Machine learning quantum phases of matter

Transcription

1 Machine learning quantum phases of matter Summer School on Emergent Phenomena in Quantum Materials Cornell, May 2017 Simon Trebst University of Cologne

2 Machine learning quantum phases of matter Summer School on Emergent Phenomena in Quantum Materials Cornell, May 2017 Simon Trebst University of Cologne

3 Collaborators Juan Carrasquilla Roger Melko Waterloo / Perimeter Peter Bröcker Cologne Fakher Assaad Würzburg

4 Supervised learning approach General setup Consider some Hamiltonian, which as a function of some parameter λ exhibits a phase transition between two phases. Supervised learning approach 1) train convolutional neural network on representative images deep within the two phases 2) apply trained network to images sampled elsewhere to predict phases + transition step 1 train here phase A phase transition phase B train here step 2 predict phases by applying neural network here What are the right images to feed into the neural network?

5 classical phases of matter Finite-temperature transition in the Ising model H = J X hi,ji S z i S z j Ernst Ising

6 classical phases of matter Finite-temperature transition in the Ising model H = J X hi,ji S z i S z j high temperature critical temperature low temperature

7 classical phases of matter Carrasquilla and Melko, Nat. Phys. (2017) LETTERS Finite-temperature transition in the Ising model a H= J X hi,ji 1.0 Siz Sjz b 0.6 Output layer L = 10 L = 20 L = 30 L = L = 60 Input Output Hidden 1.5 Pirsa: Output layer T/J

8 classical phases of matter Carrasquilla and Melko, Nat. Phys. (2017) X z z Finite-temperature transition in the Ising model H = J S Sj NATURE PHYSICS DOI: /NPHYS4035 i ETTERS L = 20 L = 30 L = b L = 60 Output L = L = L = 10 L = 20 L = 30 L = T/J 3.0 L = tl1/ 5 c f /L 0.8 L = 60 Output e L = tl1/ T /J 2.0 L = Input 0.6 Hidden 0.4 Hidden T/J Output layer 2.32 NATURE PHYSICS DOI: /NPHYS 0.6 Output layer Input hi,ji T /J 0.6 Output layer L = d c 0.8 Output layer Output layer RS.0 b T /J a /L 0.10

9 Output layer d classical L phases = 10 of matter Finite-temperature transition in the Ising model H = J X Hidden Si z Sj z hi,ji Output layer Input Output L = 20 L = 30 L = 40 L = T/J L = 10 L = 20 L = 30 L = 40 L = 60 Output layer e Output layer tl 1/ T /J T /J f Carrasquilla and Melko, Nat. Phys. (2017) /L T/J tl 1/ /L ure 1 Machine learning the ferromagnetic Ising model. a, The output layer averaged over a test set as a function of T/J for the square-lattice romagnetic Ising model. The inset in a displays a schematic of the fully connected neural network used in our simulations. b, Plot showing data collap the average Transfer output layerlearning: as a function of A tlneural 1/, wherenetwork t=(t T c )/J trained is the reduced on the temperature. square Linear lattice systemising sizes L=10,20,30,40 model applied and 60 are resented by crosses, up triangles, circles, to diamonds the triangular and squares, Ising respectively. model c, Plot correctly of the finite-size identifies scaling of its the transition. crossing temperature T /J (down ngles). d f, Analogous data to a b, but for the triangular Ising ferromagnet using the neural network trained for the square-lattice model. The vertica nge lines signal the critical temperatures of the models in the thermodynamic limit, T c /J=2/ln(1+ p 2) for the square lattice 17 and T c /J=4/ln3 for ngular lattice 19. The dashed vertical lines represent our estimates of T c /J from finite-size scaling. The error bars represent one standard deviation

10 classical phases of matter Carrasquilla and Melko, Nat. Phys. (2017) A High temperature state B Ising square ice ground state C Ising lattice gauge theory " # " #v # # p " " More interestingly, the convolutional neural network can also be trained to distinguish the high-t paramagnet from a Coulomb phase or loop gas ground state, i.e. phases without a local order parameter.

11 quantum systems

12 Dirac fermions Hubbard models on the honeycomb lattice Spinful fermions H = t X hi,ji, semi-metal Spinless fermions H = c i, c j, + U X i Gross-Neveu type fermionic quantum phase transition t X hi,ji semi-metal c i c j + c j c i n ",i n #,i spin density wave + V X hi,ji n i n j charge density wave U/t V/t no sign problem severe sign problem

13 Supervised learning approach Supervised learning approach 1) train convolutional neural network on representative images deep within the two phases 2) apply trained network to images sampled elsewhere to predict phases + transition step 1 train here phase A phase transition phase B train here step 2 predict phases by applying neural network here But what are the right images to represent a quantum state?

14 Monte Carlo for fermions Determinantal (or auxiliary field) quantum Monte Carlo for unbiased studies of strongly interacting fermions Path integral representation of partition sum Decouple quartic interaction via Hubbard-Stratonovich transformation Now integrate out free fermions moving in background field Z = X s det U(s) sample Hubbard-Stratonovich field

15 Hubbard-Stratonovich decoupling Decoupling quartic interaction via Hubbard-Stratonovich transformation introduces an Ising-type auxiliary field site site auxiliary field V i = n " i n# i V i (s)

16 Hubbard-Stratonovich decoupling real-space coupling

17 Hubbard-Stratonovich decoupling real-space coupling projection time

18 Hubbard-Stratonovich decoupling The auxiliary field has a natural interpretation has image.

19 Hubbard-Stratonovich decoupling The auxiliary field has a natural interpretation has image.

20 Supervised learning / auxiliary fields Case 1 spinful fermions The choice of Hubbard-Stratonovich transformation influences image, i.e. when coupling to... magnetization breaks SU(2) semi-metal SDW charge preserves SU(2) semi-metal SDW

21 Supervised learning / auxiliary fields coupling to magnetization

22 Supervised learning / auxiliary fields coupling to charge

23 Supervised learning / Green s functions Alternative Green s functions G(i, j) =hc i c j i Green s functions sampled as complex valued matrices. Convert into color-coded image using HSV color scheme. Hue Saturation Value (opacity) Our color mapping saturation c = c e i hue

24 Supervised learning / Green s functions Green s functions for spinful fermion model semi-metal L = 2x9x9 SDW

25 Spinful fermions Green s functions are ideal objects/images for machine learning based discrimination of quantum phases. Dirac SDW L = L = 9 prediction L = 15 L = 12 L = L = U = 1.0 interaction U U = 16.0

26 Some intermediate conclusions QMC + machine learning approach can be used to distinguish phases of interacting many-fermion systems. Green s functions are ideal images for machine learning. The ensemble of sampled Green s functions contains sufficient information to discriminate fermionic phases.

27 sign problem

28 Algorithmic power of Monte Carlo Sample configurations in high-dimensional space c 1! c 2!...c i! c i+1!... Metropolis (1953): accept new configuration with probability p acc =min 1, w(c j) w(c i ) Simultaneously measured observables converge in polynomial time. Tremendous impact across many different fields. In hard condensed matter percolation phase transitions quantum magnetism ultracold bosons

29 Quantum Monte Carlo classical Monte Carlo hoi = PC O(C) e E(C) P C e E(C) quantum Monte Carlo hoi = Tr Oe H Tr e H Map quantum to classical system Z =Tre H = X C p(c) Map to world lines of the trajectories of the particles Monte Carlo sampling of these world lines imaginary time space

30 The sign problem Expectation value for observables P P O(C)p(C) O(C) (C) p(c) hoi = P = P = ho i abs p(c) (C) p(c) h i abs when we ignore the sign of the configuration weights.... but the average sign decreases exponentially h i abs = P (C) p(c) P = Z =exp( N f) p(c) Z abs... resulting in an exponentially slow convergence of the statistical error Fundamental limit for pquantum Monte Carlo simulations of h 2 h i = i h i fermions p 2 e N f p Mh i M many-electron systems frustrated quantum magnetism, spin liquids

31 Is there a way out? The sign problem is basis dependent energy eigenbasis simulation basis exponentially hard Successful basis changes meron cluster Wiese et al., PRL (1995) fermion bag Chandrasekharan, PRD (2009) Majorana fermion basis Yao et al., PRB (2015) no general solution Troyer and Wiese, PRL (2005) the sign problem is NP-hard Change of perspective effective, sign-problem free actions Berg, Metlitski, and Sachdev, Science (2012) entanglement entropies Broecker and Trebst, PRB (2016)

32 sign problem + machine learning arxiv:

33 Can we bypass the sign problem? QMC sampling + statistical analysis hoi = P O(C)p(C) P p(c) = P O(C) (C) p(c) P (C) p(c) = ho i abs h i abs QMC sampling + machine learning Assume there exists a state function hfi abs = P F(C) p(c) P p(c) that is 0 deep in phase A and 1 deep in phase B.

34 Spinless fermions QMC + machine learning approach gives useful results even for systems with a severe sign problem w/o phases 1.0 w/ phases Dirac CDW prediction with phases no phases L = 15 L = 12 L = L = V = 0.1 interaction V V = 2.5

35 unsupervised learning Peter Broecker, Fakher Assaad, and ST, in preparation related ideas in Evert van Nieuwenburg, Ye-Hua Liu, and Sebastian Huber, Nature Physics (2017)

36 Unsupervised learning Employ ability to blindly distinguish phases to map out an entire phase diagram with no hitherto knowledge about the phases. Example: hardcore bosons / XXZ model on a square lattice H = X hi,ji S + i S j + S i S+ j + X hi,ji S z i S z j + h X i S z i 6 PRL 88, (2002) hs z i S z j i preliminary results 4 checkerboard solid ρ=1/2 V/t 2 Heisenberg point empty full ρ=0 superfluid ρ= (µ 2V)/t 12 h 12

37 Unsupervised learning Employ ability to blindly distinguish phases to map out an entire phase diagram with no hitherto knowledge about the phases. Example: hardcore bosons / XXZ model on a square lattice H = X hi,ji S + i S j + S i S+ j + X hi,ji S z i S z j + h X i S z i 6 PRL 88, (2002) hs + i S j i + hs i S+ j i preliminary results 4 checkerboard solid ρ=1/2 V/t 2 Heisenberg point empty full ρ=0 superfluid ρ= (µ 2V)/t 12 h 12

38 Topological order Assaad and Grover, PRX (2016) Gazit, Randeria & Vishwanath, Nature Physics (2017) Toy model for topological order in a fermionic system: fermions coupled to (quantum) Z2 (Ising) spins on bonds H = X! X N Z hi,ji c i, c j, + h.c. + Nh X X hi,ji hi,ji =1 hiji

39 Topological order Assaad and Grover, PRX (2016) Gazit, Randeria & Vishwanath, Nature Physics (2017) Toy model for topological order in a fermionic system: fermions coupled to (quantum) Z2 (Ising) spins on bonds H = X! X N Z hi,ji c i, c j, + h.c. + Nh X X hi,ji hi,ji =1 hiji h Z2 Dirac SDW

40 Topological order Toy model for topological order in a fermionic system: fermions coupled to (quantum) Z2 (Ising) spins on bonds H = X hi,ji Z hi,ji N X =1 c i, c j, + h.c.! + Nh X hiji X hi,ji prediction p supervised approach L = 6 L = 8 L = 10 L = prediction accuracy p unsupervised approach L = 6 L = 8 L = 10 L = h h

41 quantum state tomography Giuseppe Carleo and Matthias Troyer, Science 355, 602 (2017)

42 quantum state tomography Carleo and Troyer, Science (2017) Hidden Layer h 1 h 2 h 3 h M M hidden neurons (auxiliary spin variable) accuracy can be tuned by adjusting α=m/n z 1 z 2 z 3 Visible Layer z N restricted Boltzmann machine ( z 1, z 2,... z N ; W) = X e P j a j z j +P i b ih i + P ij W z ijh i j {h i } Variational Parameters

43 quantum state tomography Carleo and Troyer, Science (2017) ( z 1, z 2,... z N ; W) = X e P j a j z j +P i b ih i + P ij W z ijh i j {h i } Variational Parameters Feedback from variational principles Sample wavefunction with current set of parameters Change network parameters

44 quantum state tomography Variational energies. Carleo and Troyer, Science (2017) E/N D Heisenberg model =1 =2 =4 Exact # iteration # iteration 1000

45 quantum state tomography Variational energies. Carleo and Troyer, Science (2017) 1D transverse field Ising model 1D Heisenberg model 2D Heisenberg model Jastrow 10 2 rel h =1/2 h = EPS PEPS 10 7 h = very high precision, limited only by stochastic sampling compact representation ~10 2 less parameters than corresponding MPS in 1D improvements over best PEPS results

46 summary

47 Summary QMC + machine learning approaches can be used to distinguish phases of interacting quantum many-body systems (and opens opportunities to overcome the sign problem). There are a number of interesting analytical connections between neural networks and matrix product states as wells as the renormalization group. This is just the beginning. Probably, we will see, in the coming years, a similarly productive interplay between machine learning and quantum statistical physics as we have seen with quantum information. arxiv:

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