fermionic entanglement entropy - lmu
TRANSCRIPT
Ringberg, November 22nd 2016
Fermionic Entanglement EntropyRobert C. Helling (LMU) with Hajo Leschke (FAU) and Wolfgang Spitzer (FUH)
Entanglement Entropy
✤ Similarity to Black Hole Entropy: Area Law
✤ Quantum Information
✤ Quality of Numerics (Density Matrix Renormalization Group)
✤ Ryu-Takanayagi Holographic Computation
✤ Direct Computation possible
Definition
✤ Take a QFT with (quasi-local) operators
✤ Take a state (think: ground state of local Hamiltonian
✤ Restrict to operators localized in a spatial region
✤ This is also a state on but in general it is mixed:
✤ This reduced state has entropy:
✤ Scaling upon blowing up by a factor ? Area law:
! : A ! C !(A) = h |A| i
A
!|A(⌦)(A) = tr(⇢A) with ⇢⌦ = trF(L2(R\⌦))| ih |
S⌦ = �tr(⇢⌦ log ⇢⌦)
⌦ R SR⌦ = O(Rn�1)
Free Fermions (non-relativistic)
✤ Wick’s theorem: Everything determined form 2-point function
✤ Reduce to 1-particle space, projector onto Fermi sea
hc†kcki = ��(k) = hk|P�|ki
P�(x,x0) =
1
(2⇡)n
Z
�ei(x�x
0)·k dnk
1 Particle Language
✤ Restrict to by projection with
✤ 1-particle effective density operator:
✤ Entanglement entropy becomes
⌦
⌦ Q⌦ = �⌦(x)
%⌦,� = Q⌦P�Q⌦
S⌦,� = tr(%⌦,� log %⌦,� � (1� %⌦,�) log(1� %⌦,�))
� tr(%⌦,�(1� %⌦,�)
Violation of Area Law
✤ I will show you how to compute In fact, there is equality (for a slightly different coefficient).
✤ We need Reyni-entropies for k=1 and k=2.
✤ k=1 is simple:
SR⌦,� � ln 2
⇡
2
✓R
2⇡
◆n�1
lnR
Z
@⌦⇥@�d�(x)d�(p) |n
x
· np
|+ o(Rn�1 lnR)
tr(%k⌦,�)
tr(%1R⌦,�) =
✓R
2⇡
◆n Z
⌦dx
Z
�dp1 =
✓R
2⇡
◆n
|⌦||�|
k=2
✤ This is more work:
✤
✤ Since
✤ First term yields which cancels k=1 term.
P�(v) ⇠1
v(n�1)/2
✓R
2⇡
◆n
|⌦||�|
|R⌦ \ (R⌦� v)| = Rn|⌦|+Rn�1
Z
@⌦d�(x)max(0,v · n
x
) +Rn�2O(|v|2)
tr(QR⌦P�QR⌦P�) =
Z
R⌦dx
Z
R⌦dx0 |P�(x� x
0)|2 =
Z
R(⌦�⌦)dv |PG(v)|2|R⌦ \ (R⌦� v)|
k=2 (cont.)
✤ Write and use Gauß’ theorem
✤ We still need to compute
max(0,v · nx
) = ✓(v · nx
)v · nx
(2⇡)nvP�(v) = v
Z
�dp eiv·p = �i
Z
@�d�(p)npe
iv·p
Z
R(⌦�⌦)dv ✓(v · n
x
)P�(�v)eiv·p
✤ Once more Gauß:
✤ Use coordinates with and the boundary
✤ Then and
✤ Using stationary phase we find
v = (0, 0, . . . , 0, V )
Gauß curvature
Z
R(⌦�⌦)dv ✓(v · n
x
)P�(�v)eiv·p
@� 3 p0 = (t, f(t))
d�(p0) =p
1 + |rf |2dt np0 = sgn(v · p0)(�rf, 1)/p
1 + |rf |2
P�(�v) = �(2⇡)n1
v
Zdt sgn(f(t))e�ivf(t)
= �i(2⇡v)�(n+1)/2X
ka
sgn(v · ka)p| det fij(ta)|
e
�iv·ka�i⇡4 sgn(fij(ta)) + o(v�(n+1)/2)
1
(2⇡)nP�(�v) =
iv
|v|2 ·Z
@�d�(p0)np0e�iv·p0
(cont.)
✤ Use coordinates in which p is vertical, and write and .
✤ Phase in dv-integration is
✤ u-integration by stationary phase cancels Gauß curvature and leaves
Z
R(⌦�⌦)dv ✓(v · n
x
)P�(�v)eiv·p
v = �(u, h(u))@(⌦� ⌦) 3 (u, h(u))
ka((0, h(0)) = p
v · (p� ka(v)) = �h(0)(p� ka(0))n + �f�1ij (ka(0))
2h(0)uiuj
Zd�
ei�h(0)(p�ka(0))n
�
✤ This integral is over but up to an O(1) error, we can change it to
✤ Collecting everything:
Zd�
ei�h(0)(p�ka(0))n
�
Z R
1d�
ei�h(0)(p�ka(0))n
�=
(lnR+O(1) for p� ka(0)n = 0
O(1) else
� 2 [0, R] � 2 [1, R]
tr(%R⌦,�(1� %R⌦,�)) =ln 2
⇡
2
✓R
2⇡
◆n�1
lnR
Z
@⌦⇥@�d�(x)d�(p) |n
x
· np
|+ o(Rn�1 lnR)
Z R
1d�
ei�h(0)(p�ka(0))n
�=
As Quantization
✤ Instead of scaling , we can also place R more democratically in the exponent (up to an overall factor).
✤ This shows that R actually plays the role of .
✤ In an informal, semi-classical expansion
✤ so for the discontinuous symbols we find a semi-classical term at
Z
@⌦⇥@�d�(x)d�(p) |n
x
· np
| =Z
@⌦⇥@�!⌦(n�1)
R⌦
eiRx·p
1/~
tr(Q⌦P�Q⌦P�) = tr(Q⌦Q⌦P�P�) + tr(Q⌦[P�, Q⌦]P�)
= tr(Q⌦P�) + tr(Q⌦~{P�, Q⌦}P�)
{P�, Q⌦} = r�� ·r�⌦ ⇠ �(p 2 @�)�(x 2 @⌦)
O(log ~)
log(R) term
✤ Double discontinuity in phase space is essential for area law violation.
✤ There is a simple extension when or are multiplied by smooth functions.
✤ At finite temperature, the entropy has a bulk term (as we no longer start from a pure state) plus a strict surface term that goes as η(T,∂Ω) = (1/12)J(∂Γμ,∂Ω) ln(T0/T) + … and becomes the area law violating term at zero temperature
✤ The explicit form suggests there should be a more direct derivation (as an anomaly?).
✤ Holographic derivation from Fermi surface?
�⌦(x) ��(p)