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)