Local renormalization method for random systems. (English) Zbl 1360.81246
Summary: In this paper, we introduce a real-space renormalization transformation for random spin systems on two-dimensional (2D) lattices. The general method is formulated for random systems and results by merging two well-known real-space renormalization techniques, namely the strong disorder renormalization technique and the contractor renormalization technique. We analyze the performance of the method on the 2D random transverse field Ising model.
MSC:
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
References:
| [1] | Harris A B 1974 Effect of random defects on the critical behavior of Ising models J. Phys. C: Solid State Phys.7 1671-92 · doi:10.1088/0022-3719/7/9/009 |
| [2] | Imry Y and Ma S-K 1975 Random-field instability of the ordered state of continuous symmetry Phys. Rev. Lett.35 1399-401 · doi:10.1103/PhysRevLett.35.1399 |
| [3] | Cardy J 2000 Scaling and Renormalization in Statistical Physics (Cambridge: Cambridge University Press) |
| [4] | Refael G and Fisher D S 2004 Energy correlations in random transverse field Ising spin chains Phys. Rev. B 70 064409 · doi:10.1103/PhysRevB.70.064409 |
| [5] | Ma S-K, Dasgupta C and Hu C-K 1979 Random antiferromagnetic chain Phys. Rev. Lett.43 1434-7 · doi:10.1103/PhysRevLett.43.1434 |
| [6] | Fisher D S 1992 Random transverse field Ising spin chains Phys. Rev. Lett.69 534-7 · doi:10.1103/PhysRevLett.69.534 |
| [7] | Fisher D S 1995 Critical behavior of random transverse-field Ising spin chains Phys. Rev. B 51 6411-61 · doi:10.1103/PhysRevB.51.6411 |
| [8] | Igloi F and Monthus C 2005 Strong disorder RG approach of random systems Phys. Rep.412 277-431 · doi:10.1016/j.physrep.2005.02.006 |
| [9] | Griffiths R B 1969 Non-analytic behavior above the critical point in a random Ising ferromagnet Phys. Rev. Lett.23 17-9 · doi:10.1103/PhysRevLett.23.17 |
| [10] | McCoy B M 1969 Incompleteness of the critical exponent description for ferromagnetic systems containing random impurities Phys. Rev. Lett.23 383-6 · doi:10.1103/PhysRevLett.23.383 |
| [11] | Castro Neto A H, Castilla G and Jones B A 1998 Non-fermi liquid behavior and Griffiths phase in f-electron compounds Phys. Rev. Lett.81 3531-4 · doi:10.1103/PhysRevLett.81.3531 |
| [12] | Rieger H and Young A P 1996 Griffiths singularities in the disordered phase of a quantum Ising spin glass Phys. Rev. B 54 3328-35 · doi:10.1103/PhysRevB.54.3328 |
| [13] | Pich C, Young A P, Rieger H and Kawashima N 1998 Critical behavior and Griffiths-McCoy singularities in the two-dimensional random quantum Ising ferromagnet Phys. Rev. Lett.81 5916-9 · doi:10.1103/PhysRevLett.81.5916 |
| [14] | Motrunich O, Mau S-C, Huse D A and Fisher D S 2000 Infinite-randomness quantum Ising critical fixed points Phys. Rev. B 61 1160-72 · doi:10.1103/PhysRevB.61.1160 |
| [15] | Morningstar C J and Weinstein M 1996 Contractor renormalization group technology and exact Hamiltonian real-space renormalization group transformations Phys. Rev. D 54 4131 · doi:10.1103/PhysRevD.54.4131 |
| [16] | Altman E and Auerbach A 2002 Plaquette boson-fermion model of cuprates Phys. Rev. B 65 104508 · doi:10.1103/PhysRevB.65.104508 |
| [17] | Berg E, Altman E and Auerbach A 2003 Singlet excitations in pyrochlore: a study of quantum frustration Phys. Rev. Lett.90 147204 · doi:10.1103/PhysRevLett.90.147204 |
| [18] | Budnik R and Auerbach A 2004 Low-energy singlets in the Heisenberg antiferromagnet on the Kagome lattice Phys. Rev. Lett.93 187205 · doi:10.1103/PhysRevLett.93.187205 |
| [19] | Capponi S, Lauchli A and Mambrini M 2004 Numerical contractor renormalization method for quantum spin models Phys. Rev. B 70 104424 · doi:10.1103/PhysRevB.70.104424 |
| [20] | Siu M S and Weinstein M 2008 Bootstrap approximations in contractor renormalization Phys. Rev. B 77 155116 · doi:10.1103/PhysRevB.77.155116 |
| [21] | Siu M S and Weinstein M 2007 Exploring contractor renormalization: perspectives and tests on the two-dimensional Heisenberg antiferromagnet Phys. Rev. B 75 184403 · doi:10.1103/PhysRevB.75.184403 |
| [22] | Auerbach A 2005 Computing effective Hamiltonians of doped and frustrated antiferromagnets by contractor renormalization. arXiv:cond-mat/0510738v1 · Zbl 1092.82528 |
| [23] | Capponi S 2005 Numerical contractor renormalization applied to strongly correlated systems. arXiv:cond-mat/0510785v1 |
| [24] | Abendschein A and Capponi S 2007 Contractor-renormalization approach to frustrated magnets in a magnetic field Phys. Rev. B 76 064413 · doi:10.1103/PhysRevB.76.064413 |
| [25] | Ghosh S, Rosenbaum T F, Aeppli G and Coppersmith S N 2003 Entangled quantum state of magnetic dipoles Nature425 48-51 · doi:10.1038/nature01888 |
| [26] | Ancona-Torres C, Silevitch D M, Aeppli G and Rosenbaum T F 2008 Quantum and classical glass transitions in \(LiHo_{ x } Y_{1− x } F_4\)Phys. Rev. Lett.101 057201 · doi:10.1103/PhysRevLett.101.057201 |
| [27] | Fidkowski L, Refael G, Bonesteel N and Moore J E 2008 Infinite randomness phases and entanglement entropy of the disordered golden chain. arXiv:0807.1123v1 |
| [28] | Mezard M, Parisi G and Virasoro M 1987 Spin Glass Theory and Beyond (Singapore: World Scientific) · Zbl 0992.82500 |
| [29] | Young P and Young A P 1998 Spin Glasses and Random Fields (Singapore: World Scientific) |
| [30] | Das D and Chakrabarti B K (ed) 2005 Quantum Annealing and Related Optimization Methods (Berlin: Springer) · Zbl 1130.81020 · doi:10.1007/11526216 |
| [31] | Cugliandolo L F, Grempel D R and da Silva Santos C A 2001 Imaginary-time replica formalism study of a quantum spherical p-spin-glass model Phys. Rev. B 64 014403 · doi:10.1103/PhysRevB.64.014403 |
| [32] | Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) · Zbl 1049.81015 |
| [33] | Gühne O and Toth G 2008 Entanglement detection Phys. Rep.474 1 · doi:10.1016/j.physrep.2009.02.004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
