This is a small talk on the project that i did as a part of my masters thesis.

1. Anomaly and parity odd transport coecients
in 1+1 d
Subham Dutta Chowdhury
August 1, 2014
Work done under the guidance Dr. Justin David (CHEP, IISc)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 1/23

2. Outline
Introduction
What is an anomaly?
Relating anomaly to hydrodynamics
Fujikawa method
Kubo Formula
Evaluating the correlator
Setting up the formalism
Matsubara sums
Transport coeecient
Discussion
Bibliography
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 2/23

3. What is an anomaly?
An important question is whether a symmetry of classical physics is
necessarily a symmetry of quantum theory.
Consider a symmetry of classical physics ,the transformation,
! expfig ;
! exp
i
5
: (1)
This leaves the action, S( ) = i
= D , invariant.
However, the path integral
i
R
D D
exp
= D
is not invariant under
such a change.
For this symmetry to hold true, D , must remain invariant under the
transformation, which is not necessarily true.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 3/23

4. What is an Anomaly?
For a system of left handed Weyl fermions the Lagrangian is given by
L = i
= DP+ , where,
0 = 1;
1 = i2;
5 =
0
1: (2)
The signature we choose is (1,-1). The gamma matrices satisfy,
f
;
g = 2g; = 0; 1: (3)
Corresponding to chiral fermions the transformation equations are,
L ! expfig L;
R ! expf+ig R: (4)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 4/23

5. Relating anomaly to hydrodynamics
The anomaly arises when the classical symmetry is no longer a symmetry
at the quantum level. Recent studies have revealed their applications in
evaluation of hydrodynamical transport coecients.
Kharzeev and warringa [arXiv:0907.5007v1 [hep-th]] have shown the zero
external momentum and zero frequency chiral magnetic conductivity of a
parity violating
uid ,at

6. rst order in derivative, to be constrained by the
anomaly in 3+1 dimensions.
We look at the behaviour in 1+1 dimensions.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 5/23

7. Fujikawa method
The Path integral formalism gives the generating functional as,
Z =
Z
D LD L exp(iS(A)) (5)
We have evaluated the U(1) singlet anomaly using the fujikawa method.
This is dierent from the non-abelian case where both fermions and
gauge potentials are chirally transformed.
For Fermionic Grassmann variables we have,
D LD
L = (det J)1(det J')1D LD
L; (6)
where det J and det J' are the respective jacobians of the transformations.
The derivative operator = D is not hermitian. we change x0 ! ix2 and
consequently, F0 ! iF2 and i
0 =
2 where
F = @A @A: (7)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 6/23

8. To de

9. ne the path integral precisely , we change the basis from x
representation to the n representation,i.e we expand in basis of
eigenstates of the = D. Note that n are discrete.
= Dn = nn;
y
n = D = y
nn: (8)
Keeping in mind that
5n has eigenvalues n we note that the
eigenstates of the dirac operator can be arranged as,
Ln
(x)
1 +
5 p
2
n(x)(n 0);
1 +
5
2
n(x)(n = 0);
Rn
(x)
1
5 p
2
n(x)(n 0);
1
5
2
n(x)(n = 0):
(9)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 7/23

10. These form a complete set of orthonormal eigenvectors.
Z
d2xLn
(x)
y
Ln
(x) = 2(x y);
Z
d2xRn
y
Rn
(x)
(x) = 2(x y);
Z
d2xLn
y
Rn
(x)
(x) = 0:
(10)
we expand the

11. elds in a basis of eigenstates of the = D operator,
L(x)
X
n0
anLn
(x);
L(x)
X
n0
Rn
y bn: (11)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 8/23

12. The path integral measure becomes ndanmdb
m.
We apply the transformation (4) to get
0
L(x) =
X
k0
a0
kLk
(x)
a0
k =
X
(kn ckn)an;
n0
where,ckn = i
Z
d2xLk
y
Ln
(12)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 9/23

13. Similarly for 0
R(x) we have,
det J' = det(1 + c0);
where,
c0
kn = i
Z
d2xRk
y
Rn
: (13)
The jacobian of the transformation becomes,
i
det J1det J'1 = exp
Z
d2x
X
alln
k
!
:
y(x)
5k(x)
(14)
We need a proper gauge-invariant regularisation procedure to compute
this apparently in

14. nite sum. This is done by cutting the sum o at large
eigenvalues by introducing a gauge invariant cuto f( = D2
M2 ).
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 10/23

15. We also use the fourier transform of the basis eigenstates to get the
jacobian as
det J1det J'1 = lim
M!1
lim
x!y
Tr
Z
d2l
(2)2 exp(ilx + ily)
5 exp
D2
M2
ie
F
2M2 +
l2
M2
2il:D
M2
:
(15)
computing the trace we get,
@J =
e2F
4
: (16)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 11/23

16. Hydrodynamical equations
From the modern perspective, hydrodynamics is best thought of as an
eective theory, describing the dynamics at large distances and
time-scales. It is normally formulated in the language of equations of
motion instead of an action principle.
We are interested in parity violating 1+1 d hydrodynamics
T = ( + P) uu + Pg;
j = nu + vu; (17)
where u is the only possible gauge invariant parity violating term
that can be added. Note that v is the parity odd transport coecient
we are looking for.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 12/23

17. Kubo formula from hydrodynamics
Following linear response theory, we perturb the system with
uctuations,
g = + h with htx nonzero,u = (1; vx) such that
uu = 1 + O(2) and gauge

18. eld perturbations,A = (0; ax).
We apply these perturbations, obtain the value of vx from expression of
stress-tensor T and we get the relevant Kubo formula as,
lim
p0!0;p!0
hjt(p)Ttx(p)i = v: (18)
We evaluate this correlator from

19. nite temperature

20. eld theory.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 13/23

21. Evaluating the correlator
From the theory of massless fermions we have the stress-tensor T01 as,
T01(x) =
1
2
i (x)
p
g(
0@1 +
1@0) (x): (19)
The conventions used are
g = (1;1) ;
s (q) =
1
i
0 (wm ie ie5
5)
1q1
;
0 = 1;
1 = i2;
5 =
0
1: (20)
The propagator in momentum space is explicitly written as,
s(q) =
0 1
i!m+i!n+e(5)+q
1
i!m+i!n+e(+5)q 0
!
: (21)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 14/23

22. We break up the hj0(p)T0x(p0)i into two parts, hj0(p)T0x(p0)i11 and
+hj0(p)T0x(p0)i22.We have used the subscript 11 to denote contribution
from right-handed fermions and 22 to denote the left-handed
contribution.
hj0(p)T0x(p0)i11 = !m
1

23. Z 1
1
dp1
(2)
e
2
s11(p1 + p)
(p1 + i!m + 5)s11(p1)(2)22(p + p0);
(22)
hj0(p)T0x(p0)i22 = !m
e

24. Z 1
1
dp1
(2)
1
2
s22(p1 + p)
(p1 i!m 5)s22(p1)(2)22(p + p0):
(23)
Where sum is over the Fermionic Matsubara frequancy !m.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 15/23

25. Writing out explicitly in terms of propagators we

26. nd,
hj0T01i11 = !m
e

27. Z 1
0
dp1
(2)
1
2
1
(i!m + i!n + e( 5) + p1 + p)
(p1 + i!m1)
(i!m + e( 5) + p1)
+!m
e

28. Z 1
0
dp1
(2)
1
2
1
(i!m + i!n + e( 5) p1 + p)
(p1 + i!m1)
(i!m + e( 5) p1)
;
(24)
where, !m1 = !m + e( 5):
Similar expression can be written down for hj0T01i11.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 16/23

29. Matsubara Sums
We need to perform (Fermionic) Matsubara sums such as
1

30. X
!m
1
(i!m tq)
1
(i!m + i!n u(p + q))
1

31. X
i!m
i!m
(i!m tq)
1
(!m + i!n u(p + q))
:
Where, !m = (2n + 1)T and t; u = 1
(25)
We do this by noting that the sum is basically over the imaginary axis,
and introducing a function which has a pole at the values of Matsubara
frequencies, we can modify this sum to a complex integral.
Sum 1 =
1
2T
I
c
z
2T
dz tanh
(z tq)(iz + i!m u(p + q))
;
(26)
where, the contour straddles the imaginary axis in anti-clockwise
direction.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 17/23

32. We get the results as,
hj0T01i11 =
Z 1
0
dp1
2
e
(i!n + p)
[
(p + i!n 2p1)
2
n(p1 p e + e5)
+
(2p1 p i!n)
2
n(p1 + p + e e5)
+p1n(p1 e + e5) + p1n(p1 + e e5)];
(27)
hj0T01i22 =
Z 1
0
dp1
2
e
(i!n p)
[
(p i!n + 2p1)
2
n(p1 + p e e5)
+
(2p1 p + i!n)
2
n(p1 p + e + e5)
p1n(p1 e e5) p1n(p1 + e + e5)]:
(28)
Where n(q) = 1
(exp

33. (q)+1) = Fermi Dirac distribution.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 18/23

34. We perform analytic continuation i!n ! p0 + i and take the limit
p0 ! 0; p ! 0 to obtain,
hj0T01i11 =
Z 1
0
dp1
2
[
e
2
n(p1 e + e5)
e
2
n(p1 + e e5)]: (29)
We now use the following identities :-
Fs(x) = Lis+1(ex);
Li1(x) = log(1 x):
(30)
Where,
Fs(x) =
1
(s + 1)
Z 1
0
ts
(exp(t x) + 1)
dt = Fermi-Dirac integral;
Lis(x) = k=1
k=1
xk
ks = The polylogarithm function:
(31)
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 19/23

35. Applying these identities, we get,
hj0T01i11 =
(e2 e25)
4
; (32)
and
hj0T01i22 =
e2 e25
4
: (33)
Thus the contribution from left-handed weyl fermions is e2
4 ( + 5)
=
e2
4
( + 5): (34)
Note that the coecient is exactly same as the U(1) anomaly coecient
with a negative sign.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 20/23

36. Discussions
We have done our calculations in 1+1 dimensional

37. nite temperature

38. eld theory. We are interested in extending this exercise to 6 dimensions.
The relation given by equation (34) matches the results found by R.
Loganayagam and P. Surokawa.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 21/23

39. References
Landsteiner, Karl and Megias, Eugenio and Pena-Benitez,Anomalous
Transport from Kubo Formulae,Lect.Notes Phys.,vol
871(2013),[arXiv:1207.5808[hep-th]].
Loganayagam, R. and Surowka, Piotr,Anomaly/Transport in an Ideal
Weyl gas,JHEP,vol 1204 (2012),[arXiv:1201.2812[hep-th]].
Kharzeev, Dmitri E. and Warringa, Harmen J,Chiral Magnetic
conductivity,Phys.Rev,vol D80 (2009),[arXiv:0907.5007].
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 22/23

40. Thank you.
Subham Dutta Chowdhury Anomaly and parity odd transport coecients in 1+1 d 23/23