degenerate ground states
gapped with unique ground state9History of the LSM Theorem
1961 LSM S=1/2 chain
1981 [Haldane “conjecture”, dependence on 2S mod 2]
1986 Afﬂeck-Lieb LSM theorem for general S chain
1997 M.O.-Yamanaka-Afﬂeck general magnetization
1997 Yamanaka-M.O.-Afﬂeck electrons/particles
2000 M. O. “ﬂux insertion” argument for d≧2
2004 Hastings rigorous proof
2006 Nachtergale-Sims really rigorous proof…
many recent extensions!
(non-symmorphic crystal symmetry
Parameswaran et al. 2013 etc.)10Gap Closing by AB Flux?
the spectrum is identical!
EE

adiabatic evolution?

gap

gap

Φ=0Φ = 2π(= Φ0 )In principle, the ground state could evolve into anexcited state, if there is a gap closing (level crossingwith the “excited state”) at some value of Φ
11Insulator vs Conductor
Linear response theory: current induced by electric ﬁeld
Drude weight

δ→ +0

D=0 : insulator (Kohn, 1963)
D>0 : conductor

In a realistic system, the Drude peak is broadened(δ>0), but in an ideal model we can identify delta-function
Drude peak as a signature of “perfect conductor”
12Real-Time Formulation of Djx (t) ∼! tσ(t − t′ )Ex (t′ ) dt′−∞

lim σ(t) = Dt→∞

current induced by the electric ﬁeld at t=0,
that survives after an inﬁnitely long time

Initial condition at t=0: ground state|Ψ0 !

switch on an (inﬁnitesimal) constant electric ﬁeld for t>0t
Ax = A xTAx
Ex =Tadiabatic limit T →∞Axjx (t) ∼ DtT
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M. O. 2003
Watanabe-M.O. 2020

Current vs Energy
On the other hand, the current operator is!
"−1V: volume
1 ∂H
∂H
1 dAx
(Ax ) =ĵx =Vdt∂tV ∂Ax
Axtjx (t) ∼ D
TFor an adiabatic ﬂux insertion
11∆E0 =VV! T∂HAx!
" dt =∂t
T0! T0jx (t) dt ∼ D"For the adiabatic insertion of unit ﬂux quantumAx
T#2T22

Φ0Ax =LxV2∆E0 (Φ0 ) =Φ2 0 D2Lx

G. S. energy increase inthe adiabatic ﬂux insertion
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M. O. 2003

Gap Protection in Insulators (d=2)
E

the spectrum is identical!

Eadiabatic evolution?

gap
Φ=0V2ΦD∆E0 (Φ0 ) =022LxO(Lgapd−2If this happens in d=2, energy gain ≧ gap)

Φ = 2π(= Φ0 )⇒ D>0 !!

i.e. in an insulator, the groundstate must remain in thegroundstate in the adiabatic ﬂux insertion LSM
15⇒Kohn FormulaV2∆E0 (Φ0 ) =2 Φ0 D2Lx

Φ0Ax =→0Lx
2!
!
1 ∂ E0!
D=(A)x!
V ∂A 2xAx =0

Kohn’s formula for the Drude weight16

Non-Linear Conductivities
Non-linear electric conduction:
topic of current interest

AC E-ﬁelde.g. “shift current”
application to photovoltaics
DC currentjx (t) ∼" t∞!
1n=1n!

−∞...

" t " t−∞n-th order conductivity
σ (n) (t − t1 , t − t2 , . . . , t − tn )−∞Ex (t1 )Ex (t2 ) . . . Ex (tn ) dt1 dt2 . . . dtn

Nonlinear Drude weightslim∆t1 ,∆t2 ,...,∆tn →∞σ (n) (∆t1 , ∆t2 , . . . , ∆tn ) = D(n)17Non-Linear “Kohn Formula”
Consider the same adiabatic ﬂux insertionand include the non-linear Drude weights!
"ntAx = A x
Tjx(n) (t) ∼11(n+1)∆E0=VV! T1 (n)Dn!

Ax
Ttn! TAx∂H
dt =jx(n) (t) dt∂tT 00"#n+1 n+1
T1 (n) Ax
1∼ D=D(n) Ax n+1n!
Tn+1(n + 1)!

!
!
E01 ∂(n)!
D =(A)x!
V ∂A n+1n+1x

Watanabe-M.O.
2020
Watanabe-Liu-M.O. 2020

Ax =018

Sudden Flux Insertiont
Ax = A x
TAx
Ex =TT → 0: sudden insertion
delta-function electric ﬁeld pulse
In this limit, quantum state (wavefunction) does not change
but again we are in a different gauge, so need to applythe large gauge transformation to go back to the originalgauge|Ψ0 ! → |Ψ0 ! → Ux |Ψ0 !

19Energy Gain in Sudden Flux Insertion
11∆E =VV! T∂HAx!
" dt =∂t
T0! T

T→0jx (t) dt0(n)!

"nAx(0, 0, . . . , 0) 1
∼2n
n+1 T
#$n
" t" t∞!
Ax
1(n)jx (t) ∼...
σ (t − t1 , . . . , t − tn )dt1 dt2 . . . dtnn! 0T0n=1$n#(n)
σ (0, 0, . . . , 0) Axn∼t2nT
σ"1 !
!Ψ0 |Ux † H(0)Ux |Ψ0 " − !Ψ0 |H(0)|Ψ0 "V#$Φ01) − H(0) |Ψ0 "= !Ψ0 | H(Ax =VLx20cf.) LSM
variationalenergyNon-linear f-Sum Rules
Comparing both sides, we obtain the identity
instantaneous response in real-timeσ(n)!
(0, 0, . . . , 0)H(Ax ) !!
∂= !Ψ0 |n2∂A n+1 !
n+1x! ∞dω1=−∞ 2π! ∞dω2...
−∞ 2π|Ψ0 "Ax =0! ∞dωn (n)σ (ω1 , ω2 , . . . , ωn )−∞ 2π

[frequency space representation]
Watanabe-M.O. / Watanabe-Liu-M.O. 2020cf.) Shimizu 2010, Shimizu-Yuge 201121

Example: Tight-Binding Model22

Numerical Check23

Summary
Two general formulas for non-linear conductivity
f-sum rules (instantaneous response = ω-integral)σ(n)!
(0, 0, . . . , 0)H(Ax ) !!
∂= !Ψ0 |n2∂A n+1 !
n+1x|Ψ0 "Ax =0energy gain by sudden ﬂux insertion“Kohn formulas” for non-linear Drude weights(long-time response = 1/ω pole)!
!
E01 ∂(n)!
D =(A)x!
V ∂A n+1n+1xAx =0

energy gain by adiabatic ﬂux insertion
more general results are given in arXiv:2003.10390 & arXiv:2004.04561
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