## Nailing down a quantum spin liquid

Quantum spin liquids (QSLs) possess magnetic interactions that, even at absolute zero temperature, remain in a disordered liquid-like spin state. It is very difficult to prove unambiguously that a material is a QSL, because there is always a possibility that it can become ordered below the lowest measured temperature. Barkeshli *et al.* used quantum field theory to propose a direct way to identify a QSL by placing it in contact with other exotic materials, such as superconductors or magnets. The theory predicted that, at such a boundary, electrons entering the QSL would turn into excitations lacking charge or lacking spin. Future experiments may be able to detect this transmutation.

*Science*, this issue p. 722

## Abstract

Electrons have three quantized properties—charge, spin, and Fermi statistics—that are directly responsible for a vast array of phenomena. Here we show how these properties can be coherently and dynamically stripped from the electron as it enters a certain exotic state of matter known as a quantum spin liquid (QSL). In a QSL, electron spins collectively form a highly entangled quantum state that gives rise to the fractionalization of spin, charge, and statistics. We show that certain QSLs host distinct, topologically robust boundary types, some of which allow the electron to coherently enter the QSL as a fractionalized quasi-particle, leaving its spin, charge, or statistics behind. We use these ideas to propose a number of universal, conclusive experimental signatures that would establish fractionalization in QSLs.

A notable example of emergence in physics is fractionalization, where the long-wavelength, low-energy excitations of a many-body quantum phase of matter possess quantum numbers that are fractions of those of the microscopic constituents. In a fractional quantum Hall state, for example, the emergent quasi-particle excitations carry fractional electric charge and anyonic quantum statistics. In a quantum spin liquid (QSL), the electron fractionalizes at low energies into two quasi-particles—a spinon and a holon—that independently carry the spin and charge of the electron (*1*–*8*). When an electron is injected into such systems, it can decay into fractionalized components, but a direct quantum mechanical conversion of an electron to a single fractionalized quasi-particle has conventionally been thought to be impossible. Consequently, the question of how to experimentally detect fractionalization in a QSL, even in principle, has remained a major challenge. It is particularly timely to revisit this question, given the number of materials that have recently been shown to exhibit anomalous properties that may indicate that they are QSLs (*9*).

Here we show that some QSLs allow electrons to coherently enter through their boundary as a fractionalized quasi-particle, leaving behind their charge, spin, or even Fermi statistics. We show that this leads to universal experimental signatures that could provide incontrovertible evidence of fractionalization. Our considerations are based on recent theoretical breakthroughs regarding the physics of two-dimensional (2D) topologically ordered states with extrinsic line and point defects (*10*–*19*), of which robust boundary phenomena are a special case.

We focus primarily on explaining these phenomena in the context of the simplest gapped 2D QSL, the *Z*_{2} short-ranged resonating valence bond (sRVB) state, and explaining how it can be distinguished from nonfractionalized magnetic insulators, or even from QSLs with different types of fractionalization. This type of QSL has been proposed (*20*) to explain recent neutron scattering experiments in ZnCu_{3}(OH)_{6}Cl_{2} (herbertsmithite) (*21*) and also a number of experimental observations for the organic compound κ-(ET)_{2}Cu_{2}(CN)_{3} (*22*). We build on recent results that demonstrate that gapped fractionalized phases support topologically distinct types of gapped boundaries (*12*, *15*). These results imply that the *Z*_{2} sRVB state necessarily supports exactly two topologically distinct types of gapped boundaries, referred to as the *e* edge and the *m* edge (*23*), that are separated by a topological quantum phase transition.

The *Z*_{2} sRVB state is an insulating, spin-rotationally invariant gapped spin-liquid state. Its legitimacy as a state of matter has been proven by constructing model Hamiltonians with exact sRVB ground-states (*6*–*8*, *24*, *25*). At low energies, there are four topologically distinct types of elementary quasi-particle excitations: (i) topologically trivial excitations, which can be created with local operators, (ii) spinons and holons, which are topologically equivalent and carry spin 1/2 and charge 0, or spin 0 and charge 1, respectively, (iii) visons, which do not carry spin or charge and have mutually semionic statistics with respect to the spinons and holons, and (iv) the composite of a spinon or holon with a vison. Because electrons always carry both unit electric charge and spin 1/2, the spinons and holons cannot individually be created by any local combinations of electron operators and therefore must be topological excitations (*2*). That spinons and holons are topologically equivalent follows from the fact that one can be converted into the other by the local operation of adding or removing an electron. Moreover, the spinons and holons can be either bosonic or fermionic, depending on detailed energetics.

The *Z*_{2} sRVB state can be understood at low energies in terms of a *Z*_{2} lattice gauge theory. In this language, the spinons and holons carry the *Z*_{2} gauge charge, whereas the visons are the *Z*_{2} fluxes. As a result, the spinons and holons are sometimes referred to as the *e* particles and the visons as the *m* particles. This state can also be described at long wavelengths using Abelian Chern-Simons (CS) field theory (*26*, *27*)
(1) where μ, ν, λ = 0, 1, 2 are 2 + 1D space-time indices; is the Levi-Civita tensor, ; *I*, *J* = 1, 2; and describes the fractionalized quasi-particles, which are minimally coupled to the *U*(1) gauge fields *a ^{I}*. The visons carry unit charge under

*a*

^{1}, whereas the spinons and holons both carry unit charge under

*a*

^{2}. The CS term binds charges to fluxes in such a way as to properly capture the nontrivial mutual statistics between spinons or holons and visons.

Clever numerical studies of spin-1/2 frustrated Heisenberg models (*28*–*31*) provided evidence for gapped spin-liquid ground states. However, it is not yet clear whether the ground state in those models is a *Z*_{2} sRVB state or a certain competing QSL, the “doubled-semion” state, which is characterized instead by .

Each topologically ordered phase, characterized by a matrix *K*, can support topologically distinct types of gapped boundaries. A classification of such gapped edges (*12*, *15*), when applied to the *Z*_{2} sRVB state, predicts exactly two topologically distinct types of gapped edges. As we explain below, these correspond to whether the *e* or the *m* particles are condensed along the boundaries. The doubled-semion state, in contrast, possesses only one type of gapped boundary (*32*).

The edge theory can be derived by starting with the Abelian CS theory (*1*). [For an alternative explanation, see (*32*).] It is well-known (*27*) that on a manifold with a boundary, Eq. 1 is only gauge-invariant if the gauge transformations are restricted to be zero on the boundary. This implies that on the boundary, the gauge fields correspond directly to physical degrees of freedom. One can derive an edge Lagrangian (*27*) in terms of scalar fields
(2) where *V _{IJ}* is a positive-definite velocity matrix. The number of left (or right) movers is given by the number of positive (or negative) eigenvalues of the

*K*matrix. In a Hamiltonian formulation, the first term on the right hand side of Eq. 2 implies that . Quasi-particles that carry unit charge under

*a*are created with the operators .

^{I}In the *Z*_{2} sRVB state, the edge theory maps onto a single-channel Luttinger liquid, as there are two conjugate fields, and , with . A composite of two identical quasi-particles in the *Z*_{2} sRVB state always corresponds to a topologically trivial excitation and therefore corresponds to a local operator on the edge. Thus, there are two basic types of local terms, with coupling constants λ_{e,m}, that effectively backscatter counterpropagating modes and can induce an energy gap on the edge (*33*).

Because and θ are conjugate, the cosine terms cannot simultaneously pin their arguments, so there are two distinct phases. Where |λ* _{m}*| is the dominant coupling, and , implying that the

*m*particles are condensed on the edge. Conversely, if |λ

*| is dominant, the*

_{e}*e*particles are condensed on the edge: and . The two phases, referred to as the

*m*edge and

*e*edge, respectively, are topologically distinct. In the absence of any additional global symmetries, there is a single quantum critical point between these two gapped phases in the Ising universality class. [See (

*32*) for additional discussion.]

In the presence of spin rotation and charge conservation symmetries, the modes and θ must represent either low-energy spin or charge fluctuations (but not both), depending on the physical situation. If they describe charge fluctuations, then the charge density is given by , and the operator creates a holon that is bosonic for even integer *n* and fermonic for odd *n*. The operator creates a topologically trivial excitation that carries charge 2 and no spin and is therefore physically equivalent to a Cooper pair. Alternatively, if the boson modes describe spin fluctuations, then the spin density is , and . The operator creates a spin-1/2 spinon that is bosonic or fermionic for *n*, respectively, even or odd. In all cases, creates a vison.

If charge and spin are conserved, any term proportional to cos(2θ) is prohibited, because this term changes either the charge or the spin of the edge. It follows that an *e* edge is incompatible with spin and charge conservation; the *m* edge is the generic gapped boundary of a *Z*_{2} sRVB if charge and spin are conserved.

We will now explore how an edge can be realized in a physically realistic system by bringing the edge into contact with another system with one or another pattern of symmetry breaking. Our results, explained below, are summarized in Table 1. Let us begin by considering a realization of a *Z*_{2} sRVB state with easy-plane, or XXZ, spin-rotational symmetry. Again, this can be treated from the perspective of an Ising gauge theory or from the field theory perspective. In the bosonized edge theory, the bulk *U*(1) spin rotational symmetry is associated with the global transformation θ → θ + *f* (where *f* is an arbitrary constant), and therefore, so long as this symmetry is not explicitly broken, terms that would pin θ, such as cos(2θ), are disallowed. However, a magnetic field applied at the edge in an in-plane direction leads to a term –μ* _{B}B*cos(2θ) (where μ

*is the Bohr magneton and*

_{B}*B*is the magnetic field), which for strong enough magnetic field can produce a phase transition to an

*e*edge.

Now let us consider coupling a superconductor to the edge of a spin liquid. For simplicity we will consider a singlet superconductor, although the same results apply for the triplet superconductor. At low energies, the couplings between the superconductor and the spin liquid include Cooper pair tunneling of the form
(3) where Φ_{sc} is the Cooper pair operator on the edge of the superconductor, Φ_{qsl} is the Cooper pair operator on the edge of the QSL, *t*_{pair} is the pair-tunneling amplitude, and *H.c.* stands for Hermitian conjugate. Φ_{sc} can be replaced by a *c* number because the Cooper pairs are condensed in the superconductor. The first term therefore prefers to condense pairs of holons, which are the Cooper pairs on the spin-liquid edge. In the bosonized field theory, this term corresponds to a perturbation , where the charge density is . *t*_{pair} can drive a phase transition into the *e* edge, where single holons are also condensed: . The Cooper pair tunneling does not need to overcome the charge gap of the *Z*_{2} spin liquid. By tuning the chemical potential in the superconductor using a gate voltage, the energy cost to adding holons to the edge can be made much smaller than the charge gap. In this situation, even a small pair-tunneling amplitude is sufficient to condense the holons.

A similar analysis shows that strong Heisenberg exchange coupling to a noncolinear spin density wave (SDW) can also realize the *e* edge. Coupling an *SO*(3) spin rotationally invariant QSL to a colinear SDW (Néel state) is not by itself sufficient to realize the *e* edge, because the Néel state has a residual *U*(1) spin rotation symmetry that precludes spinon condensation.

Now that we have investigated the physical conditions under which the distinct edge phases can be realized, we turn to describing their physical implications. In the *Z*_{2} sRVB, the electron *c*_{α} can be thought of as a composite of a bosonic holon *b* and a fermionic spinon *f*_{α}: *c*_{α} = *bf*_{α}, where α = ↑,↓. When the *e* edge is created by strong Cooper pair tunneling from a superconductor, the bosonic holon is condensed on the edge. Consequently, any electron-tunneling term along the boundary between the QSL and the superconductor becomes
(4) Thus, at an *e* edge with a superconductor, the electron can coherently tunnel into the spin liquid as a fermionic spinon, leaving its charge behind at the edge. This would not be allowed at an *m* edge, at which the electron would tunnel into the spin liquid as a whole. Then, depending on details of the energetics of the excitations in the QSL, the electron could subsequently decay into a holon and a spinon.

There is a useful analogy here with Tomasch oscillations observed long ago in the context of superconducting films (*34*). These oscillations reflect processes in which an electron with energy in excess of the superconducting gap tunnels coherently from a metal into a superconductor, where it becomes a Bogoliubov quasi-particle (*35*). There is a well-defined sense (*36*) in which the quasi-particles in a conventional superconductor are neutral spinons, although the broken gauge symmetry of the superconductor makes this analogy somewhat subtle.

The possibility of a direct coupling between electrons and fractionalized quasi-particles opens a new realm of possible probes of QSLs. Given the existence of a material with a *Z*_{2} sRVB ground state and stable fermionic spinon quasi-particles, a direct experimental signature of such coherent fractionalization of the electron could be obtained by detecting a suitably generalized version of Tomasch oscillations: Consider the local electron tunneling density of states (LDOS), measured at the boundary of the superconductor, for a finite strip geometry (Fig. 1A). This will receive contributions from processes where superconducting quasi-particles coherently propagate into the spin liquid as fermionic spinons, reflect off the outer boundary, and propagate back. If the spinon inelastic mean free path is larger than the width *d*_{qsl} of the QSL, this leads to coherent oscillations of the LDOS as a function of the dimensionless ratio *eVd*_{qsl}/*ħ*ν_{qsl} for voltages *V* larger than both the spinon and superconducting gaps, where ν_{qsl} is the spinon velocity in the QSL and *ħ* is Planck’s constant *h* divided by 2π. Moreover, incontrovertible evidence that the oscillations are associated with fractionalized excitations can be obtained by simultaneously monitoring the bulk current (*I*_{b} in the figure), which in this case will be parametrically small and free of signatures of coherent spinon interference. For an *m* edge, there would be no such coherent oscillations, as the electron must enter into the QSL as a whole and would subsequently decay into a spinon and holon.

Similar phenomena will occur when the *e* edge is created through magnetic effects, such as through a magnetic field applied at the edge, or through coupling to a noncolinear SDW. In these cases, the bosonic spinon is condensed at the edge. We can write the electron operator as *c*_{α} = *hz*_{α}, where *z*_{α} is a bosonic spinon, and *h* is a fermionic holon. Now the electron-tunneling Hamiltonian at the edge becomes ; here we have included a spin-dependent tunneling matrix element *t*_{αβ}. Thus, the electron in this case can propagate coherently into the spin liquid as a fermionic holon. If fermionic holons are stable fractionalized quasi-particles, we again expect to observe Tomasch oscillations in finite strip geometries (Fig. 1C).

Treating the case of gapless spin liquids in a theoretically controlled manner is more difficult than the case of gapped spin liquids. Nevertheless, we expect that in gapless systems such as *Z*_{2} spin liquids with Dirac points in their spinon spectrum or *Z*_{2} chiral spin liquids with stable spinon Fermi surfaces (*37*), generalized Tomasch oscillations of the sort envisaged here would occur as well. This is because (i) in such states, the *Z*_{2} gauge field is gapped, and thus low-energy spinons can propagate coherently; and (ii) there is still a well-defined notion of whether the spinons or holons have condensed near the boundary.

At the boundary between two topologically distinct segments of edge, localized exotic topological zero modes arise that give rise to topologically protected degeneracies and projective non-Abelian statistics (*12*). In the case of the *Z*_{2} sRVB state, the domain wall between *e* and *m* edges localizes a Majorana fermion zero mode, with the following physical consequences: Let us consider the case of the superconductivity-induced *e* edge, where an electron can coherently enter the QSL as a fermionic spinon. If this process occurs in the vicinity of the domain wall between an *e* and *m* edge, then the fermionic spinon can also emit or absorb a vison from the *m* edge, thus becoming a bosonic spinon. In other words, the Majorana fermion zero mode is a source or sink of fermion parity, allowing the electron to coherently enter into the spin liquid as a bosonic spinon. If the fermionic spinon in the bulk of the QSL can decay into a vison and a bosonic spinon, then this geometry (Fig. 1B) will allow the Tomasch oscillations to be observed in the tunneling conductance. Similar considerations show that when the *e* edge is induced by magnetism, the electron can enter into the spin liquid as a bosonic holon in the vicinity of the *e*-*m* domain wall (Fig. 1D).

The considerations outlined here suggest ways to tune through the topological phase transition that separates the *e* and *m* edges, such as by applying a magnetic field to the edge of an easy-plane QSL. This can be done by taking a thin sample and shielding the bulk of the QSL by sandwiching it between two superconductors. At the critical field for the edge quantum phase transition, there will be enhanced thermal transport through the edge, leading to a nonzero intercept at low temperatures in the thermal conductance: , where *N*_{L} is the number of layers in the QSL, *c* = 1/2 is the central charge of the edge at the critical point, and *k*_{B} is Boltzmann’s constant. Because neither the trivial paramagnet nor the doubled-semion QSL have topologically distinct types of gapped boundaries, the observation of a topological quantum phase transition at the edge of a gapped insulating spin system would prove the existence of a fractionalized spin-liquid state and rule out the doubled-semion state. The present considerations are readily extended to other sorts of topologically ordered states, such as those that occur in fractional quantum Hall systems.

## Supplementary Materials

## References and Notes

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These were identified long ago in the toric code model (
*38*) in terms of rough versus smooth edges (*39*), although the generic necessity to realize one or the other in resonating valence bond states and their general independence of lattice structure has only recently been fully appreciated. - ↵
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See the supplementary materials on
*Science*Online. - ↵ If the resulting gaps are large, the field theory description of the edge is no longer controlled, but because the most important qualitative features of a gapped state do not depend on the magnitude of the gap, the field theory description of such phases should be adequate for present purposes: More formally, it gives a leading order description of the edge modes in the vicinity of an edge quantum critical point where the edge gaps, but not the bulk gaps, are vanishingly small.
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**Acknowledgments:**This work was supported by NSF-DMR grant 1265593 (S.K.), the Israel Science Foundation, the Israel-U.S. Binational Foundation, the Minerva Foundation, and the German-Israeli Foundation (E.B.). M.B. thanks X.-L. Qi for recent collaborations on related topics. We thank E. Altman, C. Nayak, M. Freedman, Z. Wang, K. Walker, M. Hastings, M. P. A. Fisher, B. Bauer, and T. Grover for discussions.