Quantum spin liquid

The simplest kind of magnetic phase is a paramagnet, where each individual spin behaves independently of the rest, just like atoms in an ideal gas. This highly disordered phase is the generic state of magnets at high temperatures, where thermal fluctuations dominate. Upon cooling, the spins will often enter a ferromagnet (or antiferromagnet) phase. In this phase, interactions between the spins cause them to align into large-scale patterns, such as domains, stripes, or checkerboards. These long-range patterns are referred to as "magnetic order," and are analogous to the regular crystal structure formed by many solids. Quantum spin liquids offer a dramatic alternative to this typical behavior. One intuitive description of this state is as a "liquid" of disordered spins, in comparison to a ferromagnetic spin state, much in the way liquid water is in a disordered state compared to crystalline ice. However, unlike other disordered states, a quantum spin liquid state preserves its disorder to very low temperatures. A more modern characterization of quantum spin liquids involves their topological order, long-range quantum entanglement properties, and anyon excitations. ==Examples==

Several physical models have a disordered ground state that can be described as a quantum spin liquid. Frustrated magnetic moments Localized spins are frustrated if there exist competing exchange interactions that can not all be satisfied at the same time, leading to a large degeneracy of the system's ground state. A triangle of Ising spins (meaning that the only possible orientation of the spins are either "up" or "down"), which interact antiferromagnetically, is a simple example for frustration. In the ground state, two of the spins can be antiparallel but the third one cannot. This leads to an increase of possible orientations (six in this case) of the spins in the ground state, enhancing fluctuations and thus suppressing magnetic ordering. Resonating valence bonds (RVB) To build a ground state without magnetic moment, valence bond states can be used, where two electron spins form a spin 0 singlet due to the antiferromagnetic interaction. If every spin in the system is bound like this, the state of the system as a whole has spin 0 too and is non-magnetic. The two spins forming the bond are maximally entangled, while not being entangled with the other spins. If all spins are distributed to certain localized static bonds, this is called a valence bond solid (VBS). There are two things that still distinguish a VBS from a spin liquid: First, by ordering the bonds in a certain way, the lattice symmetry is usually broken, which is not the case for a spin liquid. Second, this ground state lacks long-range entanglement. To achieve this, quantum mechanical fluctuations of the valence bonds must be allowed, leading to a ground state consisting of a superposition of many different partitionings of spins into valence bonds. If the partitionings are equally distributed (with the same quantum amplitude), there is no preference for any specific partitioning ("valence bond liquid"). This kind of ground state wavefunction was proposed by P. W. Anderson in 1973 as the ground state of spin liquids and is called a resonating valence bond (RVB) state. These states are of great theoretical interest as they are proposed to play a key role in high-temperature superconductor physics. File:Resonating_valence_bond1.png|One possible short-range pairing of spins in a RVB state. File:Long_range_valence_bonds.png|Long-range pairing of spins. Excitations The valence bonds do not have to be formed by nearest neighbors only and their distributions may vary in different materials. Ground states with large contributions of long range valence bonds have more low-energy spin excitations, as those valence bonds are easier to break up. On breaking, they form two free spins. Other excitations rearrange the valence bonds, leading to low-energy excitations even for short-range bonds. Something very special about spin liquids is that they support exotic excitations, meaning excitations with fractional quantum numbers. A prominent example is the excitation of spinons which are neutral in charge and carry spin S= 1/2. In spin liquids, a spinon is created if one spin is not paired in a valence bond. It can move by rearranging nearby valence bonds at low energy cost. Realizations of (stable) RVB states The first discussion of the RVB state on square lattice using the RVB picture only consider nearest neighbour bonds that connect different sub-lattices. The constructed RVB state is an equal amplitude superposition of all the nearest-neighbour bond configurations. Such a RVB state is believed to contain emergent gapless U(1) gauge field which may confine the spinons etc. So the equal-amplitude nearest-neighbour RVB state on square lattice is unstable and does not corresponds to a quantum spin phase. It may describe a critical phase transition point between two stable phases. A version of RVB state which is stable and contains deconfined spinons is the chiral spin state. Later, another version of stable RVB state with deconfined spinons, the Z2 spin liquid, is proposed, which realizes the simplest topological order – Z2 topological order. Both chiral spin state and Z2 spin liquid state have long RVB bonds that connect the same sub-lattice. In chiral spin state, different bond configurations can have complex amplitudes, while in Z2 spin liquid state, different bond configurations only have real amplitudes. The RVB state on triangle lattice also realizes the Z2 spin liquid, where different bond configurations only have real amplitudes. The toric code model is yet another realization of Z2 spin liquid (and Z2 topological order) that explicitly breaks the spin rotation symmetry and is exactly solvable. ==Experimental signatures and probes==

Since there is no single experimental feature which identifies a material as a spin liquid, several experiments have to be conducted to gain information on different properties which characterize a spin liquid. between neighboring spins in this compound. Further investigations include: • Specific heat measurements give information about the low-energy density of states, which can be compared to theoretical models. • Thermal transport measurements can determine if excitations are localized or itinerant. • Neutron scattering gives information about the nature of excitations and correlations (e.g. spinons). • Reflectance measurements can uncover spinons, which couple via emergent gauge fields to the electromagnetic field, giving rise to a power-law optical conductivity. , the mineral whose ground state was shown to have QSL behaviour ==Candidate materials==

RVB type Neutron scattering measurements of cesium chlorocuprate Cs2CuCl4, a spin-1/2 antiferromagnet on a triangular lattice, displayed diffuse scattering. This was attributed to spinons arising from a 2D RVB state. Magnetic response of this material displays scaling relation in both the bulk ac susceptibility and the low energy dynamic susceptibility, with the low temperature heat capacity strongly depending on magnetic field. it was achieved by two teams: one exploring ground state and anyonic excitations on a quantum processor and the other implementing a theoretical blueprint of atoms on a ruby lattice held with optical tweezers on a quantum simulator. ==Specific properties: topological fermion condensation quantum phase transition==

The experimental facts collected on heavy fermion (HF) metals and two dimensional Helium-3 demonstrate that the quasiparticle effective mass M* is very large, or even diverges. Topological fermion condensation quantum phase transition (FCQPT) preserves quasiparticles, and forms flat energy band at the Fermi level. The emergence of FCQPT is directly related to the unlimited growth of the effective mass M*. Near FCQPT, M* starts to depend on temperature T, number density x, magnetic field B and other external parameters such as pressure P, etc. In contrast to the Landau paradigm based on the assumption that the effective mass is approximately constant, in the FCQPT theory the effective mass of new quasiparticles strongly depends on T, x, B etc. Therefore, to agree/explain with the numerous experimental facts, extended quasiparticles paradigm based on FCQPT has to be introduced. The main point here is that the well-defined quasiparticles determine the thermodynamic, relaxation, scaling and transport properties of strongly correlated Fermi systems and M* becomes a function of T, x, B, P, etc. The data collected for very different strongly correlated Fermi systems demonstrate universal scaling behavior; in other words distinct materials with strongly correlated fermions unexpectedly turn out to be uniform, thus forming a new state of matter that consists of HF metals, quasicrystals, quantum spin liquid, two-dimensional Helium-3, and compounds exhibiting high-temperature superconductivity. ==Applications==

Materials supporting quantum spin liquid states may have applications in data storage and memory. In particular, it is possible to realize topological quantum computation by means of spin-liquid states. Developments in quantum spin liquids may also help in the understanding of high temperature superconductivity. ==References==