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 p p approximate long-time averages. 3 2 p p 4 1 x x 4 1 x x 3 2 ⟨ O ⟩ t = ∫ S O ( Γ ) d Γ ∫ S d Γ review by J. Deutsch — 1805.01616

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. 
 h ψ | O | ψ i = tr[ e − β H O ] tr[ e − β H ] [Deutsch; Srednicki] A stronger version of ETH states that all finite energy microstates reproduce thermal expectation values. There are of course violations of ETH.

Eigenstate thermalization h ψ | O | ψ i = tr[ e − β H O ] tr[ e − β H ] The notion of temperature arises when the operator is O chosen to be the Hamiltonian H . ETH proposes an ansatz for all matrix elements of the operator [Srednicki; …] ⟨ m | O | n ⟩ = g O ( E m ) δ mn + e − S ( ¯ E ) / 2 f O ( ¯ E, ω ) R mn .

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’? What is the microscopic/CFT realisation of the BTZ black hole? Is the ETH ansatz for matrix elements obeyed?

ETH for primaries? Quasi-primary expectation values in a heavy primary state disagree with thermal ones. ◆ 2 ⇣ � π 2 c ✓ 2 π h p � c h p c ⌘ h T i β = h h p | T | h p i = � L 2 = 24 β 2 . L 24 6 β 2 ◆ 4 ⇣ ✓ 2 ◆ ✓ ⌘ Λ (4) =: TT : − 3 For the level-4 quasi-primary, 10 ∂ 2 T � ◆ 2 ◆ 2 ✓ π 2 c ✓ π 2 c π 4 c + 11 h h p | Λ (4) | h p i h Λ (4) i β i = = 6 β 2 6 β 2 90 β 4 [Basu-Das-SD-Pal;…] There is a disagreement beyond the leading order 
 in a large central charge limit.

ETH for primaries? One possible resolution to this may be offered by the generalised Gibbs ensemble. [Maloney-Ng-Ross-Tsiares; Dymarsky-Pavlenko; Brehm-Das] But the reason behind this discrepancy is that primaries are not typical states. r c " # r c � 1  � ρ ( h p ) ' exp 2 π ρ ( h ) ' exp 2 π h p 6 h 6 [Cardy; Kraus-Maloney] For fixed central charge, the growth of the number of primaries is exponentially smaller than the growth of all states.

Typical states Consider a descendant at level ( h-h p ) of a primary with conformal dimension h p . " # r c � 1 Growth of primaries ρ ( h p ) ' exp 2 π h p 6 [Cardy; Kraus-Maloney] " # r h � h p Growth of descendants ρ ( h � h p ) ' exp 2 π 6 [Hardy-Ramanujan] Typical states maximize with respect to h p . ρ ( h p ) ρ ( h − h p ) c h p h = c − 1 h p = h p + c − 1 descendant contribution

The punchline We focus on stress tensor correlators in c>1 CFTs with Virasoro symmetry. States in the CFT Hilbert space are | h p , M, { N j } i ⌘ L N 1 − 1 L N 2 − 2 L N 3 − 3 · · · | h p i primary 
 partitions of 
 descendant primary action of 
 conformal dim the integer M. level state Virasoro generators Typical states which reproduce stress tensor correlators are h p = ( c − 1) L 2 L 2 1 h N j i = M = 2 πβ j 24 β 2 24 β 2 � 1 e L are Boltzmann distributed with a Bose-Einstein mean. { N j }

The punchline | h p , M, { N j } i | typ i ⌘ 1 h N j i = 2 πβ j � 1 e L L 2 M = 24 β 2 h p = ( c − 1) L 2 24 β 2 h typ | T ( w 1 ) T ( w 2 ) · · · T ( w n ) | typ i = h T ( w 1 ) T ( w 2 ) · · · T ( w n ) i β [SD-Kraus-Michel]

Current correlators

Thermal current correlators in 2d CFT We consider the 2-point function of U(1) currents on a torus as a simple example first. This is fixed by modular properties and OPEs. ℘ ( w/L, τ ) + π 2 ✓ ◆ h J ( w 1 ) J ( w 2 ) i L, β = � 1 π 3 E 2 ( τ ) � L 2 Im( τ ) [Eguchi-Ooguri] β A B L

Thermal current correlators We consider the 2-point function of U(1) currents on a torus as a simple example first. This is fixed by modular properties and OPEs. ℘ ( w/L, τ ) + π 2 ✓ ◆ h J ( w 1 ) J ( w 2 ) i L, β = � 1 π 3 E 2 ( τ ) � L 2 Im( τ ) [Eguchi-Ooguri]  � ℘ ( w, τ ) = 1 1 1 X Weierstrass-P w 2 + ( w + m + n τ ) 2 � ( m + n τ ) 2 function ( m,n ) 6 =(0 , 0) ∞ nq n Eisenstein 
 X E 2 ( τ ) = 1 − 24 series 1 − q n n =1 q = e 2 π i τ τ = i β /L

Thermal current correlators The U(1) current can be realised as a free boson. J ( w ) = � 2 π 2 π inw X [ α m , α n ] = m δ m + n, 0 α n e J ( w ) = ∂ w φ ( w, ¯ w ) L L n The thermal 2-point function is, by definition 1 h i ˜ q L 0 � 1 L 0 � 1 h J ( w 1 ) J ( w 2 ) i L, β = Z ( τ )Tr 24 q 24 J ( w 1 ) J ( w 2 ) this and the mode-expansion gives… ✓ ◆ X n ◆ 2 "X ◆# ✓ 2 π ✓ 2 π nw 2 π inw X + h α 2 = 0 i L, β + 2 h α − n α n i L, β cos h J ( w ) J (0) i L, β ne L L L n> 0 n> 0 ◆ " ◆# ✓ ✓

Thermal current correlators ✓ ◆ X n ◆ 2 "X ◆# ✓ 2 π ✓ 2 π nw 2 π inw X + h α 2 = 0 i L, β + 2 h α − n α n i L, β cos h J ( w ) J (0) i L, β ne L L L n> 0 n> 0 ◆ " ◆# ✓ ✓ We work in the occupation number eigenbasis has eigenvalues nN n α − n α n In the canonical ensemble the probability distribution of the occupation number is a Boltzmann distribution e 2 π i τ N n n ∞ 1 X P ( N n ) = h N n i L, β = P ( N n ) N n = e − 2 π i τ n � 1 P ∞ N n =0 e 2 π i τ N n n N n =0 The mean is given by a Bose-Einstein function. τ = i β /L

Thermal current correlators The final result is ✓ ◆ ✓ ◆ X X n> 0 n> 0 ◆ 2 " ◆# ✓ 2 π 1 ✓ 2 π nw L n X � + h J ( w ) J (0) i L, β = � 4 πβ + 2 e − 2 π i τ n � 1 cos 4 sin 2 � π w L L L n> 0 and this agrees with ℘ ( w/L, τ ) + π 2 ✓ ◆ h J ( w 1 ) J ( w 2 ) i L, β = � 1 π 3 E 2 ( τ ) � L 2 Im( τ ) The details of this computation gives 
 insight into what might be the typical microstates which reproduce this thermal result.

Microstate current correlators Using the mode expansion for the current again, 
 the correlator in a microstate is given by ◆ 2 " ◆# ✓ 2 π ✓ 2 π nw 1 X � + h ψ | α 2 h ψ | J ( w ) J (0) | ψ i = � 0 | ψ i + 2 N n n cos 4 sin 2 � π w L L L n> 0 h ψ | α − n α n | ψ i = N n n Total energy E = 2 π microstate is an X N n n eigenstate of . L α − n α n n Effective temperature r π L β = 12 E

Analogy with partitions of integers E = 2 π The total energy is given by X N n n L n The situation here is similar to partitioning a integer M. ∞ X M = nN n n =1 Partitions can be conveniently represented by Young diagrams.

Statistics of partitions For large integers the most Young diagrams acquire a limiting shape. [Vershik] 50 random partitions of 12000. 80 1 √ N ( x ) = e x − 1 , x = π j/ 6 M 60 edges of 
 Young diagrams 40 The ‘typical’ profile limiting curve is Bose-Einstein. 20 0 100 200 300 400 500 also [Nekrasov-Okounkov; Balasubramanian-deBoer-Jejjala-Simon]

Microstate current correlators ◆ 2 " ◆# ✓ 2 π ✓ 2 π nw 1 X � + h ψ | α 2 h ψ | J ( w ) J (0) | ψ i = � 0 | ψ i + 2 N n n cos 4 sin 2 � π w L L L n> 0 Total energy Effective temperature h ψ | α − n α n | ψ i = N n n microstate is an r E = 2 π π L X N n n β = eigenstate of . L α − n α n 12 E n If states having the above properties are chosen at random, then the occupation number is again chosen from a Boltzmann distribution. e 2 π i τ N n n P ( N n ) = P ∞ N n =0 e 2 π i τ N n n Typicality ~ randomly choosing Nn from P(Nn).

Occupation numbers 1 , L = 3 ⇥ 10 6 . β = The variance of Nn is itself large but it can be shown that the correlator has small variance. p 1 at δ h J ( w ) J (0) i L, β ⇠ L as L ! 1 , √ A sample of Boltzmann distributed occupation numbers. The mean is 
 given by the Bose-Einstein function.

Microstate v/s thermal correlator 1 , L = 3 ⇥ 10 6 . β =

Stress-tensor correlators

Stress tensor correlators A similar analysis can be performed for stress tensor correlators. We wish to establish that h typ | T ( w ) T (0) | typ i = h T ( w ) T (0) i β The thermal 2-point function is ◆ 2 ◆ 4 ✓ π 2 c ✓ 2 π + c 1 h T ( w 0 ) T ( w ) i β = β ( w 0 � w )) sinh 4 ( π 6 β 2 32 β