Quantum spin liquid

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In

The quantum spin liquid state was first proposed by physicist

Basic properties

The simplest kind of magnetic phase is a

Quantum spin liquids offer a dramatic alternative to this typical behavior. One intuitive description of this state is as a "liquid" of

Examples

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

Frustrated magnetic moments

Localized spins are

(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.

A recent research work used this concept in analyzing brain networks and surprisingly indicated frustrated interactions in the brain corresponding to flexible neural interactions. This observation highlights the generalization of the frustration phenomenon and proposes its investigation in biological systems.^[10]

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

, 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

• 
  One possible short-range pairing of spins in a RVB state.
• 
  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 ${\displaystyle S=1/2}$. 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^[11] 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.^[12][13] Later, another version of stable RVB state with deconfined spinons, the Z2 spin liquid, is proposed,^[14][15] 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,^[16] 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.^[17]

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.^[18]

Magnetic susceptibility

In a high-temperature, classical

$${\displaystyle \chi \sim {\frac {C}{T-\Theta _{CW}}}}$$

Fitting experimental data to this equation determines a phenomenological Curie–Weiss temperature, ${\displaystyle \Theta _{CW}}$. There is a second temperature, ${\displaystyle T_{c}}$, where magnetic order in the material begins to develop, as evidenced by a non-analytic feature in ${\displaystyle \chi (T)}$. The ratio of these is called the frustration parameter

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 ${\displaystyle f\to \infty }$.^[19] 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 their Curie–Weiss temperature are listed in the table below.^[7] 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[20]
Cu-(1,3-benzenedicarboxylate)	Kagome	-33[21]
Rb2Cu3SnF12	Kagome	[22]
1T-TaS2	Triangular

Other

One of the most direct evidence for absence of magnetic ordering give

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.^[25]

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.^[26] Later theoretical work challenged this picture, arguing that all experimental results were instead consequences of 1D spinons confined to individual chains.^[27]

Afterwards, it was observed in an organic Mott insulator (κ-(BEDT-TTF)₂Cu₂(CN)₃) by Kanoda's group in 2003.^[23] 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 using muon spin spectroscopy.^[28]

Herbertsmithite

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.[19] 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.^[29][30]

Synthetic, polycrystalline herbertsmithite powder was first reported in 2005, and initial magnetic susceptibility studies showed no signs of magnetic order down to 2K.

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 ${\displaystyle S=1/2}$

Large (millimeter size) single crystals of herbertsmithite were grown and characterized in 2011.

Some measurements were suggestive of

In 2020, monodisperse single-crystal

It may realize a U(1)-Dirac spin liquid.^[48]

Kitaev spin liquids

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 for the Physics of Complex Systems in Dresden, 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 honeycomb model.^[49][50]

Strongly correlated quantum spin liquid

The strongly correlated quantum spin liquid (SCQSL) is a specific realization of a possible quantum spin liquid (QSL)

with chemical structure ZnCu₃(OH)₆Cl₂.

Kagome type

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

Toric code type

In December 2021, the first direct measurement of a quantum spin liquid of the toric code type was reported,

Specific properties: topological fermion condensation quantum phase transition

The experimental facts collected on

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

Applications

Materials supporting quantum spin liquid states may have applications in data storage and memory.

References