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Exponential Thermal Tensor Network Approach for Quantum Lattice - - PowerPoint PPT Presentation
Exponential Thermal Tensor Network Approach for Quantum Lattice - - PowerPoint PPT Presentation
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
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Tensor network representation of thermal density matrix
! " # ≡ %&'(
) = + %&',-|/〉〈/| 2
= + !(4546…48), (45
<46 <…48 <)|=>=? … =@〉〈=>
A=? A … =@ A| 2
≡ => =@ =? ! =>
A
=@
A
=?
A
=> =@ =? =>
A
=@
A
=?
A
≡ ( =B = 1, … , D ) Z = tr[! " # ] = => =@ =? =>
A
=@
A
=?
A
= => =@ =?
=>
A
=@
A
=?
A
≡ Ψ Ψ
! " # = %&'
?( ) ?
= = ! #
?
∗ ! #
?
≡ Ψ ) LΨ ) always positive purification M N ≡ 1 Z tr ! "M N ≡ 1 Z tr Ψ )M NΨ ) L ≡ 1 Z 〈Ψ M N Ψ〉 Verstraete (2004) Schollwoeck (review DMRG; 2010) Thermofield approach (de Vega, Banuls; 2015; Schwarz et al, 2018) matrix product
- perator
(MPO) OB = 1, … , P
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Entanglement scaling in thermal states in 1D
q Many-body finite size spectrum (critical systems, or !" ≫ gap Δ)
finite size level-spacing !" ∼ 1/, E !" "- ". / 0 / = 2 34567|9〉〈9|
<
ℓ > ℓ ∼ 2× A 6 log (ℓ)
, ∼ G ⇒ / ≳ !" ⇒ G ≲ ,
minimal requirement for thermal simulations
9 〈9|
⇒ > ℓ ∼ A 3 log ℓ entropy of thermal state More rigorous arguments based on conformal field theory (CFT)
- J. Dubail [J. Phys. A: Math. Theor. 50 (2017) 234001]
- T. Barthel [arXiv:1708.09349 [quant-ph], 2017]
allows for efficient simulations
- f thermal states (entanglement
entropy comparable to pure states with periodic BC) >(G) ∼ A 3 log (G) ⇓ ℓ → G Calabrese (2004)
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Implication #1: Thermal correlation length and symmetries
q ! " ≲ $
% log " independent of ) for ) → ∞
} finite correlation length - ∼ " in thermal state
q Therefore as long as - ≲L
} 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)
≡ 12
3 ⋅ 15 =
e.g. spin-half site: 7 ∈ {1, 1}
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Implication #2: Exponential energy scales
q Weak growth of block entropy of thermal state ! " ∼ $
% ln "
} suggests that linear imaginary time evolution schemes are ill-suited e.g. Trotter: " → " + * with * ≪ " ,-./ ≃ ,-.1213/,-.455/ small Trotter error enforces small constant * for any "
q rather need to make bold steps when increasing " 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 → ⋯ "H = *D2H
exponential thermal tensor renormalization group (XTRG) *D 2*D 4*D 8*D 16*D
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!" 2!" 4!" 8!" 16!"
Benefits of logarithmic temperature grid
q Simple initialization of ( !"
} 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
q Maximal speed to reach large . with minimal number of truncation steps q Fine grained temperature resolution!
} using /-shifted temperature grids .0 = !"2012 with / ∈ [0,1[ } equivalent to using !" → !"22 } easy to parallelize: independent runs for logarithmically interleaved data sets
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Numerical cost of XTRG
q Naively
!" !# !$ !"
%
!#
%
!$
%
& &$ &
SVD → ) &*
q Variationally: +, ∗ +, − + /$,
$ → 012
& & & &
- verlap → ) &3
q Computational gain by using symmetries: & states → &∗ multiplets e.g. SU(2) spin-half Heisenberg: &∗ ≃ &/4
→ )( &∗ 3)
× ~100
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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
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)
2D Ising model (D=24)
!
"
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Benchmark: performance
q XTRG is most accurate q XTRG is clearly fastest speed gain (for D*=100,200) LTRG SETTN XTRG Free energy ! = −
$ % log *
L=18 spin-1/2 Heisenberg chain (PBC; + = 1) XTRG . . exponential tensor renormalization group
- → - ∗ -
LTRG . . linearized tensor renormalization group
- → - ∗ -(12)
SETTN . series expansion thermal tensor network
- → -(4) ∗ 56%7/9
×10 ×10 starting from the same - = -(12), proceed
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Block entanglement entropy
L=200 spin-1/2 Heisenberg chain (OBC; ! = 1)
- log. growth $ ∼ &
' log +
with , = 0.999 universal $ ∼ +0 behavior for very large temperatures where $ ≪ 1 (irrespective of the physics or dimensionality of the model!) 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
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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
- f the model!)
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Application
Two Temperature Scales in the Triangular Lattice Heisenberg (TLH) Antiferromagnet
arXiv:1811.01397 [cond-mat.str-el]
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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 !?
J
paramagnetic incipient 120
- order
T
??
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Roton-like excitations in the TLH
Magnon spectra [Zheng (2006)]
series expansion linear spine wave theory (LSWT) LSWT + 1/S corrections effective `massive’ quasiparticles at finite energy with Δ ≃ 0.55 Zheng et al., PRB (2006) Starykh et al., PRB(R) (2006) Zheng (2006): We have called this feature a “roton” in analogy with similar minima that
- ccur in the excitation spectra of super-fluid
4He and the fractional quantum Hall effect.
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Triangular lattice Hubbard model (DMRG @ T=0)
Szasz et al. (condmat, 2018) Non-magnetic intermediate phase q chiral and gapped q relevant in the large U limit (Heisenberg model) at finite !? ℏ = 2 YC4 data
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Experimental progress
q Ba8CoNb6O24 - 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 Ba8CoNb6O24 Ba3CoNb2O9 : 7.23Å (2 non-magnetic layers) magnetic contribution to specific heat (by ref. to non-magnetic compound Ba8ZnTa6O24)
[Rawl (2017)] J=1.66 K
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XTRG data
For sufficiently large system sizes
q consistently, two energy scales !" ≃ 0.2 and !' ≃ 0.55 q in agreement with experiment (“thermodynamic limit”) 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)] q `roton’ contribution only relevant for ! ≳ !" q strongly enhanced thermal entropy for ! ≲ !" due to frustration
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TLH spin structure factor (YC6×12 data)
S(q) ⌘ X
j
e−iq·r0j hS0 · SjiT
q S(0) measures magnetic susceptibility "# = %
&'(0)
q S(K) measures 120˚ magnetization + ≃ 0.265 vs. 0.205 [White et al, 2007] q S(M) measures nearest neighbor AF correlations ⇨ anomalous with maximum around "2 3 = 24 3 (1, 3) 8 = 24 3 (0, 3) ) b ( ) a ( ) d ( ) c ( (e)
K M
q
x
q
q
x
qx qx q
y
q
y
q
y
q
y
(f)
/ / / / /
- /
- /
- /
- T=5
T=0.54 T=0.2 T=0.1
4 1.1
q log. growth of MPO block entanglement '9 = : ln = + ? for " ≲ "A q For comparison: "A absent for square Heisenberg cylinder (SC6)
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Correlation length !
T
10 5, D*=800 D*=1000 12 6, D*=800 D*=1000 15 6, D*=600 D*=800 HTSE
T h~0.55 T l~0.2 ~T-1/2e0.1/T
ξ2 =
1 4S(K)
X
j
r2
0j e−iK·r0j hS0 · Sji
q very short correlation length! " ≲ 2 down to &' q justifies relevance of ( ≤ 6 data q " ∝ &,-
. exp(34
56) predicted by field-theoretical arguments [Chakravarty et al, PRL 1988]
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Significant chiral component in intermediate regime
q significant chiral component in intermediate regime! q chiral contribution cut off only at ! ≲ !# $ = ⟨'( ⋅ ('+×'-)⟩ i 1 2 3 For comparison: triangular lattice Hubbard model at T=0 [Szasz et al., condmat (2018)]
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Special case: YC4
q strong competition between RVB-type NN correlations (!≳!#) and 120˚ incipient order (! < !#) q maximum in chiral correlations is linked to !#
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Summary: 2D triangular Heisenberg model (S=1/2)
J
incipient 120 o order paramagnetic
Tl
Th
T
intermediate
A B
J
2
~0.55 ~0.2
W L
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Summary
q XTRG
} an extremely simple, yet efficient and accurate tensor-network approach to thermal states: ! → ! ∗ ! resulting in $ → 2$ } motivated by entanglement scaling & ∼
( ) log $