Excursion: MPS & DMRG

Transcription

1 Excursion: MPS & DMRG Acronyms for: - Matrix product states - Density matrix renormalization group Numerical methods for simulations of time dynamics of large 1D quantum lattice systems. Important for: Condensed matter physics, but even more: Atoms/molecules in optical lattices, ion traps Outline: 1. The problem 2. Matrix Product states Entanglement (von Neumann entropy) Singular value decomposition MPS construction 3. Time-evolution with MPS 4. Extensions

2 1. The problem 1 M For example: Spin-Chain/TLS = = 1 = = 0 i = c i1,i 2,,i M i 1,i 2, M i Given: (t = 0)i what is (t)i = e ith (t = 0)i i 1,i 2,,i M =0,1 Solve it numerically? Storage of state dim(h) 2 2 2=2 M complex numbers One complex number in double precision: 128 bit = 16 byte M 2^M RAM ~ 10^6 ~ 10^12 ~ 10^90 ~ 16 MB ~16 TB ~10^78 TB More bits than estimated number of particles in universe! Never possible! Exponential Hilbert space growth makes exact simulation for large systems impossible Even worse: e ith is a 2 M 2 M matrix Good news: H is usually sparse and one can exploit symmetries. (~24 spins possible without symmetries on modern clusters) The solution: t-dmrg Exact numerical simulations for 300 and more spins possible!

3 2. MPS - The idea As the name suggests, related to a product state, which is a very rough approximation (Gutzwiller ansatz/mean-field) i = c i1,i 2,,i M i 1,i 2, M i i 1,i 2,,i M =0,1 i i 1,i 2,,i M =0,1 c [1] i 1 c [2] i 2 c [M] i M i 1,i 2,i M i 2 M complex numbers 2M complex numbers Dramatically reduced complexity, but only able to represent product states=non-entangled states H non entangled states, PS MPS Matrix product states: Extension of the state representation to slightly entangled states Surprisingly, for most problems (near equilibrium dynamics, low energies, short times) only a bit of entanglement will make the simulation exact!

4 2.1 Reminder: Bipartite entanglement - Von-Neumann entropy What is bipartite entanglement? Block A Block B = i i 6= A i B i entangled State in Block A A =tr B ( ih ) S vn ( A )= tr A [ A (log 2 A )] = log 2 ( ) The entropy (i.e. mixed-ness) of rho_a is a measure for entanglement Example: Two quits 1 i = 01i = 1 ih 1 = 01ih01 A = hi ii 2 i=0,1 A = 0ih0 S vn ( A )=0 This state is pure. All information about sub-system A is in rho_a = no entanglement B i = 1 p 2 ( 00i + 11i) = 1 2 ( 00ih ih ih ih11 ) A = 1 2 ( 0ih0 + 1ih1 ) S vn ( A )=1 This state is maximally mixed. No information about sub-system A is in rho_a = max entanglement Goal: We want a general state representation with a parameter that controls the amount of bipartite VNE

5 2.2. Singular value decomposition Bipartite quantum system A B with dimension d dim(h) =d d i = d i,j=1 c i,j iji c i,j = d u i, s, v,j C = USV Singular value decomposition U,V unitary, S real & diagonal Von Neumann entropy in SVD form = c i 0,j c 0 i,j ijihi 0 j 0 A = i 0,j 0 n i,j hn ni 2 A = c i,nc 0 i,n iihi 0 i 0,i 0 v,n v A = u i,s, v,n ui 0, s, v,n iihi 0,n =, i 0,i 0 n n u,n u A = u i,(s, ) 2 u i 0, iihi 0,n =, i 0,i 0 n A = (s, ) 2 ih The SVD diagonalizes the reduced density matrix d S vn ( A )= (s, ) 2 log 2 [(s, ) 2 ] max[s vn ( A )] = log 2 (d) n U U = u i,ju i 0,j 0 jihi i0 ihj 0 = I i,j i 0,j 0 u i,ju i,j 0 jihj 0 = I i,j j 0 j 0 u i,ju i,j 0 = j,j 0 We can now make an approximation: Instead of keeping all SV, we just keep the Chi<d largest ones Then max[s vn ( A )] = log 2 ( ) and we now have a state approximation with limited entanglement This is the origin of density renormalization, since we throw away small eigenvalues of the reduced density matrix.

6 2.3. MPS construction Two make things clearer: Diagrammatic representation Number Vector Matrix Then: Singular value decomposition = Matrix-Vector product U S V = Approximation: For a larger system the MPS construction is now straightforward c i1,i 2,,i M Alternatively The name matrix product state is now obvious [1] [2] [M]!!! Matrices of kets Choosing a small Chi allows to restrict the VNE for each bipartite splitting For an exact representation Chi=1: product state a special case of a MPS = d M/2 max[s vn( A )] = log 2 ( ) which still grows exponentially with M, but often large Chi unnecessary

7 3. Time-evolution algorithm (TEBD) Consider Hamiltonian with short range interactions Ĥ = M 1 â iˆbi+1 +ĉ i + ˆd i+1 M 1 Ĥ i i=1 i Strategy: Stroboscopic evolution (t + t)i = e i tĥ (t)i Limit t kĥik e i th = e i t(ĥ1+ĥ2+ĥm ) e i tĥ1 e i tĥ2 e i tĥm + O( t 2 ) [Ĥ1, Ĥ2] 6= 0 Trotter decomposition In practice: Higher order Trotter decomposition, e.g. e i tĥ = Y Û i,i+1 + O( t 5 ) MPS language i i+1 Time-evolution <> Application of two-site Trotter gates TEBD: Contract SVD Truncation There is a controllable error in each step: = Computational time ~ 3 (contraction step) d = +1 (s, ) 2 d too big! When this error becomes large, too much entanglement is created in the time-step and we have to increase Chi.

8 4. Extensions Ground-state finding: Variational techniques local updates to minimize energy Alternative: imaginary time evolution e Ĥ i = e P E ih i!1 / GSi Long range interactions: Matrix product operators (MPO) Higher dimensions (PEPS) Ĥ = 2D Schemes for application: MPO x MPS multiplication Problem: Computational time / 12 Open systems (MPDO) or Monte-Carlo sampling techniques with MPS evolution (Quantum trajectories) Infinite systems (itebd) Further reading: Other references: Review article: U. Schollwöck ariv: F. Verstraete, V. Murg, and J. Cirac, Advances in Physics 57, 143 (2008) (Review) S. R. White, Physical Review Letters 69, 2863 (1992). (original DMRG) G. Vidal, Physical Review Letters 93, (2004). (TEBD) G. Vidal, Physical Review Letters 98, (2007). (itebd) F. Verstraete and J. I. Cirac, ariv:cond-mat/ (2004). (PEPS) B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, New Journal of Physics 12, (2010) (MPOs) R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Physical Review A 46, 4382 (1992) (Open systems). and many more