Frustrated magnets are materials in which localized magnetic moments, or spins, interact through competing exchange interactions that cannot be simultaneously satisfied, giving rise to a large degeneracy of the system ground state. Under certain conditions, this can lead to the formation of fluid-like states of matter, so-called spin liquids, in which the constituent spins are highly correlated but still fluctuate strongly down to a temperature of absolute zero. The fluctuations of the spins in a spin liquid can be classical or quantum and show remarkable collective phenomena such as emergent gauge fields and fractional particle excitations. This exotic behaviour is now being uncovered in the laboratory, providing insight into the properties of spin liquids and challenges to the theoretical description of these materials.

Spin liquids in frustrated magnets

We review recent developments in the physics of ultracold atomic and molecular gases in optical lattices. Such systems are nearly perfect realisations of various kinds of Hubbard models, and as such may very well serve to mimic condensed matter phenomena. We show how these systems may be employed as quantum simulators to answer some challenging open questions of condensed matter, and even high energy physics. After a short presentation of the models and the methods of treatment of such systems, we discuss in detail, which challenges of condensed matter physics can be addressed with (i) disordered ultracold lattice gases, (ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv) lattice gases in “artificial” magnetic fields, and, last but not least, (v) quantum information processing in lattice gases. For completeness, also some recent progress related to the above topics with trapped cold gases will be discussed. Motto: There are more things in heaven and earth, Horatio, Than are dreamt of in your...

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Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

Quantum spin liquids may be considered "quantum disordered" ground states of spin systems, in which zero point fluctuations are so strong that they prevent conventional magnetic long range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons that are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments to study quantum spin liquids, and to the diverse probes used therein.

Quantum Spin Liquids

Quantum spin liquids may be considered 'quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.

https://iopscience.iop.org/article/10.1088/0034-4885/80/1/016502/pdf

Quantum spin liquids: a review.

This is an introductory review of the physics of quantum spin liquid states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semiclassical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semiclassical approaches fail once quantum mechanics become important and then describe the alternative approaches for addressing the problem. Mainly spin-1/2 systems are discussed, and most of the time is spent in this article on one particular set of plausible spin liquid states in which spins are represented by fermions. These states are spin-singlet states and may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of so-called SU(2), U(1), or Z2 spin liquid states. A review is given of the basic theory regarding these states and the extensions of these states to include the effect of spin-orbit coupling and to higher spin (S>1/2) systems. Two other important approaches with strong influences on the understanding of spin liquid states are also introduced: (i) matrix product states and projected entangled pair states and (ii) the Kitaev honeycomb model. Experimental progress concerning spin liquid states in realistic materials, including anisotropic triangular-lattice systems [κ-(ET)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2], kagome-lattice system [ZnCu3(OH)6Cl2], and hyperkagome lattice system (Na4Ir3O8), is reviewed and compared against the corresponding theories.

Quantum spin liquid states

Exact spectra of periodic samples are computed up to N=36. Evidence of an extensive set of low-lying levels, lower than the softest magnons, is exhibited. These low-lying quantum states are degenerated in the thermodynamic limit; their symmetries and dynamics as well as their finite-size scaling are strong arguments in favor of N\'eel order. It is shown that the N\'eel order parameter agrees with first-order spin-wave calculations. A simple explanation of the low-energy dynamics is given as well as the numerical determinations of the energies, order parameter, and spin susceptibilities of the studied samples. It is shown how suitable boundary conditions, which do not frustrate N\'eel order, allow the study of samples with N=3p+1 spins. A thorough study of these situations is done in parallel with the more conventional case N=3p.

Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice

We study the exact low energy spectra of the spin 1/2 Heisenberg antiferromagnet on small samples of the kagome lattice of up to N=36 sites. In agreement with the conclusions of previous authors, we find that these low energy spectra contradict the hypothesis of Neel type long range order. Certainly, the ground state of this system is a spin liquid, but its properties are rather unusual. The magnetic (
 $$(\Delta S = 1)$$

 ) excitations are separated from the ground state by a gap. However, this gap is filled with nonmagnetic (
 $$(\Delta S = 0)$$

 ) excitations. In the thermodynamic limit the spectrum of these nonmagnetic excitations will presumably develop into a gapless continuum adjacent to the ground state. Surprisingly, the eigenstates of samples with an odd number of sites, i.e. samples with an unsaturated spin, exhibit symmetries which could support long range chiral order. We do not know if these states will be true thermodynamic states or only metastable ones. In any case, the low energy properties of the spin 1/2 Heisenberg antiferromagnet on the kagome lattice clearly distinguish this system from either a short range RVB spin liquid or a standard chiral spin liquid. Presumably they are facets of a generically new state of frustrated two-dimensional quantum antiferromagnets.

First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice

A group-symmetry analysis of the low-lying levels of the spin-1/2 kagom\'e Heisenberg antiferromagnet is performed for small samples up to $N=27$. This approach allows one to follow the effect of quantum fluctuations when the sample size increases. The results contradict the scenario of ``order by disorder'' which has been advanced on the basis of large-$S$ calculations. A large enough second-neighbor ferromagnetic exchange coupling is needed to stabilize the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ pattern: the finite-size analysis indicates a quantum critical transition at a nonzero coupling.

Order versus disorder in the quantum Heisenberg antiferromagnet on the kagomé lattice using exact spectra analysis

We study the exact low energy spectra of the spin 1/2 Heisenberg antiferromagnet on small samples of the kagom\'e lattice of up to N=36 sites. In agreement with the conclusions of previous authors, we find that these low energy spectra contradict the hypothesis of N\'eel type long range order. Certainly, the ground state of this system is a spin liquid, but its properties are rather unusual. The magnetic ($\Delta S=1$) excitations are separated from the ground state by a gap. However, this gap is filled with nonmagnetic ($\Delta S=0$) excitations. In the thermodynamic limit the spectrum of these nonmagnetic excitations will presumably develop into a gapless continuum adjacent to the ground state. Surprisingly, the eigenstates of samples with an odd number of sites, i.e. samples with an unsaturated spin, exhibit symmetries which could support long range chiral order. We do not know if these states will be true thermodynamic states or only metastable ones. In any case, the low energy properties of the spin 1/2 Heisenberg antiferromagnet on the kagom\'e lattice clearly distinguish this system from either a short range RVB spin liquid or a standard chiral spin liquid. Presumably they are facets of a generically new state of frustrated two-dimensional quantum antiferromagnets.

First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagom\'e lattice

Motivated by recent experiments on an $S=1∕2$ antiferromagnet on the kagom\'e lattice, we investigate the Heisenberg ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic ${J}_{1}$ and antiferromagnetic ${J}_{2}$, Classically the ground state displays N\'eel long-range order with 12 noncoplanar sublattices. The order parameter has the symmetry of a cuboctahedron, it fully breaks $SO(3)$ as well as the spin-flip symmetry, and we expect from the latter a ${\mathbb{Z}}_{2}$ symmetry breaking pattern. As might be expected from the Mermin-Wagner theorem in two dimensions, the $SO(3)$ symmetry is restored by thermal fluctuations while the ${\mathbb{Z}}_{2}$ symmetry breaking persists up to a finite temperature. A complete study of $S=1∕2$ exact spectra reveals that the classical order subsists for quantum spins in a finite range of parameters. First-order spin wave calculations give the range of existence of this phase and the renormalizations at $T=0$ of the order parameters associated to both symmetry breakings. This phase is destroyed by quantum fluctuations for a small but finite ${J}_{2}∕\ensuremath{\mid}{J}_{1}\ensuremath{\mid}\ensuremath{\simeq}3$, consistently with exact spectra studies, which indicate a gapped phase.

L. Pierre

Papers

Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice

First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice

Order versus disorder in the quantum Heisenberg antiferromagnet on the kagomé lattice using exact spectra analysis

First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagom\'e lattice

Twelve sublattice ordered phase in the J 1 − J 2 model on the kagomé lattice