Low-lying excitations of quantum spin-glasses Koh Yang Wei - - PowerPoint PPT Presentation

Low-lying excitations of quantum spin-glasses

Koh Yang Wei International workshop on numerical methods and simulations for materials design and strongly correlated quantum matters March 24-25, 2017. Kobe, Japan.

A ‘Personal’ Motivation

Year: 1998 Year: 2008 Size: 8 spins Size: 24 spins Di culty: Dimension of Hilbert space increases exponentially with system size

Ordered spin systems are easier

2 4 1 2 3 4 gap G/J

4 2 2 4

For spin-glasses: Ferromagnet: Conserved quantity:

Hartree-Fock and Configuration Interaction Theory

Hartree-Fock approximation Con guration Interaction

Wavefunctions of individual spins factorize.

all 1-spin excitations

...etc.

all 2-spin excitations all k-spin excitations

Hilbert space basis

}

SK model and its Hartree-Fock Approximation

Sherrington-Kirkpatrick (SK) model: H = −

• i>j

Jijσz

i σz j − Γ

• i

σx

i .

(1) Prob(Jij) = Gaussian. Hartree-Fock wavefunction: |0 =

N

• i=1

αi βi

• .

(2) We minimize E HF = 0|H|0, (3) with respect to {αi, βi}, subjected to α2

i + β2 i = 1.

HF Energy, HF Equations, and stability matrix

HF energy: E HF = −

• i>j

Jij(α2

i − β2 i )(α2 j − β2 j ) − 2Γ

• i

αiβi. (4) Stationary conditions, ∂E HF

∂αi

= 0 (HF equations): 2Γ(2α2

i − 1)

• 1 − α2

i

− 4αi

• a=i

Jia(2α2

a − 1) = 0.

(5) Paramagnetic solution: αpara = (1/ √ 2, · · · , 1/ √ 2). Its stability: ∂2E HF ∂αi∂αj

• αpara

= 8(Γδij − Jij). (6)

Transition into ordered phase: A HF description

0.4 0.8 1.2 1.6 2 10 100 1000 Critical G N 0.5 1 1.5 0.5 1 1.5 2

100 1000 10 0.4 1.2 2.0

1 2 0.5 1.0 1.5

Hartree- Fock full quantum

full quantum

transition

Comparing E HF with exact E0

0.4 0.8 2 4 excess energy Gamma

0.8 0.4 00 2 4

Generating Excitations

Since H does not involve y-direction, σy α β

• = flips the spinor

α β

• .

(7) To excite ith spin of |0, |i = σy

i |0.

(8) To excite ith and jth spins of |0, |ij = σy

i σy j |0.

(9) etc. We generate a subspace spanned by {|0, |i, |ij}.

Configuration Interaction Matrix

truncated CI matrix

• dim. of matrix: O(N )

2

Improvement of E CI over E HF

0.5 1 2 4 excess energy Gamma

2 4 0.5 1

Correction to extensive part of E0

• 4

2 4 average of smallest eigenvalue Gamma

• 4

2 4

4

Scaling of sub-extensive correction to E HF

1 10 10 100

• (mean of smallest eigenvalue)

N

10 1 10 100

Energy gap

The first excited-state is quite complex...

1000 2000 3000 4000 5000 6000 1 2 3 4 5 6 counts mu

1 2 3 4 5 6

FREQUENCY Classical SK energy:

Flip spins 1 spin 2 spins 3 spins etc.

We simplify by assuming ν = 1 for all Jij.

A Formula for the Energy Gap

Energy gap: ∆ = E1 − E0 (10) Consider an ‘excitation’ operator A: |E1 = A|E0. (11) We define a generating function: G(γ) = E0|e−iγAHeiγA|E0. (12) Expanding e±iγA, γ0C0 + γ1C1 − γ2 2 E0|HA2 + A2H − 2AHA|E0 + O(γ3) (13) Expanding G(γ), and equating: ∆(|E0, A) = 1 2 1 E0|A2|E0 ∂2G ∂γ2

• γ=0

. (14) Only |E0 is needed! Use approximate HF/CI wavefunctions. But how do we compute ∂2G/∂γ2?

Example: Let |E0 = |0. Let A = A1.

Let A = A1 =

N

• i=1

yiσy

i ,

(15) yi: parameters. We want G HF

1

(γ) = 0|e−iγA1HeiγA1|0. |¯ 0 = eiγA1|0 =

• i

eiγyiσy

i

αi βi

• =
• i

¯ αi(γ) ¯ βi(γ)

• .

(16) So G HF

1

(γ) = ¯ 0|H|¯ 0 = E HF(¯ αi(γ), ¯ βi(γ)). (17) Hence ∆HF

1

= 1 2 ∂2E HF(γ) ∂γ2 = −8

• i=j

Jijαiαjβiβjyiyj + Γ

• i

y2

i

αiβi . (18) Minimize ∆HF

1

with respect to {yi} to obtain gap.

Small N: Comparing with full quantum

1 2 3 1 2 Energy gap Gamma

1 2

3 2 1

Average HF gap

0.2 0.4 0.6 0.8 1 2 average energy gap Delta_1 Gamma

0.2 0.4 0.6 0.8 2 1

Scaling of gap near critical point

0.01 0.1 1 10 100 1000

10 100 1000 0.01 0.1 1.0

SK ferro

Some speculations...

Complexity of gap in the glass phase...

1 2 1 Energy gap Gamma

1

1 2

?

Different A’s for different regimes?

1 2 1 Energy gap Gamma

1

1 2 2 spins, 3 spins etc.? 1 spin regime e.g.:

‘Hartree-Fock’ annealing?

Simulated annealing: Quantum annealing:

thermal uctuations quantum uctuations

'Hartree-Fock' annealing:

• 2
• 1

2 Zero-point energy G/J

energy

annealing parameter

full quantum

Possible merits of HF annealing

• 1. No operators are involved. Recall that for SK model

E HF = −

• i>j

Jij(α2

i − β2 i )(α2 j − β2 j ) − 2Γ

• i

αiβi αi, βi are just numbers. Simpler than annealing ˆ H itself.

• 2. Dependence on annealing parameter (Γ) is simple.

Simpler than simulated annealing.

• 3. Hardware implementation of E HF using a classical

machine?