Study of 2D Hubbard model within the Simons Collaboration on the - PowerPoint PPT Presentation

Study of 2D Hubbard model within the Simons Collaboration on the Many Electron Problem Mingpu Qin ( 秦明普 ) Shanghai Jiao Tong University July 16, 2019, CAQMP 2019 ISSP, Kashiwa, Japan

Outline Background: Background: Hubbard model and High-Tc superconductivity Hubbard model and High-Tc superconductivity Results : Results : 1. First Simons collaboration benchmark 1. First Simons collaboration benchmark 2. Stripe phase in the 1/8 doped 2D Hubbard model 2. Stripe phase in the 1/8 doped 2D Hubbard model 3. The absent of long-range pairing 3. The absent of long-range pairing

Outline Background: Background: Hubbard model and High-Tc superconductivity Hubbard model and High-Tc superconductivity Results : Results : 1. First Simons collaboration benchmark 1. First Simons collaboration benchmark 2. Stripe phase in the 1/8 doped 2D Hubbard model 2. Stripe phase in the 1/8 doped 2D Hubbard model 3. The absent of long-range pairing 3. The absent of long-range pairing

Hubbard model and High-Tc superconductivity U Hopping: t It is now widely believed Hubbard model captures the physics of High-Tc superconductivity. Philip W. Anderson, Journal of Physics: Conference Series 449 (2013) 012001.

Hubbard model and High-Tc superconductivity Zhang-Rice singlet: from three bands to effective one band model, because of the hybridization of Cu and O orbitals.

The difficulty in correlated quantum many-body systems Exponential wall: Hilbert space dimension scales exponentially with system size and particle number. Exact Diagonalization: small system size. 1. Spin model with S = 1/2 (~ 40 sites). 2. Hubbard model (~ 20 sites). 3. For all energy levels (excited states), even smaller.

The difficulty in correlated quantum many-body systems Exponential wall: Hilbert space dimension scales exponentially with system size and particle number. Exact Diagonalization: small system size. 1. Spin model with S = 1/2 (~ 40 sites). Andreas M. Lauchli, Julien Sudan, and Roderich Moessner, 2. Hubbard model (~ 20 sites). arXiv:1611.06990 Hilbert space Dim ~2.8 x 10^14 3. For all energy levels (excited states), Largest sector Dim ~5 x 10^11 even smaller.

Outline Background: Background: Hubbard model and High-Tc superconductivity Hubbard model and High-Tc superconductivity Results : Results : 1. First Simons collaboration benchmark 1. First Simons collaboration benchmark 2. Stripe phase in the 1/8 doped 2D Hubbard model 2. Stripe phase in the 1/8 doped 2D Hubbard model 3. The absent of long-range pairing 3. The absent of long-range pairing

The Simons Collaboration on the Many Electron Problem

Ground state energy of half-filled Hubbard model on square lattice U=4 U=2 U=8 U=6 Accurate estimate of ground state energy with extrapolation and TABC Mingpu Qin , Hao Shi, and Shiwei Zhang, Phys. Rev. B 94, 085103 (2016)

Ground state energy of half-filled Hubbard model on square lattice Mingpu Qin , Hao Shi, and Shiwei Zhang, Phys. Rev. B 94, 085103 (2016)

@ 1/8 doping ✗ 1/8 is a difficult region for numerical calculations. ✗ Large discrepancy exists among different methods.

Summary of the first collaboration work ● Provide benchmark for 2D Hubbard model ● Strength and weakness for different methods ● No discussion about physical properties

Summary of the first collaboration work ● Provide benchmark for 2D Hubbard model ● Strength and weakness for different methods ● No discussion about physical properties Lead to the next collaboration work

Outline Background: Background: Hubbard model and High-Tc superconductivity Hubbard model and High-Tc superconductivity Results : Results : 1. First Simons collaboration benchmark 1. First Simons collaboration benchmark 2. Stripe phase in the 1/8 doped 2D Hubbard model 2. Stripe phase in the 1/8 doped 2D Hubbard model 3. The absent of long-range pairing 3. The absent of long-range pairing

t-j model and Heisenberg model Large U limit: no double occupancy At half-filling, the charge degree of freedom is frozen

t-j model and Heisenberg model Static magnetic structure factor can be well described by 2D Heisenberg model Matteo Calandra Buonaura and Sandro Sorella, Phys. Rev. B 57 , 11446 (1998)

t-j model and Heisenberg model Static magnetic structure factor Heisenberg model on square lattice can be well described by 2D has long-range Neel order Heisenberg model Anders W. Sandvik and Hans Gerd Evertz, Matteo Calandra Buonaura and Sandro Sorella, Phys. Rev. B 82 , 024407 (2010) Phys. Rev. B 57 , 11446 (1998)

Dope the AFM AFM order

Dope the AFM Dope a hole AFM order

Dope the AFM Dope a hole AFM order High energy hopping

Dope the AFM Dope a hole AFM order High energy hopping To lower the energy: 1. Superconducting : two holes hop together Superconducting

Dope the AFM Dope a hole AFM order High energy hopping To lower the energy: 1. Superconducting : two holes hop together 2. Stripe : holes line up in rows / columns Stripe Superconducting

1/8 doping anomaly in La 2-x Ba x CuO 4 Minimum of Tc at 1/8 doping M. Ido, N. Yabiada, M. Oda , Y. Segawa, N. Momono, A. Onodera, Y. Okajima and K. Yamaya, Physica C, 185 (1991)

Stripe phase in cuprates La 2-x Ba x CuO 4 at 1/8 doping Neutron experiment determined the stripe structure: CuO 2 layer Ref: J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura & S. Uchida, Nature 375 , 561 (1995).

Spin and charge fluctuation might be one key to understand high-Tc B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida & J. Zaanen, Nature 518 179 (2015)

Present status of the study of 1/8 doped Hubbard model Contradictory conclusions from different methods. 1. DMRG: stripe, superconductivity. D. J. Scalapino, S. R. White, Physica C, 341, 367 (2000) S. R. White, D. J. Scalapino, PRL 91, 136403 (2003) 2. Diffusion / variational MC: phase separation, Superconductivity. S. Sorella, PRB 91, 241116 (2015) L. F. Tocchio, F. Becca, S. Sorella, PRB 94, 195126 (2016) 3. Embedding methods (DMFT, DCA): superconductivity. E. Gull and A. J. Millis, PRB 86, 241106 (2012) E. Gull, O. Parcollet, and A. J. Millis, PRL 110, 216405 (2013) 4. iPEPS: coexistence of superconductivity and stripe. P. Corboz, S. R. White, G. Vidal, M. Troyer, PRB 84, 041108 (2011) P. Corboz, T. M. Rice, M. Troyer, PRL 113 , 046402 (2014) 5. constrain path Auxilliary Field Quantum Monte Carlo: SDW, stripe. Chia-Chen Chang, Shiwei Zhang, PRB 78, 165101 (2008) Chia-Chen Chang, Shiwei Zhang, PRL 104, 116402 (2010)

Ground state at 1/8 doping: stripe phase determine the stripe phase Mingpu Qin , Hao Shi, Shiwei Zhang, Phys. Rev. B 94 , 235119 (2016)

Ground state at 1/8 doping: stripe phase determine the stripe phase stripped bass Mingpu Qin , Hao Shi, Shiwei Zhang, Phys. Rev. B 94 , 235119 (2016)

Ground state at 1/8 doping: stripe phase ● Arrow direction: spin direction ● Arrow size: spin density ● Symbol size: hole density ● Agreement among different methods, with discrepancy in details. Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Ground state energy with different methods Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Why do we need different methods? Strength and limitation of different numerical methods: ✔ DMRG : “Perfect” for 1D system. For 2D, state kept need to increase exponentially with the width of the system. ✔ iPEPS : Capture the entanglement structure of 2D system. Small bond dimension available due to (D^12) cost. ✔ DMET : Deal with infinite system directly. Need an extrapolation of the unit cell size. ✔ AFQMC : Doesn't depend on the dimension of the system. Suffer from the constraint error.

An example showing the complementary of DMRG and AFQMC Self-consistent CP-AFQMC results: 6-hole stripe DMRG result: first 4-hole stripe, switches to 6-hole stripe after increasing bond dimension. Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Energy of stripe states with different wave-lengths Energies are nearly degenerate near wave- length 8 Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Outline Background: Background: Hubbard model and High-Tc superconductivity Hubbard model and High-Tc superconductivity Results : Results : 1. First Simons collaboration benchmark 1. First Simons collaboration benchmark 2. Stripe phase in the 1/8 doped 2D Hubbard model 2. Stripe phase in the 1/8 doped 2D Hubbard model 3. The absent of long-range pairing 3. The absent of long-range pairing

Conclusion ✔ Benchmarks for 2D Hubbard model are provided. ✔ We establish the stripe state as the ground state of the 1/8 doped 2D Hubbard model. ✔ Absent of long-range pairing at 1/8 doping.