Universal terms in the Entanglement Entropy of Scale
Invariant 2+1 Dimensional Quantum Field Theories
Eduardo Fradkin
Department of Physics and Institute for Condensed Matter
Theory, University of Illinois at Urbana-Champaign
Talk at the Workshop “Quantum Information in String Theory and Many-body Systems”,
at the Yukawa Institute for Theoretical Physics (YITP), Kyoto University, June 20-24, 2016
Collaborators: Xiao Chen, Tianci Zhou, Gil Young Cho,Tom Faulkner

1Entanglement2

Density Matrix and Entanglement Entropy

ϕB•Pure state in A ∪ B: Ψ[𝜑A,𝜑B]•Density Matrix

BAϕAℓAh'A , 'B |⇢A[B |'0A , '0B i =['A , 'B ]⇤['0A , '0B ]Observing only A: mixed state with areduced density matrix ρAh'A |⇢A[B |'0A i = trB ⇢A[BℓBSvN =von Neumann Entanglement Entropy:
3tr (⇢A log ⇢A )

Scaling of Entanglement•

We will consider a partition into two simply-connected regions A and B, suchthat 𝑙B≫𝑙A≫𝜉≫a; 𝜉 is the correlation length (if finite) and a is the UV cutoff•In a massive phase in d space dimensions ξ is finite and the entanglement
entropy obeys the Area Law✓ ◆d 1`ASvN = ↵+ ...
aThe prefactor of the Area Law is not universal as it depends on theshort-distance cutoff a• In a 1+1 dimensional CFT this law is replaced by the universal form•

SvN = 3c ln•`
a+ ...

In a 2d topological phase it has the scaling SvN=𝛼 (𝑙/a)- 𝛾+… (Kitaev andPreskill, Levin and Wen); 𝛾= ln 𝒟 (𝒟: effective quantum dimension).

4

Scaling of Entanglement in General Dimension•For general odd space dimension d the entanglement entropy has anexpansion of the form SvN= 𝛼 (𝑙/a0)d-1+𝛼’ (𝑙/a0)d-3+…+s ln (𝑙/a0)+…

•This expression is verified both in free field theories (Casini and Huerta,2008) and in the AdS/CFT correspondence using the Ryu-Takayanagi
ansatz (2006)•The first term is the Area Law. The coefficients 𝛼, 𝛼’, etc. are non-universal•

The coefficient s of the logarithmic term is universal•

In 3+1 dimensional CFTs, the coefficient s is expressed in terms ofintegrals of the entangling surface and of the central charges a and c5

Scaling of Entanglement in 2+1 Dimensions•Much less is known about the universal terms in the scaling ofentanglement in odd space-time dimensions at quantum criticality•

For simply-connected regions with the shape of a disk in d=2 dimensions
the entanglement entropy has the form of the F theorem: SvN=𝛼 (𝑙/a)-F+…,
where the constant term F is finite and universal (Klebanov, Pufu, Safdi(2011); Casini, Huerta and Myers (2011))•

Here we will discuss the current evidence for universal finite terms inseveral d=2 quantum critical systems of interest•

We will focus on the case in which the d=2 surface is a 2-torus and theobserved regions are cylindrical sections of the torus.

6

Quantum Dimer Models•

Quantum dimer models are simple models of 2d frustrated (or large N)
antiferromagnets (Kivelson and Rokhsar, 1988)•The space of states |C ⟩ are the dimer coverings of the lattice, each dimer
being regarded as a spin singlet (valence bond) of a pair of nearby spins•

The ground states have the RVB form

|0i =X{C}eE[C]|CiFor small enough 𝛽, on bipartite lattices (square,honeycomb, etc) these states represent quantum
critical points• On non-bipartite lattices (also for small enough 𝛽)they represent ℤ2 topological states (deconfined ℤ2gauge theories or Kitaev’s Toric Code states)• For large 𝛽 they represent ordered (VBS) states•

7The Quantum Lifshitz Model•

The Quantum Lifshitz Model is the effective continuum theory of the critical
Quantum Dimer Model (Ardonne, Fendley, Fradkin, 2004)•Free compactified boson field theory (𝜙 ~ 𝜙+ 2𝜋R) representing the coarsegrained configurations of dimers on a bipartite 2D lattice•

It has dynamic critical exponent z=2•

It describes a quantum critical point between an uniform phase and a spatiallymodulated phase•

In the uniform phase it becomes a relativistic compactified scalar field, a NGmode.

•Hamiltonian:

⇧2
2
H=+r2
222,[ (x), ⇧(y)] = i (x
20 [ (x)] = const ⇥ eZ

•Ground State Wave Functional:•The ground state wave functional is conformally invariant8y)
2d x (r (x))
2Entanglement Scaling in the QLM•

On simply connected region A with a smooth boundary the entanglement
entropy obeys the scaling SvN=𝛼 (l/a)+𝛾+…, where 𝛾 is universal (Fradkinand Moore, 2006)•

If the boundary has corners (cusps), or ends at the edge of the system,
SvN has universal logarithmic terms (Zaletel, Bardarson and Moore, 2011)•The Rényi entanglement entropy Sn is computed in terms of ratios of 2Dclassical partition functions.

•Partition function on an a “book” with 2n sheets.
Znntr⇢A = nZ

• Zn: partition function of n copies glued at theboundary or the observed region A.
• Z: partition function (norm of the wave function)for one copy.

SvN is expressed in terms of 2D classical free energies (two with Dirichlet BCs at
𝜕A and 𝜕B, and one free) and of the contribution of the winding modes, W(n)SvN [A] = FD [A] + FD [B]

F [A [ B]
9limn!1 @n W (n)Entanglement on a Torus
LxBBAB

LyALALA•

2 torus: entanglement entropy of a cylindrical region has a universal finite term
(Hsu at al; Hsu and Fradkin; Oshikawa; Stéphan, et al).

•Finite term due to the winding modes of the boson: long range entanglement.
LLπ yThin “slice” limit: SvN = 2α ay − 24LAp
LyThin torus limit: SvN = 2↵ a + 2 ln 8⇡R1012

EE in other geometries: Disk and Sphere• If A is a disk of radius r in the plane with Dirichlet BCs (and for aspherical cap of angle 2𝜃)!√" 1γ = ln 8πκR − 2• This result should be compared with the -ln (rg2) dependence foundfor a compactified relativistic scalar field (a Goldstone boson withdimensionful coupling constant g) by Agon et al.
• Annular region of radii r1<r2 in the infinite plane (and for two sphericalcaps after stereographic projection)! "$#√
1
Thick annulus : γ =2 ln
8πκR −2&%# 1"√2πr1πThin annulus : γ = −8πκR −+ ln
24 r2 − r12
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Mutual Information•

Mutual information of region A, a disk of radius r1, with B, thecomplement of a disk of radius r2>r1, in the infinite plane

I(A, B) = S(A) + S(B) − S(C) = 2!

r1r2"1/4πκR2+!

r1r2"2• Mutual information of two disks of radii RA and RB separated bya large distance r (here ζ= RA RB/r2)
I(A, B) = S(A) + S(B) − S(A ∪ B) = 2ζ 2∆1 + ζ 2∆21
∆1 =and ∆2 = 1
28πκR

scaling dimensions of the vertex operator
exp(i𝜙) and 𝜕x𝜙 of the QLM

12Torus with general aspect ratio
2 torus with aspect ratio Ly/Lx and a cylindrical region A with aspect ratiou= LA/Lx, the entanglement entropy is SvN=2𝛼 (Ly/a)+𝛽 J(u) (Stéphan et al)J(u) = log✓⌘(⌧ )✓3 ( u⌧ )✓3 ( (1 u)⌧ )2 ✓3 ( ⌧ )✓3 (⌧ / ) ⌘(2u⌧ )⌘(2(1 u)⌧ )2◆For the quantum dimer model at the RK point, 𝜆=213

𝜃3(z): Jacobi theta-function
𝜂(z): Dedekind eta-function
Ly⌧ =iLxScaling of Entanglement in the 2+1 dimensional
Ising Model in a Transverse Field•

Inglis and Melko (2013) studied numerically the second Rényi entropy S2
at the quantum critical point of the 2D TFIM on a 2 torus•This system is in the universality class of the 𝜙4 Wilson-Fisher fixed pointin 2+1 dimensions•Two different scaling functions:•S2=2𝛼 Ly+ 𝛽 log(sin (𝜋Lx/L))•S2=2𝛼 Ly+ 𝛽 J(u)+ const

The QLM scaling works surprisingly welleven though is not relativistic!

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Holographic Entanglement on the 2 torus
physical space
Chen, Cho, Faulkner and Fradkin (2015) used the RyuTakayanagi ansatz to obtain the finite term in the von Neumannminimal area
entanglement entropy for a cylindrical section of a 2 torusℓA
SvN =4GN

𝒜: area of the minimal AdS surface homologous
to the observed region where the QFT lives on
the boundary of AdS
Toroidal geometries: minimal surface falls on the“AdS4 soliton” geometry (Horowitz and Myers)•The EE is sensitive to which cycle of the torus contracts since thecut is along y•

In the thin torus limit, Ly⟪ Lx, the EE saturates•

This is natural for fermions with anti-periodic boundary conditions•

In the opposite, thin slice, limit we find universal scaling•

The von Neumann EE has the scaling form SvN=𝛼 Ly+ 𝛽 j(u), wherej(u) has two different forms depending on whether the aspect ratio of
the torus is Ly/Lx>1 or Ly/Lx<115AdS spaceParametric form of j(u) for Ly/Lx>1u( ) =31/3(12⇡P ( , ⇣) = 1⇣31/2 Z 1)0(1d⇣⇣ 21p(1⇣ 3 ) P ( , ⇣))⇣ 4

Parametric form for j(u) for Ly/Lx<1j( ) =1/3

Z 10Lx
3u=Ly
2⇡

1/3

(1!
!
p3d⇣1⇣p
1
1⇣2P ( , ⇣)Z 1d⇣⇣ 211/2pp)
1⇣ 3 P ( , ⇣)016

Tests of scaling in 2d free field fermionic models•

We computed the entanglement entropy for cylindrical regions of a 2 torus
for free massless Dirac fermions in 2+1 dimensions and also for freemassless “Lifshitz” (QBT) fermions, in both cases with anti-periodic
boundary conditions•

Dirac fermions in 2+1 dimensions are a stable phase and hence an UV
fixed point•“Lifshitz” fermions in 2+1 dimensions have z=2 and are are at themarginal dimension (asymptotically free)•We used a lattice version of these models and used standard results forthe reduced density matrix and the entanglement entropy (Peschel, 2001)•In the Dirac case we accounted for fermionic doublers

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Scaling of entanglement for free massless Dirac fermions in2+1 dimensions

numerical data
α L+β J(u) (λ=4.2)
α L+β log(sin(πu)
AdS results

SvN

365

Deviation

The Dirac model

360
0.0J(u)AdS

0.2

0.00.0u0.50.5

u

18

1.0

1.0

Conclusions•In 2+1 dimensions we expect to find universal finite corrections to theentanglement entropy•

These finite terms are determined by the operator content (compactification
radius, scaling dimensions, etc) and have topological origin•

We presented results for von Neumann EE of the QLM on different geometries
and for the mutual information•

On cylindrical regions the universal finite terms are given in terms of scaling
functions of the aspect ratios of the cylinders and of the 2 torus•

We presented expressions for the entanglement entropy derived fromholography on toroidal geometries which fit remarkably well even free massless
Dirac fermions (on anti-periodic boundary conditions)•The finite term derived by Stéphan et al for the QLM fits remarkably well even for
Dirac fermions which is a relativistic theory (although the holographic results isbetter)•How well does the holographic result fit the 2D Ising model in a transverse field?

Scaling of entanglement for free massless“Lifshitz” fermions in 2+1 dimensions