Corrections in : A Sum Rule for Thermal Conductivity and Dynamical Thermal
Transport Coefficients in Condensed Matter -I ∗
B Sriram Shastry
Physics Department, University of California, Santa Cruz, Ca 95064
(Dated: 1 March, 2007)
A list of corrections is provided below. A version of the paper with all errors removed is available at
http://physics.ucsc.edu/∼sriram/papers all/ksumrules errors etc/evolving.pdf.
−1
h̄T kx L
Z
t
0
eiωc (t−t ) dt0 h[JˆxQ (kx , t), K̂(−kx , t0 )]i
(7)
"
#
X
i 1
pn − pm
xx
Q
2
ˆ
κ(ωc ) =
hΘ i + h̄
|hn|Jx |mi| .
h̄ωc T L
− m + h̄ωc
n,m n
(13)
κ(kx , ωc ) =
−∞
"
#
X
pn − pm
i
xx
Q
ˆ
ˆ
< Φ > +h̄
hn|Jx |mihm|Jx |ni .
γ(ωc ) =
h̄ωc T L
− m + h̄ωc
n,m n
γ(ωc ) =
ih̄ X pn − pm hn|Jˆx |mihm|JˆxQ |ni
i
Dγ +
.
h̄ωc T
T L n,m m − n
n − m + h̄ωc
(26)
(27)
"
#
X pn − pm
1
xx
Q
< Φ > −h̄
Dγ =
hn|Jˆx |mihm|Jˆx |ni . (28)
L
− n
n,m m
γ(ωc ) =
i
1
Dγ +
h̄ωc T
TL
Z
0
∞
dteiωc t
Z
β
dτ hJˆx (t − iτ )JˆxQ (0)i,
(30)
0
Note added after Eq(42):
For charged systems, a small correction arises from the constraint of zero electrical current (zc) under a thermal
gradient. This correction can be included in the present formalism by using Onsager’s reciprocity relations, and leads
to modified expressions for Eq(18, 40 and 41). These may be written as
Z ∞
π
hΦxx i2
xx
Re κzc (ω) dω =
hΘ i −
,
(18.1)
2h̄T L
hτ xx i
0
hΘxx i
(40.1)
L∗ = 2 xx − (S ∗ )2 ,
T hτ i
hΦxx i2
Z∗ T =
.
(41.1).
hΘxx ihτ xx i − hΦxx i2
∗
The author is grateful to M. Peterson and S. Mukerjee for help in identifying some errors.
2
JxQ =
X
vp~x (εp~ − µ) c†p~,σ cp~,σ +
p
~,σ
JxQ (~k) =
X
vp~x (εp~ − µ) c†
p
~+ 12 ~
k,σ
p
~,σ
Θxx = Θxx
E −
o
U X n x
x
c~†
c c†~−~q,σ̄ cp~,σ̄ (61)
v~l + v~l+~
q
l+~
q ,σ ~l,σ p
2L
~l,~
p,~
q ,σ
cp~− 1 ~k,σ +
2
1
2L
X
~l,~
p,~
q ,σ,σ 0
o 1 ∂U (~q)
n
x
+
[U (~q) v~lx + v~l+~
(ε~l+~q −ε~l)]c~†
c 1 k,σ c†p~−~q,σ0 cp~,σ0 (62)
q
l+~
q + 12 ~
k,σ ~l− 2 ~
h̄ ∂qx
2µ xx µ2 xx
Φ − 2 τ , with
qe
qe
X †
−i
{(x1 − x3 )V~r2 ,~r3 − (x2 − x3 )V~r1 ,~r3 } and n~r =
c~r,σ c~r,σ ,
2
σ
1 X
≡
t(~η )t(~η 0 )t(~η 00 )(ηx + ηx0 )(ηx + ηx0 + ηx00 )c~†r+~η+~η0 +~η00 ,σ c~r,σ
2h̄
i X
−
t(~η )t(~η 0 )(ηx + ηx0 )χ(~r + ~η + ~η 0 , ~r, ~rj )c~r†+~η+~η0 ,σ c~r,σ n~rj
2h̄
1X
t(~η )χ(~r + ~η , ~r, ~rj )χ(~r + ~η , ~r, ~rl )c~r†+~η,σ c~r,σ n~rj n~rl
−
h̄
i X
−
t(~η )t(~η 0 )(ηx + ηx0 ) {χ(~r + ~η + ~η 0 , ~r + ~η 0 , ~rj ) + χ(~r + ~η , ~r, ~rj )} c~r†+~η,σ c~r,σ c~r†+~η+~η0 ,σ0 c~r,σ0 . (63)
2h̄
χ(~r1 , ~r2 , ~r3 ) ≡
Θxx
E
(ηx + ηx0 )2 t(~η ) t(η~0 ) Yσ0 ,σ (~r + ~η ) c~r†+~η+η~0 ,σ0 c~r,σ + µ2
X
h̄Θxx = µ
~ 0 ,~
η
~ ,η 0 ,σ,σ
r
+
1
4
X
X
ηx2 t(~η ) c̃~r†+~η,σ c̃~r,σ
η
~ ,σ
(ηx + ηx0 + ηx00 )(2ηx + ηx0 + ηx00 ) t(~η ) t(η~0 ) t(η~00 ) Yσ00 ,σ0 (~r + ~η + η~0 ) Yσ0 ,σ (~r + η~0 ) c̃~r†+~η+η~0 +η~00 ,σ00 c̃~r,σ
η
~ ,η~0 ,η~00 ,~
r ,σ,σ 0 ,σ 00
X
1
(ηx + ηx0 )(−ηx + ηx0 + ηx00 ) t(~η ) t(η~0 ) t(η~00 )
−
4
η
~ ,η~0 ,η~00 ,~
r ,σ
o
i
hn
o
n
†
c̃~r+η~0 ,σ c̃~r+η~0 +η~00 ,σ̄ − c̃~r†+η~0 +η~00 ,σ c̃~r+η~0 ,σ̄ c̃~r†+~η+η~0 ,σ̄ c̃~r,σ − c̃~†r+η~0 ,σ̄ c̃~r+η~0 +η~00 ,σ̄ − (h.c.) c̃~r†+~η+η~0 ,σ c̃~r,σ
X
3
∆ ∼ − Lt2
hYσ0 ,σ (~η )c̃†η~+η~0 ,σ0 c̃~0,σ i. (85)
h̄
0
σ,σ
∆=−
3 3
Lt βn(1 − n)(2 − n) + O(β 3 ). (86)
2h̄
τ xx = − lim
k→0
1 ˆ
[Jx (kx ), ρ(−kx )] (A2)
kx
"
#
X
i 1
p n − pm
xx
2
σ(ωc ) =
hτ i + h̄
|hn|Jˆx |mi| . (A5)
h̄ωc L
− m + h̄ωc
n,m n
(81)