Reference
1302.0859 with G. Mandal (TIFR, India)
Quantum quench in matrix models:
Dynamical phase transitions, equilibration and the Generalized Gibbs Ensemble
Takeshi Morita KEK (Japan)
22 Feb. Seminar at NTU
Introduction
Introduction
◆ Lot of mysteries concerning time evolutions in Gravity
• Cosmic censorship hypothesis: Do naked singularities appear?
• How the BH entropy production happens?
• Hawking radiation and Information paradox
AdS/CFT correspondence proposes that Large-N gauge theories would answer these questions (in AdS space).
HOW?
1. Solve the time evolutions in the dual large-N gauge theory.
2. Construct the precise mapping between the gauge theory and gravity.
But solving the large-N gauge theory is generally very difficult...
Even numerical computations will not work properly.
HOW?
Today, I will introduce a simple matrix model and
show interesting time evolutions, which may be related to string theory and gravity.
Introduction
1. Solve the time evolutions in the dual large-N gauge theory.
2. Construct the precise mapping between the gauge theory and gravity.
Introduction
◆ The model: unitary matrix model
U(t): N×N unitary matrix
• Integrable
• related to c=1 non-critical string theory through the double scaling limit (Today, we do not consider it.)
• quantum quench & dynamical phase transition
→ appearance of naked singularities
• equilibration & entropy production
→ black hole formation and information loss
We will see time evolutions which are potentially related to gravity
BUT do not expect too much. The connection to BH physics is unclear at all.
Introduction
◆ Integrability vs. thermodynamics
Does thermalization happen in the integrable system in which infinite conserved charges exist?
integrable system standard thermodynamics
Recently new thermal ensemble in integrable system called
"Generalized Gibbs Ensemble (GGE) " is proposed
in condensed matter physics and is confirmed in several models.
We will see our matrix model indeed obeys GGE.
Introduction
◆ Integrability vs. thermodynamics
Does thermalization happen in the integrable system in which infinite conserved charges exist?
integrable system standard thermodynamics
• Integrablity plays crucial roles in the recent developments in string theory, e.g. spin chain, BPS objects, fuzz ball conjecture.
→ GGE may be important in these studies.
• GGE may be related to the HS/CFT correspondence too, since infinite HS charges exist in this relation.
Recently new thermal ensemble in integrable system called
"Generalized Gibbs Ensemble (GGE) " is proposed
in condensed matter physics and is confirmed in several models.
We will see our matrix model indeed obeys GGE.
(Ultimate) Goals of our study
◆ Understand the time evolutions of the matrix model to reveal the time evolution of string/gravity.
◆ Study the GGE and consider the application to string and HS theories.
◆ Connect string theory to condensed matter physics through the quantum quench and GGE.
Introduction
1. Introduction
2. Review of the single trace matrix model
3. Time evolution of the single trace matrix model 4. Equilibration and Generalized Gibbs Ensemble 5. One way transition in the matrix model
6. Summary
Plan of today's talk
Review of the single trace matrix models
To analyze this model, Fermion description is convenient.
It is known that V can be gauged away and behave as N fermions on .
∵) We can rewrite the kinetic term of the Hamiltonian as
original bosonic wave function
This new wave function ψ is fermionic, since it is anti-symmetric under
Separate the diagonal component as
can be regarded as the position of the i-th fermion on .
Review of the single trace matrix models
In terms of the Fermions, the action can be rewritten as
N free fermions are in the cos potential
∵) fermi surface
second quantized fermion field.
: hamiltonian for a single fermion.
Review of the single trace matrix models
In terms of the Fermions, the action can be rewritten as
second quantized fermion field.
The eigen function is given by the Mathieu function.
Important point
Mathieu function is available in Mathematica & Maple!
→ They tell us the answer! (But some critical BUGS exist in these softwares. ) fermi surface
: hamiltonian for a single fermion.
Review of the single trace matrix models
◆ Quantum phase transition
The Gross-Witten-Wadia type 3rd order transition happens at large-N.
: The potential depth a controls the phases.
The ground state for a large a.
→ a gap exists.
gap
The ground state for a small a.
→ The gap disappears.
(If N is finite, the gap is smeared through a quantum effect.)
1. Introduction
2. Review of the single trace matrix model
3. Time evolution of the single trace matrix model 4. Equilibration and Generalized Gibbs Ensemble 5. One way transition in the matrix model
6. Summary
Plan of today's talk
◆ Quantum quench dynamics
What will happen if we consider the ground state at t<0 and change the potential from suddenly at t=0?
: The potential depth a controls the phases.
gap
Time evolution of the single trace matrix model
◆ Quantum quench dynamics
Time evolution of the single trace matrix model
Comment: Advantage of quantum quench dynamics
Generally solving the Schrödinger equation in a time dependent potential is difficult. However, in the quench case, what we need is just solving the equation with the Hamiltonian at with
the initial configuration at t=0, which is the ground state at .
→ We can avoid the time dependent potential!
=
: the ground state at
Time evolution of the single trace matrix model
◆ Quantum quench dynamics (Result at N=120)
The gap disappears.
→ A dynamical transition occurs!
Time evolution of the fermion density
gap
Initial See the movie from the following link:
http://www2.yukawa.kyoto-
u.ac.jp/~mtakeshi/MQM/index.html
Time evolution of the single trace matrix model
◆ Quantum quench dynamics (Result at N=120)
Time evolution of the fermion density
No gap appears.
→ The transition DOES NOT occur!
Initial See the movie from the following link:
http://www2.yukawa.kyoto-
u.ac.jp/~mtakeshi/MQM/index.html
Time evolution of the single trace matrix model
◆ Quantum quench dynamics (Result at N=120)
We observed two important features.
1. The initial large oscillations subtle down to the small ripples.
→ Equilibration would happen even in free and closed system.
(c.f. Black hole formulation)
2. The dynamical transition occurs only from the gapless to gapped case.
(We will argue a connection to a Gregory-Laflamme transition.)
The transition happens. No transition.
1. Introduction
2. Review of the single trace matrix model
3. Time evolution of the single trace matrix model 4. Equilibration and Generalized Gibbs Ensemble 5. One way transition in the matrix model
6. Summary
Plan of today's talk
To see the equilibration qualitatively, we evaluate the Fourier mode of ρ.
semi-classical analysis exact analysis
: Characterize the shape of the density.
Equilibration and Generalized Gibbs Ensemble
To see the equilibration qualitatively, we evaluate the Fourier mode of ρ.
: Characterize the shape of the density.
Decay to a certain value
The small oscillations of N=120 are slightly larger than N=∞.
→ Poincaré recurrence exact analysis
semi-classical analysis
Equilibration and Generalized Gibbs Ensemble
◆ Poincaré recurrence
: Characterize the shape of the density.
The late time oscillation decreases as N increase. → Poincaré recurrence
→ We expect the recurrence does not occur only at N=∞ and it really equilibrates to an asymptotic state.
Equilibration and Generalized Gibbs Ensemble
Equilibration and Generalized Gibbs Ensemble
◆ Equilibration and Generalized Gibbs Ensemble (GGE)
→ We expect the recurrence does not occur only at N=∞ and it really equilibrates to an asymptotic state.
Q. Can we predict the equilibrated observables through any ensemble?
Equilibration and Generalized Gibbs Ensemble
◆ Equilibration and Generalized Gibbs Ensemble (GGE)
Q. Can we predict the equilibrated observables through any ensemble?
◆ Integrability of the free fermion system
Since the the fermions are free, the fermion number
at each level is conserved. → Infinite conserved charges → Integrable eigen function
We can define the following infinite number of conserved charges.
Equilibration and Generalized Gibbs Ensemble
◆ Equilibration and Generalized Gibbs Ensemble (GGE)
Q. Can we predict the equilibrated observables through any ensemble?
◆ Integrability vs. thermodynamics
Since the the fermions are free, the fermion number
at each level is conserved. → Infinite conserved charges → Integrable We can define the following infinite number of conserved charges.
our system standard thermodynamics
Number of the conserved quantities is quite different!
→ Standard thermodynamics will not work.
Equilibration and Generalized Gibbs Ensemble
◆ Equilibration and Generalized Gibbs Ensemble (GGE)
Q. Can we predict the equilibrated observables through any ensemble?
◆ Integrability vs. thermodynamics
our system standard thermodynamics
Number of the conserved quantities is quite different!
→ Standard thermodynamics will not work.
Generalized Gibbs Ensemble, which was recently proposed, may work for such integrable systems.
Equilibration and Generalized Gibbs Ensemble
◆ Generalized Gibbs Ensemble (GGE)
See a review by Polkovnikov, Sengupta, Silva, Vengalattore 2010
A conjecture: GGE describes the asymptotic state of a generic quanum integrable model.
: the chemical potential for each conserved charge, which is fixed at the initial state.
: the conserved charges in a integrable system and its operator In our case
: GGE density matrix
◆ Generalized Gibbs Ensemble (GGE): example
hard-core bosons on a one-dimensional lattice
``fully constrained"=GGE
Rigol, Dunjko, Yurovsky and Olshanii 2007
Equilibration and Generalized Gibbs Ensemble
◆ Generalized Gibbs Ensemble (GGE)
Equilibration and Generalized Gibbs Ensemble
◆ Generalized Gibbs Ensemble (GGE)
Equilibration and Generalized Gibbs Ensemble
GGE works quite well in our model!
◆ Entropy production in GGE
Equilibration and Generalized Gibbs Ensemble
This agreement implies that we can approximate the asymptotic states of this system by using . (coarse graining)
Since is not a pure state, the von Neumann entropy is non-zero.
→ The equilibration causes an entropy production.
(Although the original state is pure state and the entropy=0.)
◆ Entropy production in GGE
Equilibration and Generalized Gibbs Ensemble
This agreement implies that we can approximate the asymptotic states of this system by using . (coarse graining)
Since is not a pure state, the von Neumann entropy is non-zero.
→ The equilibration causes an entropy production.
(Although the original state is pure state and the entropy=0.) Our result shows that
Entropy is proportional to N.
(N fermion system)
◆ Interpretation as a black hole formulation
Equilibration and Generalized Gibbs Ensemble
NOTE: It is unclear at all that any dual black hole exists in our model.
Suppose the time evolution of, for example, N=4 SYM is the same to our model, we can propose the following arguments.
(Pre)Black hole formulation
N=∞ achieves the GGE state.
→ A black hole is formulated.
Finite N starts the recurrence.
→ Hawking radiation + back reaction to BH (?) Approximation by GGE is
not so good in the finite N case, since it keeps on oscillating.
1. Introduction
2. Review of the single trace matrix model
3. Time evolution of the single trace matrix model 4. Equilibration and Generalized Gibbs Ensemble 5. One way transition in the matrix model
6. Summary
Plan of today's talk
◆ Quantum quench dynamics (Result at N=120)
We observed two important features.
1. The initial big oscillations subtle down to the small ripples.
→ Equilibration would happen even in free and closed system.
(c.f. Black hole formulation)
2. The dynamical transition occurs only from the gapless to gapped case.
(We will argue a connection to a Gregory-Laflamme transition.)
The transition happens. No transition.
One way transition in the matrix model
◆ Quantum quench dynamics (Result at N=120)
We observed two important features.
1. The initial big oscillations subtle down to the small ripples.
→ Equilibration would happen even in free and closed system.
(c.f. Black hole formulation)
2. The dynamical transition occurs only from the gapless to gapped case.
(We will argue a connection to a Gregory-Laflamme transition.)
The transition happens. No transition.
Why does the dynamical transition occur only in the one way?
One way transition in the matrix model
◆ One way nature
gap
One way transition in the matrix model
OK
NG
◆ One way nature
gap
One way transition in the matrix model
OK
NG To understand this issue, phase space density is useful.
◆ Phase space density analysis
One way transition in the matrix model
Let us consider a single classical particle in the cos potential.
This motion can be described in the (θ,p) phase space as
One way transition in the matrix model
Let us consider a single classical particle in the cos potential.
N fermions
At large-N, each point in the droplet(s) obeys the classical single particle equation of motion.
Droplet(s) dynamics = N fermion dynamics at large-N
◆ Phase space density analysis
This motion can be described in the (θ,p) phase space as
One way transition in the matrix model
N fermions
◆ Phase space density analysis
: The upper and lower surface of the droplet
We can calculate the fermion density ρ(θ,t) from the droplet through a projection onto the θ coordinate.
Note: Total area of the droplet is → consistent with
◆ One way nature from the phase space droplet analysis
gap
One way transition in the matrix model
OK
NG
◆ One way nature from the phase space droplet analysis
One way transition in the matrix model
NG
However, since the topology of the droplet cannot change at large-N, the gapless ground state cannot evolve to a gapped state. → no transition.
◆ One way nature from the phase space droplet analysis
gap
One way transition in the matrix model
OK
Without the change of the topology, the gap can be filled → transition OK.
◆ One way nature from the phase space droplet analysis
One way transition in the matrix model
Note that such a phase space description is valid independent of the details of the unitary matrix model as far as the kinetic term is local.
(Even non-integrable cases will be OK. )
→ The one way nature of the transition is quite universal.
◆ One way nature from the phase space droplet analysis
One way transition in the matrix model
Q. What happens if we add quite strong force to split the droplet?
The `neck' region becomes quite thin and the classical analysis would violate if the thickness achieves O(1/N).
Through a quantum effect, the gap may appear and the dynamical transition may occur.
◆ One way nature from the phase space droplet analysis
One way transition in the matrix model
Q. What happens if we add quite strong force to split the droplet?
The `neck' region becomes quite thin and the classical analysis would violate if the thickness achieves O(1/N).
Through a quantum effect, the gap may appear and the dynamical transition may occur.
Similar nature is known in the Gregory-Laflamme transition too.
◆ One way nature of the Gregory-Laflamme transition
One way transition in the matrix model
Gregory-Laflamme transition: Black string/Black hole transition on
Size of the circle
This dynamical transition can happen. This cannot happen in classical gravity.
◆ One way nature of the Gregory-Laflamme transition
One way transition in the matrix model
Gregory-Laflamme transition: Black string/Black hole transition on
Size of the circle
The horizon becomes quite thin, and a naked singularity appears when the horizon is pinched off.
This cannot happen in classical gravity.
◆ One way nature of the Gregory-Laflamme transition
One way transition in the matrix model
If we identify: GL transition horizon
thin horizon/naked singularity quantum effect in GR
Matrix Model fermion density thin (O(1/N)) fermion density
1/N effect
Our matrix model predicts a smooth transition through the quantum effect.
→ Resolution of the naked singularity in the quantum gravity.
The horizon becomes quite thin, and a naked singularity appears when the horizon is pinched off.
This cannot happen in classical gravity.
◆ One way nature of the Gregory-Laflamme transition
One way transition in the matrix model
If we identify: GL transition horizon
thin horizon/naked singularity quantum effect in GR
Matrix Model fermion density thin (O(1/N)) fermion density
1/N effect
Our matrix model predicts a smooth transition through the quantum effect.
→ Resolution of the naked singularity in the quantum gravity.
The horizon becomes quite thin, and a naked singularity appears when the horizon is pinched off.
This cannot happen in classical gravity.
quantum effect
Summary
• N=∞ : Equilibration to the GGE, and the entropy production.
finite N: Tends to equilibrate but the recurrence starts later.
• N=∞ : One way nature of the dynamical phase transition.
Gap can close but cannot appear dynamically.
finite N: Smooth transition may occur only through the 1/N effect.
Through the quantum quench dynamics, we observe several natures of the time evolution of the unitary matrix model at N=∞ and N < ∞.
N=∞ is qualitatively different from the finite N case.
In the dual gravity (if exist), these qualitative differences are connected to the differences between the classical and quantum gravity.
Similar evolution can be observed in the double trace model, which is an effective theory of N D2 brane too.
Summary
• Application to the non-critical string theory by taking the scaling limit.
• Application of GGE to integrable systems.
• Application to the HS theory.
Especially our entropy is O(N) and the HS BH may have O(N) entropy too.
•The role of the critical point in the quenched dynamics
• Understanding of the breaking of the integrablity by adding operators.
GGE will break to a standard thermodynamics.
Future directions
Summary
• Application to the non-critical string theory by taking the scaling limit.
• Application of GGE to integrable systems.
• Application to the HS theory.
Especially our entropy is O(N) and the HS BH may have O(N) entropy too.
•The role of the critical point in the quenched dynamics
• Understanding of the breaking of the integrablity by adding operators.
GGE will break to a standard thermodynamics.
Future directions
