Momentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model

Momentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model Örs Legeza Reinhard M. Noack

Collaborators Georg Ehlers Jeno Sólyom Gergely Barcza Steven R. White

Collaborators Georg Ehlers Jeno Sólyom Gergely Barcza Steven R. White Contents I. Introduction - DMRG, Entanglement, Mutual Information II. Generalized Correlation Functions III. k-dmrg applied to the D & 2D Hubbard models IV. Discussion

I. Entanglement and von Neumann Entropy Bipartite system in a pure state 0 A B Reduced density matrix A = Tr B 0 0

I. Entanglement and von Neumann Entropy Bipartite system in a pure state 0 A B Reduced density matrix A = Tr B 0 0 von Neumann entropy S( A )= Tr ( A log A )= w log w Truncation in entanglement: SVD/Schmidt decomposition 0 m dim( ) w with eigenvalues and, eigenvectors of, w A B

Density Matrix Renormalization Group Truncation in entanglement Iterative procedure to build up approximation to 0 Yields matrix product state (MPS) intrinsically D

Entropy Area Law For short-range Hamiltonians with short-range correlations, S A area of boundary D: 2D: S A const. S A L bound (criticality: multiplicative log corrections) (Wolf, Verstraete, Hastings, Cirac, PRL 00, 070502, 2008)

Entropy Area Law For short-range Hamiltonians with short-range correlations, S A area of boundary D: 2D: S A const. S A L bound (criticality: multiplicative log corrections) To obtain an MPS with fixed, bounded error: m e S A exponentially difficult in D >! How do we define range of correlations? (Wolf, Verstraete, Hastings, Cirac, PRL 00, 070502, 2008)

II. One- and Two-Site Entropies One-site entropy entanglement of a single site with rest of lattice S p = S( p ) p : density matrix for site p holds spatial information localized to site p p

II. One- and Two-Site Entropies One-site entropy entanglement of a single site with rest of lattice S p = S( p ) p : density matrix for site p holds spatial information localized to site p p Two-site entropy p q p,q : density matrix for sites p and q S p,q = S( p,q ) entanglement of sites p and q with rest of system

Mutual Information Mutual information for a bipartite system with subsystems A and B A B I AB [S AB S A S B ] When system AB is in a pure, state, S AB =0 I AB =2S A =2S B Mutual information useful when system AB is in a mixed state, or, equivalently, is embedded in a larger system: p q Two-site mutual information: I p,q = S p + S q S p,q

Construction of Density Matrices (Barcza, RMN, Sólyom, Legeza, arxiv:406.6643) Transition operators between states of the (q-dim.) local basis i i: (T (m) i ) 0 i, i = 0 iih i, m =,...q 2 One-site density matrix h 0 i i i i in local basis: i = 0 h 0 T () i 0 i 0 0 0. 0.. 0 0 0 hti m i 0 B. @.. C A 0 0 0 0 ht (q2 ) i i (diagonal if every state has a separate quantum number)

Two-site Density Matrix Expression in the two-site basis: h 0 i, 0 j ij i, j i as expectation values of the transition operators = h 0 T (m) i T (n) j 0 i, where m corresponds to the transition from i to i 0 and n from to j 0 j block-diagonal in the appropriate quantum numbers

Generalized Correlation Functions The matrix elements of the two-site density matrix h 0 T (m) i T (n) j 0 i behave like generalized correlation functions. Take the connected part: ht (m) i T (n) j i C = ht (m) i T (n) j i ht (m) i iht (n) j i asymptotic behavior for large i j determines the asymptotic behavior of the two-site mutual information I i,j can be related to usual physical correlation functions

Example: Spin-/2 Basis One-site density matrix: # " # ht () i i ht (2) i i =0 " ht (3) i i =0 ht (4) i i Two-site density matrix: where T () i S z i + 2 T (2) i T (3) i S i S + i T (4) i S z i + 2 ij # # # " " # " " # # / # " /4 2/3 " # 3/2 4/ " " 4/4 with m/n h 0 T (m) i denoting T (n) j 0 i

Example - Heisenberg Chain H = (J S i S i+ + J 2 S i S i+2 ) (S =/2) i J 2 J J 2 J =,J 2 =0 J =,J 2 =0.5 I pq 0 0 0 0 2 0 3 0 4 0 5 0 6 25 27 29 28 26 24 3 30 32 2 4 3 6 8 5 7 I pq 0 0 0 0 2 0 3 0 4 0 5 0 6 25 27 29 28 26 24 3 30 32 2 4 3 6 8 5 7 23 22 0 9 23 22 0 9 2 20 8 6 4 2 2 20 8 6 4 2 9 3 9 3 7 5 7 5 Majumdar-Gosh (dimerized)

Decay of Generalized Correlation Functions H = X i J apple 2 (S+ i S i+ + S i S+ i+ )+ Sz i S z i+ J =, = one group (n) T N/2+l (m) T N/2 T (m) N/2 T (n) N/2+l 0 0 0 0 2 0 3 0 4 0 5 (a) 32 64 28 92 fit 0 0 0 0 2 l I (2/3) (/) 0 0 0 0 2 0 3 0 4 0 5 (b) (2/3) I (/) 0 0 0 0 2 l J =, =0.5 two groups In both cases I i.j i j 2 with the exponent of the slowest decaying correlation function

III. Hubbard Model in Momentum Space Hubbard model in real space (D & 2D) H = t X c i c j +h.c. hi,ji, + U X i (Ehlers, Sólyom, Legeza, RMN) n i" n i# Hubbard model in momentum space H = X k "(k) c k c k + U N Potential advantages for DMRG: Sites are momentum eigenstates Kinetic energy term diagonal momentum conserved dimensionality, range of hopping enters diagonally MPS becomes exact at small interaction X pkq c p q" c k+q# c k# c p" U/t But: interaction nonlocal, contains N 3 terms

Systems with Nonlocal Interactions General nonlocal fermionic Hamiltonian H = i,j, t ij c i c j + i,j,k,l,, V ijkl c i c j c k c l Describes: quantum chemistry (post-hartree-fock) sites : quantum chemical basis set t ij single-electron, V ijkl two-electron integrals momentum-space Hamiltonians fractional quantum Hall long-range Hamiltonians in real space,...

DMRG for Nonlocal Hamiltonians Idea: Map orbitals onto a D lattice, carry out normal DMRG (with nonlocal, long-range interactions) General case: quantum chemistry (White & Martin, 999) E Water molecule, L=4 N =5 N =5 D chain HF orbitals Virtual states Momentum space: similar, but momentum conservation included site energy: k (Xiang, 996)

Issues/Problems N 4 interaction terms k-space Hubbard: factorization to 6N (Xiang, 996) general: factorization to O(N 2 ) Initial build up of lattice (active space) dynamically extend active space according to importance of orbitals (Legeza & Sólyom, 2003) Site-ordering problem no evident ordering at arbitrary interaction strength

Two-Site Mutual Information Measure of relative entanglement between local bases Application: quantum chemistry and other non-local Hamiltonians 4b u N 2 5a g 4a g 4b u 2b u 3a g 5a g 2a g F 2 (Rissler, Noack, White, 2006) 2b u I pq b 3g b 3g b 2u b 3u b 2g 2b 2g 2b 3g 4a g b 3g 2b 2u 3a g 3b u b 2g 2b 3u b 3u 2b 2g 2b 2u 2a g 2b 3u b 2u [Cu2O2] 2+ 5b u 3b u (Barcza, Legeza, Marti, Reiher, 20) t ij I pq optimization of algorithms

Hubbard Model: Site Ordering D Hubbard model ( "(k) = 2t cos k ) (a) j 5 0 5 0 i 0.5 0.2 0. 0.05 0.02 0.0 0.005 "k 3 2 0 2 3 0 8 8 9 6 0 4 2 9 3 2 2 3 0 0 k x Initial ordering: step through Brillouin zone Order by minimizing cost function I ij 4 sites, n=, U/t = 2 7 3 5 4 3 5 0 6 2 7 f cost = X i,j (b) j 5 0 5 0 i I i,j i j For optimal ordering, strongly entangled sites near to one another Calculations: up to m=40,000 states, 00 sites (0x0) reduced block entropy, better DMRG convergence 0.5 0.2 0. 0.05 0.02 0.0 0.005 I ij

D Hubbard at Half Filling one-dimensional Hubbard model at half filling state of the 22-site Hubbard model (a) 0.at half (b) filling for 22 U/t sites, =.0: n=, U/t = j 5 0 5 20 5 0 5 20 i 0.05 0.02 0 0.0 I ij 0.005 2 0 0.002 0.00 " 2 0.000.002 0.005 k x 0.0 I ij 0.02 0.05 0. (c) (d) Si dtot 2 0 0.5 0. 005 0 k x Strongest correlation between sites separated by umklapp vector rongest entanglement between sites where umklapp scattering is allowed two-particle umklapp process c k F + k,# c k F + k," c k F + k,"c kf + k,# i HF : k F k F rg Ehlers (Philipps-Universität Marburg) FOR 807 February 25 205 9 / 9

D Hubbard, 25% Doping (a) j 5 0 5 20 5 0 5 20 i (b) 0. 2 0.05 0.02 0 0.0 I ij 0.005 2 0 0.002 k x 0.00 0.000.002 0.005 0.0 0.02 0.05 " 22 sites, n=0.75, U/t = I ij 0. (c) (d) Si dtot 2 0 0.5 0. 005 0 k x Umklapp processes excluded by momentum conservation correlation/entanglement reduced

max(sblock) scaling with system size N:.5.0 0.5 0.0 linear fit U =0.5 U =.0 U =2.0 D Hubbard, Entropy Scaling 6 0 4 8 22 N scales linearly with system size for small to moderate U maximum value of block entropy max(sblock)/n scaling with interaction U/t: 0.4 0.3 0.2 0. 0.0 N=6 N=0 N=4 0.05 a 0 = 2.7 0 4 a =6.0 0 3 0.04 a 2 =6.0 0 2 0.03 a 3 =.0 0 2 0.02 0.0 0 5 0 5 20 U/t scales quadratically for small U saturates at large U 0.00 0.0 0.2 0.4 0.6 0.8.0 sublinear scaling with N at large U numerical effort is exponential in system size N becomes exact in limit of small U

2D Hubbard at Half Filling "(k) = 2t(cos k x cos k y ) (a) j 5 0 5 20 25 30 35 5 0 5 20 25 30 35 i 0.2 0. 0.05 0.02 0.0 0.005 I ij (b) Ky 0.0 6x6, n=, U/t = 2 0.0 k x 0.2 (c) 0. 0 0.05 I ij 0.02 (d) 0 0.0 0.005 ky ky 0 k x 0 k x 2 0 2 4 0 d tot S i strongest correlations between momentum site separated by k k 0 =(±, ± )

2D Hubbard, 25% Doping (a) j 5 0 5 20 25 30 35 5 0 5 20 25 30 35 i 0. 0.05 0.02 0.0 0.005 0.002 I ij (b) Ky 0.0 6x6, n=0.75, U/t = 2 0.0 k x 0. 0.05 0.02 0.0 I 0 ij (d) 0.005 0.002 (c) ky ky 0 0 k x 0 k x 2 0 2 4 0 d tot S i Fermi surface no longer perfectly nested Number of energetically favorable scattering channels reduced Entanglement becomes smaller

Scaling of 2-Site Mutual Information 6x6 sites 0 U/t =2.0,n=0.722 U/t =2.0,n=.0 Ii,j 0 2 0 3 0 4 U/t =0.5,n=0.722 U/t =0.5,n=.0 0 0 0 0 2 (i, j) Doping suppresses largest correlations Tails fall off more slowly for larger U

Entropy Profiles e S i, e Sblock.0 0.8 0.6 0.4 U/t =0. U/t =0.2 U/t =0.5 6x6, n= U/t =0.0 U/t =0.02 U/t =0.05 Sblock 4 3 2 0.5 (a) 6x6, n= i U/t=.0 U/t=2.0 U/t=5.0 U/t=0.0 U/t=20.0 ln(4) 0.2 0.0 0 5 0 5 20 25 30 35 i S block linear in subsystem size same form for all small U values (both S block and ) S i Si.0 0.5 (b) 0.0 0 2 4 6 8 0 2 4 6 i S block sublinear in subsystem size zigzag peaks for large U, n= S i saturates with U saturation of S block with U

2D Hubbard, Entropy Scaling scaling with system size N: scaling with interaction U/t: max(sblock) 2.0.5.0 0.5 linear fit U/t =2.0,n=.0 U/t =2.0,n 0.75 U/t =.0,n=.0 U/t =.0,n 0.75 U/t =0.5,n=.0 U/t =0.5,n 0.75 0.0 2 2 4 4 6 6 8 8 0 0 L x L y max(sblock)/n 0.25 0.20 0.5 0.0 0.05 0.00 4 4 0.05 a 0 = 6.6 0 5 a =.5 0 3 0.00 a 2 =.8 0 2 a 3 = 3.2 0 3 0.005 0.000 0.0 0.2 0.4 0.6 0.8.0 0 5 0 5 20 U/t scales linearly with system size doped system has somewhat smaller entropy scales quadratically for small U saturates at large U saturation value somewhat smaller than in D

Hybrid Algorithm momentum space in y, real space in x direction ky Comparison with real-space DMRG: x 39.6 39.8 hybrid-space real-space E0 40.0 40.2 40.4 2x6, n=, U/t = 2 0.0000 0.0002 0.0005 0.000 /M block energy converges comparably rapidly with number of block states order of magnitude faster for a given accuracy (G. Ehlers, S.R. White, RMN)

Step of 4: Extrapolation in Truncation Error E0 82.550 82.555 82.560 Lx=8, Ly=4, periodic 22k 26k 30k E0 24.9 25.0 25. Lx=8, Ly=6, anti-periodic 22k 26k 30k 82.565 0.0000 0.000 0.0002 0.0003 0.0004 trunc. error S 25.2 0.000 0.00 0.002 0.003 0.004 trunc. error S significantly better convergence for smaller cylinder width (as in real space) use every second sweep for extrapolation (noise term turned on alternate sweeps to recover QN sectors)

Step 2: Extrapolation in Cylinder Length.0 cylinder width: Ly=4 periodic anti-periodic 6.0.2 cylinder width: Ly=6 periodic anti-periodic E0/N.5 24 8 2.20 0.00 0.05 0.0 0.5 /L x E0/N.4.6 2 8 24.8 0.00 0.05 0.0 0.5 /L x 6 periodic/antiperiodic difference smaller for wider cylinders almost perfect /L x scaling

Steps 3 & 4: Boundary Condition Averaging, Extrapolation in Circumference E0/N.6.7.8.9.20 n=.0, t /t=0.0, U/t=2.0 real-space -.76 average periodic anti-periodic 0.00 0.02 0.04 0.06 /L 2 y E0/N.36.38.40 n=0.8, t /t=0.2, U/t=2.0 DMET -.40 average periodic anti-periodic.42 0.00 0.02 0.04 0.06 /L 2 y n =.0, t 0 =0.0 : perfect agreement with real-space DMRG n =0.2, t 0 =0.2 : good agreement with DMET (G. K.-L. Chan) (comparison data: Simons Collaboration on the Many Electron Problem)

IV. Discussion The two-site mutual information + generalized correlation functions are a useful tool for studying and optimizing entanglement structure. The two-site mutual information decays like most slowly decaying correlation function squared. Hubbard model in momentum space can be efficiently implemented within k-dmrg/mps with ordering optimization. accuracy best in weak-coupling limit volume scaling of entanglement entropy computationally prohibitive for thermodynamic limit Hybrid real-momentum space algorithm shows promise within the limits of the entropy area law Extrapolation process yields energies in good agreement with other methods for 2D Hubbard model.

Acknowledgements Wigner Research Centre for Physics, Budapest www.itensor.org Advancing Research in Basic Science and Mathematics DFG Research Unit 807: Advanced Computational Methods for Strongly Correlated Quantum Systems