Quantum Quench in Conformal Field Theory from a General Short ...

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

**from** a **General** **Short**-Ranged State

John Cardy

University of Oxford

GGI, Florence, May 2012

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

(Global) **Quantum** **Quench**

prepare an extended system at time t = 0 **in** a

(translationally **in**variant) pure state |ψ 0 〉 – e.g. the ground

state of some hamiltonian H 0

evolve unitarily with a hamiltonian H for which |ψ 0 〉 is not an

eigenstate and has extensive energy above the ground

state of H

how do correlation functions and entanglement evolve as a

function of t?

for a compact subsystem do they become stationary?

if so, what is the stationary state?

is the reduced density matrix thermal?

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

**Quantum** quench **in** a 1+1-dimensional CFT

P. Calabrese + JC [2006] studied this problem **in** 1+1

dimensions when H = H CFT and |ψ 0 〉 is a state with

short-range correlations and entanglement

H CFT describes the low-energy, large-distance properties

of many gapless 1d systems

1+1-dimensional CFT is exactly solvable

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Results

one-po**in**t functions **in** general decay towards their ground

state values

〈Φ(x, t)〉 ∼ e −π∆ Φt/2τ 0

for times t > |x 1 − x 2 |/2v, the correlation functions become

stationary

〈Φ(x 1 , t 1 )Φ(x 2 , t 2 )〉 ∼ e −π∆ Φ|x 1 −x 2 |/2vτ 0

for t 1 = t 2 and ∼ e −π∆ Φ|t 1 −t 2 |/2τ 0

for x 1 = x 2

the (conserved) energy density is πc/6(4τ 0 ) 2

the von Neumann entropy of a region of length l saturates

for t > l/2v at

S ∼ (πc/3(4τ 0 ))l

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

all these results are precisely those expected for the CFT

at temperature T = (4τ 0 ) −1

they accord with a simple physical picture of entangled

pairs of quasiparticles emitted **from** correlated regions

t

r

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

**Quantum** quenches **in** **in**tegrable models

however studies of quenches **in** **in**tegrable models

[(Rigol,Dunjko,Yurovsky,Olshanii),...,(Calabrese,Essler,Fagotti)]

have led to the conclusion that the steady state should be

a ‘generalised Gibbs ensemble’ (GGE) with a separate

‘temperature’ conjugate to each local conserved quantity

1+1-dimensional CFT is super-**in**tegrable: e.g. all powers

T(z) p and T(¯z)¯p of the stress tensor correspond to local

conserved currents, lead**in**g to conserved charges

so why did CC f**in**d a simple Gibbs ensemble?

this can be traced to a simplify**in**g assumption about the

form of the **in**itial state

what is the effect of relax**in**g this assumption?

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

**Quantum** quenches **in** **in**tegrable models

however studies of quenches **in** **in**tegrable models

[(Rigol,Dunjko,Yurovsky,Olshanii),...,(Calabrese,Essler,Fagotti)]

have led to the conclusion that the steady state should be

a ‘generalised Gibbs ensemble’ (GGE) with a separate

‘temperature’ conjugate to each local conserved quantity

1+1-dimensional CFT is super-**in**tegrable: e.g. all powers

T(z) p and T(¯z)¯p of the stress tensor correspond to local

conserved currents, lead**in**g to conserved charges

so why did CC f**in**d a simple Gibbs ensemble?

this can be traced to a simplify**in**g assumption about the

form of the **in**itial state

what is the effect of relax**in**g this assumption?

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Review of CC [2006,2007]

we want to compute

〈ψ 0 |e itH CFT

O e −itH CFT

|ψ 0 〉

we could get this **from** imag**in**ary time by consider**in**g

〈ψ 0 |e −τ 2H CFT

O e −τ 1H CFT

|ψ 0 〉

and cont**in**u**in**g τ 1 → it, τ 2 → −it

‘slab’ geometry with boundary condition ≡ ψ 0 , but

thickness τ 1 + τ 2 = 0 ☹

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Resolution: ‘Mov**in**g the goalposts’

resolution: replace boundary condition at τ = ±0 by

‘idealised’ bc at τ = ±τ 0

idea of ‘extrapolation length’ **in** boundary critical behaviour:

idealised bc ≡ boundary RG fixed po**in**t

τ 0

−τ0

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

we then need to compute

〈O(τ)〉 slab

and cont**in**ue the result to τ → τ 0 + it

**in** CFT, the correlations **in** the slab are related to those **in**

the upper half z-plane by z = e πw/2τ 0

power-law behaviour **in** the z-plane ⇒ exponential

behaviour **in** t and x

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

**in** particular, x + i(τ 0 + it) is mapped

to

z = i e π(x−t)/2τ 0

¯z = −i e π(x+t)/2τ0 ≠ z ∗ (!)

except for narrow regions O(τ 0 ) near

the light cone, po**in**ts are

exponentially ordered along

imag**in**ary z-axis: correlators can be

computed by successive OPEs

for t → ∞ the ¯z’s move off to −i∞

and the boundary effectively plays

no role ⇒ we have periodicity **in**

w → w + 4iτ 0 : f**in**ite temperature!

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Relax**in**g CC’s assumption

CC’s prescription is equivalent to assum**in**g

|ψ 0 〉 ∝ e −τ 0H CFT

|B〉

where |B〉 is a conformally **in**variant boundary state

**in** general we expect any translationally **in**variant state

sufficiently close to |B〉 to have the form

|ψ 0 〉 ∝ e − ∑ j λ j

∫ (b) φ j (x)dx |B〉

where φ (b)

j are all possible irrelevant boundary operators

one of the most important is the stress tensor T ττ with RG

eigenvalue 1 − 2 = −1: note that ∫ T ττ (x)dx = H CFT , so

CC’s assumption is that this is the most important one: if it

is the only one all the conclusions of CC follow exactly

a similar argument has been made **in** expla**in****in**g the

entanglement spectrum of quantum Hall states

[Dubail,Read,Rezayi]

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

so let us suppose

|ψ 0 〉 ∝ e −τ 0H CFT

e − ∑ j ′ λ j

∫

φ

(b)

j (x)dx |B〉 where ∆ j > 1

s**in**ce the φ (b)

j are irrelevant, we might expect to be able to

do perturbation theory **in** the λ j : **in** the ground state this

would lead to corrections to scal**in**g

for most simple models the only operators φ (b) which do

not explicitly break the symmetry are descendants of the

stress tensor, e.g. TT

as an example, first order correction to 〈Φ(τ)〉 slab is

∫

−λ 〈Φ(τ)TT(x)〉 slab dx

boundary

this can be computed by mapp**in**g to the UHP

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

after cont**in**u**in**g τ → τ 0 + it we f**in**d a first-order correction

e −π∆ (

Φt/2τ 0 1 + λ∆

2

Φ τ −4

0

t + · · · )

higher orders **in** λ exponentiate up to lead**in**g order, so we

get an **in**verse relaxation time

π∆ Φ

2τ 0

− λ ∆2 Φ

(2τ 0 ) 4 + O(λ2 )

note that effective temperature now depends on which

operator Φ we measure!

we get the same effective temperature shift **in** the spatial

decay of 〈Φ(x 1 , t)Φ(x 2 , t)〉 for 2vt > |x 1 − x 2 | ≫ vτ 0

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

after cont**in**u**in**g τ → τ 0 + it we f**in**d a first-order correction

e −π∆ (

Φt/2τ 0 1 + λ∆

2

Φ τ −4

0

t + · · · )

higher orders **in** λ exponentiate up to lead**in**g order, so we

get an **in**verse relaxation time

π∆ Φ

2τ 0

− λ ∆2 Φ

(2τ 0 ) 4 + O(λ2 )

note that effective temperature now depends on which

operator Φ we measure!

we get the same effective temperature shift **in** the spatial

decay of 〈Φ(x 1 , t)Φ(x 2 , t)〉 for 2vt > |x 1 − x 2 | ≫ vτ 0

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Is this a **General**ised Gibbs Ensemble?

**in** GGE an equal-time correlation function should have the

form

[

〈Φ(x 1 , t)Φ(x 2 , t)〉 = tr e −βH e − ∑ ]

p βpHp Φ(x 1 )Φ(x 2 )

where {H, H p } are an **in**f**in**ite set of commut**in**g conserved

charges.

**in** CFT a m**in**imal set are H p = ∫ [:T(x, t) p : + :T(x, t) p :]dx for

p = 2, 3, . . .

**in** terms of Virasoro operators

∑

H p ∝ :L n1 L n2 · · · L np : +c.c.

n 1 +···+n p=0

the normal order**in**g implies that n 1 ≤ n 2 ≤ · · · ≤ n p , so

H p ∝ L p 0 + terms with n p ≥ 1 + c.c.

so act**in**g on a primary operator H p ∝ ∆ p Φ

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

so for a primary operator 〈Φ(x 1 , t)Φ(x 2 , t)〉 GGE ∼ e −|x 1−x 2 |/ξ

where

ξ −1 = 2π β ∆ Φ − ∑ ( ) p 2π∆Φ

β p

β 2

p

Compare with result **from** a perturbed boundary state

ξ −1 = π∆ Φ

2τ 0

− λ ∆2 Φ

(2τ 0 ) 4 + O(λ2 )

this has exactly the same form, with β = 4τ 0 and β 2p ∝ λ p

act**in**g with other irrelevant descendants of T on the **in**itial

state gives similar results, all consistent with GGE ☺

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

More general boundary perturbations

more general irrelevant boundary perturbations φ (b)

j with

scal**in**g dimensions ∆ j ≠ **in**teger are consistent with a GGE

only if we posit the existence of bulk parafermionic

holomorphic currents φ j (z) with dimension ∆ j and **in**clude

the correspond**in**g non-local conserved charges

H j = ∫ φ j (x, t)dx **in** the GGE ?☹?

the stationary state becomes more like pure Gibbs as

T eff ↓ 0, i.e. a shallow quench

one can also add irrelevant terms like to H CFT : e.g.

TT, correspond**in**g to left-right scatter**in**g

T p + T p , correspond**in**g to curvature of dispersion relation

however perturbatively they don’t appear to change the

overall picture ?☺? ?☹?

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

More general boundary perturbations

more general irrelevant boundary perturbations φ (b)

j with

scal**in**g dimensions ∆ j ≠ **in**teger are consistent with a GGE

only if we posit the existence of bulk parafermionic

holomorphic currents φ j (z) with dimension ∆ j and **in**clude

the correspond**in**g non-local conserved charges

H j = ∫ φ j (x, t)dx **in** the GGE ?☹?

the stationary state becomes more like pure Gibbs as

T eff ↓ 0, i.e. a shallow quench

one can also add irrelevant terms like to H CFT : e.g.

TT, correspond**in**g to left-right scatter**in**g

T p + T p , correspond**in**g to curvature of dispersion relation

however perturbatively they don’t appear to change the

overall picture ?☺? ?☹?

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**

Conclusions

a quantum quench **in** 1+1-dimensional CFT **from** a more

general state leads to results consistent with a GGE, so

the conclusions of CC [2006] as predict**in**g strict

thermalisation should not be **in**terpreted too literally!

**Quantum** **Quench** **in** **Conformal** **Field** **Theory**