Typicality and thermality in 2d CFT Shouvik Datta This talk is - - PowerPoint PPT Presentation

Typicality and thermality
 in 2d CFT

Shouvik Datta

This talk is based on work with Mert Besken, Per Kraus and Ben Michel. arXiv:1904.00668 + ongoing work

Statistical mechanics & thermodynamics

Statistical mechanics provides a successful framework to describe thermodynamic behaviour. However, precise relation of the macroscopic quantities/phenomena to 
 microscopic details is often subtle. For instance, at the microscopic level a number of physical laws are reversible while thermodynamic laws aren’t. These issues are important while trying to understand
 thermalization microscopically.

Microstates & thermodynamics

There are a number of possible approaches to understand
 the emergence of macroscopic behaviour from microscopics. The most conventional way invokes the ergodic hypothesis. This states that ensemble averages approximate long-time averages.

x x x p p p p x

1 2 3 1 2 3 4 4

review by J. Deutsch — 1805.01616

⟨O⟩t = ∫S O(Γ)dΓ ∫S dΓ

Microstates & thermodynamics

Another approach is to consider typicality of states. 
 An overwhelmingly large number of microstates reproduce the same macroscopic behaviour. A single typical state may be good enough to reproduce thermodynamics.

Eigenstate thermalization

The eigenstate thermalization hypothesis (ETH) states that thermal expectation values can be reproduced by 
 a single typical microstate of finite energy density.
 There are of course violations of ETH. A stronger version of ETH states that all finite energy microstates reproduce thermal expectation values.

hψ|O|ψi = tr[e−βHO] tr[e−βH]

[Deutsch; Srednicki]

Eigenstate thermalization

The notion of temperature arises when the operator is chosen to be the Hamiltonian H.

⟨m|O|n⟩ = gO(Em)δmn + e−S( ¯

E)/2fO( ¯

E, ω)Rmn.

hψ|O|ψi = tr[e−βHO] tr[e−βH]

ETH proposes an ansatz for all matrix elements of the operator

O

[Srednicki; …]

Eigenstate thermalization

We can pose these questions in 2d CFTs which offers an arena of analytic tractability. Does a typical high-energy microstate appear thermal? What do we mean by ‘typical’? Is the ETH ansatz for matrix elements obeyed? What is the microscopic/CFT realisation of the BTZ black hole?

ETH for primaries?

Quasi-primary expectation values in a heavy primary state disagree with thermal ones.

hTiβ = π2c 6β2 ✓

2 ◆

hhp|T|hpi =

• ✓2π

L ◆2 ⇣ hp c 24 ⌘ ✓ ◆4 ⇣ ⌘

hp L2 = c 24β2 .

• =

✓π2c 6β2 ◆2 + 11 90 π4c β4

hΛ(4)iβ

i = ✓π2c 6β2 ◆2 hhp|Λ(4)|hpi

Λ(4) =: TT : − 3 10∂2T

For the level-4 quasi-primary, There is a disagreement beyond the leading order
 in a large central charge limit.

[Basu-Das-SD-Pal;…]

ETH for primaries?

One possible resolution to this may be

• ffered by the generalised Gibbs ensemble.

ρ(hp) ' exp " 2π r c 1 6 hp # ρ(h) ' exp  2π r c 6h

• For fixed central charge, the growth of the number of

primaries is exponentially smaller than the growth of all states. But the reason behind this discrepancy is that primaries are not typical states.

[Cardy; Kraus-Maloney] [Maloney-Ng-Ross-Tsiares; Dymarsky-Pavlenko; Brehm-Das]

Typical states

Consider a descendant at level (h-hp)

• f a primary with conformal dimension hp.

ρ(hp) ' exp " 2π r c 1 6 hp #

Growth of primaries

[Cardy; Kraus-Maloney]

Growth of descendants

ρ(h hp) ' exp " 2π r h hp 6 #

[Hardy-Ramanujan]

Typical states maximize with respect to hp.

ρ(hp)ρ(h − hp) h = c c − 1hp = hp + hp c − 1

descendant contribution

The punchline

We focus on stress tensor correlators in c>1 CFTs with Virasoro symmetry. Typical states which reproduce stress tensor correlators are States in the CFT Hilbert space are

hp = (c − 1) L2 24β2

are Boltzmann distributed with a Bose-Einstein mean.

M = L2 24β2 hNji = 1 e

2πβj L

1 {Nj}

primary 
 conformal dim descendant level partitions of
 the integer M.

|hp, M, {Nj}i ⌘ LN1

−1LN2 −2LN3 −3 · · · |hpi

primary state action of
 Virasoro generators

The punchline

hp = (c − 1) L2 24β2

|typi ⌘ htyp|T(w1)T(w2) · · · T(wn)|typi = hT(w1)T(w2) · · · T(wn)iβ |hp, M, {Nj}i

M = L2 24β2 hNji = 1 e

2πβj L

1

[SD-Kraus-Michel]

Current correlators

Thermal current correlators in 2d CFT

hJ(w1)J(w2)iL,β = 1 L2 ✓ ℘(w/L, τ) + π2 3 E2(τ) π Im(τ) ◆

β L

A B

[Eguchi-Ooguri]

We consider the 2-point function of U(1) currents

• n a torus as a simple example first.

This is fixed by modular properties and OPEs.

We consider the 2-point function of U(1) currents

• n a torus as a simple example first.

This is fixed by modular properties and OPEs.

hJ(w1)J(w2)iL,β = 1 L2 ✓ ℘(w/L, τ) + π2 3 E2(τ) π Im(τ) ◆ ℘(w, τ) = 1 w2 + X

(m,n)6=(0,0)

 1 (w + m + nτ)2 1 (m + nτ)2

• Weierstrass-P

function Eisenstein 
 series

E2(τ) = 1 − 24

∞

X

n=1

nqn 1 − qn

τ = iβ/L

q = e2πiτ

Thermal current correlators

[Eguchi-Ooguri]

The U(1) current can be realised as a free boson.

J(w) = 2π L X

n

αne

2πinw L

[αm, αn] = mδm+n,0

hJ(w1)J(w2)iL,β = 1 Z(τ)Tr h qL0 1

24q

˜ L0 1

24J(w1)J(w2)

i

J(w) = ∂wφ(w, ¯ w)

The thermal 2-point function is, by definition

✓ ◆ X

n

= ✓2π L ◆2 "X

n>0

ne

2πinw L

+ hα2

0iL,β + 2

X

n>0

hα−nαniL,β cos ✓2πnw L ◆# ✓ ◆ " ✓ ◆# hJ(w)J(0)iL,β

this and the mode-expansion gives…

Thermal current correlators

✓ ◆ X

n

= ✓2π L ◆2 "X

n>0

ne

2πinw L

+ hα2

0iL,β + 2

X

n>0

hα−nαniL,β cos ✓2πnw L ◆# ✓ ◆ " ✓ ◆# hJ(w)J(0)iL,β

We work in the occupation number eigenbasis has eigenvalues

α−nαn nNn

In the canonical ensemble the probability distribution

• f the occupation number is a Boltzmann distribution

P(Nn) = e2πiτNnn P∞

Nn=0 e2πiτNnn

hNniL,β =

∞

X

Nn=0

P(Nn)Nn = 1 e−2πiτn 1

The mean is given by a Bose-Einstein function.

τ = iβ/L

Thermal current correlators

The final result is

Thermal current correlators

✓ ◆ X

n>0

X

n>0

✓ ◆ = ✓2π L ◆2 "

• 1

4 sin2 πw

L

+ L 4πβ + 2 X

n>0

n e−2πiτn 1 cos ✓2πnw L ◆#

hJ(w)J(0)iL,β

The details of this computation gives
 insight into what might be the typical microstates which reproduce this thermal result. and this agrees with

hJ(w1)J(w2)iL,β = 1 L2 ✓ ℘(w/L, τ) + π2 3 E2(τ) π Im(τ) ◆

Using the mode expansion for the current again,
 the correlator in a microstate is given by

Microstate current correlators

hψ|J(w)J(0)|ψi = ✓2π L ◆2 "

• 1

4 sin2 πw

L

+ hψ|α2

0|ψi + 2

X

n>0

Nnn cos ✓2πnw L ◆#

hψ|α−nαn|ψi = Nnn

microstate is an eigenstate of .

α−nαn

Total energy

E = 2π L X

n

Nnn

β = r πL 12E

Effective temperature

Analogy with partitions of integers

E = 2π L X

n

Nnn

The total energy is given by The situation here is similar to partitioning a integer M. Partitions can be conveniently represented by Young diagrams.

M =

∞

X

n=1

nNn

Statistics of partitions

For large integers the most Young diagrams acquire a limiting shape.

edges of 
 Young diagrams limiting curve 50 random partitions of 12000.

N(x) = 1 ex − 1, x = πj/ √ 6M

The ‘typical’ profile is Bose-Einstein.

[Vershik] also [Nekrasov-Okounkov; Balasubramanian-deBoer-Jejjala-Simon]

Microstate current correlators

hψ|J(w)J(0)|ψi = ✓2π L ◆2 "

• 1

4 sin2 πw

L

+ hψ|α2

0|ψi + 2

X

n>0

Nnn cos ✓2πnw L ◆#

hψ|α−nαn|ψi = Nnn

microstate is an eigenstate of .

α−nαn

Total energy

E = 2π L X

n

Nnn

β = r πL 12E

Effective temperature

P(Nn) = e2πiτNnn P∞

Nn=0 e2πiτNnn

If states having the above properties are chosen at random, then the occupation number is again chosen from a Boltzmann distribution. Typicality ~ randomly choosing Nn from P(Nn).

Occupation numbers

A sample of Boltzmann distributed

• ccupation numbers. The mean is


given by the Bose-Einstein function.

β = 1, L = 3 ⇥ 106.

The variance of Nn is itself large but it can be shown that the correlator has small variance.

p at δhJ(w)J(0)iL,β ⇠

1 √ L as L ! 1,

Microstate v/s thermal correlator

β = 1, L = 3 ⇥ 106.

Stress-tensor correlators

Stress tensor correlators

A similar analysis can be performed for stress tensor correlators. We wish to establish that

htyp|T(w)T(0)|typi = hT(w)T(0)iβ

hT(w0)T(w)iβ = ✓π2c 6β2 ◆2 + c 32 ✓2π β ◆4 1 sinh4( π

β(w0 w))

The thermal 2-point function is

Microstate stress tensor correlators

Once again we use the mode expansion

T(w) = ✓2π L ◆2 ⇣ L0 c 24 ⌘

• ✓2π

L ◆2 X

n6=0

Lne

2πinw L

hψh|T(w)T(0)|ψhi = ✓2π L ◆4 ⇣ h c 24 ⌘2

• ✓2π

L ◆4 ✓ h 2 sin2( πw

L )

c 32 sin4( πw

L )

◆ +2 ✓2π L ◆4 X

n>0

hψh|L−nLn|ψhi cos ✓2πnw L ◆ . (3.19)

to get the microstate correlator

hψh|T(w)T(0)|ψhi = ✓2π L ◆4 ⇣ h c 24 ⌘2

• ✓2π

L ◆4 ✓ h 2 sin2( πw

L )

c 32 sin4( πw

L )

◆ +2 ✓2π L ◆4 X

n>0

hψh|L−nLn|ψhi cos ✓2πnw L ◆ . (3.19)

need to evaluate this for a typical state

The typical state is a descendant. We can replace by the thermal average, provided the variance is small.

Microstate stress tensor correlators

hψh|L−nLn|ψhi

hψh|T(w)T(0)|ψhi = ✓2π L ◆4 ⇣ h c 24 ⌘2

• ✓2π

L ◆4 ✓ h 2 sin2( πw

L )

c 32 sin4( πw

L )

◆ +2 ✓2π L ◆4 X

n>0

hψh|L−nLn|ψhi cos ✓2πnw L ◆ . (3.19)

The thermal 2-point function is indeed recovered
 by the replacing above by the thermal expectation value of the operator of a single Verma module.

1 e

2πβn L

1  c 12n3 + ✓ hp + L2 24β2 ◆ 2n

• 1

h c i

hL−nLnihp,β = 1 Zhp(q)Trhp h L−nLnqL0−c/24i ⇡

Microstate stress tensor correlators

Agreement only for real w. For complex w, there is agreement


• nly within the strip |Im(w)|< .

β

L−nLn

hψh|L−nLn|ψhi

[Maloney-Ng-Ross-Tsiares]

hψh|T(w)T(0)|ψhi = ✓2π L ◆4 ⇣ h c 24 ⌘2

• ✓2π

L ◆4 ✓ h 2 sin2( πw

L )

c 32 sin4( πw

L )

◆ +2 ✓2π L ◆4 X

n>0

hψh|L−nLn|ψhi cos ✓2πnw L ◆ . (3.19)

The replacement

Small variance

2 ✓2π L ◆4 X

n>0

hψh|L−nLn|ψhi cos ✓2πnw L ◆ ! 2 ✓2π L ◆4 X

n>0

hL−nLnihp,β cos ✓2πnw L ◆ (3.24)

is allowed only if the variance of the following operator is small

X(w) ⌘ X

n>0

Xn cos ✓2πnw L ◆ ⌘ ✓2π L ◆4 X

n>0

L−nLn cos ✓2πnw L ◆

Small variance

It can be shown by using the Virasoro algebra that
 the dominant modes contributing to the sum have

X(w) ⌘ X

n>0

Xn cos ✓2πnw L ◆ ⌘ ✓2π L ◆4 X

n>0

L−nLn cos ✓2πnw L ◆

hL−mLmL−nLnihp,β ⇡ hL−mLmihp,βhL−nLnihp,β

which implies

δX2 = hX2i hXi2 ⇠ 1 L

Sample computation

Tr[L−nLnqL0] = qn Tr[L−nqL0Ln] = qnTr[LnL−nqL0] = qnTr[ [Ln, L−n] qL0] + qnTr[L−nLnqL0]

Tr[L0qL0] = q∂qTr[qL0]

Tr[L−nLnqL0] = qn 1 − qn h 2nq∂qZ + c 12n3Z i

[Maloney-Ng-Ross-Tsiares]

LnqL0 = qL0+nLn

2nL0 + c 12(n3 − n)

Sample computation

hLmLnLpi = 1 1 qp [(p n)hLmL−mi + (p m)hL−nLni] hL−nLnL−mLmi = qn 1 qn (hLnL−m[Lm, L−n]i + hLn[L−m, L−n]Lmi + h[Ln, L−n]L−mLmi) .

Higher point functions

This analysis can be extended to higher point correlators

hψh| T(w1) . . . T(wn) |ψhi = (1)n ✓2π L ◆2n X

i1...in P ik=0

hψh| Li1 . . . Lin |ψhi e

2πi L

P

p ipwp

The fluctuations can again be shown to be 1/L . This is true when the number of stress tensors in the correlator
 is held fixed as .

L/β → ∞

Thank you