condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the ...

, a quantum spin liquid is a

phase of matter In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetizat ...

that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range

quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...

, fractionalized excitations, and absence of ordinary magnetic order. 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 antiferromagnetically with their nearest neighbors, i.e. neighboring spins seek to be aligned in opposite directions. 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.

Basic properties

The simplest kind of magnetic phase is a

paramagnet Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...

, where each individual spin behaves independently of the rest, just like atoms in an

ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is am ...

. 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 Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

(or

antiferromagnet In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...

) 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 Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

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 In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian g ...

, long-range

properties, and

anyon In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchang ...

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 spin The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent m ...

s (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.

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

spinon Spinons are one of three quasiparticles, along with holons and orbitons, that electrons in solids are able to split into during the process of spin–charge separation, when extremely tightly confined at temperatures close to absolute zero. The ...

s 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

–

Z2 topological order Z (or z) is the 26th and last letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its usual names in English are ''zed'' () and ''zee'' (), with an occas ...

. 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 The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. It is the simplest and most well studied of the quantum double models. It is also the simplest example ...

model is yet another realization of Z2 spin liquid (and

) that explicitly breaks the spin rotation symmetry and is exactly soluble.

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.

Magnetic susceptibility

In a high-temperature, classical

phase, the

magnetic susceptibility In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the ap ...

is given by the Curie–Weiss law

\chi \sim \frac

Fitting experimental data to this equation determines a phenomenological Curie–Weiss temperature,

\Theta_

. There is a second temperature,

T_c

, where magnetic order in the material begins to develop, as evidenced by a non-analytic feature in

\chi(T)

. The ratio of these is called the frustration parameter

f = \frac

In a classic antiferromagnet, the two temperatures should coincide and give

f = 1

. An ideal quantum spin liquid would not develop magnetic order at any temperature

(T_c = 0 )

and so would have a diverging frustration parameter

f \to \infty

. A large value

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. All of them are proposed spin liquid candidates.

Other

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 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 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.

s). * Reflectance measurements can uncover

s, which couple via emergent gauge fields to the electromagnetic field, giving rise to a power-law optical conductivity. Herbertsmithite-herb03a

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. Later theoretical work challenged this picture, arguing that all experimental results were instead consequences of 1D spinons confined to individual chains. Afterwards, it was observed in an organic Mott insulator (κ-(BEDT-TTF)₂Cu₂(CN)₃) by Kanoda's group in 2003. It may correspond to a gapless spin liquid with spinon Fermi surface (the so-called uniform RVB state). The peculiar phase diagram of this organic quantum spin liquid compound was first thoroughly mapped using

muon spin spectroscopy Muon spin spectroscopy, also known as µSR, is an experimental technique based on the implantation of spin-polarized muons in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. ...

Herbertsmithite

Herbertsmithite is one of the most extensively studied QSL candidate materials. It is a mineral with chemical composition ZnCu₃(OH)₆Cl₂ and a

rhombohedral In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be us ...

crystal structure. Notably, the copper ions within this structure form stacked two-dimensional layers of kagome lattices. Additionally,

superexchange Superexchange, or Kramers–Anderson superexchange, is the strong (usually) antiferromagnetic coupling between two next-to-nearest neighbour cations through a non-magnetic anion. In this way, it differs from direct exchange, in which there is coupl ...

over the oxygen bonds creates a strong antiferromagnetic interaction between the

S=1/2

copper spins within a single layer, whereas coupling between layers is negligible. 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. Synthetic, polycrystalline herbertsmithite powder was first reported in 2005, and initial magnetic susceptibility studies showed no signs of magnetic order down to 2K. In a subsequent study, the absence of magnetic order was verified down to 50 mK,

inelastic neutron scattering Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...

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

S=1/2

excitations. A broad array of additional experiments, including ¹⁷O NMR, and neutron spectroscopy of the dynamic magnetic structure factor, reinforced the identification of herbertsmithite as a gapless spin liquid material, although the exact characterization remained unclear as of 2010. Large (millimeter size) single crystals of herbertsmithite were grown and characterized in 2011. 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. Follow-up experiments (using ¹⁷O NMR 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. Some measurements were suggestive of quantum critical behavior. 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. This scaling is seen in certain quantum antiferromagnets, heavy-fermion metals, and two-dimensional ³He as a signature of proximity to a quantum critical point. In 2020 monodisperse single-crystal

nanoparticles A nanoparticle or ultrafine particle is usually defined as a particle of matter that is between 1 and 100 nanometres (nm) in diameter. The term is sometimes used for larger particles, up to 500 nm, or fibers and tubes that are less than 10 ...

of herbertsmithite (~10 nm) were synthesized at room temperature, using gas-difusion electrocrystallization, showing that their spin liquid nature persists at such small dimensions. Herb4

It may realize a U(1)-Dirac spin liquid.

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 Oak Ridge National Laboratory (ORNL) is a U.S. multiprogram science and technology national laboratory sponsored by the U.S. Department of Energy (DOE) and administered, managed, and operated by UT–Battelle as a federally funded research and ...

, 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 Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure.

. Their experimental results successfully matched with one of the main theoretical models for a quantum spin liquid, known as a Kitaev honeycomb model.

Strongly correlated quantum spin liquid

The ''strongly correlated quantum spin liquid'' (''SCQSL'') is a specific realization of a possible quantum spin liquid (QSL) representing a new type of strongly correlated electrical insulator (SCI) that possesses properties of heavy fermion metals with one exception: it resists the flow of

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

. At low temperatures ''T'' the

specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of 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 whose

heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...

is proportional to ''T''³. When a

magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

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 In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. ...

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, 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 In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

Helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...

demonstrate that the

quasiparticle In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For ex ...

effective mass ''M''* is very large, or even diverges.

Topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

fermion condensation

quantum phase transition In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physic ...

(FCQPT) preserves quasiparticles, and forms flat

energy band In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...

at the

Fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remov ...

. The emergence of FCQPT is directly related to the unlimited growth of the effective mass ''M''*. Near FCQPT, M* starts to depend on

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

''T'',

number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects ( particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...

''x'',

B and other external parameters such as

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and ...

''P'', etc. In contrast to the

Landau Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990) ...

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 Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...

, relaxation, scaling and

transport Transport (in British English), or transportation (in American English), is the intentional movement of humans, animals, and goods from one location to another. Modes of transport include air, land ( rail and road), water, cable, pipelin ...

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 In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal, ...

that consists of HF

metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typica ...

quasicrystals A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...

, quantum spin liquid,

, and compounds exhibiting

high-temperature superconductivity High-temperature superconductors (abbreviated high-c or HTS) are defined as materials that behave as superconductors at temperatures above , the boiling point of liquid nitrogen. The adjective "high temperature" is only in respect to previo ...

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

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Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the '' Jour ...

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