Level Spectroscopy in Quantum Antiferromagnets Using Monte Carlo Simulations

  • Arnab Sen Indian Association for the Cultivation of Science
Keywords: Quantum antiferromagnets, Deconfined quantum critical points, Spin waves, Spinons, Quantum Monte Carlo methods

Abstract

The low-energy spectrum of antiferromagnets reveal valuable information
about the nature of the phase or phase transition in such systems. I review
the recent works done in collaboration with Suwa and Sandvik
[Phys. Rev. B 92, 195145 (2015), Phys. Rev. B 94, 144416 (2016)]
on how to probe the dispersion of the excitations in a
variety of SU(2) symmetric S=1/2 spin systems using
quantum Monte-Carlo methods.
Various applications are discussed in both one and two
dimensions, which include probing the critical excitations for
conventional and unconventional quantum critical points. In
the latter case, the excitation spectrum is highly unusual and has
additional gapless modes and possible emergent symmetries
which are not present otherwise.

Author Biography

Arnab Sen, Indian Association for the Cultivation of Science

Assistant Professor

Department of Theoretical Physics

References

Anderson, P W (1987), “The resonating valence bond state in la2cuo4 and superconductivity,” Science 235 (4793), 1196–1198,
http://science.sciencemag.org/content/235/4793/1196.full.pdf.
Block, Matthew S, Roger G. Melko, and Ribhu K. Kaul (2013), “Fate of CpN−1 fixed points with q monopoles,”
Phys. Rev. Lett. 111, 137202.
Canali, C M, S. M. Girvin, and Mats Wallin (1992), “Spin-wave velocity renormalization in the two-dimensional heisenberg
antiferromagnet at zero temperature,” Phys. Rev. B 45, 10131–10134.
Chandra, P, and B. Doucot (1988), “Possible spin-liquid state at large s for the frustrated square heisenberg lattice,”
Phys. Rev. B 38, 9335–9338.
Chen, Kun, Yuan Huang, Youjin Deng, A. B. Kuklov, N. V. Prokof’ev, and B. V. Svistunov (2013), “Deconfined criticality
flow in the heisenberg model with ring-exchange interactions,” Phys. Rev. Lett. 110, 185701.
Chubukov, Andrey (1991), “First-order transition in frustrated quantum antiferromagnets,” Phys. Rev. B 44, 392–394.
Chubukov, Andrey V, and Dirk K. Morr (1995), “Phase transition, longitudinal spin fluctuations, and scaling in a two-layer
antiferromagnet,” Phys. Rev. B 52, 3521–3532.
Chubukov, Andrey V, Subir Sachdev, and Jinwu Ye (1994), “Theory of two-dimensional quantum heisenberg antiferromagnets
with a nearly critical ground state,” Phys. Rev. B 49, 11919–11961.
des Cloizeaux, Jacques, and J. J. Pearson (1962), “Spin-wave spectrum of the antiferromagnetic linear chain,”
Phys. Rev. 128, 2131–2135.
Dagotto, Elbio, and Adriana Moreo (1989), “Phase diagram of the frustrated spin-1/2 heisenberg antiferromagnet in 2 dimensions,”
Phys. Rev. Lett. 63, 2148–2151.
Dyson, Freeman J, Elliott H. Lieb, and Barry Simon (1978), “Phase transitions in quantum spin systems with isotropic and
nonisotropic interactions,” Journal of Statistical Physics 18 (4), 335–383.
Eggert, Sebastian (1996), “Numerical evidence for multiplicative logarithmic corrections from marginal operators,”
Phys. Rev. B 54, R9612–R9615.
Fazekas, P (1999), Lecture Notes on Electron Correlation and Magnetism (World Scientific).
Fisher, Daniel S (1989), “Universality, low-temperature properties, and finite-size scaling in quantum antiferromagnets,”
Phys. Rev. B 39, 11783–11792.
Fisher, Matthew P A, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher (1989), “Boson localization and the superfluidinsulator
transition,” Phys. Rev. B 40, 546–570.
Gelfand, Martin P, Rajiv R. P. Singh, and David A. Huse (1989), “Zero-temperature ordering in two-dimensional frustrated
quantum heisenberg antiferromagnets,” Phys. Rev. B 40, 10801–10809.
Ghioldi, E A, M. G. Gonzalez, L. O. Manuel, and A. E. Trumper (2016), “Rvb signatures in the spin dynamics of the
square-lattice heisenberg antiferromagnet,” EPL (Europhysics Letters) 113 (5), 57001.
Gong, Shou-Shu, Wei Zhu, D. N. Sheng, Olexei I. Motrunich, and Matthew P. A. Fisher (2014), “Plaquette ordered phase and
quantum phase diagram in the spin- 1
2 J1−J2 square heisenberg model,” Phys. Rev. Lett. 113, 027201.
Harada, Kenji, Takafumi Suzuki, Tsuyoshi Okubo, Haruhiko Matsuo, Jie Lou, Hiroshi Watanabe, Synge Todo, and Naoki
Kawashima (2013), “Possibility of deconfined criticality in su(n) heisenberg models at small n,” Phys. Rev. B 88, 220408.
Hasenfratz, P, and F. Niedermayer (1993), “Finite size and temperature effects in the af heisenberg model,”
Zeitschrift f¨ur Physik B Condensed Matter 92 (1), 91–112.
Huh, Yejin, Philipp Strack, and Subir Sachdev (2013), “Vector boson excitations near deconfined quantum critical points,”
Phys. Rev. Lett. 111, 166401.
Jarrell, M, and J. E. Gubernatis (1996), “Bayesian inference and the analytic continuation of imaginary-time quantum monte
carlo data,” Physics Reports 269 (3), 133 – 195.
Jiang, F-J (2011), “Method of calculating the spin-wave velocity of spin- 1
2 antiferromagnets with o(n) symmetry in a monte
carlo simulation,” Phys. Rev. B 83, 024419.
Jiang, F-J, M. Nyfeler, S. Chandrasekharan, and U-J. Wiese (2008), “From an antiferromagnet to a valence bond solid:
evidence for a first-order phase transition,” Journal of Statistical Mechanics: Theory and Experiment 2008 (02), P02009.
Jiang, F-J, and U.-J. Wiese (2011), “High-precision determination of low-energy effective parameters for a two-dimensional
heisenberg quantum antiferromagnet,” Phys. Rev. B 83, 155120.
Kaul, Ribhu K (2011), “Quantum criticality in su(3) and su(4) antiferromagnets,” Phys. Rev. B 84, 054407.
Kaul, Ribhu K, and Roger G. Melko (2008), “Large-n estimates of universal amplitudes of the CpN−1 theory and comparison
with a s = 1
2 square-lattice model with competing four-spin interactions,” Phys. Rev. B 78, 014417.
Kennedy, Tom, Elliott H. Lieb, and B. Sriram Shastry (1988), “Existence of n´eel order in some spin-1/2 heisenberg antiferromagnets,”
Journal of Statistical Physics 53 (5), 1019–1030.
Kivelson, Steven A, Daniel S. Rokhsar, and James P. Sethna (1987), “Topology of the resonating valence-bond state: Solitons
and high-Tc superconductivity,” Phys. Rev. B 35, 8865–8868.
Lhuillier, C (2005), “Frustrated Quantum Magnets,” eprint arXiv:cond-mat/0502464 cond-mat/0502464.
Li, Tao, Federico Becca, Wenjun Hu, and Sandro Sorella (2012), “Gapped spin-liquid phase in the J1 − J2 heisenberg model
by a bosonic resonating valence-bond ansatz,” Phys. Rev. B 86, 075111.
27
Manousakis, Efstratios (1991), “The spin-1/2 heisenberg antiferromagnet on a square lattice and its application to the cuprous
oxides,” Rev. Mod. Phys. 63, 1–62.
Melko, Roger G, and Ribhu K. Kaul (2008), “Scaling in the fan of an unconventional quantum critical point,”
Phys. Rev. Lett. 100, 017203.
Millis, A J, and H. Monien (1993), “Spin gaps and spin dynamics in la2−xsrxCuo4 and Yba2cu3o7−,”
Phys. Rev. Lett. 71, 210–210.
Motrunich, Olexei I, and Ashvin Vishwanath (2004), “Emergent photons and transitions in the O(3) sigma model with hedgehog
suppression,” Phys. Rev. B 70, 075104.
Nahum, Adam, P. Serna, J. T. Chalker, M. Ortu˜no, and A. M. Somoza (2015), “Emergent so(5) symmetry at the n´eel to
valence-bond-solid transition,” Phys. Rev. Lett. 115, 267203.
Neuberger, Herbert, and Timothy Ziman (1989), “Finite-size effects in heisenberg antiferromagnets,”
Phys. Rev. B 39, 2608–2618.
Neves, E J, and J. F. Perez (1986), “Long range order in the ground state of two-dimensional antiferromagnets,”
Physics Letters A 114 (6), 331 – 333.
Nomura, K (1995), “Correlation functions of the 2d sine-gordon model,” Journal of Physics A: Mathematical and General 28 (19), 5451.
Pollock, E L, and D. M. Ceperley (1987), “Path-integral computation of superfluid densities,” Phys. Rev. B 36, 8343–8352.
Powalski, M, G. S. Uhrig, and K. P. Schmidt (2015), “Roton minimum as a fingerprint of magnon-higgs scattering in ordered
quantum antiferromagnets,” Phys. Rev. Lett. 115, 207202.
Pujari, Sumiran, Fabien Alet, and Kedar Damle (2015), “Transitions to valence-bond solid order in a honeycomb lattice
antiferromagnet,” Phys. Rev. B 91, 104411.
Read, N, and Subir Sachdev (1989), “Valence-bond and spin-peierls ground states of low-dimensional quantum antiferromagnets,”
Phys. Rev. Lett. 62, 1694–1697.
Read, N, and Subir Sachdev (1991), “Large-n expansion for frustrated quantum antiferromagnets,”
Phys. Rev. Lett. 66, 1773–1776.
Reger, J D, and A. P. Young (1988), “Monte carlo simulations of the spin-(1/2 heisenberg antiferromagnet on a square lattice,”
Phys. Rev. B 37, 5978–5981.
Sachdev, S (2011), Quantum Phase Transitions (Cambridge University Press).
Sandvik, A W (1992), “A generalization of handscomb’s quantum monte carlo scheme-application to the 1d hubbard model,”
Journal of Physics A: Mathematical and General 25 (13), 3667.
Sandvik, A W (2010a), “Computational studies of quantum spin systems,” AIP Conference Proceedings 1297 (1), 135–338,
https://aip.scitation.org/doi/pdf/10.1063/1.3518900.
Sandvik, A W, S. Daul, R. R. P. Singh, and D. J. Scalapino (2002), “Striped phase in a quantum xy model with ring exchange,”
Phys. Rev. Lett. 89, 247201.
Sandvik, A W, and D. J. Scalapino (1994), “Order-disorder transition in a two-layer quantum antiferromagnet,”
Phys. Rev. Lett. 72, 2777–2780.
Sandvik, AndersW(1999), “Stochastic series expansion method with operator-loop update,” Phys. Rev. B 59, R14157–R14160.
Sandvik, Anders W (2005), “Ground state projection of quantum spin systems in the valence-bond basis,”
Phys. Rev. Lett. 95, 207203.
Sandvik, Anders W (2007), “Evidence for deconfined quantum criticality in a two-dimensional heisenberg model with four-spin
interactions,” Phys. Rev. Lett. 98, 227202.
Sandvik, Anders W (2010b), “Continuous quantum phase transition between an antiferromagnet and a valence-bond solid in
two dimensions: Evidence for logarithmic corrections to scaling,” Phys. Rev. Lett. 104, 177201.
Sandvik, Anders W (2010c), “Ground states of a frustrated quantum spin chain with long-range interactions,”
Phys. Rev. Lett. 104, 137204.
Sandvik, Anders W, and Hans Gerd Evertz (2010), “Loop updates for variational and projector quantum monte carlo simulations
in the valence-bond basis,” Phys. Rev. B 82, 024407.
Sandvik, Anders W, and Rajiv R. P. Singh (2001), “High-energy magnon dispersion and multimagnon continuum in the
two-dimensional heisenberg antiferromagnet,” Phys. Rev. Lett. 86, 528–531.
Sen, Arnab, Hidemaro Suwa, and Anders W. Sandvik (2015), “Velocity of excitations in ordered, disordered, and critical
antiferromagnets,” Phys. Rev. B 92, 195145.
Senthil, T, Leon Balents, Subir Sachdev, Ashvin Vishwanath, and Matthew P. A. Fisher (2004a), “Quantum criticality beyond
the landau-ginzburg-wilson paradigm,” Phys. Rev. B 70, 144407.
Senthil, T, and Matthew P. A. Fisher (2000), “Z2 gauge theory of electron fractionalization in strongly correlated systems,”
Phys. Rev. B 62, 7850–7881.
Senthil, T, Ashvin Vishwanath, Leon Balents, Subir Sachdev, and Matthew P. A. Fisher (2004b), “Deconfined quantum critical
points,” Science 303 (5663), 1490–1494, http://science.sciencemag.org/content/303/5663/1490.full.pdf.
Shao, Hui, Wenan Guo, and Anders W. Sandvik (2016), “Quantum criticality with two length scales,”
Science 352 (6282), 213–216, http://science.sciencemag.org/content/352/6282/213.full.pdf.
Shastry, B S, and B. Sutherland (1981), “Exact ground state of a quantum mechanical antiferromagnet,”
Physica B+C 108 (1), 1069 – 1070.
Singh, Rajiv R P, and Martin P. Gelfand (1995), “Spin-wave excitation spectra and spectral weights in square lattice antiferromagnets,”
Phys. Rev. B 52, R15695–R15698.
28
Singh, Rajiv R P, Martin P. Gelfand, and David A. Huse (1988), “Ground states of low-dimensional quantum antiferromagnets,”
Phys. Rev. Lett. 61, 2484–2487.
Spanu, Leonardo, Federico Becca, and Sandro Sorella (2006), “Theoretical constraints for the magnetic-dimer transition in
two-dimensional spin models,” Phys. Rev. B 73, 134429.
Suwa, Hidemaro, Arnab Sen, and Anders W. Sandvik (2016), “Level spectroscopy in a two-dimensional quantum magnet:
Linearly dispersing spinons at the deconfined quantum critical point,” Phys. Rev. B 94, 144416.
Suwa, Hidemaro, and Synge Todo (2015), “Generalized moment method for gap estimation and quantum monte carlo level
spectroscopy,” Phys. Rev. Lett. 115, 080601.
Sylju°asen, Olav F, and Anders W. Sandvik (2002), “Quantum monte carlo with directed loops,” Phys. Rev. E 66, 046701.
Trebst, Simon, Eddy Ardonne, Adrian Feiguin, David A. Huse, AndreasW.W. Ludwig, and Matthias Troyer (2008), “Collective
states of interacting fibonacci anyons,” Phys. Rev. Lett. 101, 050401.
Troyer, Matthias, and Uwe-Jens Wiese (2005), “Computational complexity and fundamental limitations to fermionic quantum
monte carlo simulations,” Phys. Rev. Lett. 94, 170201.
Wang, L, and A. W. Sandvik (2017), “Critical level crossings and gapless spin liquid in the square-lattice spin-1/2 J 1-J 2
Heisenberg antiferromagnet,” ArXiv e-prints arXiv:1702.08197 [cond-mat.str-el].
Wang, Ling, K. S. D. Beach, and Anders W. Sandvik (2006), “High-precision finite-size scaling analysis of the quantum-critical
point of s = 12 heisenberg antiferromagnetic bilayers,” Phys. Rev. B 73, 014431.
Wang, Ling, Zheng-Cheng Gu, Frank Verstraete, and Xiao-Gang Wen (2016), “Tensor-product state approach to spin- 1
2 square
J1−J2 antiferromagnetic heisenberg model: Evidence for deconfined quantum criticality,” Phys. Rev. B 94, 075143.
Wen, X G (1991), “Mean-field theory of spin-liquid states with finite energy gap and topological orders,”
Phys. Rev. B 44, 2664–2672.
Wietek, A, M. Schuler, and A. M. L¨auchli (2017), “Studying Continuous Symmetry Breaking using Energy Level Spectroscopy,”
ArXiv e-prints arXiv:1704.08622 [cond-mat.str-el
Published
2018-07-09
Section
Review Articles