Deep Learning and Physics 2018, Osaka Reverse Engineering Hamiltonian from Spectrum Hiroyuki Fujita, Insitute for Solid State Physics Phys. Rev. B 97 , 075114 (2018) � 1
Yuya. O. Nakagawa (PhD @ ISSP -> Fintech company) S. Sugiura (PD @ ISSP -> PD @ Harvard) M. Oshikawa (Prof. @ ISSP, now @ Wien) Collaborators � 2
something NOT deep Deep to be interesting? I am going to talk about � 3
512×512 256×256 Simple but not efficient → principal component analysis (PCA) (1) (2) Optimization maximizing the data variance e.g. (1) is better in the variance of the projected data PCA is to find a subspace onto which e.g. Simple average of nearby pixels the data are projected. Image compression � 4
2 states/site AFM Heisenberg model half-filling, large U 4 states/site What is “PCA” for this problem? Low-energy model = “compressed image” Hubbard model “Image compression” in cond. mat. ? 4 L ✓ t 2 ◆ 2 L O U 2 Perturbation theory 2 L 4 L � 5
??????? from sklearn.decomposition import PCA pca = PCA(n_components = ncomp) pca.fit(Hubbard_H) pca_res = pca.transform(Hubbard_H) Spin_H = pca.inverse_transform(pca_res) ??????? e.g. L = 10 Hubbard chain trillion pixel image million pixel image We cannot simply use the (variant of) existing algorithms. Totally unrealistic! Straightforward(?) approach � 6
Reconstruct the matrix Hermitian matrix based on the low-energy properties of the Hubbard model The “PCA” has to construct the effective model = low-energy spectrum “PCA” for low-energy physics of Hubbard N × e e A = N N ≤ e N A Can we solve the “ inverse problem ” of diagonalization? � 7
Data Number of independent params of Parameters # of params in # of eigenvalues Number counting Exact Diagonalizaiton (ED) N × e e A = N N × e is e A N Number of eigenvalues available is at most e N A = O ( e N × e N ) ∞ A = O ( e e N ) N → ∞ Too many parameters…… � 8
“Being physical” is a powerful constraint on the Hamiltonian Short-ranged interactions Few-body interactions Symmetries such as U(1), SU(2), … # of params in # of eigenvalues Dimension of Hilbert space is exponential in the system size Data Parameters (up to n-body) Solvable because it’s physics! Physicists point of view L → ∞ 0 H = O ( L n ) � 9
Original Hamiltonian ED Low-energy spectrum (ED) -1 Low-energy Hamiltonian e.g. Hubbard model @ half-filling e.g. AFM Heisenberg model Quantum state (w.f.) ED entanglement spectrum (ED) -1 Entanglement Hamiltonian What can we do with (Exact Diagonalization) -1 � 10
Spin model with parameters to be optimized What we have to do is to 1. find a proper form of the ansatz 2. find the proper parameters for the fixed ansatz Correlated electrons in 1D Make the problem concrete E 1 , ......, E N (Small number of eigenvalues) ED X H = c i H i ED -1 i =1 ,...,M Heisenberg, four-spins interactions, … � 11
Formulate as a supervised learning problem ED E 0 1 , ......, E 0 E 1 , ......, E N X H = c i H i N i =1 ,...,M = prediction = data = model Gradient descent algorithm N 1 (1) Define cost function X i − E i ) 2 Cost( E, E 0 ( { c j } )) = ( E 0 2 N i =1 (2) Calculate the gradient in terms of the parameters c j = c j − α ∂ (3) Update the parameters as Cost( E, E 0 ( { c j } )) ∂ c j E 1 , ......, E n cost evaluation Cost E 0 1 , ......, E 0 n gradient descent diagonalization X H = c i H i c j i =1 ,...,M � 12
What we have to do is to trivial How? ∴ For a given ansatz, we can optimize its parameters 2. find the proper parameters for the fixed ansatz 1. find a proper form of the ansatz “Quantum computation” of the gradient ✓ ∂ ◆ ∂ ◆ ∂ Cost( E, E 0 ( { c j } )) = E 0 ∂ E 0 Cost( E, E 0 ( { c j } )) ∂ c j ∂ c j Luckly, we know the perturbation theory of quantum mechanics: X c i H i + δ c j H j E i ! E i + δ c j h Ψ i | H j | Ψ i i H → i =1 ,...,M X E 0 1 , ......, E 0 H = c i H i N i =1 ,...,M � 13
are nonzero Both and L1 norm regularization Model selection by regularization X Cost( E, E 0 ) → Cost( E, E 0 ) + λ | c j | Prefer small parameters j =1 ,...,M contours of MES contours of MES contours of reg. term contours of reg. term Cost( E, E 0 ) Cost( E, E 0 ) “Sparse” nature of the estimation → Model selection � 14
“Important” terms should survive under strong λ optimized A with regul. Spin model ansatz w/o regul. trapped by local minima Model selection based on the insensitivity to the regularizationλ ? Effective model of optimized A w/o regul. Demonstration: Hubbard chain at half-filling ( S i · S i +1 )( i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 ( S i · S i +1 )( S i +2 · S i +3 ) i i + 1 i + 2 i + 3 K 1 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 K 2 i i + 1 i + 2 i + 3 i i + 1 K 3 i + 2 i + 3 i i + 1 i + 2 i + 3 � 15
λ Simplified model Hierarchical structure ～ order of importance? insensitivity to the regul. → Estimate parameters again WITHOUT regul. Hierarchical structure L=10, total Sz = 2, U = 8, n = 50, α = 0.01 ( S i · S i +1 )( i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 K 2 i i + 1 i + 2 i + 3 K 3 i i + 1 i + 2 i + 3 i i + 1 i + 2 i + 3 � 16
Estimation is correct up to Deviation from the perturb. Cross validation error Difference betw. ML & Macdonald et. al. Comparison with perturbation theory Perturbation theory A. H. MacDonald, S. M. Girvin, and D. Yoshioka, PRB (1988). A. Rej, D, Serban, and M. Staudacher, JHEP (2006) e J i = J i − J p i ∝ U − 7 e K i = K i − K p i ∝ U − 14 � 17 L=10, α = 0.1, Sz = 2, n=50
preserving the 99.9999% of the essential information (for U=10). trillion pixel image million pixel image We demonstrated the following “image compression” original (30 MB) 5KB image cf) something From the viewpoint of image compression � 18
Quantum state (w.f.) ED entanglement spectrum (ED) -1 Entanglement Hamiltonian Physical Hamiltonian ED DMRG etc. = spectrum of RDM quantum state reduced density matrix entanglement spectrum Entanglement Hamiltonian Next application: entanglement � 19
B A A Bath ・Gibbs ensemble ・Reduced density matrix ・Thermodynamic entropy ・Entanglement entropy thermal fluctuation from the bath quantum fluctuation from B e.g. entangled two-spins A B Thermodynamics from pure quantum state � 20
Pushing forward the analogy Entanglement Hamiltonian e.g. entangled two-spins A B Physical Hamiltonian free spin spectrum of (ED) -1 “log (matrix)” Entanglement Hamiltonian � 21
defined for “physical” cut B defined for “virtual” cut e.g. B A A As long as the ent-edge corresp. holds, we can estimate EH A EH is local & few-body? Similar to physical edge Hamiltonian Entanglement-edge correspondence A Symmetries such as U(1), SU(2), … Few-body interactions Short-ranged interactions Construction of Entanglement Hamiltonian True for ? see Phys. Rev. B 97 , 075114 (2018) � 22
w/o transl. symm. e.g. magnetization conservation Hilbert space dimension # of evals. > # of params. We can reduce dim. as long as # of up spins RDM → ES ES → EH w/ transl. symm. ・use of symmetry High-spin sector → small Hilb. space ・only a small number of eigenvalues → memory efficient algos. (e.g. Lanczos) Our method: approximate but computationally cheap Computational cost RDM → EH → strong constraint: memory size < matrix dimension ・need all the eigenvectors of the RDM Full diag. of RDM: exact but computationally hard chain Comparison with other method 10 5 10 4 1000 100 10 1 5 10 15 20 � 23
arXiv:1803.10856 arXiv:1712.03557 Not just a “Hello, World” ! � 24
Levin-Wen (2005) Materials design e.g. string-net condensation (lattice gauge) e.g. Q. dimer model with exotic quantum state e.g. Honeycomb iridates e.g. Kitaev model e.g. Majorana fermion ED (ED) -1 unknown parent Hamiltonian = low energy spectrum of Energy spectrum physical d.o.f (electrons, spins) Parent Hamiltonian written with Model of emergent d.o.f Outlook: Materials design for exotic quantum states Models written with emergent d.o.f (e.g. Majoranas, dimers, lattice gauge fields) � 25
Summary We propose a scheme to construct Hamiltonians from a given spectrum H E 1 , E 2 , ......, E N Low energy Hamiltonian Energy spectrum Parent Hamiltonian Entanglement spectrum Entanglement Hamiltonian Phys. Rev. B 97 , 075114 (2018) � 26
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