Matrix Quantum Mechanics from Qbits

Sean Hartnoll (Stanford)
Yukawa Institute for Theoretical Physics
Kyoto — 2016
Based on: arXiv/1606.xxxx

Collaborators

Liza Huijse(Stanford)

Edward Mazenc(Stanford)

Preliminary comments•

Deep aspects of quantum gravity (Bekenstein-Hawking,Ryu-Takayanagi, …) have been gleaned from the semiclassical path integral.

•Nonetheless, ultimately, understanding the emergence ofspacetime, ‘It from Qbit’, will involve a characterization ofan underlying microscopic entanglement structure thatsupports emergent local dynamics.

•Emergent locality is somewhat different from MPS,
MERA, PEPS, etc, whose role is to put manifest locality
to a powerful use.

“It from Qbit”, explicit version•This talk is about an explicit realization of emergent local
dynamics from a quantum spin system.

•Logic: map the spin system onto a matrix quantum
mechanics that is known to lead to emergent local
dynamics at a quantum critical point.

Ising spins

Matrix quantum mechanics

Spacetime“It from Qbit”, explicit version•The emergent local dynamics will be rather simple for themodel we solve: a scalar field in 1+1 dimensions.

•However, start to get a feel for various nontrivial steps
that arise in obtaining a matrix quantum mechanics
structure from ‘atoms of entanglement’ (Qbits/spins).

•Previously shown how the accumulation of entanglement
necessary for emergent locality arises in this case: it iscontained in the Fermi Sea structure of the eigenvalue
dynamics [1504.07985, SAH + Mazenc; Das ’95].

Transverse field Ising models•The usual transverse field Ising model isXX
H= hSix + v2Siz Sjz
i| !! · · · !i<ij>| "#" · · · "iZ2 symmetry broken

Continuous quantum phase transition
Gapless excitations Continuum limit described by CFT.

Transverse field Ising models•~AB , interacting nonlocally:
A symmetric matrix of N2 spins, SX
Xv
4zzzz
x
SAB SBC SCD SDA
H= hSAB +NABABCD•

Symmetries:(1) flipping the spin of rows and columns:(2) permuting rows and columns: SN•

Classical model (h=0) studied by[Cugliandolo-Kurchan-Parisi-Ritort, cond-mat/9407086]
has ‘crystalline’ order at low T, preempted by glassiness.

Transverse field Ising models
Our results for this model at large N:•

First order symmetry breaking transition at critical v4/h.

•

Disordered large N state is described by a matrix
quantum mechanics for all couplings.

•Disordered large N state undergoes a continuous
quantum phase transition as a function of v4/h.

•

At the quantum critical point, the low energy excitations
are described by massless scalar field in 1+1 dimensions.

Transverse field Ising models
TThermallydisordered

GlassyregimeQuantum
critical point

Quantum
disordered00h/v4

Step 1: Suzuki-Trotter decomposition
Path integral for spins:
Z = Tr e=X

1 2{±1}⇡=X

1 2{±1}X

1 2{±1}H

N2N2

N2·········X

M 2{±1}X

M 2{±1}X

M 2{±1}

N2N2N

M
Ym=1M
Yh m |e ✏H | m+1 i ,e ✏ trV ( m )

m=18
<YA,B

M
X

0✏⌘

M

⌧1xh m |1 + ✏ h SAB| m+1 ie X
K@tr V ( m ) +✏exp:
22m=1A,BABm19
AB 2 =m+1A .
2;✏Step 2: Constrained bosonsIntroduce continuous boson field:X

1 2{±1}

N2···XM 2{±1}

F( ) =N2=ZZYdm,A,BYm,A,BdABm⇣⌘
1 F( )AB 2m

ABABdµmm eiµABm⇣(AB 2m)1⌘F( ),

Step 3: Continuum limit (in time)
Send time step to zero, but remember time step was
introduced by hand, not a microscopic parameter:e X
K2A,BAB 2m+1ABm

✏2K!
2✓dd⌧

◆2

✓ 2 ◆2K 1 d++ ··· .
222 hd⌧0Microscopic energyscale to flip a spin
Derivative expansion can be truncated when excitations withenergy ΔE ≪ h are present: quantum critical point.
Crucially, can also truncate at weak coupling (free spins)
and strong coupling (classical spins).

Step 4: Matrix Quantum Mechanics Saddle!
So far:
Z=Z

D Dµ exp( Z"d⌧ tr0K2✓dd⌧◆2+V( )!

+iXABµAB

⇣

AB 2⌘
1#)

.

Not SO(N) invariant
Import a trick from [Cugliandolo-Kurchan-Parisi-Ritort, ’94]:•At large N, a consistent saddle point has μAB = μ.

•In classical spin model, this saddle was shown todescribe the high T disordered phase.

•Perhaps captures the quantum disordered wavefunction?

Step 4: Matrix Quantum Mechanics Saddle!
Thus:
Z=Z

D Dµ exp( Zd⌧ tr0K2✓

dd⌧◆2+µ2+V( )µN!)

Expectation: If there is continuous QPT in the disordered state,
its critical dynamics is described by the above MQM.
Remainder:•Find the ground state of the above.
Show that it correctly describes the disordered state.

•Show that there is a continuous QPT.

•Show that low energy excitations: 1+1 scalar field.

•Step 5: Hamiltonian and ground state•

Hamiltonian: Exercise in constrained quantization.
[Dirac brackets, secondary constraints etc.]•Go to eigenvalue basis (non-singlet modes decouple atthe quantum critical point [Gross-Klebanov ’91]):= OT ⇤O ,•⇤ij =i ij

Change variables to collective field [Jevicki-Sakita ’80]:⇢( , t) =NX
i=1(i (t))

Step 5: Hamiltonian and ground state
The collective field Hamiltonian and constraints are:

•H=Q1 =Q2 =•ZZZd ⇢( )

21⇡@ ⇡( )@ ⇡( ) +⇢( )2 + V ( )2K
12!

12KN 2✓Zd ⇢( ) @ ⇡( )d ⇢( ) = N
d2Ensures dQ2/dt = 0⇢( ) = N 2

Ground state eigenvalue distribution has π=0.
0.5

0.4

0.3
ρ
N
0.2

0.1

0.0
-1.5

-1.0

-0.5

0.0
λ
N

0.5

1.0

1.5

◆2

Step 6: Existence of continuous QPT•As v4 → 0 (quantum disordered spins), EMQM matches thespin system ground state energy to 3rd order in
perturbation theory [Expect disagreement at 4th order]:1 E0=216h N1v4+21616h8⇣ v ⌘2416h+ 256⇣ v ⌘3416h14848⇣ v ⌘4416h+ ···•Confirms MQM describes the disordered state.

•As v4 → ∞, EMQM matches the energy of (subdominant)classical disordered state of Cugliandolo et al.:
E0= v4 + · · ·2N

Step 6: Existence of continuous QPT•MQM eigenvalue distribution is connected at small v4 anddisconnected at large v4.
‘Topological’ QPT at intermediate coupling.
0.5

0.40.3
ρ
N
0.2

0.1

0.0
-1.5

-1.0-0.50.0λN0.5

1.01.5

Step 7: Emergent local dynamics•Critical dynamics of topological phase transitions in MQM
are well understood [eg. Das-Jevicki ’90], ripples of theeigenvalue distribution:⇢ = ⇢( ) + ⇢( , t)•

Key step: canonical transformation on the fluctuations@•⇡ ⇠ ⇡ .

Quadratic Hamiltonian becomes:H•⇢ ⇠ @q ,(2)

⇡=4K

Z

dq (P̄ ⇡ )2 + (@q )2

Excitations described by free 1+1 scalar field!

Step 7: Emergent local dynamics•At weak coupling, fluctuations of the collective field canbe related to particular spin excitations.

•Given by SN singlet states with n << N spins flipped along
diagonal:!
!
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!

!
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! ! !
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! ! !

+

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+

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! !
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+ ···

Conclusions•

Shown that Matrix Quantum Mechanics can arise atquantum critical points in transverse field Ising models.

•Analogous to how Ising CFTs arise, but dynamics is notin space but in the large N matrix interactions.

•Critical dynamics of these systems: free scalar field in
emergent 1+1 dimensional spacetime.

•In the disordered phase, N2 constraints can be relaxed toa single constraint:
Xz
2z
22(SAB ) = 1!
(SAB ) = N .
AB