Tensor Network Renormalization of Quantum Spin Liquids Haijun Liao, - - PowerPoint PPT Presentation

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Tensor Network Renormalization of Quantum Spin Liquids Haijun Liao, - - PowerPoint PPT Presentation

Jun 09, 2023 185 likes •438 views

Tensor Network Renormalization of Quantum Spin Liquids Haijun Liao, IOP, China Tao Xiang Lei Wang txiang@iphy.ac.cn Bruce Normand PSI, Swiss Institute of Physics Federico Becca SISSA, Italy Chinese Academy of Sciences Juraj Hasik

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Tensor Network Renormalization of Quantum Spin Liquids

Tao Xiang txiang@iphy.ac.cn Institute of Physics Chinese Academy of Sciences Haijun Liao, IOP, China Lei Wang Bruce Normand PSI, Swiss Federico Becca SISSA, Italy Juraj Hasik

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Quantum Spin Liquid

➢ Novel quantum states probably with topological orders ➢ Mott insulators without magnetic orders ➢ Might be the parent compounds of high-Tc superconductivity

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✓ Insulators with spin interactions ✓ Odd number of electrons per unit cell ✓ Strong geometric or quantum fluctuations

Kagome Heisenberg Materials Route I: Geometrical frustration Route II: Proximity to Mott transition High-Tc cuprates: doped Mott insulators

Routes to Quantum Spin Liquid States

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Idealized model Hamiltonian

Herbertsmithite: ZnCu3(OH)6Cl2

Shores, et al., J. Am. Chem. Soc. (2005)

ZnCu3(OH)6FBr

Feng Z, et al., Chin. Phys. Lett. (2017)

Possible Quantum Spin Liquid Materials: Kagome Lattice

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Herbertsmithite ZnCu3(OH)6Cl2 Along the (H, H, 0) direction, a broad excitation continuum is observed over the entire range measured

Nature 492 (2012) 406

Neutron Scattering: Gapless Spin Liquid

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NMR Knight shift Science 360 (2016) 655

NMR: Gapped Spin Liquid

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Valence-bond Crystal

Marston et al., J. Appl. Phys. 1991 Zeng et al., PRB 1995 Nikolic et al., PRB 2003 Singh et al., PRB 2008 Poilblanc et al., PRB 2010 Evenbly et al., PRL 2010 Schwandt et al., PRB 2011 Iqbal et al., PRB 2011 Poilblanc et al., PRB 2011 Iqbal et al., New J. Phys. 2012 ……

Gapped

Jiang, et al., PRL 2008 Yan, et al., Science 2011 Depenbrock, et al., PRL 2012 Jiang, et al., Nature Phys. 2012 Nishimoto, Nat. Commu. 2013 Gong, et al., Sci. Rep. 2014 Li, arXiv 2016 Mei, et al., PRB 2017 ……

Gapless

Hastings, PRB 2000 Hermele, et al., PRB 2005 Ran, et al., PRL 2007 Hermele, et al., PRB 2008 Tay, et al., PRB 2011 Iqbal, et al., PRB 2013 Hu, et al., PRB 2015 Jiang, et al., arXiv 2016 Liao, et al., PRL 2017 He, et al., PRX 2017 ……

Quantum Spin Liquid

Theoretical Predictions

Not Spin Liquid

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✓ Upper bound: iDMRG (cylinder Ly=12), D=5000, E=-0.4332 ✓ Yan et al: -0.4379(3) ✓ Depenbrock et al: -0.4386(5) D=16000, Ly=17

DMRG Results: Ground State Energy

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Possible Quantum Spin Liquid Materials: Square Lattice

Li2VOSiO4 Li2VOGeO4 VOMoO4 BaCdVO(PO4)2

  • O. Mustonen, et al., Nature Communication 2018
  • R. Melzi, et al., PRL 2000

Sr2Cu(Te0.5W0.5)O6 Idealized Hamiltonian: Square J1-J2 Model

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S=1/2 Antiferromagnetic Square J1-J2 Model

Jc1 Jc2 What are the intermediate phases? ✓ How many phases in between ✓ Is there a quantum spin liquid? ✓ Is there a deconfined quantum critical point?

Senthil, et. al., Science (2004)

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Possible Intermediate Phases

Columnar Dimer State Plaquette Bond Crystal Quantum Spin Liquid

Read & Sachdev, PRL 1989 Zhitomirsky & Ueda, PRB 1996 Doretto, PRB 2014 Mambrini et. al., PRB 2006 Gong, Sheng, et. al., PRL 2014 ...... Chandra & Doucot, PRB 1988 Gochev, PRB 1993 Figueirido, Kivelson, et al., PRB 1989 Richter & Schulenburg, PRB 2010 Li, Becca, Hu, Sorella, PRB 2012 Jiang, Yao, Balents, PRB 2012 Wang & Sandvik, PRL 2018 …… Gelfand, Singh, Huse, PRB 1989 Sachdev & Bhatt, PRB 1990 Valeri, et al., PRB 1999 Murg, Verstraete, Cirac, PRB 2009 Haghshenas & Sheng, PRB 2018 Wang, Gu, Verstraete, Wen, PRB 2016 ......

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Variance of Critical Points: No Agreement

0.3 0.4 0.5 0.6 0.7

J2/J1

Gelfand, Singh, Huse, PRB 1989 Wang, Gu, Verstraete, Wen, PRB 2016 Haghshenas &Sheng, PRB 2018 Murg, Verstraete, Cirac, PRB 2009 Li, Becca, Hu,Sorella, PRB 2012 Wang & Sandvik, PRL 2018 Jiang, Yao, Balents, PRB 2012 Chandra &Doucot, PRB 1988 Gochev, PRB( 1993)

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Possible Evidence of Spin Liquid

L Wang & A W Sandvik, PRL 121, 107202 (2018)

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➢ No good (controllable) analytic methods ➢ Quantum Monte Carlo: severe sign problem ➢ DMRG: strong finite size effect

Difficulty in the Study of Frustrated Quantum Antiferromagnets

Methods commonly used

  • 1. Series expansion
  • 2. Spin-wave theory
  • 3. Large-N expansion
  • 4. Exact Diagonalization
  • 5. DMRG
  • 6. Variational Monte Carlo
  • 7. Tensor Network States

The method we use Tensor Network Renormalization

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➢ Variational wave function satisfying the entanglement area law ➢ Exact in the limit D → 

Tensor Network Renormalization

𝒚 𝒚′

𝑈𝑦𝑦′𝑧𝑧′ [𝑛] =

y' 𝒏

Physical state Virtual state

D

Niggemann & Zittarz, Z. Phys. B 101, 289 (1996) Verstraete & Cirac, cond-mat/0407066

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➢ Simple update

Jiang, Weng, Xiang, PRL 101, 090603 (2008)

Fast and can access large D tensors

➢ Full update

Jordan et al PRL 101, 250602 (2008)

more accurate than simply update cost high

➢ Variational minimization with automatic differentiation

Liao, Liu, Wang, Xiang, arXiv:1903.09650, PRX in press

most accurate and reliable method cost high

Methods for Determining Local Tensors

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Automatic Differentiation (AD)

➢ a cute technique which computes exact derivatives, whose errors are limits only floating point error ➢ a powerful tool successfully used in deep learning Computation Graph Chain Rule of differentiation Haijun Liao’s talk 3-4 pm 26 July

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Strategy of solving this problem:

➢ Gain insight by making comparison with a reference system: ✓ Husimi Lattice, which can be almost exactly solved by TRG ✓ Husimi Lattice: locally similar to but less frustrated than Kagome lattice ➢ Find gap information from the D-dependence of the ground state energy ✓ Converge exponentially with D if the ground state is gapped ✓ Converge algebraically with D if the ground state is gapless

Kagome Heisenberg Model

Liao, et al, PRL 118, 137202 (2017)

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✓ Tree Structure ✓ D ~ 1000 accessible, quasi-exact ✓ Highly frustrated ✓ D is generally small

Reference System: Husimi Lattice

Kagome Lattice Husimi Lattice

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Both energy and magnetization converge algebraically with D

𝛽 =0.588

Magnetizatoin M

S=1/2 Husimi Heisenberg: Gapless Spin Liquid

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power law behavior

Ground State Energy E0

S=1/2 Kagome Heisenberg: Ground State Energy

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Ground State Energy E0

S=1/2 Kagome Heisenberg: Ground State Energy

power law behavior

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𝑵𝑳𝒃𝒉𝒑𝒏𝒇 < 𝑵𝑰𝒗𝒕𝒋𝒏𝒋

S=1/2 Kagome Heisenberg: Magnetic Order Free

Magnetization

Kagome Heisenberg model

Kagome Heisenberg is a spin liquid

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➢ Ground state of the Kagome Heisenberg model is a

gapless spin liquid

➢ Spin liquid phase has not bee found in the intermediate

phase of the square J1-J2 model, more works needed

Summary