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