"Quantum spin liquids and fractionalization "
unpublished (pdf)
Notes for some lectures given at the school on "Highly frustrated magnetism" (Trieste, August 2007). 29 pages.
Abstract: This chapter discusses quantum antiferromagnets which do not break any symmetries at zero temperature -- also called spin liquids -- and focuses on lattice spin models with Heisenberg-like (i.e. SU(2)-symmetric) interactions in dimensions larger than one. We begin by discussing the Lieb-Schultz-Mattis theorem and its recent extension to D>1 by Hastings (2004), which establishes an important distinction between spin liquids with an integer and with a half-integer spin per unit cell. Spin liquids of the rst kind, band insulators, can often be understood by elementary means, whereas the latter, Mott insulators, are more complex (featuring topological order) and support spin-1/2 excitations (spinons). The fermionic formalism (Afleck and Marston, 1988) is described and the effect of fluctuations about mean-field solutions, such as the possible creation of instabilities, is discussed in a qualitative way. In particular, we explain the emergence of gauge modes and their relation to fractionalization. The concept of the projective symmetry group (X.-G. Wen, 2002) is introduced, with the aid of some examples. Finally, we present the phenomenology of (gapped) short-ranged resonating-valence-bond spin liquids, and make contact with the fermionic approach by discussing their description in terms of a uctuating Z2 gauge field. Some recent references are given to other types of spin liquid, including gapless ones.
"Shannon and entanglement entropies of one- and two-dimensional critical wave functions "
arXiv:0906.1153
Abstract: We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant S0$. In c=1 critical systems (Tomonaga-Luttinger liquids) , we find that $S_0$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 XXZ chain. In a massive (crystal) phase, $S_0$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a c=1/2 critical point.
"Comment on "Regional Versus Global Entanglement in Resonating-Valence-Bond States" "
Phys. Rev. Lett. 101, 248901 (2008) (pdf)
Abstract: In a recent Letter [Phys. Rev. Lett. 99, 170502 (2007); quant-ph/0703227], Chandran and coworkers study the entanglement properties of valence bond (VB) states. Their main result is that VB states do not contain (or only an insignificant amount of) two-site entanglement, whereas they possess multi-body entanglement. Two examples ("RVB gas and liquid") are given to illustrate this claim, which essentially comes from a lower bound derived for spin correlators in VB states. We show in this Comment that (i) for the "RVB liquid" on the square lattice, the calculations and conclusions of Chandran et al. are incorrect. (ii) A simple analytical calculation gives the exact value of the correlator for the "RVB gas", showing that the bound found by Chandran et al. is tight. (iii) The lower bound for spin correlators in VB states is equivalent to a celebrated result of Anderson dating from more than 50 years ago.
"Quantum spin liquids "
arXiv:0809.2257
Lectures given at the Les Houches summer school on "Exact Methods in Low-dimensional Statistical Physics and Quantum Computing" (July 2008)
Abstract: These notes are an introduction to a few selected theoretical ideas in the field of quantum spin liquids: classical zero modes and breakdown of the 1/S expansion, the Lieb-Schultz-Mattis-Hastings theorem and Oshikawa's argument, the short-ranged resonating valence-bond picture, large-N limit (Schwinger bosons) and Z2 gauge theory.
"Correlations and order parameter at a Coulomb-crystal phase transition in a three-dimensional dimer model "
Gregoire Misguich, Vincent Pasquier, Fabien Alet .
Phys. Rev. B 78, 100402(R) (2008) (pdf)
Abstract: The three-dimensional classical dimer model with interactions shows an unexpected continuous phase transition between an ordered dimer crystal and a Coulomb liquid. A detailed analysis of the critical dimer and monomer correlation functions points to a subtle interplay between the fluctuations of the crystal order parameter and the "magnetic" degrees of freedom present in the Coulomb phase. The distribution probability of the crystal order parameter suggests an emerging continuous O(3) symmetry at the critical point.
"Thermal destruction of chiral order in a two-dimensional model of coupled trihedra "
Laura Messio, Jean-Christophe Domenge, Claire Lhuillier, Laurent Pierre, Pascal Viot, and Gregoire Misguich .
Phys. Rev. B 78, 054435 (2008) (pdf)
Abstract: We introduce a minimal model describing the physics of classical two-dimensional (2D) frustrated Heisenberg systems, where spins order in a nonplanar way at T=0. This model, consisting of coupled trihedra (or Ising-RP3 model), encompasses Ising (chiral) degrees of freedom, spin-wave excitations, and Z2 vortices. Extensive Monte Carlo simulations show that the T=0 chiral order disappears at finite temperature in a continuous phase transition in the 2D Ising universality class, despite misleading intermediate-size effects observed at the transition. The analysis of configurations reveals that short-range spin fluctuations and Z2 vortices proliferate near the chiral domain walls, explaining the strong renormalization of the transition temperature. Chiral domain walls can themselves carry an unlocalized Z2 topological charge, and vortices are then preferentially paired with charged walls. Further, we conjecture that the anomalous size effects suggest the proximity of the present model to a tricritical point. A body of results is presented, which all support this claim: (i) first-order transitions obtained by Monte Carlo simulations on several related models, (ii) approximate mapping between the Ising-RP3 model and a dilute Ising model (exhibiting a tricritical point), and, finally, (iii) mean-field results obtained for Ising-multispin Hamiltonians, derived from the high-temperature expansion for the vector spins of the Ising-RP3 model.
"Quantum Dimer Model on the triangular lattice : Semiclassical and variational approaches to vison dispersion and condensation "
Gregoire Misguich, Frederic Mila .
Phys. Rev. B 77, 134421 (2008) (pdf)
"Editors' Suggestions"
Abstract: After reviewing the concept of vison excitations in Z2 dimer liquids, we study the liquid-crystal transition of the Quantum Dimer Model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a "soft-dimer" version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and \sqrt{12}x\sqrt{12} symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the RK point which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave-functions at the RK point.
"Magnetic-Field Induced Bose-Einstein Condensation of Magnons and Critical Behavior in Interacting Spin Dimer System TlCuCl3 "
Fumiko Yamada, Toshio Ono, Hidekazu Tanaka, Gregoire Misguich, Masaki Oshikawa, Toshiro Sakakibara .
J. Phys. Soc. Jpn. 77, 013701 (2008) (pdf)
"Paper of Editors' Choice"
Abstract: Magnetization measurements were performed to investigate the critical behavior of the field-induced magnetic ordering in gapped spin system TlCuCl3. The critical density of the magnons was obtained as a function of temperature and the magnon-magnon interaction constant was evaluated. The experimental phase boundary for T<5K agrees almost perfectly with the magnon BEC theory based on the Hartree-Fock approximation with realistic dispersion relations. The phase boundary can be described by the power law [HN(T)-Hc] ~ Tphi. With decreasing fitting temperature range, the critical exponent ${phi}$ decreases and converges at phiBEC=3/2 predicted by the magnon BEC theory.
"Magnetic susceptibility and specific heat of the spin-1/2 Heisenberg model on the kagome lattice and experimental data on ZnCu3(OH)6Cl2 "
Gregoire Misguich, Philippe Sindzingre .
Eur. Phys. J. B 59, 305 (2007) (pdf)
Abstract: We compute the magnetic susceptibility and specific heat of the spin-1/2 Heisenberg model on the kagome lattice with high-temperature expansions and exact diagonalizations. We compare the results with the experimental data on ZnCu3(OH)6Cl2 obtained by Helton et al. [Phys. Rev. Lett. 98, 107204 (2007)]. Down to kBT/J~0.2, our calculations reproduce accurately the experimental susceptibility, with an exchange interaction J~190K and a contribution of 3.7% of weakly interacting impurity spins. The comparison between our calculations of the specific heat and the experiments indicate that the low-temperature entropy (below ~20K) is smaller in ZnCu3(OH)6Cl2 than in the kagome Heisenberg model, a likely signature of other interactions in the system.
"Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice "
Shunsuke Furukawa, Gregoire Misguich .
Phys. Rev. B 75, 214407 (2007) (pdf)
Abstract: A characterization of topological order in terms of entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid, 110405]. It was argued that, in a topological phase, there is a universal additive constant in the entanglement entropy, called topological entanglement entropy, determined by the underlying gauge theory of topological order. In the present paper we evaluate numerically the topological entanglement entropy in the ground states of a quantum dimer model on the triangular lattice, which is known to have a Z2 topological ordered phase. We examine the two original constructions to measure the topological entanglement entropy, and we observe that they both approach the value expected for Z2 topological order, in the large-area limit. We also consider the entanglement entropy on a ``zigzag'' topologically non-trivial area and propose to use it as a more accurate way to measure topological entanglement entropies.
"Detecting spontaneous symmetry breaking in finite-size spectra of frustrated quantum antiferromagnets "
Gregoire Misguich, Philippe Sindzingre .
J. Phys.: Condens. Matter 19, 145202 (2007). (pdf)
Proceedings of the International Conference "Highly Frustrated Magnets", Osaka (Japan), August 2006.
Abstract: Exact diagonalization is a powerful numerical technique to analyze static and dynamical quantities in quantum many-body lattice models. It provides unbiased information concerning quantum numbers, energies and wave-functions of the low-energy eigenstates for almost any kind of microscopic model. The information about energies and quantum numbers is particularly useful to detect possible spontaneous symmetry breaking at T=0. We review some of the advances in the field of frustrated quantum magnets which have been possible thanks to detailed symmetry analysis of exact diagonalizations spectra. New results concerning the kagome and star lattice Heisenberg antiferromagnets are presented.
"Classical dimers with aligning interactions on the square lattice "
Fabien Alet, Yacine Ikhlef, Jesper Lykke Jacobsen, Gregoire Misguich, Vincent Pasquier .
Phys. Rev. E 74, 041124 (2006) (pdf)
Abstract: We present a detailed study of a model of close-packed dimers on the square lattice with an interaction between nearest-neighbor dimers. The interaction favors parallel alignment of dimers, resulting in a low-temperature crystalline phase. With large-scale Monte Carlo and Transfer Matrix calculations, we show that the crystal melts through a Kosterlitz-Thouless phase transition to give rise to a high-temperature critical phase, with algebraic decays of correlations functions with exponents that vary continuously with the temperature. We give a theoretical interpretation of these results by mapping the model to a Coulomb gas, whose coupling constant and associated exponents are calculated numerically with high precision. Introducing monomers is a marginal perturbation at the Kosterlitz-Thouless transition and gives rise to another critical line. We study this line numerically, showing that it is in the Ashkin-Teller universality class, and terminates in a tricritical point at finite temperature and monomer fugacity. In the course of this work, we also derive analytic results relevant to the non-interacting case of dimer coverings, including a Bethe Ansatz (at the free fermion point) analysis, a detailed discussion of the effective height model and a free field analysis of height fluctuations.
"Reduced density matrices and topological order in a quantum dimer model "
Shunsuke Furukawa, Gregoire Misguich, Masaki Oshikawa .
J. Phys.: Condens. Matter 19, 145212 (2007)
Proceedings of the International Conference "Highly Frustrated Magnets", Osaka (Japan), August 2006.
Abstract: Resonating valence bond (RVB) liquids in two dimensions are believed to exhibit topological order and to admit no local order parameter of any kind. This is a defining property of "liquids" but it has been explicitly confirmed only in a few exactly solvable models. In this paper, we investigate the quantum dimer model on the triangular lattice. It possesses an RVB-type liquid phase, however, for which the absence of the local order parameter has not been proved. We examine the question numerically with a measure based on reduced density matrices. We find a scaling of the measure which strongly supports the absence of any local order parameter.
"Unconventional continuous phase transition in a three dimensional dimer model "
Fabien Alet, Gregoire Misguich, Vincent Pasquier, Roderich Moessner, Jesper Lykke Jacobsen .
Phys. Rev. Lett. 97, 030403 (2006) (pdf)
Abstract: Phase transitions occupy a central role in physics, due both to their experimental ubiquity and their fundamental conceptual importance. The explanation of universality at phase transitions was the great success of the theory formulated by Ginzburg and Landau, and extended through the renormalization group by Wilson. However, recent theoretical suggestions have challenged this point of view in certain situations. In this Letter we report the first large-scale simulations of a three-dimensional model proposed to be a candidate for requiring a description beyond the Landau-Ginzburg-Wilson framework: we study the phase transition from the dimer crystal to the Coulomb phase in the cubic dimer model. Our numerical results strongly indicate that the transition is continuous and is compatible with a tricritical universality class, at variance with previous proposals.
"Local spin spirals in the Neel phase of La2-xSrxCuO4 "
Andreas Luscher, Gregoire Misguich, Alexander I. Milstein and Oleg P. Sushkov .
Phys. Rev. B 73, 085122 (2006) (pdf)
Abstract: Experimental observations of lightly doped La2-xSrxCuO4 x<0.02, revealed remarkable magnetic properties such as the incommensurate noncollinear ordering (additional to the Neel ordering) and a tremendous doping dependence of the uniform longitudinal susceptibility. We show that the spiral solution of the t-t'-t''-J model obtained by taking into account the Coulomb trapping of holes by Sr ions describes these puzzling data perfectly well. Our solution firstly explains why the incommensurate structure is directed along the orthorhombic b-axis, and secondly allows a numerical calculation of the positions and shapes of the incommensurate neutron scattering peaks. Thirdly, we calculate the doping dependence of the spin-wave gap, and lastly, we study the longitudinal magnetic susceptibility and show that its doping dependence is due to the noncollinearity of the spin spiral.
"Systematic derivation of order parameters through reduced density matrices "
Shunsuke Furukawa, Gregoire Misguich and Masaki Oshikawa .
Phys. Rev. Lett. 96, 047211 (2006) (pdf)
Abstract: A systematic method for determining order parameters for quantum many-body systems on lattices is developed by utilizing reduced density matrices. This method allows one to extract the order parameter directly from the wave functions of the degenerate ground states without aid of empirical knowledge, and thus opens a way to explore unknown exotic orders. The applicability of this method is demonstrated numerically or rigorously in models which are considered to exhibit dimer, scalar chiral, and topological orders.
"Detection of Exotic Order Parameters of Quantum Antiferromagnets through Reduced Density Matrices "
Shunsuke Furukawa, Gregoire Misguich and Masaki Oshikawa .
Progress of Theoretical Physics Supplement 159, 143 (2005)
"Interacting classical dimers on the square lattice "
F. Alet, J. L. Jacobsen, G. Misguich, V. Pasquier, F. Mila, and M. Troyer .
Phys. Rev. Lett. 94, 235702 (2005) (pdf)
Abstract: We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)]. By means of Monte Carlo and transfer matrix calculations, we show that this system undergoes a Kosterlitz-Thouless transition separating a low temperature ordered phase where dimers are aligned in columns from a high temperature critical phase with continuously varying exponents. This is understood by constructing the corresponding Coulomb gas, whose coupling constant is computed numerically. We also discuss doped models and implications on the finite-temperature phase diagram of quantum dimer models.
"Quantum dimer model with Z2 liquid ground-state: interpolation between cylinder and disk topologies and toy model for a topological quantum-bit "
G. Misguich, V. Pasquier, F. Mila and C. Lhuillier .
Phys. Rev. B 71, 184424 (2005) (pdf)
Abstract: We consider a quantum dimer model (QDM) on the kagome lattice which was introduced recently [Phys. Rev. Lett. 89, 137202 (2002)]. It realizes a Z2 liquid phase and its spectrum was obtained exactly. It displays a topological degeneracy when the lattice has a non-trivial geometry (cylinder, torus, etc). We discuss and solve two extensions of the model where perturbations along lines are introduced: first the introduction of a potential energy term repelling (or attracting) the dimers along a line is added, second a perturbation allowing to create, move or destroy monomers. For each of these perturbations we show that there exists a critical value above which, in the thermodynamic limit, the degeneracy of the ground-state is lifted from 2 (on a cylinder) to 1. In both cases the exact value of the gap between the first two levels is obtained by a mapping to an Ising chain in transverse field. This model provides an example of solvable Hamiltonian for a topological quantum bit where the two perturbations act as a diagonal and a transverse operator in the two-dimensional subspace. We discuss how crossing the transitions may be used in the manipulation of the quantum bit to optimize simultaneously the frequency of operation and the losses due to decoherence.
"Specific heat of the S=1/2 Heisenberg model on the kagome lattice: high-temperature series expansion analysis "
G. Misguich and B. Bernu .
Phys. Rev. B 71, 014417 (2005) (pdf)
Abstract: We compute specific heat of the antiferromagnetic spin-1/2 Heisenberg model on the kagome lattice. We use a recently introduced technique to analyze high-temperature series expansion based on the knowledge of high-temperature series expansions, the total entropy of the system and the low-temperature expected behavior of the specific heat as well as the ground-state energy. In the case of kagome-lattice antiferromagnet, this method predicts a low-temperature peak at T/J<0.1.
"Bose-Einstein Condensation of Magnons in TlCuCl3: Phase diagram and specific heat from a self-consistent Hartee-Fock calculation with a realistic dispersion relation "
G. Misguich and M. Oshikawa .
J. Phys. Soc. Jpn. 73, 3429 (2004) (pdf)
Abstract: We extend the self-consistent Hartree-Fock-Popov calculations by Nikuni et al. [Phys. Rev. Lett. 84, 5868 (2000)] concerning the Bose-Eistein condensation of magnons in TlCuCl3 to include a realistic dispersion of the excitations. The result for the critical field as a function of temperature behaves as Hc(T)-Hc(0)~T3/2 below 2K but deviates from this simple power-law at higher temperature and is in very good agreement with the experimental results. The specific heat is computed as a function of temperature for different values of the magnetic field. It shows a lambda-like shape at the transition and is in good qualitative agreement with the results of Oosawa et al. [Phys. Rev. B 63, 134416 (2001)].
"Soliton binding and low-lying singlets in frustrated odd-legged S=1/2 spin tubes "
A. Luscher, R. M. Noack, G. Misguich, V. N. Kotov and F. Mila .
Phys. Rev. B 70, 060405(R) (2004) (pdf)
Abstract: Motivated by the intriguing properties of the vanadium spin tube Na2V3O7, we show that an effective spin-chirality model similar to that of standard Heisenberg odd-legged S=1/2 spin tubes can be derived for frustrated inter-ring couplings, but with a spin-chirality coupling constant alpha that can be arbitrarily small. Using density matrix renormalization group and analytical arguments, we show that, while spontaneous dimerization is always present, solitons become bound into low-lying singlets as alpha is reduced. Experimental implications for strongly frustrated tubes are discussed.
"Cyclic exchange, isolated states, and spinon deconfinement in an XXZ Heisenberg model on the checkerboard lattice "
N. Shannon, G. Misguich, and K. Penc .
Phys. Rev. B 69, 220403(R) (2004) (pdf)
Abstract: The antiferromagnetic Ising model on a checkerboard lattice has an ice-like ground state manifold with extensive degeneracy. and, to leading order in Jxy, deconfined spinon excitations. We explore the role of cyclic exchange arising at order Jxy2/Jz on the ice states and their associated spinon excitations. By mapping the original problem onto an equivalent quantum six-vertex model, we identify three different phases as a function of the chemical potential for flippable plaquettes.a phase with long range Néel order and confined spinon excitations, a nonmagnetic state of resonating square plaquettes, and a quasicollinear phase with gapped but deconfined spinon excitations. The relevance of the results to the square-lattice quantum dimer model is also discussed.
"Interaction between static holes in a quantum dimer model on the kagome lattice "
G. Misguich, D. Serban and Vincent Pasquier .
J. Phys.: Cond. Mat. 16, 823 (2004) (pdf)
Abstract: A quantum dimer model (QDM) on the kagome lattice with an extensive ground-state entropy was recently introduced [Phys. Rev. B 67, 214413 (2003)]. The ground-state energy of this QDM in presence of one and two static holes is investigated by means of exact diagonalizations on lattices containing up to 144 kagome sites. The interaction energy between the holes (at distances up to 7 lattice spacings) is evaluated and the results show no indication of confinement at large hole separations.
"Two-dimensional quantum antiferromagnets "
G. Misguich and C. Lhuillier .
Review chapter in the book ``Frustrated spin systems'', edited by H. T. Diep, World-Scientific (2005). [cond-mat/0310405]
Abstract: This review presents some theoretical advances in the field of quantum magnetism in two-dimensional systems. Among the subjects which are discussed: the spin-1/2 J1-J2 Heisenberg model on the square lattice, valence-bond crystals, large-N methods, quantum dimer models, the physics of multiple-spin exchange interactions, RVB spin liquids as well as the spin-1/2 Heisenberg model on the kagome lattice.
"Ising transition driven by frustration in a 2D classical model with continuous symmetry "
C. Weber, L. Capriotti, G. Misguich, F. Becca, M. Elhajal and F. Mila .
Phys. Rev. Lett. 91, 177202 (2003) (pdf)
Abstract: We study the thermal properties of the classical antiferromagnetic Heisenberg model with both nearest (J1) and next-nearest (J2) exchange couplings on the square lattice by extensive Monte Carlo simulations. We show that, for J2/J1 > 1/2, thermal fluctuations give rise to an effective Z2 symmetry leading to a finite-temperature phase transition. We provide strong numerical evidence that this transition is in the 2D Ising universality class, and that Tc->0 with an infinite slope when J2/J1->1/2.
"Determination of the exchange energies in Li2VOSiO4 from a high-temperature series analysis of the square lattice J1-J2 Heisenberg model "
G. Misguich, B. Bernu, L. Pierre .
Phys. Rev. B 68, 113409 (2003) (pdf)
Abstract: We present a high-temperature expansion (HTE) of the magnetic susceptibility and specific heat data of Melzi et al. on Li2VOSiO4 [Phys. Rev. B 64, 024409 (2001)]. The data are very well reproduced by the J1-J2 Heisenberg model on the square lattice with exchange energies J1=1.25+-0.5 K and J2=5.95+-0.2 K. The maximum of the specific heat Cvmax(Tmax) is obtained as a function J2/J1 from an improved method based on HTE.
"Quantum dimer model with extensive ground-state entropy on the kagome lattice "
G. Misguich, D. Serban, V. Pasquier .
Phys. Rev. B 67, 214413 (2003) (pdf)
Abstract: We introduce a quantum dimer model on the kagome lattice with kinetic terms allowing from three to six dimers to resonate around hexagons. Unlike the models studied previously, the different resonance loops appear with different signs (given by the parity of the number of dimers involved). These signs naturally appear when performing the lowest-order overlap expansion (Rokhsar and Kivelson 1988) of the Heisenberg model. We demonstrate that the quantum dimer model has an extensive zero-temperature entropy and has very short-range dimer-dimer correlations. We discuss the possible relevance of this kind of quantum dimer liquid to the physics of the spin-(1/2) Heisenberg model on the kagome lattice.
"Quantum dimer model on the kagome lattice: solvable dimer liquid, and Ising gauge theory "
G. Misguich, D. Serban, V. Pasquier .
Phys. Rev. Lett. 89, 137202 (2002) (pdf)
Abstract: We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices includes the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer-liquid. These models offer a very natural - and maybe the simplest possible - framework to illustrate general concepts such as fractionalization, topological order and relation to Z2 gauge theories.
"Degeneracy of the ground-state of antiferromagnetic spin-1/2 Hamiltonians "
G. Misguich, C. Lhuillier, M. Mambrini, P. Sindzingre .
Eur. Phys. J. B. 26, 167 (2002) (pdf)
Abstract: In the first part of this paper, the extension of the Lieb-Schultz-Mattis theorem to dimensions larger than one is discussed. A counter example to the original formulation of Lieb-Schultz-Mattis and Affleck is exhibited and a more precise statement is formulated. The degeneracy of the ground-state in symmetry breaking phases with long-range order is analyzed. The second and third parts of the paper concern resonating valence-bond (RVB) spin liquids. In these phases the relationship between various authors approaches: Laughlin-Oshikawa, Sutherland, Rokhsar and Kivelson, Read and Chakraborty and the Lieb-Schultz-Mattis-Affleck proposal is studied. The deep physical relation between the degeneracy property and the absence of stiffness is explained and illustrated numerically. A new conjecture is formed concerning the absolute absence of sensitivity of the spin liquid ground-states to any twist of the boundary conditions (thermodynamic limit). In the third part of the paper the relations between the quantum numbers of the degenerate multiplets of the spin liquid phases are obtained exactly. Their relationship with a topological property of the wave functions of the low lying levels of this spin liquid phase is emphasized. In spite of the degeneracy of the ground-state, we explain why these phases cannot exhibit spontaneous symmetry breaking.
"Frustrated quantum magnets "
C. Lhuillier and G. Misguich .
cond-mat/0109146
Contribution to the lecture notes of the Cargese summer school on "Trends in high magnetic field science" (may 2001). Published in Lecture Notes in Physics (Springer Series) Vol 595. "High Magnetic Fields Applications in Condensed Matter Physics and Spectroscopy" (C. Berthier, L. P. Levy and G. Martinez Eds.).
Abstract: In these lectures we sketch a rapid survey of recent theoretical advances in the study of frustrated quantum magnets with a special emphasis on two dimensional magnets.
"Magnetization plateaus of SrCu2(BO3)2 from a Chern-Simons theory "
G. Misguich, T. Jolicoeur, S. M. Girvin .
Phys. Rev. Lett. 87, 097203 (2001) (pdf)
Abstract: The antiferromagnetic Heisenberg model on the frustrated Shastry-Sutherland lattice is studied by a mapping onto spinless fermions carrying one quantum of statistical flux. Using a mean-field approximation these fermions populate the bands of a generalized Hofstadter problem. Their filling leads to the magnetization curve. For SrCu2(BO3)2 we reproduce plateaus at 1/3 and 1/4 of the saturation moment and predict a new one at 1/2. Gaussian fluctuations of the gauge field are shown to be massive at these plateau values.
"Magnetization process from Chern-Simons theory and its application to SrCu2(BO3)2 "
Th. Jolicoeur, G. Misguich and S. M. Girvin .
Progr. Theor. Phys. Supp. 145, 76 (2002)
Proceedings of the 16th Nishinomiya-Yukawa Memorial Symposium, Nishinomiya, Japan, Nov. 2001
Abstract: In two-dimensional systems, it is possible transmute bosons into fermions by use of a Chern-Simons gauge field. Such a mapping is used to compute magnetization processes of two-dimensional magnets. The calculation of the magnetization curve then involves the structure of the Hofstadter problem for the lattice under consideration. Certain features of the Hofstadter butterfly are shown to imply the appearance of magnetization plateaus. While not always successfull, this approach leads to interesting results when applied to the 2D AF magnet SrCu2(BO3)2.
"Specific heat and high-temperature series of lattice models: interpolation scheme and examples on quantum spin systems in one and two dimensions "
B. Bernu, G. Misguich .
Phys. Rev. B 63, 134409 (2001) (pdf)
Abstract: We have developed a new method for evaluating the specific heat of lattice spin systems. It is based on the knowledge of high-temperature series expansions, the total entropy of the system and the low-temperature expected behavior of the specific heat as well as the ground-state energy. By the choice of an appropriate variable (entropy as a function of energy), a stable interpolation scheme between low and high temperature is performed. Contrary to previous methods, the constraint that the total entropy is log(2S+1) for a spin S on each site is automatically satisfied. We present some applications to quantum spin models on one- and two- dimensional lattices. Remarkably, in most cases, a good accuracy is obtained down to zero temperature.
"From Néel long-range order to spin-liquids in the multiple-spin exchange model "
W. LiMing, G. Misguich, P. Sindzingre and C. Lhuillier .
Phys. Rev. B 62, 6372 (2000) (pdf)
Abstract: The phase diagram of the multiple-spin exchange model on the triangular lattice is studied using exact diagonalizations. The two-spin (J2) and four-spin (J4) exchanges have been taken into account for 12, 16, 19, 21, 24, and 27 site samples in the parameter region J4=0-0.25 (for a fixed J2=1). It is found that the three-sublattice Néel ordered state built up by the pure two-spin exchange can be destroyed by the four-spin exchange, forming a spin-liquid state. The different data suggest that the phase diagram in this range of parameters exhibits two phases. The pure J2 phase is a three-sublattice Néel ordered phase, a small J4 drives it into a spin-liquid state with a spin gap filled of a large number of singlets. This spin-liquid phase is not of the same generic kind as the phase studied by Misguich et al. [Phys. Rev. B 60, 1064 (1999)]. It is observed on the finite-size samples that the Spin Liquid phase, as the Neel ordered phase, exhibits a magnetization plateau at m=1/3, and for J4>0.15 a second plateau at m=1/2. These two plateaus are associated respectively to the semi-classical orderings uud and uuud.
"Magneto-thermodynamics of the spin-1/2 Kagome antiferromagnet "
P. Sindzingre, G. Misguich, C. Lhuillier, B. Bernu, L. Pierre, C. Waldtmann and H. U. Everts .
Phys. Rev. Lett. 84, 2953 (2000) (pdf)
Abstract: In this paper, we use a new hybrid method to compute the thermodynamic behavior of the spin-1/2 Kagome antiferromagnet under the influence of a large external magnetic field. We find a T2 low-temperature behavior and a very low sensitivity of the specific heat to a strong external magnetic field. We display clear evidence that this low temperature magneto-thermal effect is associated to the existence of low-lying fluctuating singlets, but also that the whole picture (T2 behavior of Cv and thermally activated spin susceptibility) implies contribution of both non magnetic and magnetic excitations. Comparison with experiments is made.
"Modèle d'échange multiple sur le réseau triangulaire: Liquide de spins quantiques en deux dimensions et magnétisme des films d'3He solide "
G. Misguich and C. Lhuillier(supervisor) .
Thèse de doctorat de l'Université Paris-6 (June 1999) (pdf)
Abstract:
English. The multiple-spin exchange model is an effective description of the magnetic properties of quasilocalized fermions. We study this model for spins-1/2 on a triangular lattice and include exchanges involving 2, 3, 4, 5 and 6 particles. This hamiltonian is now recognized as relevant to the description of the nuclear magnetism of solid $^3$He films adsorbed on graphite. Our analysis is based on exact diagonalizations of finite size systems up to 36 sites. In a large parameter range, finite size effects, structures of spectra and correlation functions lead to the conclusion that quantum fluctuations destroy magnetic long-range order at zero temperature. The ground state is characterized by a finite correlation length; it is a spin-liquid with gapped magnetic excitations. We compute the heat capacity and find that the low-temperature entropy is significant, in agreement with experimental data. We investigate the response to an external field, and find a magnetization plateau at M/Msat=1/2. We discuss the conditions for such a plateau to appear and compare them to the better understood situations of one-dimensional systems and of classical spins. The largest sample we studied (36 sites) has a nearly fourfold degenerate ground state. This degeneracy turns out to be incompatible with a spontaneous symmetry breaking but leads us to a topological interpretation. We propose a short-ranged ``resonating valence-bond'' (RVB) picture of the ground state wave-function and discuss the possible relation to the Haldane phase or ``valence-bond solid'' states. The multiple-spin exchange model on the triangular lattice seems to be one of the very first examples of two-dimensional spin-1/2 model without any broken symmetry at zero temperature.
French. L'échange multiple d'écrit de manière effective les propriétés magnétiques d'un système de fermions quasiment localisés. Nous considérons ce modèle pour des spins-1/2 sur le réseau triangulaire avec des processus à 2, 3, 4, 5 et 6 particules. Cet hamiltonien est aujourd'hui reconnu comme étant un bon candidat pour décrire le magnétisme nucléaire des films d'3He solide adsorbés sur du graphite. Nous l'étudions par diagonalisations exactes sur des échantillons jusqu'à 36 sites. Dans un large domaine de paramètres, l'analyse des effets de taille finie sur les énergies propres, la structure des spectres et les fonctions de corrélation amènent à conclure que les fluctuations quantiques détruisent l'ordre magnétique à température nulle. L'état fondamental est caractérisé par une longueur de corrélation finie, c'est un liquide de spins où les excitations magnétiques possèdent un gap. Nous calculons la chaleur spécifique et trouvons une entropie importante à basse température, en accord avec les mesures expérimentales. La réponse à un champ extérieur révèle un plateau d'aimantation à M/Msat=1/2. Nous discutons les conditions d'apparition d'un tel plateau et les comparons avec les situations, mieux comprises, de la dimension un ainsi que celle de spins classiques. Pour le plus grand système étudié (36 sites), le fondamental possède une quasi-dégénérescence quatre. Cette dégénérescence ne s'explique pas par une brisure spontanée de symétrie mais elle conduit à une interprétation topologique. Nous suggérons une image de ``resonating valence-bond'' (RVB) à courte portée pour la fonction d'onde du fondamental et discutons du lien éventuel avec la phase de Haldane et les états de type ``valence-bond solid''. Le modèle d'échange multiple sur le réseau triangulaire semble un des tous premiers exemples d'hamiltonien bidimensionnel de spins-1/2 dont le fondamental ne brise aucune de symérie à température nulle.
"Spin-Liquid phase of the Multiple-Spin Exchange Hamiltonian on the Triangular Lattice "
G. Misguich, C. Lhuillier, B. Bernu and C. Waldtmann .
Phys. Rev. B 60, 1064 (1999) (pdf)
Abstract: We performed an exact diagonalization study of the spin liquid phase of the Multiple-Spin Exchange model on the triangular lattice. It is characterized by no Néel Long Range Order (NLRO), short-ranged magnetic correlations and a spin gap. We found no LRO in any local order parameter we investigated (chiral, dimer,...). The probable asymptotic ground state degeneracy is discussed. We argue that it could be of topological origin and that the system is probably not a chiral spin liquid. A possible relation to the Affleck-Kennedy-Lieb-Tasaki (AKLT) phase is discussed.
"Spin Liquid in the Multiple-Spin Exchange Model on the triangular lattice: 3He on graphite "
G. Misguich, B. Bernu, C. Lhuillier and C. Waldtmann .
Phys. Rev. Lett. 81, 1098, (1998) (pdf)
Abstract: Using exact diagonalizations, we investigate the T = 0 phase diagram of the multiple-spin exchange (MSE) model on the triangular lattice, we find a transition separating a ferromagnetic phase from a nonmagnetic gapped spin liquid phase. Systems far enough from the ferromagnetic transition have a metamagnetic behavior with magnetization plateaus at M/Msat = 0 and 1/2 . The MSE has been proposed to describe solid 3He films adsorbed onto graphite, thus we compute the MSE heat capacity for parameters in the low density range of the 2nd layer and find a double-peak structure.
"The Multiple-Spin Exchange Phase Diagram on the Triangular Lattice: Schwinger-Boson analysis "
G. Misguich, B. Bernu and C. Lhuillier .
J. of Low Temp. Phys. vol. 110, 327 (1998)
Proceedings of the QFS'97 conference
Abstract: We present the classical phase diagram of a multiple-spin exchange hamiltonian involving up to five-particle exchange. Schwinger-boson (mean field theory) calculations confirm the classical picture of competing ferromagnetic and antiferromagnetic phases, and suggest that the frustration induced by four-particle exchange might destroy long range order.
"Geometrically Frustrated Quantum Antiferromagnets and Spin Liquids "
C. Lhuillier, B. Bernu and G. Misguich .
Int. Journal of Mod. Phys. B. vol.13 Nos. 5 & 6, 687 (1999)
Proceedings of the 9th International Conference on recent progress in Many Body Theories, Sydney, August 1997
Abstract: We briefly review experimental and theoretical results related to the problem of long range magnetic order in geometrically frustrated 2-dimensional quantum antiferromagnets. We show that 2-dimensional 3He at low coverage is a good candidate as a Spin-Liquid. We underline that sophisticated numerical studies of the spectra of such systems point at least to two different kinds of Spin-Liquids.
"Alkali suboxides as weaker binding substrates for helium "
J. Dupont-Roc, G. Misguich and L. Girlanda .
Czech. J. Phys. 46, 419 (1996)
Abstract: Suboxides of heavy alkali metals are known for their particularly low work function. This property should make the surface of those materials less binding for helium than that of pure metals. A semi-quantitative assessment of their wetting properties for liquid helium is proposed. Preparation of such surfaces has been undertaken for rubidium suboxides. The wetting properties obtained for liquid helium will be reported.
