Probing quantum spin liquids in equilibrium using the inverse spin Hall effect

We propose an experimental method utilizing a strongly spin orbit coupled metal to quantum magnet bilayer that will probe quantum magnets lacking long range magnetic order, e.g., quantum spin liquids, via a dc resistance measurement across the metal. The bilayer is held in thermal and chemical equilibrium, and spin fluctuations arising across the single interface are converted into voltage fluctuations in the metal as a result of the inverse spin Hall effect. In thermal equilibrium, changes to the voltage noise in the metal are measurable as changes to the resistance via the fluctuation dissipation theorem. In this article, we elucidate the theoretical workings of the proposed bilayer system, and provide precise predictions for the temperature scaling of the enhancement to the dc resistance measured across the metal layer for three candidate quantum spin liquid models. Application to the Heisenberg spin-$1/2$ kagom\'e lattice model should allow for the extraction of any spinon gap present. If the magnet is well-captured by the Kitaev model in the gapless spin liquid phase, the result is characteristic $T^3$ scaling of the dc resistance enhancement across the metal. Finally, a quantum spin liquid consisting of fermionic spinons coupled to a $U(1)$ gauge field should cause subdominant $T^{4/3}$ scaling of the dc resistance enhancement of the coupled metal. We therefore show that a dc resistance measurement across the metal in a single interface, equilibrium bilayer can test the relevance of a quantum spin liquid model to a given candidate material.


I. INTRODUCTION
Quantum spin liquids (QSLs) refer to intriguing states of insulating spin systems in which strong quantum fluctuations prevent spins from ordering down to zero temperature and the prototypical wavefunction exhibits extensive many-body entanglement [1,2].They are endowed with fascinating physical properties like non-local excitations [3,4] and non-trivial topology [5,6], and substantial pioneering work has been accomplished in the pursuit of a physical instantiation of this long-sought-after phase of matter.The most promising candidates to date include the mineral herbertsmithite [7], certain organic salt compounds [8], and the so-called Kitaev materials [9], and their experimental studies have ranged from nuclear magnetic resonance [10][11][12][13], to susceptibility and heat capacity investigations [14][15][16], to thermal transport studies [17, 18, 20? ], and to neutron scattering [21][22][23][24].Theoretical works have shown that QSL ground states are realized in the antiferromagnetic S = 1/2 Heisenberg model on the kagomé lattice [25][26][27][28], in the exactly solvable Kitaev model [29] in two [30,31] and three [32,33] dimensions, as well as in models that couple low-energy fermionic spin excitations to an emergent U(1) gauge field [34][35][36][37][38][39].However, while the number and types of potential QSL models have proliferated, the experimental methods available for studying them have not [40].It is therefore important to the search for compounds possessing QSL ground states to propose previously unknown techniques by which to probe and categorize candidate materials.
The inverse spin Hall effect (ISHE) [41] is an essential component of the spintronics repertoire [42], and provides an opportunity to develop probes of unconventional quantum magnets.This phenomenon has found utility in detecting spin currents induced by thermal gradients [43][44][45] and in performing non-local spin transport experiments through magnetic insulators [46][47][48][49][50].In recent years, proposals to probe certain QSL states via nonequilibrium spin transport -and therefore spin Hall effects -have also been made [51,52].However, no extant probe of QSLs exploits the effect in thermal equilibrium.The prime function of the ISHE is that it acts as a transducer between spin and charge current densities in metals with strong spin orbit coupling.Thus thermal fluctuations arising in the spin sector of a heavy-element metal (e.g., Pt, Ta, W) can be transferred into electrical fluctuations via the ISHE.One may then envisage a bilayer system in which a strongly spin orbit coupled metal is deposited on top of a quantum magnet, i.e., consider coupling the metal to a spin dissipating subsystem that freezes out charge degrees of freedom to focus solely on the spin sector, while holding the entire bilayer in thermal and chemical equilibrium.The addition of a spin dissipating subsystem will result in equilibrium spin current fluctuations across the interface [53,54], which, as a result of the ISHE, are converted into charge fluctuations inside the metal -charge fluctuations that may encode information about the microscopic structure of the insulator.The QSL-to-normalmetal bilayer therefore presents a relatively straightforward table-top setup that directly probes the low energy density of states of the QSL material.This type of bilayer system has garnered some study with respect to ferromagnetic insulators hosting magnons [55]; however, the utility of the ISHE as a noise conversion mechanism has not yet been explored in the context of exotic quantum magnets in general and the search for QSLs in particular.
In this work, we propose a relatively simple mechanism that can probe QSL ground states via equilibrium measurements.We will develop the theory underpinning the aforementioned QSL-to-normal-metal bilayer that utilizes the QSL material as a spin dissipating subsystem.Spin dissipation via the QSL material results in spin noise generation due to the fluctuation dissipation theorem, and we show that any spin noise generated must affect the voltage noise -or equivalently the resistance -measured in the adjacent normal metal, if the normal metal possesses strong spin orbit coupling.We then use our theory to examine three current QSL models: the Heisenberg S = 1/2 kagomé lattice model, the Kitaev model in the gapless spin liquid phase, and a model consisting of gapless fermionic spin excitations coupled to an emergent U(1) gauge field.Utilizing a bosonic parton approach to the kagomé lattice model, we show that our proposed bilayer can successfully quantify any spin gap present; we further show that a material possessing a gapless Kitaev spin liquid state, when coupled to a normal metal, should produce a characteristic T 3 temperature scaling of the dc resistance measured across the metal; and finally we show that a QSL model coupling gapless fermions to an emergent U(1) gauge field produces a subdominant T 4/3 temperature scaling correction to the dc resistance measured across the normal metal that our proposed system can educe.The heterostructure we outline marries concepts from spintronics and quantum magnetism in order to use the well-understood physics of equilibrium noise in a new context, namely the search for, and categorization of, QSL ground states in various materials.We believe that this opens the door to previously unknown experimental possibilities.

II. PROPOSED SETUP
The fluctuation-dissipation theorem (FDT) establishes a strict relationship between the resistance through a conductor and voltage noise in thermal equilibrium, S V (T ) = 4k B T R(T ), where S V is the voltage noise, T is the temperature of the conductor, and R(T ) is its (potentially) temperature dependent resistance.This is the starting point for the concept behind our proposal.We may now consider coupling an insulating spin system to the conductor, as depicted in Fig. 1.The addition of the spin system gives rise to increased spin dissipation, which, as a result of the FDT, contributes additional spin fluctuations inside the normal metal.The key point is that if strong spin orbit interactions are present in the conductor (as in, e.g., Pt, Ta, W), then the FDT requires that spin fluctuations across the interface will give rise to voltage fluctuations in the conductor via the ISHE, S V = S (0) V + δS V , where S V is the total voltage noise in the metal in thermal equilibrium, S (0)  V is the thermal noise present in the bare metal, and δS V is the portion of the noise that arises due to the presence of the quantum magnet.We can connect this to the resistance measured in the metal straightforwardly, i.e., where δR(T ) is an enhancement to the resistance of the conductor arising due to the presence of the spin system.One instance of the more general phenomenon we propose here is what is known as spin Hall magnetoresistance (SMR).If the heterostructure were formed of a strongly spin orbit coupled metal and a ferromagnetic insulator with an inplane magnetization, i.e., magnetization parallel to the interface, then δR(T ) in Eq. (1) represents SMR.This was first worked out from a microscopic perspective in Ref. 51, and later from the perspective of equilibrium thermal noise and the FDT in Ref. 56 (we elucidate the equivalence between these ⊗ I s < l a t e x i t s h a 1 _ b a s e 6 4 = " q / 0 3 D j 0 y 2 8 a t O e n K 0 s d b e X N t 4 f a f 5 a O v n 4 j X o D e s H 6 0 f r t t W 2 7 l u P r I 7 1 y j q 0 P K t v / W r 9 Y f 2 5 4 q 3 8 s v L b y u 8 F 9 f q 1 8 t X p d 1 b l s / L X v 2 s F e m I = < / l a t e x i t > I c < l a t e x i t s h a 1 _ b a s e 6 4 = " s 9 w S D g d o A 8 I u C A o l 3 j N 0 s 6 a h q j H E Z d r d P o b 8 w T K Z s e x 6 I 1 3 4 s 5 2 S o l 7 0 8 j K 6 V 8 G T 8 S j y 0 m 4 D m I L 3 9 S 0 g 2 j 0 Z D H z 9 I x m T O h 0 x y e f k 7 p x 8 j G T r j / c / T l / D X r N + t 7 6 w b p l N a x 7 1 k O r b b 2 0 D i 3 X G l i / W n 9 Y f 6 5 5 a 7 + s / b b 2 e 0 6 9 e q V 4 d f q t V f q s / f U v 1 u x 6 n g = = < / l a t e x i t > ⌘ < l a t e x i t s h a 1 _ b a s e 6 4 = " r 6 5 9 r 1 j z 7 + 5 N P P b n x e + + L L r 7 7 + 5 u a t b 4 9 V P J I u P 3 V H e z 9 n r 0 F v W D 9 Y P 1 q 3 r b p 1 3 3 p k t a x X 1 r H l W t z 6 1 f r D + n P D 2 f h l 4 7 e N 3 z P q 9 W v 5 q 9 P v r M J n 4 6 9 / A S L V e f g = < / l a t e x i t > z < l a t e x i t s h a 1 _ b a s e 6 4 = " S N 8 k 3 / 9 N B L W j W U 1 C M B P K w y 1 e Z I j p P F 6 L E d I H / T S N z p p P r P X O q S C c p j 0 F G 8 / R 2 g / u u L p + 4 u l A W 0 T 7 9 e 5 9 2 u C 7 O U I W W j e u x x 5 h x H d n t 8 4 y V r R 8 + 0 L H Y P D Z F q S T J b U I T F e X L b 9 Z L 5 C K 5 W z m 9 U 6 f t V J T 0 5 2 t u q 7 W z u H d 6 q P 9 n 7 O X o P e s H 6 w f r R u W 3 X r v v X I a l m v r G P L t b j 1 q / W H 9 e e G s / H L x m 8 b v 2 f U 6 9 f y V 6 f f W Y X P x l / / A j h R e f k = < / l a t e x i t > FIG. 1. (Color online) A cartoon of the proposed system: a quantum spin liquid to normal metal bilayer in equilibrium, where fluctuating spin current I s across the interface becomes fluctuating charge current I c (or fluctuating voltage) in the normal metal via the inverse spin Hall effect and results in measurable modifications to the resistance of the metal.two approaches in Appendix A).Our proposed bilayer system expands on these ideas by extending the theory to insulating magnets in general.
In the subsequent sections, we will compute the correction to the electrical resistance in a strongly spin orbit coupled metal interfaced with a general quantum magnet with no long-ranged order, and then apply our results to some wellknown QSL models.This requires that we first compute the voltage noise correction present in the metal, and then use the FDT to extract the portion of the resistance that arises due to the presence of the spin system.Characteristic features in the resistance will then allow for discriminating between the QSL ground states of the various candidate materials via a relatively straightforward near-equilibrium resistance measurement.In the next section, we provide the technical calculations necessary to extract the correction to the equilibrium voltage noise, δS V , that arises in the metal layer due to the QSL, in order to extract δR(T ) via Eq.(1).

III. THEORY: VOLTAGE FLUCTUATIONS
Let us consider a bilayer system as shown in Fig. 1, comprised of a thin normal metal layer (thickness d) affixed atop a QSL material.The interface is set at the xz plane, and the QSL is treated as a 2d lattice of fixed quantum spins S i .The particular lattice structures of the QSLs come into play later in the discussion.Coupling between the metal and the QSL occurs via the spin density in a volume v 0 in the metal interacting with the fixed spins of the QSL; v 0 here is the volume in the metal per spin of the QSL.We further assume that the entire heterostructure is thermalized to a temperature T in order to eliminate the effects of nonequilibrium drives.
The QSL and the normal metal are coupled via a spinconserving exchange interaction where J is the exchange constant and r i is a vector of the interfacial coordinates at the y = 0 plane specifying the position of S i .The local spin density of the metal is given as usual by s(x) = (1/2)ψ † s (x)τ ss ψ s (x), where τ is the vector of Pauli matrices and s labels the electron spin quantum number.
The interfacial spin current operator I s is defined by considering the total z polarized spin entering the metal, i.e., where ι = √ −1.Then the spin current noise across the interface can then be computed to lowest non-trivial order in J using the Keldysh formalism and the result formally takes the form [57] s − (y = 0, r i , t)s + (y = 0, r j , 0) where • • • represents correlation functions with respect to the unperturbed Hamiltonian.The spectrum of the interfacial spin current noise is then defined via We model the isolated metal as a 3d free fermion bath, held at temperature T and chemical potential µ, with a finite spatial extent d along the y direction, such that We assume periodic boundary conditions in the interfacial plane and zero flux boundary conditions at the top and bottom surfaces, in which case it is convenient to expand the electron eigenstates in the mixed Fourier representation, i.e., where (k x , k z ) = (2π/L)(n x , n z ) with n x , n z ∈ Z and k y = n y π/d with n y = 1, 2, 3, . . . .Utilizing this expansion in Eq. ( 4) to compute the spin correlation function for the metallic sector and inserting Eq. ( 4) into Eq.( 5), we arrive at [57] S s (Ω, where k F is the Fermi wavevector of the metal, and we have introduced the dynamical spin correlation function of the QSL, to account for the portion of the noise that arises due to spin fluctuations in the QSL.However, we note that for large Fermi wavevectors, such that k F r i − r j 1 for all i j, Eq. ( 8) can be approximated in spatially local terms as We emphasize here that the derivation so far assumes no particular form for the QSL Hamiltonian.Instead, Eq. ( 10) is constructed in order to apply to noise generated in a strongly spin orbit coupled normal metal due to the proximity of a general quantum magnet (albeit with no long-ranged magnetic order), such that extracting a usable quantity depends only on characterizing χ ±∓ ii (ν) for a given QSL model.Finally, the ISHE converts the z polarized spin current density in the y direction into a charge current density along the x direction (see Fig. 1).The conversion ratio between the interfacial spin current fluctuations and the measurable voltage fluctuations may depend non-trivially on various properties of the metal, e.g., its spin Hall angle, geometry, and spin diffusion length.However, as shown in, e.g., Refs.[56,57], these details may be lumped into a single phenomenological spin-to-voltage noise conversion constant Θ so that the ac voltage fluctuations generated by the quantum magnet can be expressed in terms of the spin fluctuations in Eq. (10) as Therefore, we note that what we have calculated in Eq. ( 10) is directly related to the ac voltage noise in the metal generated by the presence of the quantum magnet.We may then extract the modification to the resistance acquired by the presence of the quantum magnet from this quantity using Eq.(1).

IV. APPLICATIONS TO QUANTUM SPIN LIQUID MODELS
In this section, we consider three types of QSL models in the context of our proposed bilayer system: the antiferromagnetic Heisenberg S = 1/2 kagomé lattice model (HKLM); the Kitaev honeycomb model in the gapless spin liquid phase; and a model involving a spinon Fermi surface coupled to an emergent U(1) gauge field.First, we utilize Schwinger bosons [58] rather than Abrikosov fermions [39] in our approach to the HKLM in order to immediately acquire a bosonic spinon that can more transparently account for condensation that drives the phase transition out of the QSL and into an ordered phase.Next, we examine the Kitaev spin liquid as it is exactly solvable and therefore compelling theoretically.Finally, motivated by Ref. 59 and the search for observables that can identify signatures of gauge fields in spin liquid states, we perform a real-frequency, finite temperature Keldysh calculation in our treatment of fermionic spinons with a Fermi surface coupled to an emergent U(1) gauge field.We ultimately show that the bilayer system we propose will allow the extraction of characteristic temperature dependencies in each case.All three models above are believed to be of relevance for the descriptions of some QSL candidate materials.Detailed discussion of these connections will be presented in Sec.V.

A. Heisenberg antiferromagnet on the kagomé lattice
The HKLM is an important prototypical model that supports a QSL ground state, and is believed to be a relevant model for the mineral compound ZnCu 3 (OH) 6 Cl 2 (herbertsmithite), one of the prime contenders for an experimental realization of a QSL.Recent neutron scattering experiments suggest that the relevant theoretical model for the QSL state observed in this compound consists of the so-called Z 2 spin liquids [25,60,61].These QSL states, in principle, possess two types of excitations, i.e., the S = 1/2 spinon and a gapped vortex, also known as a vison, corresponding to an emergent Z 2 gauge field [5].It was shown recently that these visons can act as a momentum sink for the spinons and can flatten the dynamic spin structure factor [61] in accordance with experimental observations of herbertsmithite.However, for the purposes of the heterostructure we are proposing in this work, we note that only local spin fluctuations are important and visons do not themselves carry spin.Therefore, we expect that visons should not significantly renormalize spin current fluctuations, and that the re-shuffling of the spin quasiparticle spectral weight in momentum space should not drastically affect local quantities.We therefore ignore the effects of the visons in this subsection, follow the calculations performed in Ref. 25 for the HKLM, and apply the result to the observable we are proposing as a probe of the QSL state.
We begin with the Hamiltonian for the bare HKLM where i, j indicates nearest neighbor pairings and J is the exchange coupling.Mean-field approaches to characterizing the HKLM ground states include bosonic representations [25] of the gapped spinons and fermionic representations, where a gap arises due to pairing in the Hamiltonian [62].In what follows, we will focus on the bosonic representation of the QSL phase as done in Ref. 25.
The goal is to compute the local dynamic spin susceptibility χ ±∓ ii (ν), which is ultimately inserted into Eq.( 10) for the evaluation of the spin current fluctuations.We begin with the standard Schwinger boson representation of the spin operators , where b iσ represents a bosonic spinon.We then use the mean-field decoupling Q i j = ε σσ b iσ b jσ , where ε σσ is the completely antisymmetric tensor of S U(2) and the field satisfies Q i j = −Q ji .Substituting the Schwinger bosons into Eq.( 12), the above mean-field decoupling results in the Hamiltonian where h.c.denotes the hermitian conjugate and λ is a Lagrange multiplier that constrains the model to one boson per site, i.e., σ b † iσ b iσ = 1.For ease of calculation, as we are primarily interested in illustrating the workings of our proposed heterostructure, we do not here consider Dzyaloshinskii-Moriya interactions [63,64].
Figure 2 shows a depiction of the kagomé lattice, where the arrows between sites in a unit cell correspond to the direction dependent Q i j , i.e., Q 1 and Q 2 are the two distinct expectation values of Q i j .In the figure, solid arrows correspond to Q i j = Q 1 and wire arrows correspond to Q i j = Q 2 .The two possible locally stable QSL mean-field solutions [25,60,61,63] occur when Q 1 = Q 2 = Q, corresponding to π flux through the hexagonal plaquette, and Q 1 = −Q 2 , corresponding to no flux through the plaquette, both with Q 1 , Q 2 ∈ R. Here, we will specifically consider the π flux case in order to utilize mean-field theoretical values put forward in Ref. 61 for our purposes.
Substituting the Schwinger bosons into Eq.( 9) and diagonalizing Eq. ( 13) (see Appendix B), we can solve for the spin susceptibility χ(ν We re-express the sum over all positions in the lattice i by reformulating it as , where N u is the number of unit cells and n indexes each position within a single unit cell.We then obtain where U k ml is the 6 × 6 unitary matrix that diagonalizes Eq. ( 13) (see Appendix B), n = n + 3, km are the energy eigenvalues, and γ km represents the rotated boson operators.With this, we can express the noise created by the proximate QSL by using Eq. ( 14) in Eq. ( 10), We now analyze this expression numerically.Figure 3 shows our primary result for the HKLM, i.e., that in the low temperature regime it is possible to extract an estimate of the spin gap using our proposed bilayer.We use the variable values proffered in Refs.52 and 61 for the HKLM, taking as a reasonable estimate for the spin gap ∆ s = 0.055J.In the HKLM, the gap energy can be found by setting k = 0 and finding the minimal eigenvalue [63], with the outcome ∆ s = λ − √ 3Q.Therefore, rather than computing the values of λ and Q variationally, we set λ = 0.695J and tune Q such that ∆ s = 0.055J, resulting in Q = 0.37J.Note that Q quantifies antiferromagnetic correlations and is restricted to |Q| ≤ 1/ √ 2, at which point nearest neighbor spins form singlets and the model experiences a phase transition into an ordered phase.
The HKLM also includes quantum fluctuations, so when examining the low temperature regime, quantum fluctuations must be removed in order to focus solely on the interfacial equilibrium noise generated thermally as a result of the SMRlike mechanism we are considering.We therefore subtract out the T = 0 component, i.e., S s (0, 0), in constructing Fig. 3.In the dc limit, where Ω → 0, the excess thermal noise becomes S s (0, T ) − S s (0, 0) ∼ e −∆ s /k B T for k B T ∆ s , so that we expect a gap mediated exponential suppression of the noise enhancement for temperatures well below the spinon gap.A linear fit to the low temperature regime should give a direct estimate for the spin gap ∆ s .Indeed, (the negative of) the slope of the linear fit above gives 0.054J, which is very close to the expected

B. The Kitaev Honeycomb Model
A second example of a Z 2 QSL is the Kitaev model on the honeycomb lattice (see Fig. 4), where exchange frustration arising due to the inability to simultaneously satisfy all Kitaev interactions along neighboring bonds can drive the system into a QSL phase.The Kitaev spin liquid is exactly solvable, and we select this example for consideration in our proposed system for that reason, in addition to the fact that there are potential material candidates available currently [13,24].Furthermore, we restrict our investigation to the gapless phase, where the Kitaev exchange couplings are equal, and note that the vison excitations reflecting the emergent Z 2 gauge field are gapped out.We give a short overview of the approach to solving the Kitaev model following Refs.29 and 65, and then apply the solution to extracting an observable out of our proposed heterostructure.
The Kitaev model on the honeycomb lattice is given by [29] where γ = {x, y, z} represents the different nearest neighbor bond directions (see Fig. 4) at each lattice point with interaction strength K γ .In understanding the gapless spin liquid phase of the Kitaev model, parton mean-field theories have been proposed [29,66] that characterize the emergent excitations as Dirac fermions [65] arising in tandem with an emergent gapped flux.This can be seen by first representing the spin operators in terms of four Majorana fermions, i.e., ii , and { f iγ , c i } = 0, which then gives S γ i S γ j = −ιû i j c i c j , where ûi j = ι γ f iγ f jγ is the bond operator.Noting that the bond operators commute with each other and with any bilinear operator containing c i , it is possible to replace them with their eigenvalues ±1.One thus obtains S γ i S γ j = ±ιc i c j , and H K becomes bilinear in the Majorana fermions.Second, we note that the so-called flux operator on a plaquette W p = f 1x f 2y f 3z f 4x f 5y f 6z , where the subscript p labels the plaquette number, commutes with the Hamiltonian and is therefore an integral of motion.When a plaquette has an even number of bonds, as Fig. 4 makes clear is the case for the honeycomb lattice, its eigenvalues are ±1.Finally, we note that the spin representation in terms of four Majorana fermions enlarges the Hilbert space from two to four, and must therefore be constrained in order to recover the physical Hilbert space.This constraint is enforced via a projection operator P i = (1/2)(1 + f ix f iy f iz c i ) for each site i, which requires that the initial spin algebra be conserved.Effectively, what has been done is to reduce the initial Hamiltonian to a noninteracting Dirac fermion hopping Hamiltonian in a static Z 2 gauge field, where the choice of values for ûi j amounts to fixing a gauge, and the gauge invariant quantities are the plaquette operators W p .
A theorem by Lieb [67] guarantees that if the number of sites per plaquette is 2 mod 4, then the ground state is in the flux-free sector, i.e., where it is possible to set ûi j = 1.The ground state is therefore described by the free Majorana hopping Hamiltonian, where K x = K y = K z ≡ K, and the subscripts i and j are now labeling A and B sites respectively.Equation ( 17) can be diagonalized with a standard procedure [31].We take r as a unit cell coordinate, comprised of an A site and a B site as shown in Fig. 4, and combine the two Majorana species into a single complex fermion species, b r = (c Ar + ιc Br )/2.Putting the system on a torus with N u unit cells, going to reciprocal space with b r = N −1/2 u k e ιkr b k , and performing a Bogoliubov transformation b k = cos θ k a k + ι sin θ k a † −k , where tan 2θ k = −Im(s k )/Re(s k ) [31], Eq. ( 17) diagonalizes to where s k = K(1 + e ιkn 1 + e ιkn 2 ) and the primitive vectors n 1 and n 2 are defined in Fig. 4. Our goal is to use Eq. ( 18) to derive the susceptibility in order to extract the noise correction emerging in our proposed bilayer system when the QSL material is a gapless Kitaev spin liquid compound.From Eq. ( 9) we see that the required correlation functions are S ± m (t)S ∓ m (0) , where m now indexes the lattice points; however, in the case of the isotropic Kitaev spin liquid, S x m (t)S x m (0) = S y m (t)S y m (0) , and S γ m (t)S γ m (0) = 0 for γ γ [68].As a result, we may write the total susceptibility where N = 2N u is the total number of lattice points.Following Baskaran et al. [68], the above dynamical spin correlation function can be written as where the expectation value on the right is taken with respect to Eq. ( 18), and the bond potential reads V x = −2ιKc i c j with i and j, again, representing the A and its adjacent x-bond connected B site, respectively [31,68].Here, m = A0 corresponds to the A site of the r = 0 unit cell.The form of the correlator Eq. ( 20) can be understood by taking into consideration the effect of operator S γ m on an eigenstate.In addition to adding a single Majorana fermion c m at site m, the spin operator also adds one π flux each to the two plaquettes sharing the γ-bond emanating from m.The bond potential V x represents the insertion of this flux pair, and Eq. ( 20) calculates the dynamic rearrangement of the Majorana fermions at time t following a sudden appearance of the fluxes at t = 0. Incidentally, Baskaran et al. showed that this quench problem is equivalent to an exactly solvable x-ray edge problem [68].
It was later shown by Knolle et al. [31] that due to a vanishing Majorana density of states in the Kitaev model, slowly switching on the bond potential in the infinite past and then switching it off in the infinite future -what may be called the adiabatic approximation -exactly replicates the quench dynamics in the low energy limit.Thus, as we desire a probe of the low energy density of states, we perform the calculation under this adiabatic approximation and evaluate using the real-time (real-frequency) Keldysh diagrammatic formalism.Here, the Keldysh s-matrix reads the time integrals are performed over the Keldysh time-loop contour c K , and T K is the time ordering operator on the contour.The superscripts on the time arguments + and − label the forward and backward portions of the Keldysh contour, respectively.Under these conditions, the spin correlator S xx (t) ≡ S x A0 (t)S x A0 (0) can be expressed as a sum of four point correlation functions in the Bogoliubov rotated fermion basis, where V qq pp = e ιθ q −ιθ q −ιθ p −ιθ p −ι p n 2 .The total susceptibility, Eq. ( 19), then becomes where n F (ν) = (e β ν + 1) −1 is the Fermi-Dirac distribution.
The ac noise correction generated in the metal by proximity to a gapless Kitaev spin liquid can now be calculated by utilizing Eq. (24) in Eq. (10).As we are primarily interested in the dc component, we set Ω = 0 and present where β = (k B T ) −1 is the inverse temperature.Figure 5 displays our primary result for the gapless Kitaev spin liquid compound, namely, that the dc noise correction generated in the metal is suppressed algebraically as T 4 for the entire region of validity, i.e., for k B T below the flux gap.Equivalently, via the FDT, we can say the dc resistance enhancement in the metal should scale as δR(0, T ) ∼ T 3 when measured utilizing the proposed heterostructure.In Fig. 5 we depict our numerical evaluation of Eq. ( 25) for N u = 5000 unit cells, resulting in a characteristic T 4 suppression of low energy dc noise.This T 4 behavior is physically understandable by recalling that in the low energy regime most of the scattering occurs near the two Dirac points, k ± = (2π/3a)(±1/ √ 3, 1), where the fermion density of states is linear in the energy.From Eq. ( 23), we see that the application of Wick's theorem on the four point correlation functions leads to a product of two fermion Green functions in each term, and therefore to two powers of the fermion density of states.This fact, combined with a density of states for S = 1 particle-hole excitations in the metal also linear in the energy and an integral over all energy, produces an expectation that the overall temperature scaling should be quartic.
C. Spinon Fermi surface coupled to an emergent U(1) gauge field Some QSL candidate materials, e.g., organic salt compounds [34][35][36] and YbMgGaO 4 [69][70][71], may be described as spinon Fermi seas, where the spin susceptibility is strongly renormalized by an emergent U(1) gauge field.In this subsection, we compute, using the Keldysh functional integral formalism, first the thermal spin current fluctuations arising in the metal due to the presence of fermions in the QSL material, and second how the gauge field renormalization affects those thermal fluctuations in our proposed bilayer system at finite temperature.While the finite temperature calculation can also be performed using the more familiar Matsubara formalism (since the system is in thermal equilibrium), we opt for the Keldysh real-frequency calculation to avoid the need for analytic continuation at a later point.
We begin with the real-time action for N flavors of fermionic spinons coupled to a compact U(1) gauge field in 2 + 1 dimensions [35,72,73] where c σ (t, x) is a spinon Grassmann field, σ is the flavor index, m s is the spinon effective mass, µ is the spinon chemical potential, and a(t, x) is the gauge field.The (real-frequency) retarded and advanced propagators for the gauge fluctuations can be obtained in the Coulomb gauge by analytically continuing the result from the Matsubara formalism [72,74] where ξ, ξ label the Cartesian components, F s = k F s /m s is the spinon Fermi velocity, k F s is the spinon Fermi wavevector, E F s is the spinon Fermi energy, and χ d = (24 m s ) −1 is the Landau diamagnetic susceptibility of the fermions.Lastly, we define the interaction portion of the Keldysh action by placing Eq. ( 26) on the Keldysh contour and extracting the term dependent on the gauge field to first order: where η = ± represents the Keldysh time-loop forward and backward branches, ks = k/m s , and A is the total area of the QSL.The spin operators of the QSL are represented using Abrikosov fermions of the form S iσ = (1/2)c † iσ τ σσ c iσ , where σ c † σ c σ = 1 is enforced.Starting from the calculation of the bare bubble, without gauge field renormalization, i.e., the zeroth order calculation using Eq. ( 28) as our perturbation, we can find the uncorrected dc noise enhancement generated across the interface by first calculating the total susceptibility, ii (ν) , given by (29) where on the right hand side + (−) indicates the forward (backward) Keldysh contour, and T K is the Keldysh time ordering operator.Equation ( 29) quantifies the portion of the susceptibility that arises due simply to the presence of the fermions in the QSL material, without the additional corrections from emergent gauge photons, and evaluates to where N is the total number of lattice points and a s is the area occupied by each QSL spin.Therefore we present the ac voltage noise correction due to the fermions, In the dc limit the integral is straightforward, and the zeroth order noise correction becomes

Gauge field correction to the noise
The next non-zero correction to the susceptibility arises due to gauge field renormalization and is given by the second order expansion using Eq. ( 28) as the perturbation, Expanding this gives five possible diagrams, only three of which contribute and must therefore be considered; these three are diagrammatically depicted in Fig. 6.It is important when calculating gauge invariant quantities to sum over all three contributing diagrams, because only then do the divergences cancel exactly [59,74].Accounting for the required diagrams, we take χ (2) (ν ii (ν)] to arrive at the expression where We can then express the second order ac voltage noise correction as Thus we can express the total ac correction to the voltage noise across the heavy metal sample as S s (Ω, T ) = S (0) s (Ω, T ) + δS (2)  s (Ω, T ).We now move into the dc limit, i.e., when Ω → 0, where we can simplify Eq. ( 36) by performing the ν integral, The total correction to the dc voltage noise arising due to fermions coupling to an emergent U(1) gauge field is there-fore Sectioning off the q integral, completing it, and expanding the result for ω E F s , to lowest order in ω we find where the spinon Fermi energy can be expressed in terms of the spinon Fermi temperature as E F s = k B T F s .Finally, performing the ω integral we find that the dc noise correction due to gauge fluctuations for T T F s becomes δS (2)  s (0, T ) = 8.498 Thus we can see from the zeroth and second order calculations that the total dc voltage fluctuations generated across the normal metal sample when interfaced with a QSL material possessing an emergent U(1) gauge field will have a T 3 lowest order temperature dependence modified by a T 4/3 subdominant component.The total dc noise correction generated is given by the sum of the two terms, S s (0, T ) = S (0) s (0, T ) + δS (2)  s (0, T ), such that by Eq. ( 1) the modification to the resistance is when considering the regime T T F s , and we note that T T F s should always pertain in the QSL phase.This serves to enhance the voltage noise generated in the metal contact by the proximity of a quantum magnetic material well characterized by a model consisting of a spinon fermi sea that includes an emergent U(1) gauge field.That a noise enhancement accrues due to the presence of an emergent gauge field is physically understandable as the collective fermionic spin excitations present, i.e., the spinons, acquiring a higher scattering probability due to the added presence of gauge fluctuations.Additional scattering must result in additional noise generation.This enhancement to the voltage noise -or equivalently, via the FDT, enhancement of the resistance measured in the normal metal -should be extricable in the low temperature regime, i.e., T T F s , by first plotting δR(T )/T 2 and then subtracting off the intercept, thus revealing the bare T 4/3 correction term characteristic of the gauge field renormalization.
V. DISCUSSION AND CONCLUSIONS Equation ( 10) is a general relationship that quantifies the noise enhancement present in a normal metal adjacent to an insulating magnet lacking long ranged magnetic order as a result of the coupling to that magnet, i.e., as a result of spin fluctuations across the interface.It connects to resistance through the normal metal in the linear response regime via the FDT, as shown in Eq. ( 1).We assume that for temperatures T < 10 K the base resistance is essentially constant and known in a given strongly spin orbit coupled normal metal, e.g., Pt [75], Ta, and W [76], and as a result can be subtracted off.This exposes a temperature dependent portion of the resistance, δR(T ), that arises due to the presence of the insulating magnet, or more specifically the QSL candidate material.Additionally, many of these models account for quantum fluctuations, e.g., the HKLM, which appear as a constant enhancement to the noise that must be removed as well, in order to isolate the portion that exposes quantities of interest.Then, by sweeping temperature, it should be possible to compare the various QSL candidate materials in order to examine gapped and gapless models, and the low energy temperature scaling of these exotic phases of matter in general.Now we will connect the models we have examined to real materials: herbertsmithite, Kitaev compounds, and κ-type organic compounds.
The compound herbertsmithite, ZnCu 3 (OH) 6 Cl 2 , evinces no long range magnetic order down to temperatures of at least 50 mK, which is four orders of magnitude lower than its exchange coupling J/k B ∼ 200 K [21], suggesting the presence of a QSL phase.Additionally, the Cu 2+ ions form a perfect kagomé lattice, and therefore herbertsmithite is modeled, to a first approximation, by the HKLM.Experimental treatments of herbertsmithite include NMR [11,12], neutron scattering [22,23], and susceptibility [14] studies, most of which indicate that ground state excitations should be gapless.However, numerical DMRG studies show that the ground state of herbertsmithite should be a gapped spin liquid with Z 2 topological order [27], and the more recent NMR measurements indicate a gapped ground state as well [12].Additional complications arise due to the fact that the low energy spectrum appears to be dominated by impurity spins [77] that occur as a result of Cu 2+ ions replacing some non-magnetic Zn 2+ in the transition metal sites between kagomé layers [78].We have shown that our proposed heterostructure can address some of these issues.For instance, the presence of a gap energy will be indicated by a lack of temperature scaling in the resistance measured across the heavy metal layer for T < 10 K. Technical issues notwithstanding, herbertsmithite remains one of the most promising QSL candidate material on offer, and our proposed technique adds a tool for further investigation.
Next, we mention the iridate honeycomb materials [79] and α-RuCl 3 [80] in the context of Kitaev compounds accessible to our bilayer technique.Initially, iridates like α-Na 2 IrO 3 and α-Li 2 IrO 3 were realized to have Ir 4+ ions, with effective j = 1/2, arranged in a honeycomb lattice.While these particular compounds were found to order at low temperatures [81][82][83], this class of materials does exhibit bond-directional Kitaev interactions [84].Thus the discovery of a variant that does not magnetically order, H 3 LiIr 2 O 6 , is interesting [13]; in particular, it would be an excellent test of this material to see if the measured resistance enhancement in a coupled heavy metal film in fact scales as we predict should be the case for Kitaev spin liquid compounds.A second material, α-RuCl 3 , is also known to evince similar structural properties to the iridates, with magnetic Ru 3+ ions forming a honeycomb lattice.As with most iridates, α-RuCl 3 orders at low temperatures [85], though interesting Kitaev spin liquid-like behavior occurs before ordering [24].However, while α-RuCl 3 exhibits Kitaev-like physics prior to freezing, questions have arisen as to whether or not Kitaev physics are actually responsible for that appearance, rather than more conventional physics from the perspective of magnon-like excitations [86].
We have shown that the proposed bilayer could perhaps address this controversy directly, as the presence of Kitaev spin liquid physics in the α-RuCl 3 layer should result in characteristic T 3 temperature scaling of the resistance enhancement in the coupled metal.We therefore believe that our proposed QSL-to-metal bilayer could be helpful to the search for a material exhibiting a true Kitaev spin liquid state.Finally, the primary organic salt candidate QSL material that has been modeled as a spinon fermi sea coupled to an emergent U(1) gauge field is κ-(BEDT-TTF) 2 -Cu 2 (CN) 3 , henceforth κ-ET [34,35].It is an antiferromagnetic weak Mott insulator, where structural dimers possessing a single spin-1/2 degree of freedom arise and form an approximately triangular lattice that becomes geometrically frustrated when cooled.The primary experimental evidence of QSL like behavior comes from thermal and NMR measurements; in the specific heat versus temperature, for example, a large linear term is observed [15][16][17], and NMR measurements show a lack of long range magnetic ordering down to temperatures of 32 mK, approximately 4 orders of magnitude lower than the exchange coupling of J/k B ∼ 250 K [10].The necessity of working at very low temperatures when probing the low energy density of states of κ-ET and similar compounds has caused some doubt as to the reliability of specific heat measurements, however, due to the difficulty of properly accounting for nuclear contributions.We have shown that in its QSL state, a compound that can be modeled as a spinon fermi sea coupled to a U(1) gauge field should evince a temperature dependent correction to the resistance measured across a coupled heavy metal that goes as T 4/3 , which would therefore shed light on the low energy density of states directly.The heterostructure proposed here would avoid the issue of contamination by signals due to nuclear spins entirely.
In conclusion, we have proposed a heterostructure composed of a QSL candidate material overlaid with a strongly spin orbit coupled heavy metal film as a viable probe of QSL ground states that can perhaps alleviate some of the currently outstanding controversies.The theory we have advanced here indicates that an equilibrium measurement, i.e., a measurement taken within the linear response regime, of the dc resistance across the normal metal layer will provide information about the low energy density of states of the QSL material under observation.The system we propose will have wide ranging applicability to probing insulating quantum magnets in general, and should provide an interesting method of examining and categorizing QSL candidate materials in particular.We show that the proposed bilayer should be able to extract any gap energy present in the HKLM, that the Kitaev model in the spin liquid phase should evince a characteristic T 3 temperature dependence of the electrical resistance across the coupled metal layer, and that fermions coupled to an emergent U(1) gauge field should see a sub-dominant temperature dependent correction to the resistance that goes as T 4/3 .In future works, it would be interesting to expand this analysis to include extensions to the QSL models such as Dzyaloshinskii-Moriya interactions, or Heisenberg and Γ terms in the Kitaev model, and especially to include disorder in the analysis, as disorder seems to be highly influential when attempting to discriminate between QSL and non-QSL ground states.
Here, ρ is the base electric resistivity of the metal, ρ 1 is the portion of the resistivity arising due to the presence of the insulating material, the spin Hall angle of the metal is θ S H , the spin diffusion length is λ, the thickness of the metal layer is d N , the electric conductivity of the metal is σ, and the real part of the spin mixing conductance is G r .Another approach, however, models the same effect utilizing the FDT and noise.Importantly, note that using the FDT, no assumptions need be made with respect to the specific spin current conversion process occurring in the Pt layer; it assumes simply that spin to charge current conversion takes place.That this method achieves the same result was shown by Kamra et al. using a bilayer formed of YIG and Pt, where the magnetization angle in the YIG layer measurably affected the resistivity of the Pt layer in equilibrium [56].Their findings confirmed that the spin-current based physics of SMR provided earlier in fact implied the existence of spin Hall noise, and that these processes could be understood directly in equilibrium using the FDT.In the same situation, that is, where the imaginary portion of the spin mixing conductance vanishes, Ref. 56 quotes-in their Eq.( 15)-a similar result to our Eq.(A1): where gr is the modified spin mixing conductance.We can combine terms in Eq. (A1) such that Gr = G r 1 + 2 λG r σ coth d N λ −1 , note that ρ = σ −1 , and write gr = 4π Gr /2e 2 , which in combination make it clear that Eq. (A1) and Eq.(A2) are in fact identical.Thus there exists a correspondence between SMR from a spin-current perspective and from the perspective of the FDT.
Appendix B: HKLM Calculation The three basis vectors for the unit cell shown in Fig. 2 are: where a is the lattice spacing around the equilaterial triangle.These unit vectors allow us to Fourier transform the Hamiltonian into momentum space, where we write the momenta as k i = êi • k for i = 1, 2, 3. Then from Eq. ( 13) the Hamiltonian for a single unit cell becomes noting that because we have three sites in the unit cell, C is a three by three matrix and therefore H is a six by six matrix.The matrix C is traceless and has components: where Q 1 and Q 2 are the two distinct expectation values of Q i j (see Fig. 2).We then diagonalize the coefficient matrix such that H = UDU † and Eq.(B2) becomes where the rotated boson operators, γ(k) = U † Ψ(k), are required to obey the typical bosonic commutation rules.This is the form of the Hamiltonian we use to derive Eq. ( 14) of the main text.
H S 7 p p i J U e m P j n 2 v X P / r 4 k 0 8 / u / F 5 4 4 s v v / r 6 m 5 u 3 v j 1 U yV B 6 / M B L R C K P X K a 4 C G N + o E M t + F E q O Y t cw d + 6 p 4 8 N / v a c S x U m 8 b 4 e p f w 4 Y o M 4 7 I c e 0 8 b k c M 1 O b j Y 3 1 j f y j 0 1 P W u V J 0 y o / r 0 5 u f f + 3 4 y f e M O K x 9 g R T q t f a S P X x m E k d e o J n D W e o e M q 8 U z b g P

FIG. 2 .
FIG. 2. (Color online) The kagomé lattice, with positions on a representative unit cell marked.The êi represent the unit vectors of a unit cell, and a is the length of a bond.Arrows between sites correspond to the direction dependent Q i j .Solid arrows around upward facing triangles depict Q i j = Q 1 , and wire arrows around downward facing triangles depict Q i j = Q 2 .

FIG. 3 .
FIG.3.A numerical plot showing ln(S s (0, T ) − S s (0, 0)) against inverse temperature, 1/T , in order to expose the spin gap in the HKLM.Note that we have subtracted off quantum fluctuations.The solid line is the computed sum, while the dashed line is a linear fit to the low temperature data.The slope of the fit serves as an estimate of the gap energy, and gives −0.054J.

FIG. 4 .
FIG. 4. (Color online) The Kitaev honeycomb lattice, where the two sub-lattices are shown by the purple (sub-lattice A) and orange (sublattice B) dots.Representative x-, y-, and z-links are depicted.The flux operator on plaquette p, W p , is shown, n 1 and n 2 are the primitive lattice vectors, and a is the lattice spacing.Also depicted is a representative unit cell enclosed in the dashed box.

FIG. 5 .
FIG.5.The dc noise correction obtained when applying our theory to the gapless Kitaev spin liquid model in the isotropic limit.The plot is our numerical output computed using N u = 5000 unit cells, with the result that S s (0, T ) ∝ T 4 .

1 .
Noise correction due to bare susceptibility

FIG. 6 .
FIG. 6.Three diagrams contributing to the O(1/N) correction to the spin susceptibility generated by the presence of U(1) gauge field fluctuations.
B2) with Ψ(k) = [b 1 (k), b 2 (k), b 3 (k), b † 1 (−k), b † 2 (−k), b † 3 (−k)]T the vector of particle-hole boson operators for each of the three sites of the unit cell, H representing the coefficient matrix, and the momentum now summed over the first Brillouin zone.The coefficient matrix is