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Incondensed matter physics, aquantum spin liquid is aphase of matter that can be formed by interactingquantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-rangequantum entanglement,fractionalizedexcitations, and absence of ordinary magneticorder.^[1]

The quantum spin liquid state was first proposed by physicistPhil Anderson in 1973 as the ground state for a system of spins on atriangular lattice that interactantiferromagnetically with their nearest neighbors, i.e. neighboring spins seek to be aligned in opposite directions.^[2] Quantum spin liquids generated further interest when in 1987 Anderson proposed a theory that describedhigh-temperature superconductivity in terms of a disordered spin-liquid state.^[3][4]

The simplest kind of magnetic phase is aparamagnet, where each individual spin behaves independently of the rest, just like atoms in anideal 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 aferromagnet (orantiferromagnet) phase. In this phase, interactions between the spins cause them to align into large-scale patterns, such asdomains, 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.^[5]

Quantum spin liquids offer a dramatic alternative to this typical behavior. One intuitive description of this state is as a "liquid" ofdisordered spins, in comparison to aferromagnetic spin state,^[6] 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.^[7] A more modern characterization of quantum spin liquids involves theirtopological order,^[8] long-rangequantum entanglement properties,^[1] andanyon excitations.^[9]

Several physical models have a disordered ground state that can be described as a quantum spin liquid.

Localized spins arefrustrated 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 ofIsing 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.

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 aremaximally entangled, while not being entangled with the other spins. If all spins are distributed to certain localized static bonds, this is called avalence 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 byP. W. Anderson in 1973 as the ground state of spin liquids^[2] and is called aresonating 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.^[4]

• 
  One possible short-range pairing of spins in a RVB state.
• 
  Long-range pairing of spins.

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 supportexotic excitations, meaning excitations with fractional quantum numbers. A prominent example is the excitation ofspinons which are neutral in charge and carry spin${\displaystyle S=1/2}$. 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.

The first discussion of the RVB state on square lattice using the RVB picture^[10] 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${\displaystyle U(1)}$ 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.^[11][12] Later, another version of stable RVB state with deconfined spinons, the Z2 spin liquid, is proposed,^[13][14] which realizes the simplesttopological 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,^[15] where different bond configurations only have real amplitudes. Thetoric code model is yet another realization of Z2 spin liquid (andZ2 topological order) that explicitly breaks the spin rotation symmetry and is exactly solvable.^[16]

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.^[17]

In a high-temperature, classicalparamagnet phase, themagnetic susceptibility is given by theCurie–Weiss law$${\displaystyle \chi \sim \frac{C}{T-{\mathrm{\Theta }}_{CW}}}$$

Fitting experimental data to this equation determines a phenomenological Curie–Weiss temperature,${\displaystyle {\mathrm{\Theta }}_{CW}}$. There is a second temperature,${\displaystyle T_c}$, where magnetic order in the material begins to develop, as evidenced by anon-analytic feature in${\displaystyle \chi (T)}$. The ratio of these is called thefrustration parameter$${\displaystyle f=\frac{|{\mathrm{\Theta }}_{cw}|}{T_c}}$$

In a classic antiferromagnet, the two temperatures should coincide and give${\displaystyle f=1}$. An ideal quantum spin liquid would not develop magnetic order at any temperature${\displaystyle (T_c=0)}$ and so would have a diverging frustration parameter${\displaystyle f\to \mathrm{\infty }}$.^[18] A large value${\displaystyle f>100}$ is therefore a good indication of a possible spin liquid phase. Some frustrated materials with different lattice structures and theirCurie–Weiss temperature are listed in the table below.^[7] All of them are proposed spin liquid candidates.

Material	Lattice	${\displaystyle {\mathrm{\Theta }}_{cw}[\mathrm{K}]}$
κ-(BEDT-TTF)2Cu2(CN)3	anisotropic triangular	-375
ZnCu3(OH)6Cl2 (herbertsmithite)	Kagome	-241
BaCu3V2O8(OH)2 (vesignieite)	Kagome
Na4Ir3O8	Hyperkagome	-650
PbCuTe2O6	Hyperkagome	-22[19]
Cu-(1,3-benzenedicarboxylate)	Kagome	-33[20]
Rb2Cu3SnF12	Kagome	[21]
1T-TaS2	Triangular

One of the most direct evidence for absence of magnetic ordering giveNMR orμSR experiments. If there is a local magnetic field present, the nuclear or muon spin would be affected which can be measured.¹H-NMR measurements^[22] on κ-(BEDT-TTF)₂Cu₂(CN)₃ have shown no sign of magnetic ordering down to 32 mK, which is four orders of magnitude smaller than thecoupling constant J≈250 K^[23] 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 uncoverspinons, which couple via emergent gauge fields to the electromagnetic field, giving rise to a power-law optical conductivity.^[24]

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.^[25] Later theoretical work challenged this picture, arguing that all experimental results were instead consequences of 1D spinons confined to individual chains.^[26]

Afterwards, it was observed in an organic Mott insulator (κ-(BEDT-TTF)₂Cu₂(CN)₃) by Kanoda's group in 2003.^[22] It may correspond to a gapless spin liquid with spinon Fermi surface (the so-called uniform RVB state).^[2] The peculiar phase diagram of this organic quantum spin liquid compound was first thoroughly mapped usingmuon spin spectroscopy.^[27]

Herbertsmithite is one of the most extensively studied QSL candidate materials.^[18] It is a mineral with chemical composition ZnCu₃(OH)₆Cl₂ and arhombohedral crystal structure. Notably, the copper ions within this structure form stacked two-dimensional layers ofkagome lattices. Additionally,superexchange over the oxygen bonds creates a strong antiferromagnetic interaction between the${\displaystyle S=1/2}$ copper spins within a single layer, whereas coupling between layers is negligible.^[18] Therefore, it is a good realization of the antiferromagnetic spin-1/2 Heisenberg model on the kagome lattice, which is a prototypical theoretical example of a quantum spin liquid.^[28][29]

Synthetic, polycrystalline herbertsmithite powder was first reported in 2005, and initial magnetic susceptibility studies showed no signs of magnetic order down to 2K.^[30] In a subsequent study, the absence of magnetic order was verified down to 50 mK,inelastic neutron scattering measurements revealed a broad spectrum of low energy spin excitations, and low-temperature specific heat measurements had power law scaling. This gave compelling evidence for a spin liquid state with gapless${\displaystyle S=1/2}$spinon excitations.^[31] A broad array of additional experiments, including¹⁷ONMR,^[32] and neutron spectroscopy of thedynamic magnetic structure factor,^[33] reinforced the identification of herbertsmithite as a gapless spin liquid material, although the exact characterization remained unclear as of 2010.^[34]

Large (millimeter size) single crystals of herbertsmithite were grown and characterized in 2011.^[35] These enabled more precise measurements of possible spin liquid properties. In particular, momentum-resolved inelastic neutron scattering experiments showed a broad continuum of excitations. This was interpreted as evidence for gapless, fractionalized spinons.^[36] Follow-up experiments (using¹⁷ONMR and high-resolution, low-energy neutron scattering) refined this picture and determined there was actually a small spinon excitation gap of 0.07–0.09 meV.^[37][38]

Some measurements were suggestive ofquantum critical behavior.^[39][40] Magnetic response of this material displays scaling relation in both the bulk acsusceptibility and the low energy dynamic susceptibility, with the low temperature heat capacity strongly depending on magnetic field.^[41][42] This scaling is seen in certain quantumantiferromagnets,heavy-fermion metals, and two-dimensional³He as a signature of proximity to a quantum critical point.^[43]

In 2020, monodisperse single-crystalnanoparticles of herbertsmithite (~10 nm) were synthesized at room temperature, usinggas-diffusion electrocrystallization, showing that their spin liquid nature persists at such small dimensions.^[44]

It may realize a U(1)-Dirac spin liquid.^[47]

Another evidence of quantum spin liquid was observed in a 2-dimensional material in August 2015. The researchers ofOak Ridge National Laboratory, collaborating with physicists from the University of Cambridge, and the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, measured the first signatures of these fractional particles, known asMajorana fermions, in a two-dimensional material with a structure similar tographene. Their experimental results successfully matched with one of the main theoretical models for a quantum spin liquid, known as aKitaev honeycomb model.^[48][49]

Thestrongly correlated quantum spin liquid (SCQSL) is a specific realization of a possible quantum spin liquid (QSL)^[7][39] representing a new type of strongly correlated electricalinsulator (SCI) that possesses properties ofheavy fermion metals with one exception: it resists the flow ofelectric charge.^[46][50] At low temperaturesT thespecific heat of this type of insulator is proportional toTⁿ, withn less or equal 1 rather thann=3, as it should be in the case of a conventional insulator whoseheat capacity is proportional toT³. When amagnetic fieldB is applied to SCI the specific heat depends strongly onB, contrary to conventional insulators. There are a few candidates of SCI; the most promising among them isHerbertsmithite,^[50] amineral with chemical structure ZnCu₃(OH)₆Cl₂.

Ca₁₀Cr₇O₂₈ is a frustratedkagome bilayer magnet, which does not develop long-range order even below 1 K, and has a diffuse spectrum of gapless excitations.

In December 2021, the first direct measurement of a quantum spin liquid of the toric code type was reported,^[51][52] it was achieved by two teams: one exploring ground state and anyonic excitations on aquantum processor^[53] and the other implementing a theoretical blueprint^[54] of atoms on a ruby lattice held withoptical tweezers on aquantum simulator.^[55]

The experimental facts collected onheavy fermion (HF) metals andtwo dimensionalHelium-3 demonstrate that thequasiparticleeffective massM* is very large, or even diverges.Topological fermion condensationquantum phase transition (FCQPT) preservesquasiparticles, and forms flatenergy band at theFermi level. The emergence of FCQPT is directly related to the unlimited growth of theeffective massM*.^[43] Near FCQPT, M* starts to depend ontemperatureT,number densityx,magnetic fieldB and other external parameters such aspressureP, etc. In contrast to theLandau paradigm based on the assumption that the effective mass is approximately constant, in the FCQPT theory the effective mass of new quasiparticles strongly depends onT,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 thethermodynamic,relaxation,scaling andtransport properties of strongly correlated Fermi systems and M* becomes a function ofT,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 newstate of matter that consists of HFmetals,quasicrystals, quantum spin liquid,two-dimensionalHelium-3, andcompounds exhibitinghigh-temperature superconductivity.^[39][43]

Materials supporting quantum spin liquid states may have applications in data storage and memory.^[56] In particular, it is possible to realizetopological quantum computation by means of spin-liquid states.^[57] Developments in quantum spin liquids may also help in the understanding ofhigh temperature superconductivity.^[58]

• Correlated electrons
• Liquids
• Phases of matter
• Condensed matter physics
• Quasiparticles

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