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Title: 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

2
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

3
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)
4
  • 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 ?
6
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)

7
Speicial cases U0 free electron,
U t and general filling Hirsch derived an
effective Heff
8
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
9
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 ?

10
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.
11
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
12
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
13
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.

14
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

15
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
16
Spin-wave energy at k(?,0) and (?/2,?/2)
?(?,0)- ?(?/2,?/2), changes sign at t/U0.053
17
Fit the spin-wave dispersion for La2CuO4R.
Coldea, et. Al., PRL 86, 5377 (2001)
U3.9eV t0.39eV Jeff155meV
18
The spin-wave intensity in neutron scattering
19
The spin-wave intensity in neutron scattering
20
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
21
Momentum distribution function n(k)
Red curves analytic res. Of Oleg for
large-U. No Fermi surface, System is in
insulator at half-filling
22
1-hole structure factor
Flat S1h along antiferromagnetic zone boundary
23
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.
24
t-J model Comparison between series and SCBA
(self-consistent Born approximation) for square
lattice
25
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)
26
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)
27
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).

28
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.
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