Study of tJ and Hubbard Models via Series Expansions

1
Study of t-J and Hubbard Models via Series
Expansions

• Weihong Zheng
• UNSW
• J. Oitmaa, C.J. Hamer, O. Sushkov, R. Singh, et
  al

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Outline

• Motivation
• Models Hubbard, t-J and t-t-J models
• 1D Hubbard model (comparison between series and
  exact results)
• 2D Hubbard model (spin-wave and single hole
  dispersion on square lattice)
• 2D t-J model (single hole dispersion on square
  lattice)
• 2D t-t-J model (single hole dispersion on square
  lattice)
• Summary

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Motivation
Neutron scattering (CuDCOO)2 4D2O (CFTD)
2D quantum S1/2 Heisenberg antiferromagnet on
a square lattice (J6.13meV)
E(?,0) lt E(?/2,?/2)
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• High-Tc Superconductor
• Parent Compound La2CuO4

Model ?
Heisenberg model With ring exchange?
Hubbard model ?
5
Electronic dispersion for Sr2CuO2Cl2 (measured
by Angle-resolved photoemission spectroscopy)
ARPES data and self-consistent Born results for
the t-t-t-J model (t0.35 eV, t-0.12 eV,
t0.08 eV, and J0.14 eV) Can t-J or Hubbard
model explain this? will t and t necessary ?
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Hubbard Model
Coulomb repulsion term
Electron hopping term

• ci?y (ci?) electron creation (destruction)
  operators
• t electron hoping term (non-interacting
  electrons)
• U Coulomb repulsion between electrons
• In real material U/t 10-50
• the simplest generic model for strongly
  correlated electron systems
• Exact solution only available in 1D (n1)

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Speicial cases U0 free electron,
U t and general filling Hirsch derived an
effective Heff
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t-J model

• Ignore above third and fourth terms

another much studied model for strongly
correlated electrons
At half-filling limit, reduce to Heisenberg
antiferromagnet
t-t-J model
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1D Hubbard model comparision of series
expansion/extrapolation with exact results

• Exact soluable via Bethe ansatz (Lieb and Wu,
  1968)

Expand the exact result in series for small t
(set U1)

• Naive sum of series converge up to t0.25 only.
• Badly behaviour series
• oscillate
• coef. increase rapidly with order
• change variable from t2 to t for large t (large
  t E0/N-4t/ ?1/4 ).
• How reliable of series extrapolation to this
  short series ?

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Naive extrapolation

• Naive extrapolation of 10 terms series by
• integrated differential approximant (xt2)

PK, QL, RM and ST are polynomials of order K, L,
M, T. Converge up to t 2.5, xt2, the range
of convergence is 100 times larger than that for
naive sum. This series extrapolation fail at
t2.5 due to the change of variable from t2 to t
for large t.
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Better extrapolation
Change of variable (xt2)

• Transformations
• Euler transformation ?x/(1x)
• Transformation ?21-(1- ?)1/2
• Extrapolate series (1- ?2)-1E0/t2
• using IDA as before

Converge up to t8 ! Only 10 error at t8 from
a series with 10 terms
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Compare exact results with results of Ising and
Dimer expansions
Ising expansions
Dimer expansions (t 0) Better convergent for
small t/U Better convergent for
large t/U
t t t t t
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Ising expansions for Hubbard Model

• J4t2/U, h varied to improve convergence
• ? is the expansion parameter.
• ?0 Neel order G.S. ?1 original H
• Series up to order ?11 ?sw and ?1h
• Series converge better for larger U.

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Ising expansions for t-J Model

• Field r to improve convergence
• ? is the expansion parameter.
• ?0 Neel order G.S. ?1 original H
• Series for 1-hole and 2-hole dispersions are
  computed up to order ?13
• Series converge better for large J

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Spin-wave excitation spectrum for square lattice
Hubbard model
Jeff4t2/U high energy spin-spectra at the
antiferromagnetic zone-boundary are sensitive to
charge fluctuations
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Spin-wave energy at k(?,0) and (?/2,?/2)
?(?,0)- ?(?/2,?/2), changes sign at t/U0.053
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Fit the spin-wave dispersion for La2CuO4R.
Coldea, et. Al., PRL 86, 5377 (2001)
U3.9eV t0.39eV Jeff155meV
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The spin-wave intensity in neutron scattering
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The spin-wave intensity in neutron scattering
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1-hole dispersion for square lattice Hubbard
model
Disagree with dispersion for Sr2CuO2Cl2
(measured by Angle-resolved photoemission
spectroscopy)
Disagreemet exists with extrended Hubbard model
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Momentum distribution function n(k)
Red curves analytic res. Of Oleg for
large-U. No Fermi surface, System is in
insulator at half-filling
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1-hole structure factor
Flat S1h along antiferromagnetic zone boundary
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t-J model 1-hole doped dispersion (square
lattice)
Red curves t-J model Blue curves
Hubbard model Bandwidth for Hubbard model is
much larger than t-J model dispersion disagree
with ARPES, second-neighbour hopping t is needed
for both t-J and Hubbard model.
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t-J model Comparison between series and SCBA
(self-consistent Born approximation) for square
lattice
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t-t-J model 1-hole dispersion (square
lattice)effect of t
dispersion with t/t-0.2 E(0,0)E(?
,0) E(0,0)-E(? /2, ? /2)2E(0,0)-E(?
/2,0) agree with ARPES
SCBA (t/t-0.34, J/t0.4)
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t-t-J model 1-hole dispersion (square lattice)
dispersion with t/t-0.2 E(0,0)E(?
,0) E(0,0)-E(? /2, ? /2)2E(0,0)-E(?
/2,0) agree with ARPES
SCBA (t/t-0.34, J/t0.4)
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Other calculations

• Hubbard model on honeycomb lattice, locate the
  transition between semimetal phase and AF at
  U/t4, (no nesting of Fermi surface here).
• t-J model on honeycomb lattice (larger
  discrepancy between series and SCBA is found).
• t-t-U model on square lattice, effect of t for
  spin-wave excitation and charge excitation
  (spin-wave dispersion do not depend on sign of
  t).
• 2-hole dispersion for t-J, t-t-J, Hubbard and
  t-t-U models (d-wave and p-wave pairing, effect
  of t).

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Conclusions

• For half-filling Hubbard model on square lattice,
  spin-wave dispersion along AF zone boundary is
  sensitive to charge fluctuation.
• Spin-wave dispersion agree with neutron
  scattering results. We got U3.9 eV, t0.39 eV
  for La2CuO4.
• 1-hole dispersion for both t-J and Hubbard models
  has very flat charge excitation spectrum along
  (?,0) to (?/2, ?/2) , disagreeing with ARPES.
• Including of negative second-neighbor hoping t
  can resolve this disagreement, we got t/t-0.2,
  much smaller than t/t-0.34 obtained by SCBA.
• Series expansion is good to study the
  half-filling, 1 and 2 hole systems, but it still
  have problem to handle the system with finite
  doping.