Numerical Simulations of Frustrated Systems

In many fields of science, computational approaches have become one of the central cornerstones alongside experimental and theoretical approaches. This is also the case in the field of frustrated magnetic systems, where theory, simulations, and experiments drive each other mutually, thus advancing our overall understanding. This chapter aims to provide an overview of the most common numerical techniques available for strongly correlated lattice models, with an emphasis on frustrated spin systems. While it is not possible to treat all methods in sufficient depth to provide a comprehensive introduction, here the key ideas are presented and special issues regarding their application to frustrated systems are discussed. The power of these techniques is illustrated by examples taken from the literature, and specific references to appropriate detailed presentations are included where possible.

[1]  D. Ivanov Vortexlike elementary excitations in the Rokhsar-Kivelson dimer model on the triangular lattice , 2004 .

[2]  Numerical contractor renormalization method for quantum spin models , 2004, cond-mat/0404712.

[3]  White Spin Gaps in a Frustrated Heisenberg Model for CaV4O9. , 1996, Physical review letters.

[4]  Frustrated ferromagnetic spin-12chain in a magnetic field: The phase diagram and thermodynamic properties , 2006, cond-mat/0605204.

[5]  I. Affleck,et al.  Spectral function for theS=1Heisenberg antiferromagetic chain , 2007, 0712.3785.

[6]  P. Sindzingre,et al.  Detecting spontaneous symmetry breaking in finite-size spectra of frustrated quantum antiferromagnets , 2006, cond-mat/0607764.

[7]  Simulations of pure and doped low-dimensional spin-1/2 gapped systems , 2004, cond-mat/0408363.

[8]  Wolff,et al.  Collective Monte Carlo updating for spin systems. , 1989, Physical review letters.

[9]  Roger G. Melko,et al.  Simulations of quantum XXZ models on two-dimensional frustrated lattices , 2006, cond-mat/0608038.

[10]  G. Uhrig,et al.  Dynamic structure factor of the two-dimensional Shastry-Sutherland model. , 2003, Physical review letters.

[11]  Steven R White,et al.  Neél order in square and triangular lattice Heisenberg models. , 2007, Physical review letters.

[12]  Leon Balents,et al.  Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets , 2006, cond-mat/0612001.

[13]  D. Landau,et al.  Spin dynamics simulations of classical ferro- and antiferromagnetic model systems: comparison with theory and experiment , 1999 .

[14]  Matthias Troyer,et al.  Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations , 2004, Physical review letters.

[15]  G. Wellein,et al.  The kernel polynomial method , 2005, cond-mat/0504627.

[16]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[17]  Weihong Zheng,et al.  Anomalous excitation spectra of frustrated quantum antiferromagnets. , 2005, Physical review letters.

[18]  A. Sandvik Stochastic series expansion method with operator-loop update , 1999, cond-mat/9902226.

[19]  Coulomb and liquid dimer models in three dimensions. , 2003, Physical review letters.

[20]  T. Ziman,et al.  Magnetic Order and Disorder in the Frustrated Quantum Heisenberg Antiferromagnet in Two Dimensions , 1994, cond-mat/9402061.

[21]  Christopher L. Henley From classical to quantum dynamics at Rokhsar–Kivelson points , 2003 .

[22]  A. Laeuchli,et al.  Exact diagonalization study of the antiferromagnetic spin-1/2 Heisenberg model on the square lattice in a magnetic field , 2008, 0812.3420.

[23]  Y. Shimizu,et al.  Spin liquid state in an organic Mott insulator with a triangular lattice. , 2003, Physical review letters.

[24]  Ceperley,et al.  Calculation of exchange frequencies in bcc 3He with the path-integral Monte Carlo method. , 1987, Physical review letters.

[25]  A. Läuchli,et al.  Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelson points , 2007, 0711.0752.

[26]  K. Hukushima,et al.  Exchange Monte Carlo Method and Application to Spin Glass Simulations , 1995, cond-mat/9512035.

[27]  S. Chandrasekharan,et al.  Meron-Cluster Solution of Fermion Sign Problems , 1999, cond-mat/9902128.

[28]  E. Berg,et al.  Singlet excitations in pyrochlore: a study of quantum frustration. , 2002, Physical review letters.

[29]  Toshiyuki Imamura,et al.  16.14 TFLOPS Eigenvalue Solver on the Earth Simulator: Exact Diagonalization for Ultra Largescale Hamiltonian Matrix , 2005, ISHPC.

[30]  C. Lanczos An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[31]  Creutz,et al.  Overrelaxation and Monte Carlo simulation. , 1987, Physical review. D, Particles and fields.

[32]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

[33]  P. Sindzingre,et al.  Planar pyrochlore: A valence-bond crystal , 2001, cond-mat/0108070.

[34]  D. Huse,et al.  Ground state of the spin-1/2 kagome-lattice Heisenberg antiferromagnet , 2007, 0707.0892.

[35]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[36]  T. Pruschke,et al.  Quantum cluster theories , 2004, cond-mat/0404055.

[37]  M. Cross,et al.  Magnetism in solid He-3: Confrontation between theory and experiment , 1985 .

[38]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[39]  Lin,et al.  Exact diagonalization of quantum-spin models. , 1990, Physical review. B, Condensed matter.

[40]  Variational treatment of the Shastry-Sutherland antiferromagnet using projected entangled pair states. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  S. Cheong,et al.  Correlation density matrix : an unbiased analysis of exact diagonalizations , 2008, 0809.0075.

[42]  Matthias Troyer,et al.  Feedback-optimized parallel tempering , 2006 .

[43]  R. Bishop,et al.  FRUSTRATED QUANTUM ANTIFERROMAGNETS: APPLICATION OF HIGH-ORDER COUPLED CLUSTER METHOD , 2006, cond-mat/0612146.

[44]  Salvatore R. Manmana,et al.  Diagonalization‐ and Numerical Renormalization‐Group‐Based Methods for Interacting Quantum Systems , 2005 .

[45]  Contractor renormalization group method: A new computational technique for lattice systems. , 1994, Physical review letters.

[46]  Hallberg Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems. , 1995, Physical review. B, Condensed matter.

[47]  Ke-ren Dynamical simulation of spins on kagomé and square lattices. , 1994, Physical review letters.

[48]  J. Chalker,et al.  Hidden order in a frustrated system: Properties of the Heisenberg Kagomé antiferromagnet. , 1992, Physical review letters.

[49]  Holger Fehske,et al.  Exact Diagonalization Techniques , 2008 .

[50]  F. Mila,et al.  Frustrated three-leg spin tubes: from spin 1/2 with chirality to spin 3/2 , 2005, cond-mat/0509217.

[51]  Trivedi,et al.  Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study. , 1990, Physical review. B, Condensed matter.

[52]  Michel Ferrero,et al.  Zero-temperature properties of the quantum dimer model on the triangular lattice , 2005, cond-mat/0502294.

[53]  S. Furukawa,et al.  Systematic derivation of order parameters through reduced density matrices. , 2005, Physical review letters.

[54]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[55]  Damian J. J. Farnell,et al.  The coupled cluster method applied to quantum magnetism , 2004 .

[56]  White,et al.  Real-space quantum renormalization groups. , 1992, Physical review letters.

[57]  T. Park,et al.  Unitary quantum time evolution by iterative Lanczos reduction , 1986 .

[58]  Rajiv R. P. Singh,et al.  High-order convergent expansions for quantum many particle systems , 2000 .

[59]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[60]  R. Moessner,et al.  LOW-TEMPERATURE PROPERTIES OF CLASSICAL GEOMETRICALLY FRUSTRATED ANTIFERROMAGNETS , 1998, cond-mat/9807384.

[61]  Zeng,et al.  Quantum dimer calculations on the spin-1/2 kagome-acute Heisenberg antiferromagnet. , 1995, Physical review. B, Condensed matter.

[62]  Leung,et al.  Percolation properties of the Wolff clusters in planar triangular spin models. , 1991, Physical review. B, Condensed matter.