Numerical Studies of Magnetism in Heavy Fermion System

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Numerical Studies of Magnetism in Heavy Fermion
System
Frontiers of Theoretical and Computational
Physics and Chemistry

• H. Q. LinPhysics Department, The Chinese
  University of Hong Kong
• CollaboratorH.Y. Shik, Y.Q. Wang (CUHK)C. D.
  Batista, J.E. Gubernatis, Los Alamos National
  Lab, USA
• Work supported in part by the Earmarked Grant for
  Research from the Research Grants Council of the
  HKSAR

Yantai, May 24, 2005
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Outline

• Introduction
• Periodic Anderson Lattice Model (PAM)
• Numerical Approach to Strongly Correlated
  Systems Mean-field, Exact Diagonalization, and
  Quantum Monte Carlo Simulation
• Numerical Studies of PAM
• Summary and Discussions

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Types of Magnetic Structure

• Ferromagnetic (FM) Ordering
• All (almost) moments parallel to each other.
• Antiferromagnetic (AFM) Ordering
• FM on sublattices, but with net moment zero
• Ferrimagnetic Ordering
• MA gt MB.

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Hubbard Model

• HHubbard t ??i,j?? (ci? cj? h.c.)
  Kinetic U ?i ni? ni? Coulomb
• Hk ?k? ?k ck? ck?
• HI ?i ( 2U/3)(Si)2 NeU/6
• Parameters U/t, Ne/N

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Phase Diagram of the 2D Hubbard Model

• Hartree-Fock solution
• However, quantum Monte Carlo simulations (etc.)
  show that there exists no ferromagnetic phase in
  the Hubbard model.
• What is missing?

The phase diagram for the paramagnetic,
ferromagnetic, antiferromagnetic, and
ferrimagnetic states. Solid lines indicate
second-order phase transformations and dashed
lines indicate first-order transformation.
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Multi-band Effect

• Mixed bands when turn on hybridization.
• An effective one-band Hubbard model with a double
  shell dispersion which exhibits unsaturated FM
  state.

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Anderson Lattice Model

• where td hopping of the d-band tf hopping of
  the f-band V hybridization U on-site
  interaction of the f-band ?f chemical
  potential of the f-band

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Methodology

• Hartree-Fock method
• Exact diagonalization studies
• Quantum Monte Carlo

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Mean-Field Approximation
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Exact Diagonalization Method

• It is an eigenvalue problems
• Hij ?iHj?,i, j 1, 2, , M
• H?? ???,?? ?i ?i i?
• ?O? ??O??
• Gives exact solutions of the model on finite
  lattice

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Orthonomal Basis

• Hubbard model 0, ?, ?, ??M 4N, N number of
  Lattice
• Spin model Siz ?S, ?S 1, , S.M (2S
  1)N. e.g., Spin-½ , M 2N
• Binary representation and bit operationI? ?i
  n? 2i?1.I ?i s(i) (2S 1)i?1 , s(i) ? Siz
  S 0, 1, , 2S.cicj h.c. or SiSj? h.c.
  (1,0) ? (0,1)

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Sparse Matrix

• There exists many zero matrix elements
• Number of operations N, not N3
• Ground state properties (low dimensions)
• No finite temperature phase transitions
  (Mermin-Wagner)
• Ef ? 105TLow lying excitations determine basic
  physics

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Lanczös Algorithm

• Recursion RelationH?j? ?j?1?j?1? ?j?j?
  ?j?j1?
• Tridiagonal matrix
• Orthogonality ??i?j? ?ij
• ?i ??jH?j? ?i ??j1H?j?

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Symmetry and Conservation Laws

• If H, Os 0, and ?1? is in a subspace of
  Os,then all ?j? are in the subspace of Os and
  so is ?0? .
• ?j1,2,,m? belongs to an invariant subspace of
  H.
• Choose different initial state.
• 4?4 Hubbard, Ne 8 8, M 416 4 294 967 296
  ? (16)!/(8!)2 601 080 390 ? 1 310 242 (90
  sec.)
• For 1D, L16 or 8 x 2.

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Anderson Lattice Model

• where td hopping of the d-band tf hopping of
  the f-band V hybridization U on-site
  interaction of the f-band ?f chemical
  potential of the f-band

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Quantum Monte Carlo

• Quantum Monte Carlo
• d dimensional d1 dimensionalquantum
  statistical classical statistical
  problem problem
• Ground state Monte Carlo
• Numerical realization

PathIntegral
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Quantum Monte Carlo

• Trotter approximation (H KV)
• Hubbard-Stratonovich transformation (HST)

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Hubbard-Stratonovich Transformation

• Continuous HST
• More general
• where nT (n1, n2, ) is a vector of electron
  occupation numbers. All the eigenvalues of M must
  be positive

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Discrete HST
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Quantum Monte Carlo
Iteration equation becomes
Sample x from P(x), then propagate by B(x).
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Sign Problem

• Basic Monte Carlo
• where xi samples from p(x).
• If, p(x) lt 0 for some x, then we define
  p(x) s(x) p(x) where s(x) is the sign of p(x).

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Sign Problem

• Sign weighted average
• where xi' samples from p(x).

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Sign Problem

• Sign problem
• Variance becomes exponentially large with
  increasing systems size and the the lowering of
  the temperature.
• For the 2D Hubbard model, non-1/2 filled,it goes
  like exp(cbU).
• Constrained path and fixed node methods eliminate
  the sign problem.

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Constrained Path Monte Carlo
Constrained Path approximation
Expectation values
Mixed estimator
Back propagation estimator
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Constrained Path Monte Carlo

• Advantages
• Get around the sign problem.
• Very accurate energy.
• Reliable correlations.
• Good for one-dimensional problems.
• Disadvantages
• Not good for open-shell cases.
• Not easy to get time correlations.

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PC Clusters
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Numerical Studies of PAM

• Fermi surface effects, effect of tf. (0, 0.1,
  -0.1)
• Exact diagonalization studies
• CPMC studies
• One dimension (ED/CPMC)
• Two dimensions (check mean field)
• Ground state energy as function of total spin S
• Spin-spin correlation

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Non-interacting Dispersion
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1D Fermi Surface for ?f - 2
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1D Fermi Surface for ?f - 2, tf -0.5
EF
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1D Fermi Surface for ?f - 5
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2D Fermi Surface for ?f - 2, tf -0.2, -0.5
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2D Fermi Surface for ?f - 2, positive tf
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2D Fermi Surface for ?f - 5, positive tf
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Numerical Studies of PAM

• Ground state energy as function of total spin S,
  plot E(S)/E(S0) and E(S)/E(S1/2) for
  even/odd number of electrons
• Band-filling NeN, ¼ -filled, Ne2N, ½-filled
• Observed partial magnetism

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ED E vs S for 2D v8 v8-site lattice, Ne 8
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CPMC E vs S for 4 ? 4 lattice, Ne 16
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CPMC E vs S for 4 ? 4, Ne 22, Ne 24
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Numerical Studies of PAM

• Spin-spin correlation (f-electron)

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ED Spin-spin correlation for 8 ? 1 chain, Ne 8
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ED Spin-spin correlation for 2D 8-site lattice,
Ne 8
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CPMC Spin-spin correlation for 4 ? 4 lattice, Ne
16
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Phase Diagram
Representation

• Red Ferromagnetic
• Blue Antiferromagnetic
• Purple RSDW
• S(q) peaks at q (p, 0) or (0,p)
• White Uncertain

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Summary of current study

• Ferromagnetism seem to appear in a narrow range,
  e.g., for (2/8,3/8), small hybridization.
• Antiferromagnetism appears near 1/2 filled. tf
  enhances AFM correlation.
• Due to multi-scales in the model, detail analysis
  of data yet to be done. Example, RSDW state,
  non-uniform density distribution (phase
  separation?), etc.

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Discussions

• Multi-band structure is essential for the
  existence of ferromagnetism
• Periodic Anderson lattice model seems to be an
  minimum model accounts for magnetic materials
• f-band dispersion and fermi surface structure
  have rather profound effects on the magnetic
  properties
• Due to multi-scales in the model,
  non-perturbative approach such QMC must be used
• There are many unsolved issues