￼

Drude weights and f-sum rules forlinear and nonlinear conductivities

Masaki Oshikawa (ISSP)
1

ISSP Theory Seminar
26 May 2023 (Fri)n-th order conductivities<latexit sha1_base64="FKC6ew+KLhpThRgDFLqUaRfGld0=">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</latexit>

j⇠

(1)

E+

(2)

EE +

(3)

EEE + . . .

renewed interest thanks to developments inlaser technology, theory, ……
AC E- eld

Non-linear electric conduction:
topic of current interest
e.g. “shift current”<latexit sha1_base64="oMX78k5ZE4KvbBDBHWOnj9FpCoQ=">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</latexit>j/E

2

application to photovoltaics

DC current
2fi￼Non-Linear Conductivities

General constraints on nonlinear conductivities- nonlinear f-sum rules- nonlinear Kohn formula for Drude weights
in a uni ed formulation

3
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￼

This Talk

￼

References
W. Kohn, Phys. Rev. 133, A171 (1964)
M. O. PRL 90, 236401; 90 109901 (E) (2003)
T. Oka, R. Arita, and H. Aoki, PRL 91, 066406 (2003), …
T. Oka and H. Aoki, PRB 81, 033103 (2010) etc.
(Dielectric breakdown of Mott insulators)
H. Watanabe and M.O., Phys. Rev. B 102, 165137 (2020)
H. Watanabe,Y. Liu, and M. O., J. Stat. Phys. 181, 2050 (2020)
Y. Tanikawa, K. Takasan, and H. Katsura,
Phys. Rev. B 103, L201120 (2021)
A. Urichuk, A. Klumper, and J. Sirker,
Phys. Rev. Lett. 129, 096602 (2022)
K. Takasan, H. Watanabe, and M. O., Phys. Rev. B 107, 075141 (2023)
4

Linear response theory: current induced by electric eld
Drude weight

δ→ +0

D=0 : insulator (Kohn, 1964)
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”
5fi￼

Insulator vs Conductor

particles do not touch
the magnetic eld directly
no effect within classical mechBut quantum interference isstill affectedx

Aharonov-Bohm effect

Quantum system de ned on the annulus does dependon the ux, except when the Aharonov-Bohm phase is= 2⇡ ⇥ integer
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fi

6

fl

￼

Aharonov-Bohm EffectI have implicitly chosen the units so thate=1~=1
Covariant derivative
✓

e
exp i
~

I

@S

~ r) · d~rA(~

kinetic momentum

⇣

◆

~
i~r

~ r)
eA(~


= exp 2⇡i

⌘

(~r)

(S)
0

h
15
= 4.136 ⇥ 10
Wb
0 =
e

(twice the “unit ux quantum” commonly used insuperconductivity literature) 7
fl￼

Unit Flux Quantum￼Spectrum of the Hamiltonian
the spectrum
is identical!

Energy=0

Neverthelessgenerally depends
on Φ (AB effect)H(= 2⇡) 6= H(
8= 0)= 2⇡(=

0)If the Aharonov-Bohm ux is an integral multiple ofthe unit ux quantum it can be eliminated by
a topologically nontrivial (“large”) gauge transformation(~r) ! (~r)e

i✓(~
r)

x
✓(~r) = 2⇡Lxphase is multivalued
but wavefunctionis unique

For a many-body Hamiltonian on a latticeH(1= 2⇡) = Ux

H(= 0)UxUx = exp

2⇡i X
xn~r
Lx
~
r

fl

9
fl

￼

Large Gauge Transformation!

(i) Increase Aharonov-Bohm ux Φ adiabatically from
0
|
i
!
|
0 to Φ0(=2π)
00iΦHamiltonian for the nal state isdifferent from the original one, but
we can(ii) eliminate the unit ux quantum by
the large gauge transformation
Ux H(= 2⇡)Ux

Ux = exp

1

Final state may be different
from the original ground state10
fl0i ! |

0
0i
fl

|

0
0 i ! Ux |= H(! = 0)

2⇡i X
xn~r
Lx
~
r

fi

￼

Adiabatic Unit Flux Quantum Insertion

particles do not touch
the magnetic eld directly
static ux has no effect withinclassical mech<latexit sha1_base64="uAfj+ACTNV9Moz1rY4is+4VSz1g=">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</latexit>

vector potential Ax =LxHowever, a time-dependent ux doesx induce an electric eld, either in aclassical or quantum system
Time-dependent ux<latexit sha1_base64="18WFaeA+CUSSmoHEAIKYGZnLt/8=">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</latexit>

induced electric eld

1 d
dAx
=
= Ex
Lx dt
dt
fi

fl

fi

fl

fi

11
fl

￼

Time-Dependent Flux and Electric Fieldjx (t) ⇠Z tt0 )Ex (t0 ) dt0(t1

current
inducedbytheelectric
eld
at
t=0,
lim (t) = D
t!1
that survives after an in nitely long time
Initial condition at t=0: ground state|0i

switch on an (in nitesimal) constant electric eld for t>0
Ax
Ex =
T

adiabatic limit T →∞Ax
jx (t) ⇠ D
t for large t
T

M. O. 2003
Watanabe-M.O. 2020
fi

fi

12

fi

t
Ax = A xTfi￼

Real-Time Formulation of D

On the other hand, the current operator is✓◆ 1
V:volume
1 @H
1 dAx
@H
ĵx =
(Ax ) =
V
dt
@t
V @Ax
Ax
jx (t) ⇠ DtTFor an adiabatic ux insertion
1
V

1
E0 =
V

Z T

@H
Ax
h
i dt =
@t
T
0

Z T
0

jx (t) dt ⇠ D✓

For the adiabatic insertion of unit ux quantumV
E0 ( 0 ) =2
2Lx

G. S. energy increase in
the adiabatic ux insertion

fl

fl

Ax
T

◆2

Ax =
0

2

T2
2
0

Lx

D

M. O. 2003

13
fl

￼

Current vs Energy￼

Kohn FormulaV
E0 ( 0 ) =
2
2Lx

0

2

D

Ax =

0

Lx

!0

1 @ 2 E0
D=
(Ax )
2
V @Ax
Ax =0

Kohn’s formula for the Drude weight14

￼Nonlinear Drude Weights
Nonlinear conductivities (real-time form)jx (t) ⇠

Z t
1
X
1

n=1

n!

1

...

Z t Z t1(n)(t

t1 , tt2 , . . . , t

tn )

1

Ex (t1 )Ex (t2 ) . . . Ex (tn ) dt1 dt2 . . . dtn

Nonlinear Drude weights
lim

t1 , t2 ,..., tn !1(n)( t1 ,

15

t2 , . . . ,tn ) = D(n)Consider the same adiabatic ux insertionand include the non-linear Drude weights✓◆ntAx = A xT1V

jx(n) (t) ⇠

Z T

1 (n)
D
n!

Ax
T

tn

Z T

@H
Ax
(n)
dt =
jx (t) dt
V 0 @tT 0✓◆n+1 n+1
1 (n) Ax
T
1
n+1
(n)
⇠ D
=
D Ax
n!
T
n+1
(n + 1)!

1
(n+1)
E0
=

n+1
1
@
E0
(n)
D =
(Ax )
n+1
V @Ax
Ax =0

Watanabe-M.O.
2020
Watanabe-Liu-M.O. 2020
16

fl

￼

Non-Linear “Kohn Formula”

tAx = A x
T

Ax
Ex =
T

T → 0: sudden insertion
delta-function electric eld pulse
In this limit, quantum state (wavefunction) does not change
but again we are in a different gauge, so need to apply
the large gauge transformation to go back to the original
gauge|

0i ! |

0 i ! Ux |

17
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￼

Sudden Unit Flux Quantum Insertion0i￼Energy Gain in Sudden Flux Insertion
1
V

1
E=
V

Z T

@H
Ax
h
i dt =
@t
T
0

Z T

T→0

jx (t) dt

0

✓

(n)

⇠
jx (t) ⇠

Z t
1
X
1

⇠

⇣
1

V
1
= h
V

(n)

(t

(0, 0, . . . , 0)
2n

Ax
T

◆n

tn

0i

h

n!
n=1
(n)

0

0

✓

†

h

0 |Ux H(0)Ux |0|(0, 0, . . . , 0) 1
Ax
2n
n+1 T
✓
◆n
Ax
t1 , . . . , t t n )dt1 dt2 . . . dtn
T

Z t

...



H(Ax =

0

Lx

)

◆n

0 |H(0)|H(0) |
18

0i

0i

⌘

cf.) LSM
variationalenergy￼Non-linear f-Sum Rules
Comparing both sides, we obtain the identity
instantaneous response in real-time
(n)

=

Z 1

(0, 0, . . . , 0)
=
h
2n

d!1
1 2⇡

Z 1

d!2
...
1 2⇡

@ n+1 H(Ax )
|
0|
n+1
@Ax
Ax =0

Z 1

d!n
1 2⇡

(n)

0i

(!1 , !2 , . . . , !n )[frequency space representation]

Watanabe-M.O. / Watanabe-Liu-M.O. 2020cf.) Shimizu 2010, Shimizu-Yuge 201119￼(Unexpected) Symmetries

hold without any symmetry of the system!
Watanabe-M.O. 202020

￼

Direct “Kubo Theory” Calculation21

￼

Explicit Proof for 2nd Kohn formula

Similar explicit proof for up to 3rd order
Kohn formula & f-sum rule Watanabe-Liu-M.O. 202022

￼

Example: Tight-Binding Model23￼

Numerical Check24

can be mapped to spinless fermion withdensity-density interaction
exactly solvable by Bethe Ansatz technique;effective eld theory (Tomonaga-Luttinger Liquid)
Ground-state energy density as a function of the ux

fl

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￼

S=1/2 XXZ Chain￼

Exact 3rd-order Drude Weight<latexit sha1_base64="KiJiYxljZ5kTjKvqZqJTi5weFag=">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</latexit>=cos

M.O.-Watanabe 2020
based on Lukyanov 1998
Good agreement with numerics
Divergent for Δ<-1/2 ?!
26

￼

Recent Extension
Tanikawa-Takasan-Katsura 2021

(sign of Δ is reversed in their paper)

5-th order Drude weight
diverges for Δ<0(antiferromagnetic region)
27

Tanikawa-Takasan-Katsura 2021

Vanishes at zero ux by symmetry,
but can be non-vanishing under a non-zero ux and in fact
diverges for a range of Δ!
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￼

Even-order Drude Weights

￼Does the Current Diverge??
Diverging Drude weight does NOT imply
a particularly enhanced current due to nonlinear effects
So far, the divergent nonlinear Drude weights are negative,
rather representing the limitation in the linear response
current growing linearly in time
The divergent nonlinear Drude weights can be understoodin terms of Conformal Field Theory with perturbations
Tanikawa-Takasan-Katsura 2021

29

Tanikawa-Takasan-Katsura 2021

Effect of irrelevant perturbations (“Umklapp terms”)
to the CFT (Tomonaga-Luttinger Liquid):source of the divergence of the non-linear Drude weights!
magnetic eld suppresses
the Umklapp terms
and thus divergence of thenonlinear Drude weights!
30
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￼

Perturbed Conformal Field Theory

K. Takasan, H. Watanabe & M. O. PRB 2023

(with periodic b.c.)Exactly solvable in terms of the scattering at the defect

L: system size
θ: AB ux (=A×L)

Nonlinear Drude weight (according to Kohn formula)diverge if dN+1 ≠ 0 !
31

fl

￼

Free Electrons in 1D w/ DefectKohn formula: adiabatic limit for a given system size
thermodynamic limit of the Kohn formula:L→∞ after ω→ 0
let us call the result as “Kohn Drude weight”
But the adiabatic dynamics may require extremely longtime scale as the system size is enlarged and may not
be physical“Bulk Drude weight” may be de ned by
ω→0 limit after the thermodynamic limit L→∞
the distinction would be also relevant for
linear Drude weights, although the difference ismore dramatic for nonlinear32 Drude weightsfi￼

Why divergent?

(Linear conductivity)
low-frequency contributionsat ω〜1/L(→ zero frequency in thethermodynamic limit)
Kohn Drude weight: count only strictly ω=0 response
for nite L

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￼

Low-Frequency Contributions

Kohn formula(Kohn formula)“bulk Drude”
f-sum✴bulk Drude = f-sum forsingle-band tight-binding model

Kohn formula: strong dependence on AB ux θ,although the bulk properties should be independent of θ
“bulk Drude weight” including |ω|≦1/L : independent of θ!
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￼

Linear Drude Weight: Kohn vs Bulk

e.g.) Second-order conductivity<latexit sha1_base64="dD+BBEc2JpjCwkiX7kNM787gJmw=">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</latexit>

L,✓(2) (!1 , !2 )Again we de ne the “bulk Drude weight” by

35
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Nonlinear Conductivities

Kohn Drude weight: divergent, depends on AB ux
Bulk Drude weight: nite, independent of AB ux[actually vanishes cf.) symmetry ]K. Takasan, H. Watanabe & M. O. PRB 2023
fl

fl

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￼

2nd order Drude Weights: Kohn vs Bulk

H. Watanabe and M.O., Phys. Rev. B 102, 165137 (2020)
H. Watanabe,Y. Liu, and M. O., J. Stat. Phys. 181, 2050 (2020)

f-sum rules (instantaneous response = ω-integral)(n)(0, 0, . . . , 0)
=h
n
2

0|

@

n+1

H(Ax )
|
n+1
@Ax
Ax =0

0i

energy gain by sudden ux insertion“Kohn formulas” for non-linear Drude weights(long-time response = 1/ω pole)
D

(n)

n+1

1 @
E0
=
(Ax )
n+1
V @Ax
Ax =0

energy gain by adiabatic ux insertion
fl

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￼

Summary

However, the “Kohn formula” sometimes leads to“pathological” properties unrelated to bulk transport
D

(n)

n+1

1 @
E0
=
(Ax )
n+1
V @Ax
Ax =0

(several works on the linear conductivity, but
the problem ampli ed in the nonlinear conductance)
To probe the bulk property (“bulk Drude weight”)
Thermodynamic limit L →∞ rst, then consider“adiabatic limit” T → ∞ω〜0 [coef cient of δ(ω) ] afterwards
K. Takasan, H. Watanabe & M. O. PRB 107, 075141 (2023)
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Summary II￼Summary II
M. O. 2003

39

￼Open Problems- When Kohn Drude weight = bulk Drude weight ?
- Calculation of bulk Drude weights forinteracting systems- How to utilize the nonlinear f-sum rules
etc. etc.

40