The Coupled Cluster Method Applied to Quantum Spin Systems

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The Coupled Cluster Method Applied to Quantum
Spin Systems

• Damian JJ Farnell
• Academic Department of Radiation Oncology,
• University of Manchester, c/o The Christie NHS
  Foundation Trust,
• Wilmslow Road, Manchester M20 4BX. TEL [phone]
  3314
• http//www.medicine.manchester.ac.uk/staff/dfarnel
  l
• damian.farnell_at_manchester.ac.uk

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Many thanks to

• Raymond F. Bishop
• Johannes Richter
• Joerg Schulenburg
• John B. Parkinson
• Ronald Zinke
• Klaus A. Gernoth
• Sven Krüger
• Chen Zeng
• Rachid Darradi

• Peggy Li
• Nedko Ivanov
• Manfred Ristig
• Ulrich Schollwöck
• Yang Xian
• Roger Hale
• Josey Rosenfeld
• Niels Walet
• And many more

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This Talk

• CCM Formalism for Quantum Spin Systems
• High-Order CCM The CCCM Code
• The Square and Triangular Lattice
  Antiferromagnets
• Tables of ground-state properties
• Response to external magnetic fields
• New code test for spin-spin correlation
  functions
• Spontaneous Symmetry Breaking
• Majumdar-Ghosh (1D J1-J2) Model
• Shastry-Sutherland Model - SrCu2(BO3)2
• Testing new CCM code for excitations s1/2
• Initial finite-lattice results
• Infinite-lattice results

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The Coupled Cluster Method (CCM)
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CCM Ground State
Schrödinger Equation
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The Bra State
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Bra-State Equations Expectation Values
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The Excited State
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High-Order CCM Flowchart
www-e.uni-magdeburg.de/jschulen/ccm/index.html
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Approximation Schemes

• LSUBm connected and disconnected clusters of m
  contiguous sites or less.
• SUBn-m connected and disconnected clusters of m
  contiguous sites or less with no more than n spin
  flips
• LPSUBm connected and disconnected clusters for a
  longest path that equals m(jk in 2D), but no
  restriction on numbers of spin flips.
• SUBm all possible m-body clusters.
• DSUBm, PSUBm etc.

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Define Fundamental Clusters
LSUB4 s1/2 Square Lattice HAF

• Find all contiguous (connected) clusters for
  LSUBm, SUBn-m, LPSUBm, etc.
• E.g., via a simple recursive routine.
• Then decompose the connected clusters into
  disconnected subclusters and list them all too.
• Then, restrict to those clusters w.r.t. various
  quantum numbers, e.g., STZ
• And keep only those clusters that are distinct
  under the symmetries of the lattice and
  Hamiltonian

Connected
Disconnected
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High-Order CCM Formalism
Which is essentially a pattern-matching
exercise that is perfectly suited to parallel
computational methods.
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Obtain CCM Equations

• E.g., for term
• Choose the sites i1 i2, , iL to be those for a
  given cluster in the fundamental set.
• Enumerate all ways of partitioning the cluster
  into the two Fs
• Each site may thus belong to either the first or
  second operator, F, above -- but not to both
• Aftrer each partition of the cluster, see if the
  sub-clusters in the two Fs belong to the
  fundamental set
• If so then remember this contribution to the CCM
  equations by storing it in local memory or on
  disk.

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Solving the CCM Equations

• Ket-state Newton-Raphson
• Bra-state LU decomposition
• Excited-state via exact diagonalisation (EISPACK)
• Expectation values can be found directly once bra
  ket equations solved.

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Direct Iteration

• Re-arrange the ket- and bra-state equations so
  that ket- and bra-state correlations coefficients
  are
• Linear on LHS
• Everything else on the RHS
• Iterate to convergence often 103 to 104
  iterations
• Excited-state gap use the shifted power
  iteration for lowest eigenvalue
• Note gap generally gt 0 (at given LSUBm level)
  where model state is good.

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Naïve Parallelisation e.g., ket state
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Extrapolations

• As we can carry out calculations to finite order
  of approximation, we extrapolate to the limit
  m?8.
• Although no exactly known extrapolation rules
  exist, we find that the ground-state energy (in
  some cases!) scales with m-2 and the excitation
  energy as m-1.
• E.g., for the 1D, 2D, 3D bipartite Heisenberg
  antiferromagnets.
• Let xm1/m, so that two common heuristic
  schemes are

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The CCCM Code Recent Advances

• Generalised expectation values e.g., lattice
  magnetisation, spin-spin correlation functions
  (DJJF)
• Excited states for s1/2 direct iteration
  (DJJF)
• Vastly improved routines for memory handling,
  MPI, and minima of energy w.r.t. params in H.
  (JS)
• Number of high-order terms
• Ground state has 68 terms
• Excited state has 81 terms.

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The Square and Triangular Lattice Antiferromagnets
Square Lattice a) (UC2 sites) Triangular
Lattice b), c), d) (UC3 sites) (? ?0 UC1 site)
Rotate all spins to down spins so that all
sites are treated equivalently.
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Square Lattice, s1/2 (??0)
See Bishop et al. J. Phys. Condens. Matter 12
(2000) 116 and Quantum Magnetism Schollwöck et
al. (eds.) Lecture Notes in Physics 645, p.
103. S. Krueger et al. Phys. Rev. B 73, 094404
(2006) cond-mat/0601691. Bishop et al. Phys Rev
B 44, 9425 (1991). Farnell et al -- submitted
to J. Phys. Condens. Matt. (May 2009)
1000 CPUs 15 hours
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Extrapolations
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Triangular Lattice, s1/2 (??0)
See Zeng et al J Stat Phys 90, 327 (1998)
Quantum Magnetism Schollwöck et al. (eds.)
Lecture Notes in Physics 645, p. 108/9. S.
Krueger et al. Phys. Rev. B 73, 094404 (2006)
cond-mat/0601691. Farnell et al -- submitted to
J. Phys. Condens. Matt. (May 2009)
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Extrapolations
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Square Lattice Magnetisation and Susceptibility
Similar increases in ? with the external field
have been observed experimentally, e.g., for the
quasi-two-dimensional AF Ba2CuGe2O7 via QMC,
SWT etc.
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Triangular Lattice Magnetisation and
Susceptibility
The peaks in ? at the end of the plateau are
indeed observed in experiments on
triangular-lattice antiferromagnets, see e.g. T.
Ono et al. Phys. Rev. B 67, 104431 (2003) or M.F.
Collins and O.A. Petrenko, Can. J. Phys. 75, 605
(1997).
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Spin-Spin Correlation FunctionsSquare Lattice
(??0)
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Spin-Spin Correlation FunctionsTriangular
Lattice (??0)
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Spontaneous Symmetry Breaking in The Spin-Half 1D
J1-J2 Model
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Correlation Coefficients LSUB14
Bifurcation point indicates onset of dimer phase
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Ground-State Energies
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Excitation Energy Gap
ED Gap at J2/J11/2 is 0.234 (Caspers, Spin
Systems) Gap opens up at J2/J10.2411(1)
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Shastry-Sutherland Model SrCu2(BO3)2

• Classical ground state (middle) is a spiral for
  J2/J1gt1
• Quantum spiral for J2/J1gt1.5
• Quantum Phase Transition 1st order.
• However, quantum dimer state has lower energy
  than the quantum spiral in this regime.
• Thus, no intermediate quantum spiral!
• See Darradi, Farnell, Richter Phys. Rev. B 72,
  104425 (2005) Farnell et al., J. Stat. Phys.
  135, 175 (2009).

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Testing new code for excitations s1/2
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Finite-Sized Systems (Periodic BCs)
J1 J0 J1
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s1 0D/1D AFs
J1 Jvary J0
GS Energy per site
Gap
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Infinite-Lattice Results 1D Heisenberg Model
DMRG 0.41050(2)
DMRG -1.401484038971(4)
Gap for s1 SUB10-10 to SUB14-14 equals
0.490103762, 0.447560432, and 0.421436022.
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Conclusions

• The CCM can
• Provide accurate ground-state expectation values
  and excited-state energy gap
• Detect phase transitions
• Show spontaneous symmetry breaking
• The CCM code
• Is now highly versatile (DJJF/JS)
• Runs in parallel to very high orders (JS)
• Has been optimised (JS) still room for
  improvement in solve equations?
• Recent new code needs to be validated more
  stringently
• Improvements regarding our heuristic
  extrapolation schemes and/or much higher orders
  of approximation are needed (esp. in 2D).

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Future CCM Calculations

• More Frustrated 2D Magnets
• High-Order Excitation Spectra
• High-Order CCM for Lattice Bosons?
• Spin Dynamics ? Time-Dependent CCM M.D. Prasad,
  Int. J. Mol. Sci. 3, 447458 (2002)
• Non-zero temperatures and phase transitions.
• Finite-Size CCM? Not Sure ?