Exponential Thermal Tensor Network Approach for Quantum Lattice Models Chen et al., PRX 8 , 031082 (2018); arXiv:1811.01397 Andreas Weichselbaum Collaboration Bin-Bin Chen, Lei Chen, Ziyu Chen, Dai-Wei Qu, Han Li, Shou-Shu Gong, Wei Li (Beihang University, Beijing), Jan von Delft (LMU, Munich) Supported by German Research Foundation (WE4819/3-1) National Natural Science Foundation of China (NSFC) Department of Energy (DE-SC0012704)
Outline q XTRG (exponential thermal tensor network renormalization group) } Tensor network representation of thermal states [Chen et al., for quasi-1D systems [in the spirit of 2D-DMRG, yet for finite ! ] PRX 8 , 031082 (2018)] } Entanglement in thermal states } Exponential energy scales and logarithmic " grid q Benchmark: performance (numerical cost and accuracy) [arXiv:1811.01397] q Application: 2D spin-half triangular Heisenberg model } Two temperature scales ! # and ! $ } `Roton-like’ excitations in intermediate regime ! # ≲ ! ≲ ! $ with significant chiral component q Summary & outlook
Tensor network representation of thermal density matrix ) = + % &', - |/〉〈/| A … = @ " # ≡ % &'( A = ? A | ! = + ! (4 5 4 6 …4 8 ), (4 5 < ) |= > = ? … = @ 〉〈= > ( = B = 1, … , D ) < 4 6 < …4 8 2 2 A A A A A A = > = @ = > = @ = ? = ? matrix product ≡ ! ≡ operator = > = ? = @ = > = ? = @ (MPO) O B = 1, … , P A A A = > = @ = ? ∗ ! # ? ? " # = % &' ) L Ψ ) ≡ Ψ ) ?( ! = = ! # ? always positive = > = ? = @ Z = tr[! " # ] = = A A A = ? = @ ≡ Ψ Ψ = > purification = > = ? = @ N ≡ 1 N ≡ 1 ) L ≡ 1 )M NΨ N Ψ〉 M Z tr ! "M Z tr Ψ Z 〈Ψ M Verstraete (2004) Schollwoeck (review DMRG; 2010) Thermofield approach (de Vega, Banuls; 2015; Schwarz et al, 2018)
Entanglement scaling in thermal states in 1D Many-body finite size spectrum (critical systems, or !" ≫ gap Δ ) q E minimal requirement for thermal simulations 0 / = 2 3 456 7 |9〉〈9| < ⇒ / ≳ !" ⇒ G ≲ , / " . !" finite size level-spacing !" ∼ 1/, , ∼ G " - entropy of thermal state ⇒ > ℓ ∼ A 9 〈9| > ℓ ∼ 2× A ℓ 3 log ℓ 6 log (ℓ) ⇓ ℓ → G Calabrese (2004) >(G) ∼ A 3 log (G) allows for efficient simulations More rigorous arguments based on conformal field theory (CFT) of thermal states (entanglement • J. Dubail [J. Phys. A: Math. Theor. 50 (2017) 234001] entropy comparable to pure • T. Barthel [arXiv:1708.09349 [quant-ph], 2017] states with periodic BC)
Implication #1: Thermal correlation length and symmetries ! " ≲ $ % log " independent of ) for ) → ∞ q } finite correlation length - ∼ " in thermal state Therefore as long as - ≲ L q } can use finite system with open BC to simulate thermodynamic limit } can exploit all symmetries (abelian and non-abelian) in an optimal way (note that thermal state / can never be symmetry broken) 3 ⋅ 1 5 = ≡ 1 2 e.g. spin-half site: 7 ∈ {1, 1}
Implication #2: Exponential energy scales Weak growth of block entropy of thermal state ! " ∼ $ % ln " q } suggests that linear imaginary time evolution schemes are ill-suited e.g. Trotter: " → " + * with * ≪ " , -./ ≃ , -. 1213 / , -. 455 / small Trotter error enforces small constant * for any " rather need to make bold steps when increasing " q to see a significant change in physical properties within a critical regime natural choice: " → Λ" ( Λ > 1 ) ⇒ :! ∼ const. simple choice: Λ = 2 B C * D → B C * D ∗ B C * D = B C 2* D → B C 2* D ∗ B C 2* D = B C 4* D → ⋯ exponential thermal " H = * D 2 H tensor renormalization * D 2* D 4* D 8* D 16* D group (XTRG)
Benefits of logarithmic temperature grid Simple initialization of ( ! " q } can start with exponentially small ! " such that ( ! " = 1 − ! " , simply use the MPO of , ⇒ up to minor tweak, same MPO for ( ! " } q No Trotter error } no swap gates to deal with Trotter steps } simply applicable to longer range Hamiltonians } including (quasi-) 2D systems Maximal speed to reach large . with minimal number of truncation steps q Fine grained temperature resolution! q using / -shifted temperature grids . 0 = ! " 2 012 with / ∈ [0,1[ } ! " 2! " 4! " 8! " 16! " equivalent to using ! " → ! " 2 2 } } easy to parallelize: independent runs for logarithmically interleaved data sets
Numerical cost of XTRG q Naively % % % ! " ! $ ! # & $ SVD → ) & * & & ! " ! $ ! # $ → 012 Variationally: + , ∗ + , − + / $, q & & × ~100 overlap → ) & 3 → )( & ∗ 3 ) & & Computational gain by using symmetries: & states → & ∗ multiplets q e.g. SU(2) spin-half Heisenberg: & ∗ ≃ &/4
Brief comparison to coarse graining renormalization q Xie et al. (PRB 2012) Coarse-graining renormalization by higher-order singular value decomposition q starting point: Trotter gates q infinite tensor network } no clean orthogonal vector spaces } no symmetries used 2D Ising model ( D= 24) q no interleaved temperatures } „However, the number of temperature points that can be studied with this approach is quite limited […], since the temperature is reduced by a factor of 2 at each contraction along the Trotter direction.” } linearized imaginary time evolution largely favored ! " Similarly for Czarnik et al. (PRB 2015)
Benchmark: performance $ Free energy ! = − % log * L=18 spin-1/2 Heisenberg chain (PBC; + = 1 ) q XTRG is most accurate q XTRG is clearly fastest speed gain (for D*=100,200) ×10 ×10 LTRG SETTN XTRG starting from the same - = -(1 2 ) , proceed - → - ∗ - XTRG . . exponential tensor renormalization group LTRG . . linearized tensor renormalization group - → - ∗ -(1 2 ) - → -(4) ∗ 5 6%7/9 SETTN . series expansion thermal tensor network
Block entanglement entropy L=200 spin-1/2 Heisenberg chain (OBC; ! = 1 ) log. growth $ ∼ & ' log + with , = 0.999 This offers an alternative to compute central charge via finite- 2 simulation using open BC! In comparison to Calabrese (2004) for obtaining , from ground states with periodic BC • comparable block entropy scaling • no system size dependence as long as 3 ≳ 5 universal $ ∼ + 0 behavior for very large temperatures where $ ≪ 1 (irrespective of the physics or dimensionality of the model!)
Specific heat and critical exponents L=300 spin-1/2 Heisenberg chain ( ! = 1 ) Specific heat at low temperatures $ % = &' () * with + = & , ⇒ $ = 0.996 at large temperatures: universal 1/* , behavior ( irrespective of the physics or dimensionality of the model!)
Application Two Temperature Scales in the Triangular Lattice Heisenberg (TLH) Antiferromagnet arXiv:1811.01397 [cond-mat.str-el]
2D triangular Heisenberg model (S=1/2) q What is known (theory) } 120˚ magnetically ordered state at T=0: ! " = 0.205(15) , [White et al. (2007)] } paramagnetic at large T } problem [Kulagin et al (2013) using `sign-blessed’ BDQMC]: data extrapolates to disordered, i.e., non -magnetic state for + → 0 !? incipient 120 o order paramagnetic T J ??
Zheng et al., PRB (2006) Roton-like excitations in the TLH Starykh et al., PRB(R) (2006) Magnon spectra [Zheng (2006)] linear spine wave theory (LSWT) LSWT + 1/S corrections series expansion effective `massive’ quasiparticles at finite energy with Δ ≃ 0.55 Zheng (2006): We have called this feature a “roton” in analogy with similar minima that occur in the excitation spectra of super-fluid 4 He and the fractional quantum Hall effect.
Szasz et al. Triangular lattice Hubbard model (DMRG @ T=0) (condmat, 2018) YC4 data Non-magnetic intermediate phase q chiral and gapped ℏ = 2 q relevant in the large U limit (Heisenberg model) at finite ! ?
Experimental progress q Ba 8 CoNb 6 O 24 - close to ideal 2D triangular Heisenberg material! } Perovskite, first synthesized [Mallinson et al. (Angew. Chem. Int. Ed., 2005)] } equilateral effective spin-1/2 Co2+ triangular layers separated by six nonmagnetic layers. } [Rawl et al., 2017] A spin-1/2 triangular Heisenberg antiferromagnet in the 2D limit } [Cui et al., 2018] Mermin-Wagner physics, (H,T) phase diagram, and candidate quantum spin-liquid phase in the spin-1/2 triangular antiferromagnet Ba 8 CoNb 6 O 24 magnetic contribution to specific heat (by ref. to non-magnetic compound Ba 8 ZnTa 6 O 24 ) Ba 3 CoNb 2 O 9 : 7.23Å (2 non-magnetic [Rawl (2017)] J=1.66 K layers)
XTRG data For sufficiently large system sizes q consistently, two energy scales ! " ≃ 0.2 and ! ' ≃ 0.55 q in agreement with experiment (“thermodynamic limit”) q `roton’ contribution only relevant for ! ≳ ! " q strongly enhanced thermal entropy for ! ≲ ! " due to frustration TPO . . . tensor product operator method (complimentary to XTRG) RSBMF . reconstructed Schwinger boson mean field [Mezio et al, NJP (2012)] Roton . . roton contribution only [Zheng et al, PRB (2006)] HTSE . . high temperature series expansion (Elstner et al, PRL (1993)] Pade . . a particular way to deal with the low-T divergence of the partition function in HTSE [Rawl (2017)]
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