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

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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

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Outline

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing
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Outline

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing
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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.

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Hubbard model and High-Tc superconductivity

Zhang-Rice singlet: from three bands to effective one band model, because

  • f the hybridization of Cu and O
  • rbitals.
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Exponential wall: Hilbert space dimension scales exponentially with system size and particle number.

The difficulty in correlated quantum many-body systems

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.

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Exponential wall: Hilbert space dimension scales exponentially with system size and particle number.

The difficulty in correlated quantum many-body systems

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.

Andreas M. Lauchli, Julien Sudan, and Roderich Moessner, arXiv:1611.06990

Hilbert space Dim ~2.8 x 10^14 Largest sector Dim ~5 x 10^11

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Outline

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing
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The Simons Collaboration on the Many Electron Problem

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Ground state energy of half-filled Hubbard model on square lattice

Mingpu Qin, Hao Shi, and Shiwei Zhang, Phys. Rev. B 94, 085103 (2016)

U=2 U=4 U=6 U=8

Accurate estimate of ground state energy with extrapolation and TABC

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Ground state energy of half-filled Hubbard model on square lattice

Mingpu Qin, Hao Shi, and Shiwei Zhang, Phys. Rev. B 94, 085103 (2016)

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@ 1/8 doping

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

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Summary of the first collaboration work

  • Provide benchmark for 2D Hubbard model
  • Strength and weakness for different methods
  • No discussion about physical properties
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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

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Outline

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing
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t-j model and Heisenberg model

Large U limit: no double occupancy At half-filling, the charge degree of freedom is frozen

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t-j model and Heisenberg model

Matteo Calandra Buonaura and Sandro Sorella,

  • Phys. Rev. B 57, 11446 (1998)

Static magnetic structure factor can be well described by 2D Heisenberg model

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t-j model and Heisenberg model

Matteo Calandra Buonaura and Sandro Sorella,

  • Phys. Rev. B 57, 11446 (1998)

Static magnetic structure factor can be well described by 2D Heisenberg model Heisenberg model on square lattice has long-range Neel order

Anders W. Sandvik and Hans Gerd Evertz,

  • Phys. Rev. B 82, 024407 (2010)
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Dope the AFM

AFM order

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Dope the AFM

AFM order Dope a hole

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Dope the AFM

AFM order Dope a hole hopping High energy

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Dope the AFM

AFM order Dope a hole hopping High energy To lower the energy:

  • 1. Superconducting: two holes hop together

Superconducting

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Dope the AFM

AFM order Dope a hole hopping High energy To lower the energy:

  • 1. Superconducting: two holes hop together
  • 2. Stripe: holes line up in rows / columns

Superconducting Stripe

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1/8 doping anomaly in La2-xBaxCuO4

  • M. Ido, N. Yabiada, M. Oda , Y. Segawa, N. Momono, A. Onodera, Y. Okajima and K. Yamaya,

Physica C, 185 (1991)

Minimum of Tc at 1/8 doping

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Stripe phase in cuprates La2-xBaxCuO4 at 1/8 doping

Neutron experiment determined the stripe structure:

Ref: J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura & S. Uchida, Nature 375, 561 (1995).

CuO2 layer

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Spin and charge fluctuation might be

  • ne key to understand high-Tc
  • B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida & J. Zaanen, Nature 518 179 (2015)
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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)

Present status of the study of 1/8 doped Hubbard model

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Ground state at 1/8 doping: stripe phase

determine the stripe phase

Mingpu Qin, Hao Shi, Shiwei Zhang, Phys. Rev. B 94, 235119 (2016)

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Ground state at 1/8 doping: stripe phase

determine the stripe phase

Mingpu Qin, Hao Shi, Shiwei Zhang, Phys. Rev. B 94, 235119 (2016)

stripped bass

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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)

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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)

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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

  • f the unit cell size.

✔ AFQMC: Doesn't depend on the dimension of the system.

Suffer from the constraint error.

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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)

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Energy of stripe states with different wave-lengths

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)

Energies are nearly degenerate near wave- length 8

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Outline

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing

Background:

Hubbard model and High-Tc superconductivity

Results:

  • 1. First Simons collaboration benchmark
  • 2. Stripe phase in the 1/8 doped 2D Hubbard model
  • 3. The absent of long-range pairing
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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.

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Acknowledgment

Simons Collaboration on the Many Electron Problem Shiwei Zhang, Hao Shi, Ettore Vitali, William & Mary, Flatiron Institute Steven R White, Chia-Min Chung, UC Irvine Boxiao Zheng, Garnet Chan, Caltech Philippe Corboz, University of Amsterdam Georg Ehlers, Reinhard M. Noack, University of Marburg Claudius Hubig, Ulrich Schollowock, LMU Munich

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Thanks