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 socalled 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 spinliquid state.^{[3]}
A quantum spin liquid state was first discovered in an organic Mott insulator with a triangular lattice (κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} ) by Kanoda's group in 2003.^{[4]} It may correspond to a gapless spin liquid with spinon Fermi surface (the socalled 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 ZnCu_{3}(OH)_{6}Cl_{2} 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 2dimensional 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 twodimensional 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
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 CurieWeiss temperature are listed in the table.^{[2]} All of them are proposed spin liquid candidates.
Material  Lattice  

κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}  anisotropic triangular  375  
ZnCu_{3}(OH)_{6}Cl_{2} (herbertsmithite)  Kagome  241  
BaCu_{3}V_{2}O_{8}(OH)_{2} (vesignieite)  Kagome  
Na_{4}Ir_{3}O_{8}  Hyperkagome  650  PbCuTe_{2}O_{6}  Hyperkagome  22 
Cu(1,3benzenedicarboxylate)  Kagome  33 ^{[11]}  
Rb_{2}Cu_{3}SnF_{12}  Kagome  ^{[12]} 
^{[13]}
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 nonmagnetic. 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 longrange 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 hightemperature superconductor physics.^{[14]}

One possible shortrange pairing of spins in a RVB state.

Longrange 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 lowenergy 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 lowenergy excitations even for shortrange 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 . 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 sublattices. The constructed RVB state is an equal amplitude superposition of all the nearestneighbour bond configurations. Such a RVB state is believed to contain emergent gapless gauge field which may confine the spinons etc. So the equalamplitude nearestneighbour 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 order – Z2 topological order. Both chiral spin state and Z2 spin liquid state have long RVB bonds that connect the same sublattice. 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 , which is defined as
where is the CurieWeiss temperature and 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. ^{1}HNMR measurements ^{[4]} on κ(BEDTTTF)_{2}Cu_{2}(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 lowenergy 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 powerlaw optical conductivity.^{[23]}
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 spinstate fractionalization, a predicted property of quantum spinliquid 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 T^{n} 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 T^{3}. 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 ZnCu_{3}(OH)_{6}Cl_{2}.
Specific properties
Exotic SCQSL's are formed with such hypothetical particles as fermionic spinons carrying spin 1/2 and no charge. The experimental studies of Herbertsmithite ZnCu_{3}(OH)_{6}Cl_{2} 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 Cu^{2+} ions can be considered as twodimensional 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 twodimensional ^{3}He.^{[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 particlesspinons, nonexisting as free, replace the initial particles appearing in the Hamiltonian and define the thermodynamic and relaxation properties at low temperatures. Because of the spincharge separation, heat transport, thermodynamic and relaxation properties at low temperatures of the SCI Herbertsmithite are similar to those of heavyfermion metals rather than of insulators.^{[33]}^{[38]}
Fermion condensation quantum phase transition
The experimental facts collected on heavy fermion (HF) metals and two dimensional ^{3}He 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 welldefined quasiparticles determine the thermodynamic, relaxation, scaling and transport properties of strongly correlated Fermisystems 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, ZnCu_{3}(OD)_{6}Cl_{2}.^{[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 heavyfermion metals as a signature of proximity to a quantum critical point. The lowtemperature 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 αRuCl_{3}, which is a proximate Kitaev quantum spin liquid producing Majorana fermions,^{[10]} and then in Ca_{10}Cr_{7}O_{28}, 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 spinliquid states.^{[44]} Developments in quantum spin liquids may also help in the understanding of high temperature superconductivity.^{[45]}
References
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 ↑ In literature, the value of J is commonly given in units of temperature () instead of energy.
 ↑ T. Ng & P. A. Lee (2007). "PowerLaw Conductivity inside the Mott Gap: Application to κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}". Phys. Rev. Lett. 99 (15). arXiv:0706.0050. Bibcode:2007PhRvL..99o6402N. doi:10.1103/PhysRevLett.99.156402.
 ↑ Anthony, Sebastian. "MIT discovers a new state of matter, a new kind of magnetism". Extremetech. Retrieved 23 August 2013.
 ↑ "Third State Of Magnetism Discovered By MIT Researchers". Red Orbit. December 21, 2012. Retrieved 24 December 2012.
 ↑ Han, TiangHeng; Young S. Lee; et al. (2012). "Fractionalized excitations in the spinliquid state of a kagomelattice antiferromagnet". Nature. 492 (7429). arXiv:1307.5047. Bibcode:2012Natur.492..406H. doi:10.1038/nature11659. Retrieved 24 December 2012.
 ↑ Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). "Theory of HeavyFermion Compounds  Theory of Strongly Correlated FermiSystems". Springer. ISBN 9783319108254.
 ↑ Bert, F.; Mendels, P. (2010). "Quantum Kagome antiferromagnet ZnCu_{3}(OH)_{6}Cl_{2}". Journal of the Physical Society of Japan. 79: 011001. arXiv:1001.0801. Bibcode:2010JPSJ...79a1001M. doi:10.1143/JPSJ.79.011001.
 1 2 Stewart, G. R. (2001). "NonFermiliquid behavior in d and felectron metals". Reviews of Modern Physics. 73 (4): 797–855. Bibcode:2001RvMP...73..797S. doi:10.1103/RevModPhys.73.797.
 1 2 Löhneysen, H. V.; Wölfle, P. (2007). "Fermiliquid instabilities at magnetic quantum phase transitions". Reviews of Modern Physics. 79 (3): 1015. arXiv:condmat/0606317. Bibcode:2007RvMP...79.1015L. doi:10.1103/RevModPhys.79.1015.
 ↑ Gegenwart, P.; et al. (2006). "Highfield phase diagram of the heavyfermion metal YbRh_{2}Si_{2}". New Journal of Physics. 8 (9): 171. Bibcode:2006NJPh....8..171G. doi:10.1088/13672630/8/9/171.
 1 2 3 4 Helton, J. S.; et al. (2010). "Dynamic Scaling in the Susceptibility of the Spin1/2 Kagome Lattice Antiferromagnet Herbertsmithite". Physical Review Letters. 104 (14): 147201. arXiv:1002.1091. Bibcode:2010PhRvL.104n7201H. doi:10.1103/PhysRevLett.104.147201. PMID 20481955.
 1 2 3 Shaginyan, V. R.; Msezane, A.; Popov, K. (2011). "Thermodynamic Properties of Kagome Lattice in ZnCu_{3}(OH)_{6}Cl_{2} Herbertsmithite". Physical Review B. 84 (6): 060401. arXiv:1103.2353. Bibcode:2011PhRvB..84f0401S. doi:10.1103/PhysRevB.84.060401.
 ↑ Helton, J. S.; et al. (2007). "Spin Dynamics of the Spin1/2 Kagome Lattice Antiferromagnet ZnCu_{3}(OH)_{6}Cl_{2}". Physical Review Letters. 98 (10): 107204. arXiv:condmat/0610539. Bibcode:2007PhRvL..98j7204H. doi:10.1103/PhysRevLett.98.107204. PMID 17358563.
 ↑ Green, D.; Santos, L.; Chamon, C. (2010). "Isolated flat bands and spin1 conical bands in twodimensional lattices". Physical Review B. 82 (7): 075104. arXiv:1004.0708. Bibcode:2010PhRvB..82g5104G. doi:10.1103/PhysRevB.82.075104.
 ↑ Heikkilä, T. T.; Kopnin, N. B.; Volovik, G. E. (2011). "Flat bands in topological media". JETP Letters. 94 (3): 233. arXiv:1012.0905. Bibcode:2011JETPL..94..233H. doi:10.1134/S0021364011150045.
 1 2 3 4 Shaginyan, V. R.; Amusia, M. Ya.; Msezane, A. Z.; Popov, K. G. (2010). "Scaling Behavior of Heavy Fermion Metals". Physics Reports. 492 (2–3): 31. arXiv:1006.2658. Bibcode:2010PhR...492...31S. doi:10.1016/j.physrep.2010.03.001.
 1 2 Shaginyan, V. R.; et al. (2012). "Identification of Strongly Correlated Spin Liquid in Herbertsmithite". EPL. 97 (5): 56001. arXiv:1111.0179. Bibcode:2012EL.....9756001S. doi:10.1209/02955075/97/56001.
 ↑ Coleman, P. (2007). "Heavy Fermions: Electrons at the edge of magnetism". Handbook of Magnetism and Advanced Magnetic Materials. John Wiley & Sons. pp. 95–148. arXiv:condmat/0612006.
 ↑ Han, TianHeng; et al. (2012). "Fractionalized excitations in the spinliquid state of a kagomelattice antiferromagnet". Nature. 492: 406–410. arXiv:1307.5047. Bibcode:2012Natur.492..406H. doi:10.1038/nature11659.
 1 2 de Vries, M. A.; et al. (2008). "The magnetic ground state of an experimental S=1/2 kagomé antiferromagnet". Physical Review Letters. 100 (15): 157205. arXiv:0705.0654. Bibcode:2008PhRvL.100o7205D. doi:10.1103/PhysRevLett.100.157205. PMID 18518149.
 ↑ "Physical realization of a quantum spin liquid based on a complex frustration mechanism". Retrieved 25 July 2016.
 ↑ Aguilar, Mario (December 20, 2012). "This Weird Crystal Demonstrates a New Magnetic Behavior That Works Like Magic". Gizmodo. Retrieved 24 December 2012.
 ↑ Fendley, Paul. "Topological Quantum Computation from nonabelian anyons" (PDF). University of Virginia. Retrieved 24 December 2012.
 ↑ Chandler, David (December 20, 2012). "New kind of magnetism discovered: Experiments demonstrate 'quantum spin liquid'". Phys.org. Retrieved 24 December 2012.
Books
Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). Theory of HeavyFermion Compounds  Theory of Strongly Correlated FermiSystems. Springer. ISBN 9783319108254.