Entanglement in CFTs at Finite Chemical
Potential

John Cardy

University of California Berkeley
University of Oxford

YKIS 2016

Outline

Statement of problem.
Spin-1 currents in 1+1 dimensions.
Higher spin currents in 1+1 dimensions.
Higher dimensions.

CFT with a global symmetry.
Conserved spin-1 Noether currents {Jaν }.
RCharges Qa = Ja0 d d−1x commute with HCFT .
Finite chemical potentialρ(β, {µa }) ∝ e−βHCFT +β[β → ∞ gives ground state]

Pa µa Qa

Block entanglement: divide space into regions A and B andsimilarly the Hilbert space HA ⊗ HB .
ρA (β, {µa }) = TrHB ρ(β, {µa })Rényi entropies(n)

SA (β, {µa }) = (1 − n)−1 log TrHA ρA (β, {µa })n(n)Entanglement entropy SA (β, {µa }) = limn→1 SA (β, {µa }).
How does SA behave for µa 6= 0?

Other (un)related work

Holographic Charged Renyi Entropies, Belin, Hung, Maloney,
Matsuura, Myers, Sierens, arXiv:1310.4180: consideredchemical potential coupling to charge in region A, gives Wilsonline around entangling surface.
Charged Renyi entropies and holographic superconductors,Belin, Hung, Maloney, Matsuura, arXiv:1407.5630
Higher Spin Entanglement and WN Conformal Blocks, de Boer,Castro, Hijano, Jottar, Kraus, arXiv:1412.7520
A new spin on entanglement entropy, Hijano, Kraus,
arXiv:1406.1804

Expectations(1) µ has dimensions of mass so is a relevant perturbation.
Based on entropic c-theorems, SA should decrease.
RBut adding µ J 0 d d−1x to the action breaks Lorentz invariance.

(2) When β d−1  Vol(A)  Vol(B), SA ∼ SGibbs .
Grand partition function Ξ = Tr e−βH+βµQTo O(µ2 )δSGibbs = µ2 (β∂β − 1) β 2 hQ 2 i

For a CFT,hQ 2 i ∝Ld−1
β d−1

δSGibbs ∝ −(d − 2)µ2 β 2 (L/β)d−1 < 0,except in d = 2, where it vanishes.

Main result

For a 1+1-dimensional CFT(n)SA (β, {µa }) is independent of the {µa }(and therefore of hQa i)
Argument is in two parts:
show this holds for a free scalar fieldargue that it is also true more generally

ReplicasAA

TrHA ρnA can be computed by taking n replicas of euclidean
space-time and sewing them together cyclically alongA ∩ {τ = 0}
ZnTr ρnA = n
Z1Note that we should regularize the behavior near the conicalsingularities, e.g. by removing a small disc around them: thisnecessitates choosing a boundary condition, which we take to beconformal.

Free field theory in 1+1 dimensions
Free scalar field, euclidean action
ZZ β
1(∂x φ)2 + (∂τ φ)2 dxdτACFT = 20[We can think of this as describing the phase fluctuations of acomplex scalar Φ = |Φ|eiφ , or coming from bosonization.]

Symmetry φ → φ+const., current J µ = ∂ µ φZ
H = HCFT − µ π(x)dx
ZZ βZZ β221(∂x φ) +(∂τ φ+iµ) dxdτ = ACFT +iµJ0 dxdτ −βLµ2
A= 200φ(x, β) = φ(x, 0) so the second term integrates to zero, apart fromboundary terms.
δΩ = −Lµ2 , so δSthermo = 0.

Replicated action
A = 12ZZ βX
n−10(∂x φj )2 + (∂τ φj )2 + iµ∂τ φj dxdτ − nβLµ2j=0φj (x, 0−) = φj+1(mod n) (x, 0+) for x ∈ A,φj (x, 0−) = φj (x, 0+) for x ∈ B.
Define linear combinations φ̃k = √1nPn−1j=0 e2πikj/n φ .
jThese fields decouple in the quadratic part of the action.
They satisfy φ̃k (x, 0−) = e2πik /n φ̃k (x, 0+) across A and arecontinuous across B.
They obey the same (homogeneous) boundary conditions near
the conical singularities.

√The chemical potential µ couples only tonφ̃0 .

e0 (√nµ) Qn−1 Zee0 (√nµ) Qn−1 ZeZZZn (µ)
k =1 k (µ)
k =1 k (0)==Z1 (µ)nZ1 (µ)nZ1 (µ)n√e0 (√nµ) Zn (0)ZZn (0) Z1 ( nµ)/Z1 (0)==e0 (0)Z1 (0)n Z1 (µ)/Z1 (0) nZ1 (µ)n Z
2But Z1 (µ)/Z1 (0) = e−f (β)µ because the path integral isgaussian.
√22So the 2nd factor is e−f (β)( nµ) / e−f (β)µ

n= 1. QED.

Extension to general 1+1-dimensional CFTs
Rotate in internal space so thatννa µa Ja = µJ .

PIn 2d CFT J(z) = 12 (J 1 − iJ 0 ),J(z̄) = 21 (J 1 + iJ 0 )are respectively holomorphic and antiholomorphic.
ZA = ACFT + µ(J + J)d 2 x + µ2ZOdxdτ + · · ·But O must have dimension zero and therefore be a constant ina unitary CFT.
Zn (µ)/Z1 (µ)n can be expanded as a sum of integrals ofcorrelators of J and J on the n-fold cover.

The OPEs
J(z) · J(z 0 ) =κ+ O(1) ,(z − z 0 )2J(z̄) · J(z̄ 0 ) =κ+ O(1)(z̄ − z̄ 0 )2

completely fix the form of the 2N-point correlators in the planehJ(z1 )J(z2 )J(z3 )J(z4 ) · · · i
to be the same as for the U(1) current of a free scalar field, up
to the constant κ.
This extends to any conifold of genus zero, which includes
those used to compute the Rényi entropies in simple cases.
Zn (µ)/Z1 (µ)n is therefore independent of µ for any CFT, as longas suitable boundary conditions are chosen at the conicalsingularities. QED.

Perturbative calculation
Consider the case where A = (− 2` , 2` ) with `  β.
In that case the dimensionless expansion parameter is µ`.
The coupling between the replicas along A may be viewed as
the insertion of twist operators at the ends of A.

TnTnA

These behave like local operators of dimension (c/24)(n−1/n).

Short interval expansion [Headrick; Calabrese, JC, Tonni]
n−1XY
1T n (− 2` ) · Tn ( 2` ) = c (n−1/n) 1 +
Φkj (0)
C{kj }` 12
j=0{k }jΦkj is a local operator in the jth replica.
The C{kj } mayQ be evaluated by taking the correlation function of bothsides with j Φkj (∞).
P

C{kj } ∝ `j ∆kjso this gives a series of increasing powers of `.

We can use this to compute Tr ρnA in any stateTr ρnA = hT n (− 2` )Tn ( 2` )i =1`c(n−1/n)12

1+X{kj }C{kj }

n−1YhΦkj (0)ij=0where hΦkj (0)i is evaluated in the decoupled jth replica.

In our case the interesting terms in the OPE areXX
1 + `2
cj (Tj + T j ) + `2bjk (Jj Jk + J j J k ) + · · ·jj6=kWe find
cj =n2 − 1,12nbjk ∝1) 2[2 − 2 cos( 2π(j−k)]nhTj i = hT j i has two pieces:– πc/12β 2 which gives the usual O((`/β 2 )) term when µ = 0;
– a term due to the O(µ2 ) change in the energy density
On the other hand hJi2 = O(µ2 ).
P

j6=k bjk ∝ (nall n !!

2 − 1), and the last two pieces cancel exactly forIn retrospect this cancellation had to happen, otherwise wewould have found a super-volume termδSA ∼ µ2 `2at β = ∞.
This argument also implies the cancellation holds for cases
when A is several intervals of lengths ∝ `.
One way this might fail is if there is a phase transition at µ = 0(but there are no IR divergences in the perturbation expansion).

Higher spin currents in 2d

Many 2d CFTs possess holomorphic and antiholomorphic
currents (W , W ) of integer spins ±s withR s > 2, andcorresponding conserved charges Q = (W + W )dx.
If we add a coupling µQ to the hamiltonian how does theentanglement change?

In this case the important terms in the OPE areXX
1 + `2
cj (Tj + T j ) + `2sbjk (Wj Wk + W j W k ) + · · ·jj6=kwhere now
cj =n2 − 1,
12nbjk ∝1) 2s[2 − 2 cos( 2π(j−k)]n

The change in hTj i and hT j i is now O(µ2 β 2−2s ).
On the other hand hW i2 = O(µ2 β 4−4s ).
So the 2 terms do not cancel, and δSA ∝ µ2 `2 /β 2s−2 .
[AlsoPjk bjk is a polynomial in n of degree 2s 6= 2.]

Higher dimensions

Free massless bosons Bose condense when µ 6= 0.
Free massless fermions form a Fermi surface and there is an
extra log R term in the area law.
If we add repulsive interaction λ|Φ|4 to the bosons and go into
the condensed phase where Φ ∼ |Φ|eiφ , we get a similar result
as in 1+1 dimensions, but this is no longer typical.
If A is a region of size `  β there is an analog of the smallinterval expansion:

(n)If ΣA denotes the operator which sews together the replicas
on the entangling surface,XX(n)cj Tjτ τ + `2(d−1)
bjk Jj0 Jk0 + · · · Σ = e−Area(∂A) 1 + `dAjjk

The coefficients are in principle computable for a sphere.
hT τ τ i contains a thermal term σ/β d , and a correction due to thechemical potential ∼ µ2 /β d−2 . On the other hand hJi ∼ µ/β d−2so the second term is higher order in `/β.
So the leading correction for `/β  1 is entirely given by theequation of state (also for `  β). In this case one can showthat δSA < 0.
Direct perturbation theory theory also suggests a universalterm ∼ µ2 `2 log ` for d = 4.

Other approaches– Entanglement hamiltonian (work in progress with M. Smolkin):in principle this allows computation of SA in any state, but there
is a subtlety since the perturbation breaks Lorentz invariance.

Summary
Adding a chemical potential to a 1+1-dimensional CFT has noeffect on the entanglement.
For d > 2 there are no universal perturbative corrections atzero temperature (besides a possible log correction to area lawin 4d), and the leading corrections in for `  β are determinedby the equation of state (as they are for `  β).
Exceptions to this can occur if turning on µ totally changes thephysics.
Similiar ideas apply to entanglement in non-equilibrium steadystates where Jx 6= 0.