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

Ordered spin systems are easier Ferromagnet: Conserved quantity: 4 4 gap 2 2 0 0 0 2 4 1 2 3 4 G/J For spin-glasses:

Hartree-Fock and Configuration Interaction Theory Hartree-Fock approximation Wavefunctions of individual spins factorize. Con guration Interaction Hilbert space } all k-spin excitations all 2-spin excitations all 1-spin ...etc. basis excitations

SK model and its Hartree-Fock Approximation Sherrington-Kirkpatrick (SK) model: � J ij σ z i σ z � σ x H = − j − Γ i . (1) i > j i Prob ( J ij ) = Gaussian . Hartree-Fock wavefunction: N � α i � � | 0 � = . (2) β i i =1 We minimize E HF = � 0 | H | 0 � , (3) with respect to { α i , β i } , subjected to α 2 i + β 2 i = 1.

HF Energy, HF Equations, and stability matrix HF energy: E HF = − � J ij ( α 2 i − β 2 i )( α 2 j − β 2 � j ) − 2Γ α i β i . (4) i > j i Stationary conditions, ∂ E HF = 0 (HF equations): ∂α i 2Γ(2 α 2 i − 1) � J ia (2 α 2 − 4 α i a − 1) = 0 . (5) � 1 − α 2 a � = i i √ √ Paramagnetic solution: α para = (1 / 2 , · · · , 1 / 2). Its stability: ∂ 2 E HF � � = 8(Γ δ ij − J ij ) . (6) � ∂α i ∂α j � α para

Transition into ordered phase: A HF description 2.0 2 1.5 1.5 full quantum 1.6 1.0 1 Hartree- Critical G 1.2 1.2 Fock 0.5 0.5 transition 0 0 0.8 0 1 2 0 0.5 1 1.5 2 full quantum 0.4 0.4 10 1000 100 10 100 1000 N

Comparing E HF with exact E 0 0.8 0.8 excess energy 0.4 0.4 00 0 2 4 0 2 4 Gamma

Generating Excitations Since H does not involve y -direction, � α � � α � σ y = flips the spinor . (7) β β To excite i th spin of | 0 � , | i � = σ y i | 0 � . (8) To excite i th and j th spins of | 0 � , | ij � = σ y i σ y j | 0 � . (9) etc. We generate a subspace spanned by {| 0 � , | i � , | ij �} .

Configuration Interaction Matrix truncated CI matrix 2 dim. of matrix: O(N )

Improvement of E CI over E HF 1 1 excess energy 0.5 0.5 0 0 4 2 0 0 2 4 Gamma

Correction to extensive part of E 0 0 0 average of smallest eigenvalue -4 -4 0 0 4 0 2 4 0 2 4 Gamma

Scaling of sub-extensive correction to E HF 10 10 -(mean of smallest eigenvalue) 1 1 10 100 10 100 N

Energy gap

The first excited-state is quite complex... Classical SK energy: 6000 Flip FREQUENCY 1 spin 5000 spins 4000 counts 3000 2 spins 3 spins etc. 2000 1000 0 1 2 3 4 5 6 1 2 3 4 5 6 mu We simplify by assuming ν = 1 for all J ij .

A Formula for the Energy Gap Energy gap: ∆ = E 1 − E 0 (10) Consider an ‘excitation’ operator A : | E 1 � = A | E 0 � . (11) We define a generating function: G ( γ ) = � E 0 | e − i γ A He i γ A | E 0 � . (12) Expanding e ± i γ A , γ 0 C 0 + γ 1 C 1 − γ 2 2 � E 0 | HA 2 + A 2 H − 2 AHA | E 0 � + O ( γ 3 ) (13) Expanding G ( γ ), and equating: ∂ 2 G � ∆( | E 0 � , A ) = 1 1 � . (14) � � E 0 | A 2 | E 0 � ∂γ 2 2 � γ =0 Only | E 0 � is needed! Use approximate HF/CI wavefunctions. But how do we compute ∂ 2 G /∂γ 2 ?

Example: Let | E 0 � = | 0 � . Let A = A 1 . Let N � y i σ y A = A 1 = i , (15) i =1 y i : parameters. ( γ ) = � 0 | e − i γ A 1 He i γ A 1 | 0 � . We want G HF 1 � α i � � ¯ α i ( γ ) � e i γ y i σ y | ¯ � � 0 � = e i γ A 1 | 0 � = = . (16) i ¯ β i β i ( γ ) i i So α i ( γ ) , ¯ G HF ( γ ) = � ¯ 0 | H | ¯ 0 � = E HF (¯ β i ( γ )) . (17) 1 Hence ∂ 2 E HF ( γ ) y 2 = 1 � � ∆ HF i = − 8 J ij α i α j β i β j y i y j + Γ . (18) 1 2 ∂γ 2 α i β i i � = j i Minimize ∆ HF with respect to { y i } to obtain gap. 1

Small N : Comparing with full quantum 3 3 2 2 Energy gap 1 1 0 0 0 1 2 1 2 Gamma

Average HF gap 0.8 0.8 0.6 average energy gap Delta_1 0.6 0.4 0.4 0.2 0.2 0 0 0 1 2 1 2 Gamma

Scaling of gap near critical point ferro 1.0 1 0.1 0.1 SK 0.01 0.01 10 100 1000 10 100 1000

Some speculations...

Complexity of gap in the glass phase... 2 2 Energy gap 1 1 ? 0 0 0 1 1 Gamma

Different A ’s for different regimes? 2 2 1 spin 2 spins, 3 spins etc.? regime e.g.: Energy gap 1 1 0 0 0 1 1 Gamma

‘Hartree-Fock’ annealing? Simulated annealing: thermal uctuations quantum Quantum annealing: uctuations 'Hartree-Fock' annealing: energy -1 Zero-point energy full quantum -2 2 annealing parameter G/J

Possible merits of HF annealing 1. No operators are involved. Recall that for SK model E HF = − � � J ij ( α 2 i − β 2 i )( α 2 j − β 2 j ) − 2Γ α i β i i > j i α i , β i 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?