Entanglement & dynamics in 2+1D CFTs

William Witczak-KrempaHarvard University

AdS/CFT workshop at CRM, Montréal, 19/10

S. Sachdev@Harvard / PI

E. Sorensen@McMaster

E. Katz@Boston U.

R.C. Myers@PI

P. Bueno@Leuven

Punch lines❖ Super-universality of entanglement entropy from corners

❖ Non-perturbative results for dynamics near quantum critical phase transitions (CFTs)

❖ Asymptotics and sum rules of observables (conductivity)

❖ Concrete statements for the superfluid insulatorQCP in 2+1D [incl. Monte Carlo]

Plan

❖ Brief intro to quantum criticality

❖ Entanglement entropy (focus: corners)

❖ Quantum dynamics at finite T

• quasi-particles: excitations with ∞ lifetime at T=0

• systems without qp’s:

‣ quantum critical phase transition (incl. metals)

‣ some gapless spin liquids (gauge theory), etc

• focus on Conformal Field Theories (2+1D)

Quantum critical fluids

❖ O(2) universality class: XY spins, arrays of Josephson junctions, ultra-cold atoms, etc

t

U

Superfluid-insulator transitionH = �t

X

hi,ji

b†i bj + UX

i

ni(ni � 1)

t/UQCP

insulator superfluid

order param. ρs

QCP t/U

T

superfluidinsulator

QCfluid

T

strongly coupled fixed point in 2+1D (Wilson-Fisher CFT)

qp

SSB of O(2) order parameter

φ1φ2

Cheaper down there!

L = (@⌧ ~�)2 + c2s(r~�)2 +m2�2 + u�4

[Bloch, Nature 05]

superfluid

insulator

[Endres et al., Nature 12]

87Rb

Cold atoms

EntanglementT = 0

P. Bueno, R. Myers, me, PRL 15

P. Bueno, R. Myers, me, JHEP 15

Counting non-quasiparticles

❖ Entanglement entropy➝ how much entanglement between inside & outside?

❖ RG monotone for CFTs (analog of c of 1+1D CFTs):[Myers et al ; Casini, Huerta]

S = �Tr (⇢V ln ⇢V )V

`

S = B`

�� F +O(�/`)

“World isn’t a smooth place”

❖ hard to compute F in many-body simulations

❖ lattice: can’t avoid corners!

❖ their contribution to entanglement entropy?

V

Corner entanglement❖ V has corner, angle θ

❖ EE has universal log:

❖ a(θ) good measure of dof-s:

• free fields: a(θ) = N ascalar (θ) [Casini, Huerta]

• large-N super-conformal gauge th.: a(θ) ∝ N3/2

[Hirata, Takayanagi]

❖ numerics for O(N) Wilson-Fisher QCPs [Melko et al, Helmes et al ; Devakul et al ; etc]

ℓ

θV

CFT3

zAdS4 γ

θ

V

a) b)

S = B`

�� a(✓) ln(`/�) + const

Smooth limit❖ smooth limit

❖ related to energy density ε:

❖ smooth limit entanglement → local correlator

a(✓ ! ⇡) = � (✓ � ⇡)2

� =⇡2

24CT

θ

h"(~x)"(0)i / CT

|~x|6

[Bueno, Myers, WK, PRL 15]

Evidence for conjecture❖ holds for free theories➞ numerical check [proof: Elvang, Hadjiantonis]

❖ AdS/CFT:holds in strongly coupled holographic models (incl. higher curvature terms)[Bueno, Myers, WK; proof: Bueno, Myers; Miao]

� =⇡2

24CT

�scalar =1

256= �Dirac/2

ℓ

θV

CFT3

zAdS4 γ

θ

V

a) b)

Beyond smooth limit❖ Nearly super-universal behavior for different CFTs

θ

π/4 π/2 3π/4 π

ÛÛÊÊıı

‡

‡

‡

‡

‡

‡02468

101214

aH�LêC T

‡

‡

‡

‡‡

‡

ÛÛ

ÊÊ

ıı

ÊÊ

ÊÊ

��

0 � ê4 � ê2 3� ê4 �1

1.021.041.061.081.1

1.12

@aH�LêC TDê@

aH�LêC TD

holo

AdS/CFT Free scalar Free fermion Ising (N=1) XY (N=2) Heisenberg (N=3) Extensive

ÊÊ

ııÛ

‡

‡

θ

Thermodynamics from groundstate

❖ Rényi entropy

❖ n-dependence of smooth-limit coefficient: σn

❖ Extracting the thermal entropy:

❖ “Groundstate knows a lot”

Sn =

1

1� nlog Tr ⇢nV

�n!0 =1

12⇡3

csn2

s = cs T2

[Bueno, Myers, WK, JHEP 15]

CFT dynamicsat finite T

Universal dynamical conductivity

Universal scaling function:SuperfluidInsulator

Quantumcritical

CFTtêU

T

�(!) = �⇣!T

⌘?

[early work: M.P. Fisher et al., Damle, Sachdev]

�(!) =1

i!hJ

x

(!)Jx

(�!)i

Short & long times

❖ Characteristic time scale: 1 / T

❖ Short times ω ≫ T : probing near vacuum

❖ Long times ω ≪ T : excitations interact strongly with thermal background

Short times viaOperator Product Expansion

❖ Scalar (primary) operator O(x), scaling dim Δ: ⟨O(x) O(0)⟩ = 1 / x2Δ

❖ OPE:

[Wilson ; Polyakov; Ferrara et al ; etc]

Current OPE

❖ Get OPE coefficients from ⟨JJO⟩T=0

O(2) CFT:relevant scalar

O ~ φ2

stress tensor

Jµ(x)J⌫(0) =Iµ⌫1

x

4+ CJJO Iµ⌫ O(0)

x

4��+ CJJT Tµ⌫(0)

|x| + · · ·

hOiT = BT�thermal average

Asymptotic conductivity

❖ fourier & take expectation value of OPE:

stress tensor

�(i!n)!n�T= �1 + bO

✓T

!n

◆�

+ bT

✓T

!n

◆3

+ · · ·

dynamics reveal critical exponents

[Katz, Sachdev, Sorensen, WK]

� = 3� 1/⌫

⇡ 1.51

Ask the computer

Quantum Monte Carlo

❖ Large-scale simulations of O(2) QCP

❖ loop-current model (Villain)

❖ quantum rotors

❖ Finite-T but imaginary time...

❖ Analytic continuation difficult!

[WK, Sorensen, Sachdev, Nat. Phys. 2014]

iωn

ω + i0+

0 5 10 15 20ωn/(2πT)

0.35

0.36

0.37

0.38

0.39

0.40

σ(iω

n)/σQ

Quantum Rotor ModelVillain Model

[also Chen et al]

[WK, Sorensen, Sachdev, Nat. Phys. 2014]

Universal quantum critical dynamics

Quantum Monte Carlo II[WK, Sorensen, Sachdev, Katz]

0 2 4 6 8 10 12 14 16 18 200.36

0.37

0.38

0.39

0.40 Fit: 0.36038+0.053/n1.516-0.01/n3

QMC Villain Model

2⇡�(i!n)/�Q

n = !n/(2⇡T )

Fit : 0.36038 + 0.053/n1.516 � 0.01/n3

QMC Villain Model

Δ = 1.516

�(i!n)!n�T= �1 + bO

✓T

!n

◆�

+ bT

✓T

!n

◆3

+ · · ·

�1 = 0.36

❖ determined after 25 years

❖ Conformal bootstrap : σ∞ = 0.3554(8)[Kos et al 2015]

�1 = 0.36

[also Chen et al; Gazit et al]

Exact conductivity at QCP with emergent SUSY

Superconducting instability of Dirac fermion ➝ chiral Wess-Zumino SCFT !=2

[in prep. with J. Maciejko]

�1 =5(16⇡ � 9

p3)

243⇡

Sum rules

Building sum rules❖ 2 ingredients:

1. Kramers-Kronig (retarded)

2. Asymptotics via OPE

Im ω

Re ω

Sum rule w/out quasi-particles

❖ general proof via OPE

❖ check:

Z 1

0d! [Re�(!)� �1] = 0

[WK, Sachdev ; Gulotta et al ; Katz, Sachdev, Sorensen, WK]

�(i!n)!n�T= �1 + bO

✓T

!n

◆�

+ bT

✓T

!n

◆3

+ · · ·

✔ SUSY gauge theories at large-N✔ O(N) CFT at N = ∞

✔ Dirac fermion

dual sum ruleσ ↔ 1/σ

Long times &analytic continuation

★AdS/CFT ➝ generate family ofphysically motivated (constrained) variational functions σ(ω/T)

Why dabble w/ black holes?❖ Physical properties:

✦ Tailored holographic description to match asymptotics

✦ Sum rule

✦ Practical: real-time, small number of params, fast numerics

Z 1

0d!Re[�(!/T )� �(1)] = 0

σ via AdS/CFT

❖ classical EoM for Aµ → JJ-correlator of CFT

[Maldacena etc]

u = r0êr1 0

CFTBH

JmAmblack hole in AdS4

Aµ(t, x, y;u) $ J

CFTµ (t, x, y)

u

�(!/T ) =T

!

@uAy(!;u)

Ay(!;u)

����u=0

Bulk action

❖ scalar field φ (dilaton)

❖ simplest Ansatz: fix φ using OPE of O(2) Wilson-Fisher QCP

CFT AdS

O φ

'(u) = u�

[Katz, Sachdev, Sorensen, WK]

Sbulk[Aµ

] =

Z

x

1g

24[1 + ↵'(x)]F

ab

F

ab

Holographic fit

Ê

Ê

ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê

0 5 10 150.36

0.38

0.40

0.42

wnê2pT

sHiwL

2pês Q � = 1.5 ; ↵ = 0.6

[Katz, Sachdev, Sorensen, WK]

Real frequencies

0 2 4 6 80.300.350.400.450.500.550.60

wê2pT

Re@sHwLD2pês Q

[Katz, Sachdev, Sorensen, WK]

Conclusions 1❖ corner entanglement is a useful measure of DoF

❖ smooth limit: a(θ) fixed by Tμν 2-point function

❖ small Rényi index ➞ thermal entropy from groundstate!

a(✓ ! ⇡) = � (✓ � ⇡)2 � =⇡2

24CT

θ

Conclusions II❖ Quantum critical dynamics at T > 0

❖ OPE to constrain large frequency physics

❖ conductivity

❖ Use holography to analytically continue Monte Carlo data forconductivity at superfluid/insulator QCP

�(i!n)!n�T= �1 + b1

✓T

!n

◆�

+ b2

✓T

!n

◆3

+ · · ·

Outlook❖ physics of corner entanglement beyond the

smooth limit? [Bueno, WK, in prep.]

❖ corner entanglement at deconfined QCPs?

❖ deform CFT by relevant operator[in prog. w/ Gazit, Podolsky, Sachdev]

PMFM

Quantumcritical

CFTh

T