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