# Quantum spin liquid

In condensed matter physics, quantum spin liquid is a state that can be achieved in a system of interacting quantum spins. The state is referred to as a "liquid" as it is a disordered state in comparison to a ferromagnetic spin state,[1] 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.[2]

The quantum spin liquid state was first proposed by physicist Phil Anderson in 1973 as the ground state for a system of spins on a triangular lattice that interact with their nearest neighbors via the so-called antiferromagnetic interaction. Quantum spin liquids generated further interest when in 1987 Anderson proposed a theory that described high temperature superconductivity in terms of a disordered spin-liquid state.[3]

A quantum spin liquid state was first discovered in an organic Mott insulator with a triangular lattice (κ-(BEDT-TTF)2Cu2(CN)3 ) by Kanoda's group in 2003.[4] It may correspond to a gapless spin liquid with spinon Fermi surface (the so-called uniform RVB state).[5] The peculiar phase diagram of this organic quantum spin liquid compound was first thoroughly mapped using muon spin spectroscopy.[6] A second quantum spin liquid state in herbertsmithite ZnCu3(OH)6Cl2 was discovered in 2006 by Young Lee's group at MIT.[7] It may realize a U(1)-Dirac spin liquid.[8]

Another evidence of quantum spin liquid was observed in a 2-dimensional material in August 2015. The researchers of Oak Ridge National Laboratory, collaborating with physicists from the University of Cambridge, and the Max Planck Institute in Germany, measured the first signatures of these fractional particles, known as Majorana fermions, in a two-dimensional material with a structure similar to graphene. Their experimental results successfully matched with one of the main theoretical models for a quantum spin liquid, known as a Kitaev model.[9] The results are reported in the journal Nature Materials.[10]

## Examples

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

### Frustrated magnetic moments

Frustrated Ising spins on a triangle.

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 the only possible orientations of the spins are "up" and "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.

Some frustrated materials with different lattice structures and their Curie-Weiss temperature are listed in the table.[2] All of them are proposed spin liquid candidates.

Material Lattice ${\displaystyle \Theta _{cw}[\mathrm {K} ]}$
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
Cu-(1,3-benzenedicarboxylate) Kagome -33 [11]
Rb2Cu3SnF12 Kagome [12]

[13]

### Resonating valence bonds (RVB)

Valence bond solid. The bonds form a specific pattern and consist of pairs of entangled spins.

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[5] 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.[14]

#### Excitations

Spinon moving in spin liquids.

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. 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 ${\displaystyle S=1/2}$. 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.

#### Realizations of (stable) RVB states

The first discussion of the RVB state on square lattice using the RVB picture[15] 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.[16][17] Later, another version of stable RVB state with deconfined spinons, the Z2 spin liquid, is proposed,[18][19] which realizes the simplest topological orderZ2 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,[20] 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 soluble.[21]

## Identification in Experiments

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. An indication is given by a large value of the frustration parameter ${\displaystyle f>100}$, which is defined as

${\displaystyle f={\frac {|\Theta _{cw}|}{T_{c}}}}$

where ${\displaystyle \Theta _{cw}}$ is the Curie-Weiss temperature and ${\displaystyle T_{c}}$ is the temperature below which magnetic order begins to develop.

One of the most direct evidence for absence of magnetic ordering give NMR or µSR experiments. If there is a local magnetic field present, the nuclear or muon spin would be affected which can be measured. 1H-NMR measurements [4] 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 the coupling constant J≈250 K[22] 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.[23]
Herbertsmithite, the mineral whose ground state was shown to have QSL behaviour

### Observation of fractionalization

In 2012, Young Lee and his collaborators at MIT and the National Institute of Standards and Technology artificially developed a crystal of herbertsmithite, a crystal with kagome lattice ordering, on which they were able to perform neutron scattering experiments.[24] The experiments revealed evidence for spin-state fractionalization, a predicted property of quantum spin-liquid type states.[25] The observation has been described as a hallmark for the quantum spin liquid state in herbertsmithite.[26] Data indicate that the strongly correlated quantum spin liquid, a specific form of quantum spin liquid, is realized in Herbertsmithite.[27]

## Strongly correlated quantum spin liquid

Strongly correlated quantum spin liquid (SCQSL) is a specific realization of a possible quantum spin liquid (QSL)[2][28] representing a new type of strongly correlated electrical insulator (SCI) that possesses properties of heavy fermion metals[29][30] with one exception: it resists the flow of electric charge. At low temperatures T the specific heat of this type of insulator is proportional to Tn with n less or equal 1 rather than n=3, as it should be in the case of a conventional insulator when the heat capacity is proportional to T3. When a magnetic field B is applied to SCI the specific heat depends strongly on B, contrary to conventional insulators. There are a few candidates of SCI; the most promising among them is Herbertsmithite, a mineral with chemical structure ZnCu3(OH)6Cl2.

### Specific properties

Fig. 1: T-dependence of the electronic specific heat C/T of YbRh2Si2 at different magnetic fields [31] as shown in the legend. The values of (C/T)max and Tmax at B=8 Tesla are shown. The maximum (C/T)max decreases with growing magnetic field B, while Tmax shifts to higher T reaching 14 K at B=18 Tesla. Observing that C/T~χ~M*, one concludes that SCQSL in ZnCu3(OH)6Cl2 shown in Fig. 2 exhibits the similar behavior as heavy fermions in YbRh2Si2.
Fig.2: T-dependence of the magnetic susceptibility χ at different magnetic fields for ZnCu3(OH)6Cl2.[32] The values of χmax and Tmax at B=7 Tesla are shown. T-dependence T(-2/3) at B=0 is depicted by the solid curve. The maximum χmax(T) decreases as magnetic field B grows, while Tmax(B) shifts to higher T reaching 15 K at B=14 Tesla. Observing that χ~C/T~M*, one concludes that the specific heat of YbRh2Si2 shown in Fig. 1 exhibits the similar behavior as χ does. Thus, SCQSL in ZnCu3(OH)6Cl2 behaves as heavy fermions in YbRh2Si2.[33]

Exotic SCQSL's are formed with such hypothetical particles as fermionic spinons carrying spin 1/2 and no charge. The experimental studies of Herbertsmithite ZnCu3(OH)6Cl2 single crystal have found no evidence of long range magnetic order or spin freezing indicating that Herbertsmithite is the promising system to investigate SCQSL. The planes of the Cu2+ ions can be considered as two-dimensional layers with negligible magnetic interactions along the third dimension. Experiments have found neither long range magnetic order nor glassy spin freezing down to temperature 50 mK[32][34] making Herbertsmithite the best candidate for QSL realization. Frustration of a simple kagome lattice leads to dispersionless topologically protected flat bands.[35][36] In that case fermion condensation quantum phase transition (FCQPT)[37] can be considered as quantum critical point (QCP) of Herbertsmithite. FCQPT creates SCQSL composed of chargeless fermions with spin=1/2 occupying the corresponding Fermi sphere with a finite Fermi momentum. Herbertsmithite's thermodynamic and relaxation properties are similar to those of heavy fermion metals and two-dimensional 3He.[37] The key features of the findings are the presence in Herbertsmithite of spin–charge separation and SCQSL formed with itinerant spinons. Herbertsmithite represents a fascinating example of SCI where particles-spinons, non-existing as free, replace the initial particles appearing in the Hamiltonian and define the thermodynamic and relaxation properties at low temperatures. Because of the spin-charge separation, heat transport, thermodynamic and relaxation properties at low temperatures of the SCI Herbertsmithite are similar to those of heavy-fermion metals rather than of insulators.[33][38]

### Fermion condensation quantum phase transition

The experimental facts collected on heavy fermion (HF) metals and two dimensional 3He demonstrate that the quasiparticle effective mass M* is very large, or even diverges.[29][30][39] Fermion condensation quantum phase transition (FCQPT) preserves quasiparticles and is directly related to the unlimited growth of the effective mass M*.[37] Near FCQPT, M* starts to depend on temperature T, 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 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.[37]

### Identification in Experiments

Quantum spin liquid - the new state of matter - is realized in Herbertsmithite, ZnCu3(OD)6Cl2.[40] 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.[32][41] This scaling is seen in certain quantum antiferromagnets and heavy-fermion metals as a signature of proximity to a quantum critical point. The low-temperature specific heat follows the linear temperature dependence.[32][41] These results suggest that a SCQSL state with essentially gapless excitations is realized in Herbertsmithite.[33][38]

In 2016, two different groups reported the observation of characteristic features matching a quantum spin liquid in two different materials - first in α-RuCl3, which is a proximate Kitaev quantum spin liquid producing Majorana fermions,[10] and then in Ca10Cr7O28, which is a frustrated Kagome bilayer magnet.[42]

## Applications

Materials supporting quantum spin liquid states may have applications in data storage and memory.[43] In particular, it is possible to realize topological quantum computation by means of spin-liquid states.[44] Developments in quantum spin liquids may also help in the understanding of high temperature superconductivity.[45]

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### Books

Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). Theory of Heavy-Fermion Compounds - Theory of Strongly Correlated Fermi-Systems. Springer. ISBN 978-3-319-10825-4.