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
Quantum Spin Liquid ➢ Novel quantum states probably with topological orders ➢ Mott insulators without magnetic orders ➢ Might be the parent compounds of high-Tc superconductivity
Routes to Quantum Spin Liquid States ✓ Insulators with spin interactions ✓ Odd number of electrons per unit cell ✓ Strong geometric or quantum fluctuations Route I: Geometrical frustration Route II: Proximity to Mott transition High-Tc cuprates: doped Mott insulators Kagome Heisenberg Materials
Possible Quantum Spin Liquid Materials: Kagome Lattice Idealized model Hamiltonian ZnCu 3 (OH) 6 FBr Herbertsmithite: ZnCu 3 (OH) 6 Cl 2 Feng Z, et al. , Chin. Phys. Lett. (2017) Shores, et al. , J. Am. Chem. Soc. (2005)
Neutron Scattering: Gapless Spin Liquid Nature 492 (2012) 406 Along the (H, H, 0) direction, a broad excitation continuum is observed over the entire range measured Herbertsmithite ZnCu 3 (OH) 6 Cl 2
NMR: Gapped Spin Liquid Science 360 (2016) 655 NMR Knight shift
Theoretical Predictions Not Spin Liquid Quantum Spin Liquid Valence-bond Crystal Gapped Gapless Hastings, PRB 2000 Marston et al. , J. Appl. Phys. 1991 Jiang, et al. , PRL 2008 Hermele, et al., PRB 2005 Zeng et al. , PRB 1995 Yan, et al. , Science 2011 Ran, et al., PRL 2007 Nikolic et al. , PRB 2003 Depenbrock, et al. , PRL 2012 Hermele, et al., PRB 2008 Singh et al. , PRB 2008 Jiang, et al. , Nature Phys. 2012 Tay, et al. , PRB 2011 Poilblanc et al. , PRB 2010 Nishimoto, Nat. Commu. 2013 Iqbal, et al. , PRB 2013 Evenbly et al. , PRL 2010 Gong, et al. , Sci. Rep. 2014 Hu, et al. , PRB 2015 Schwandt et al. , PRB 2011 Li, arXiv 2016 Jiang, et al. , arXiv 2016 Iqbal et al. , PRB 2011 Mei, et al. , PRB 2017 Liao, et al. , PRL 2017 …… Poilblanc et al. , PRB 2011 He, et al. , PRX 2017 Iqbal et al., New J. Phys. 2012 …… ……
DMRG Results: Ground State Energy ✓ 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
Possible Quantum Spin Liquid Materials: Square Lattice Idealized Hamiltonian: Square J 1 -J 2 Model Sr 2 Cu(Te 0.5 W 0.5 )O 6 Li 2 VOSiO 4 Li 2 VOGeO 4 VOMoO 4 O. Mustonen, et al. , Nature Communication 2018 BaCdVO(PO 4 ) 2 R. Melzi, et al. , PRL 2000
S=1/2 Antiferromagnetic Square J1-J2 Model 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) J c1 J c2
Possible Intermediate Phases Columnar Dimer State Plaquette Bond Crystal Quantum Spin Liquid Chandra & Doucot, PRB 1988 Gelfand, Singh, Huse, PRB 1989 Read & Sachdev, PRL 1989 Gochev, PRB 1993 Sachdev & Bhatt, PRB 1990 Zhitomirsky & Ueda, PRB 1996 Valeri, et al. , PRB 1999 Figueirido, Kivelson, et al., PRB 1989 Doretto, PRB 2014 Richter & Schulenburg, PRB 2010 Murg, Verstraete, Cirac, PRB 2009 Mambrini et. al., PRB 2006 Li, Becca, Hu, Sorella, PRB 2012 Haghshenas & Sheng, PRB 2018 Gong, Sheng, et. al. , PRL 2014 Jiang, Yao, Balents, PRB 2012 Wang, Gu, Verstraete, Wen, PRB 2016 ...... Wang & Sandvik, PRL 2018 ...... ……
Variance of Critical Points: No Agreement Wang & Sandvik, PRL 2018 Haghshenas &Sheng, PRB 2018 Wang, Gu, Verstraete, Wen, PRB 2016 Jiang, Yao, Balents, PRB 2012 Li, Becca, Hu,Sorella, PRB 2012 Murg, Verstraete, Cirac, PRB 2009 Gochev, PRB( 1993) Gelfand, Singh, Huse, PRB 1989 Chandra &Doucot, PRB 1988 0.3 0.4 0.5 0.6 0.7 J 2 /J 1
Possible Evidence of Spin Liquid L Wang & A W Sandvik, PRL 121, 107202 (2018)
Difficulty in the Study of Frustrated Quantum Antiferromagnets ➢ No good (controllable) analytic methods Methods commonly used 1. Series expansion ➢ Quantum Monte Carlo: severe sign problem 2. Spin-wave theory ➢ DMRG: strong finite size effect 3. Large-N expansion 4. Exact Diagonalization 5. DMRG The method we use 6. Variational Monte Carlo Tensor Network Renormalization 7. Tensor Network States
Tensor Network Renormalization ➢ Variational wave function satisfying the entanglement area law ➢ Exact in the limit D → 𝒏 Physical state 𝑈 𝑦𝑦 ′ 𝑧𝑧 ′ [𝑛 ] = 𝒚 𝒚′ D y' Virtual state Niggemann & Zittarz, Z. Phys. B 101, 289 (1996) Verstraete & Cirac, cond-mat/0407066
Methods for Determining Local Tensors ➢ 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
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 Haijun Liao’s talk Computation Graph 3-4 pm 26 July Chain Rule of differentiation
Kagome Heisenberg Model Liao, et al, PRL 118, 137202 (2017) 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
Reference System: Husimi Lattice Kagome Husimi Lattice Lattice ✓ Highly frustrated ✓ Tree Structure ✓ D is generally small ✓ D ~ 1000 accessible, quasi-exact
S=1/2 Husimi Heisenberg: Gapless Spin Liquid Magnetizatoin M 𝛽 = 0.588 Both energy and magnetization converge algebraically with D
S=1/2 Kagome Heisenberg: Ground State Energy Ground State Energy E 0 power law behavior
S=1/2 Kagome Heisenberg: Ground State Energy Ground State Energy E 0 power law behavior
S=1/2 Kagome Heisenberg: Magnetic Order Free 𝑵 𝑳𝒃𝒉𝒑𝒏𝒇 < 𝑵 𝑰𝒗𝒕𝒋𝒏𝒋 Magnetization Kagome Heisenberg model Kagome Heisenberg is a spin liquid
Summary ➢ 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
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