Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface

Significance

In a quantum spin liquid, the spins remain disordered down to zero temperature, and yet, it displays topological order that is stable against local perturbations. The Kitaev model with anisotropic interactions on the bonds of a honeycomb lattice is a paradigmatic model for a quantum spin liquid. We explore the effects of a magnetic field and discover an intermediate gapless spin liquid sandwiched between the known gapped Kitaev spin liquid and a polarized phase. We show that the gapless spin liquid harbors fractionalized neutral fermionic excitations, dubbed spinons, that remarkably form a Fermi surface in a charge insulator.

Abstract

The Kitaev model with an applied magnetic field in the $H\parallel \left[111\right]$ direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at $H_{c1}\simeq 0.2K$ and a second transition at a higher field $H_{c2}\simeq 0.35K$ to a gapped partially polarized phase, where $K$ is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function $S\left(\mathbf{k}\right)$ and the Fermi surface ${ϵ}_F^S\left(\mathbf{k}\right)$ of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.

The inspired exact solution of the Kitaev model on a two-dimensional honeycomb lattice with bond-dependent spin exchange interaction (1) (Eq. 1) has emerged as a paradigmatic model for quantum spin liquids (QSLs). For equal bond interactions ($K$), the ground state of the Kitaev model is known to be a topologically nontrivial gapless QSL with fractionalized excitations. The multiple ground-state degeneracy reflects the topology of the lattice manifold (twofold degeneracy on a cylinder and fourfold degeneracy on a torus). The promise of using fractionalized excitations for applications in robust quantum computing (1–5) has led to considerable excitement and activity in the field from diverse directions, all of the way from fundamental developments to applications. On breaking time-reversal symmetry (for example, by applying a magnetic field), the exact solvability of the Kitaev model is lost, but perturbatively, it can be shown that the Kitaev gapless QSL becomes a gapped QSL phase with nonabelian anyonic excitations (1).

To realize the exotic properties of the Kitaev model, there has been keen interest to discover Kitaev physics in materials. To this end, Mott insulators with magnetic frustration and large spin-orbit coupling have been proposed (6), leading to the discovery of $\alpha -{\mathrm{R}\mathrm{u}\mathrm{C}\mathrm{l}}_3$ and ${\mathrm{A}}_2{\mathrm{I}\mathrm{r}\mathrm{O}}_3$ (A = Na, Li). These are candidate QSL materials with the desired honeycomb geometry. Although given additional interactions beyond the Kitaev interaction in materials, there is still controversy about the regimes where Kitaev physics can be accessed (7–12). Additionally, triangular and Kagome lattice with spin frustration has also been proposed as a candidate for a QSL (13–19). In this context, organics, such as $\kappa$-(BEDT-TTF)₂Cu₂ (CN)₃ (20) and EtMe₃Sb_[Pd(dmit)2]2 (21), the transition metal dichalcogenides 1T-${\mathrm{T}\mathrm{a}\mathrm{S}}_2$ (19), and a large family of rare earth dichalcogenides ${\mathrm{A}\mathrm{R}\mathrm{e}\mathrm{X}}_2$ (A = alkali or monovalent ions, Re = rare earth, X = O, S, Se) (22) have been explored. Remarkably, recent NMR and thermal Hall conductivity experiments on $\alpha -{\mathrm{R}\mathrm{u}\mathrm{C}\mathrm{l}}_3$ demonstrate that one can drive the magnetically ordered phase into a QSL phase using an external magnetic field (23, 24). To this end, various numerical tools were used to explore the possible QSL phases in the models relevant to candidate honeycomb materials with an external magnetic field (25–29).

In this article, we theoretically address the following question: what is the fate of the Kitaev QSL with increasing magnetic field (Eq. 1) beyond the perturbative limit? Previous studies, using a variety of numerical methods (30–33), have pushed the Kitaev model solution to larger magnetic fields outside the perturbative regime. At high magnetic fields, one would expect a polarized phase. What is surprising is the discovery of an intermediate phase sandwiched between the gapped QSL at low fields and the polarized phase at high fields when a uniform magnetic field is applied along the [111] direction (Fig. 1C). The main question that we focus on is as follows: what is the nature of this intermediate phase? A study based on exact diagonalization on small clusters claims to get an intermediate QSL phase with a stable spinon Fermi surface (SFS) (30). An independent analysis based on the central charge and symmetry arguments also deduce an SFS (34, 35). In this article, we present exact calculations of the spin structure factor $S\left(\mathbf{k}\right)$ on large systems and reconstruct the SFS ${ϵ}_F^S\left(\mathbf{k}\right)$ in the intermediate phase, details of which are distinct from other recent proposals. These are the primary significant results of this work.

Fig. 1.

(A) Spin susceptibility ($\chi$) for $L_1=\mathrm{4,16}$ and (B) different measures of magnetization as a function of magnetic field strength ($H$) for $L_1=16$. The vertical dashed lines show critical magnetic fields where phase transitions occur. (C) Corresponding phase diagram that shows transitions from gapless QSL to gapped QSL to an intermediate gapless QSL with an SFS into a PPM phase ($H=0$ to large $H$, respectively). (D) Average spin-spin correlations between sites at a fixed distance of $R$. The dashed black line represents the power law decay with power −1.5 as a guide to the eye. All results were obtained using DMRG at zero temperature.

Model and Method

We consider the Kitaev model on a honeycomb lattice in the presence of an external magnetic field along the [111] direction. The model Hamiltonian is defined as

$$H_K=K\sum _{\gamma =x,y,z}\sum _{{⟨ij⟩}_{\gamma }}{S}_i^{\gamma }{S}_j^{\gamma }-\sum _i\mathbf{H}\cdot {\mathbf{S}}_i,$$ [1]

where $\gamma =x,y,z$ is the nearest neighbor bond index of each site and $S^{\gamma }$ is the $\gamma$ projection of spin-$1/2$ degrees of freedom on each site. The first term is the Kitaev model where $K=1$ eV, and the second term represents Zeeman field with a uniform magnetic field $\mathbf{H}=\left|H\right|\left({\widehat{e}}_x+{\widehat{e}}_y+{\widehat{e}}_z\right)$ (in units of $K$) applied in the $\left[111\right]$ direction, i.e., perpendicular to the two-dimensional honeycomb lattice. This model is solved using density matrix renormalization group (DMRG) and Lanczos on a lattice with $L_1\times L_2$ number of unit cells and with two sites per unit cell labeled as $A$ or $B$ (the total number of sites equals $2L_1{L}_2$) (see Fig. 3A). All DMRG simulations are performed with fixed $L_1=16$ and cylindrical boundary conditions using up to 1,200 DMRG states per block, ensuring the truncation error $\le 5\times 10^{-6}$ for $L_2\le 5$ ($\le 10$ sites in ${\mathbf{a}}_{\mathbf{2}}$ direction) within the two-site grand canonical DMRG (36–43). Additionally, we also use finite-temperature Lanczos method (FTLM) for calculations of specific heat and thermodynamic entropy at finite temperatures $T$ (in units of $K$) on small clusters (39, 44). SI Appendix presents definitions of all observables and convergence analysis of DMRG and FTLM calculations.

Fig. 3.

The spin structure factor and deduced SFS in the intermediate phase. A shows the honeycomb lattice with cylindrical boundary conditions with L₂ = 3, 4, 5 (6, 8, 10 sites in the ${\mathbf{a}}_{\mathbf{2}}$ direction), shown here for $L_2=5$. The unit vectors are ${\mathbf{a}}_{\mathbf{1}}=\left(\mathrm{0,1}\right)$ and ${\mathbf{a}}_{\mathbf{2}}=\left(0.5,\sqrt{3}/2\right)$. The spin structure factor $S_{tot}\left(\mathbf{k}\right)$ in C shown along the crystal momentum cuts in B within the extended BZ at fixed $H=0.3$ in the intermediate phase. These cuts are defined using the reciprocal lattice vectors ${\mathbf{k}}_{\mathbf{1}}=\left(2\pi ,-2\pi /\sqrt{3}\right)$ and ${\mathbf{k}}_{\mathbf{2}}=\left(0,4\pi /\sqrt{3}\right)$, where circles, squares, and triangles represent cuts obtained using L₂ = 3, 4, 5, respectively. The high-symmetry points of the BZ and extended BZ are also labeled in B. D shows a contour plot of $S_{tot}\left(\mathbf{k}\right)$, with blue representing low intensity and red representing high intensity. The diagonal lines are special cuts $C_1,C_2$, and $C_3$ defined in C. (E) The peak locations (red points) of the intrasublattice structure factor $S_{tr}\left(\mathbf{k}\right)$, where gray circles at $M$ points represent large peaks. (F) Proposed SFS $A\left(\mathbf{k},\omega =0\right)=\delta \left(\omega -{ϵ}_F^S\left(\mathbf{k}\right)\right)$ with pockets around $\mathrm{\Gamma }$ and $M$ points, assuming that sublattices $A$ and $B$ are equivalent as in the Kitaev model. (G) Scattering spectrum of the proposed SFS calculated using $\mathcal{F}\left(\mathbf{k}\right)\sim {\sum }_{\mathbf{q}}A\left(\mathbf{q},\omega =0\right)A\left(\mathbf{q}+\mathbf{k},\omega =0\right)$. The proposed SFS reproduces robust peaks at $M$ points of the BZ and circular patterns around the $\mathrm{\Gamma }$ points, similar to the DMRG results in E.

Results

We begin with the magnetic and thermodynamic properties of the model. Fig. 1 shows the spin susceptibility ($\chi =\partial S_{tot}/\partial H$) and various measures of magnetization as a function of field strength $H$. The clear two-peak structure of $\chi$ demonstrates the presence of two-phase transitions at finite field strengths. The corresponding magnetization also shows kinks at the critical fields $H\simeq 0.2$ and 0.35 (dashed lines in Fig. 1 A and B). The zero field limit is known to be a QSL with a gapless energy spectrum and fourfold topological ground-state degeneracy that is associated with a symmetry-protected topological phase (1). At finite fields $H\lesssim 0.2$, the ground state is twofold degenerate on a cylinder, and the energy spectrum becomes gapped (1, 32). The gapped phase is also understood to be a $p+ip$ state of the emergent fermions. For high fields $H\gtrsim 0.35$, we find a partially polarized magnetic (PPM) phase, where the magnetization monotonically increases toward its saturation value ($M_{111}=1.5$) in the trivially field-polarized product state. Remarkably, the two-peak structure of $\chi$ indicates that an intermediate phase is sandwiched between the gapped Kitaev QSL and polarized phases (Fig. 1C).

What are the properties of the intermediate phase? We study the decay of spin-spin correlations $C\left(R\right)$ shown as a function of the distance $R$ between the two spins in Fig. 1D. As expected, in the gapped nonabelian QSL phase ($H=0.15$), the $C\left(R\right)$ decays exponentially, consistent with a finite gap to flux excitations (4, 30, 33, 45). In the PPM phase, the behavior of $C\left(R\right)$ is consistent with an exponential decay due to short-ranged spin-spin correlations arising from the energy cost for a spin flip that is proportional to $H$. However, in the intermediate regime, the behavior of $C\left(R\right)\propto R^{-m}$ is approximately fit as a power law with $m=1.5$ shown by the dashed black line in Fig. 1D. SI Appendix shows more detained fitting analysis of $C\left(R\right)$, where $m$ is weakly dependent on the field in the intermediate phase.

We provide further evidence of the gapless nature of excitations in Fig. 2 through the temperature-dependent specific heat $C_v\left(T\right)$ and the thermodynamic entropy $S\left(T\right)$. We capture the characteristic two-peak structure of $C_v$ and two hump structures in the entropy at zero field. As expected, $S$ saturates to $\ln \left(2\right)$ at large temperatures for all field values. The two-peak/hump feature is also found in the gapped QSL phase ($H=0.15$), where low-temperature peak/hump shifts to even lower temperatures compared with $H=0$. The behavior of the low-temperature $S\left(T\right)$ and $C_v\left(T\right)$ is consistent with a gapped spectrum both at low and high fields. However in the intermediate phase, the peak at low $T/K$ is suppressed, leading to an algebraic behavior that is distinct compared with both gapped phases. In fact, the low-temperature $C_v$ at $H=0.33$ shows linear behavior in temperature (shown in SI Appendix with a linear temperature scale) that is consistent with a gapless phase. As the FTLM is limited to small system sizes, severe size effects prevent us from obtaining an exact dependence of $C_v$ on temperature, and therefore, we only discuss the qualitative trends. The analysis of the statistical error in calculations of $C_v$ and $S$ is also shown in SI Appendix along with $C_v$ at low temperatures. Finally, with increasing $H$ across $H_{c2}\simeq 0.35$ into the PPM phase ($H=0.45$), the two peaks in $C_v$ merge, and similarly, only one hump feature is visible in the entropy.

Fig. 2.

Specific heat $C_v\left(T\right)$ and thermodynamic entropy $S\left(T\right)$ as a function of temperature $T$. Shown results are using a $3\times 3$ unit cell (18 sites) cluster solved using FTLM averaged over 50 different random runs (39, 44) for each fixed magnetic field strength $H$ (key label).

We also analyze the low-temperature missing part of the entropy ($S_{low}$) resulting from finite size effects (shown in SI Appendix). The $H=0$ and gapped phases have a quadratic increase in $S_{low}$ with temperature. On the contrary, the intermediate phase shows almost a linear behavior of $S_{low}$ at low temperatures. This is consistent with our earlier arguments of gapless nature of the intermediate phase. To summarize, we observe the following features in the intermediate phase: (i) a slow power law decay of the $C\left(R\right)$ (Fig. 1D) that is indicative of a gapless phase and (ii) a $C_v\left(T\right)$ consistent with linear behavior at low $T$ indicating a presence of itinerant emergent fermions. All of this evidence taken together provides strong evidence for an intermediate phase that is indeed gapless. Additionally, recently calculated $T=0$ spin dynamics on small systems also show a continuum in energy, suggesting that the intermediate phase is indeed a $U\left(1\right)$ gapless QSL (30). In fact, such gapless spin liquid phase on a triangular lattice was recently shown to have stable SFS (46, 47). Therefore, it is likely that even in our case, on a honeycomb lattice, we may have an SFS as discussed before.

To explore this possibility, we calculate the spin structure factor along various cuts of the Brillouin zone (BZ) in Fig. 3. For two sites per unit cell ($A$ and $B$), we define structure factor using Fourier transform of inter- and intrasublattice spin-spin correlations. The calculation of (trace) $S_{tr}\left(\mathbf{k}\right)$ is performed using only intrasublattice correlations, while (total) $S_{tot}\left(\mathbf{k}\right)$ sums over both inter- and intrasublattice correlations (SI Appendix). Note that the sublattices $A$ and $B$ are located on different positions within a unit cell (unlike orbitals), and therefore, $S_{tot}\left(\mathbf{k}\right)$ must be considered in the extended BZ scheme (Fig. 3 B–D), while the $S_{tr}\left(\mathbf{k}\right)$ may be considered only using the first BZ (Fig. 3E). In short, $S_{tr}\left(\mathrm{\Gamma }\right)=S_{tr}\left({\mathrm{\Gamma }}_e\right)$, but $S_{tot}\left(\mathrm{\Gamma }\right)\ne S_{tot}\left({\mathrm{\Gamma }}_e\right)$, where $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_e$ refer to high-symmetry points in the first and extended BZs, respectively, labeled in Fig. 3B. We shall use subscript e to distinguish between high-symmetry points of the extended BZ vs. the first BZ.

In Fig. 3 C and D, we show cuts and contour of the $S_{tot}\left(\mathbf{k}\right)$ at fixed $H=0.3K$ in the intermediate phase [SI Appendix has cuts of the $S_{tr}\left(\mathbf{k}\right)$]. There is suppression of intensity around the $\mathrm{\Gamma }$ point and peaks around the $M_e$ points. The cuts between $C_2$ and $C_3$ pass through two $M_e$ points, showing robust peaks visible in Fig. 3C, green curve. $S_{tot}\left(\mathbf{k}\right)$ at all of the $M_e$ points is distinctly visible as high-intensity red circles in Fig. 3D. Apart from the total spin structure factor $S_{tot}\left(\mathbf{k}\right)$, we also analyze the peak locations of the trace spin structure factor $S_{tr}\left(\mathbf{k}\right)$ that respects symmetries within the first BZ. Fig. 3E shows large peaks at the $M$ points of the first BZ, with soft peaks around the $K$ and $K\prime$ points forming three petal-shaped patterns and peaks around $\mathrm{\Gamma }$ forming a circle. In fact, comparing Fig. 3 D and E, it can be observed that the dominant peaks at the $M$ points in $S_{tr}\left(\mathbf{k}\right)$ simply shift to the $M_e$ points when considering the $S_{tot}\left(\mathbf{k}\right)$ that also retains the intersublattice correlations.

Given the information provided by $S_{tot}\left(\mathbf{k}\right)$ and $S_{tr}\left(\mathbf{k}\right)$, we propose the corresponding SFS ${ϵ}_F^S\left(\mathbf{k}\right)$ with pockets around $\mathrm{\Gamma }$ and $M$ points of the BZ (Fig. 3F). Using the spinon spectral function, defined by $A\left(\mathbf{k},\omega \right)=\delta \left(\omega -{ϵ}_F^S\left(\mathbf{k}\right)\right)$, we construct the scattering function in Fig. 3G, $\mathcal{F}\left(\mathbf{k}\right)\sim {\sum }_{\mathbf{q}}A\left(\mathbf{q},\omega =0\right)A\left(\mathbf{q}+\mathbf{k},\omega =0\right)$, that picks up the momentum transfer $\mathbf{k}$ from $\mathbf{q}$ to $\mathbf{q}+\mathbf{k}$ scattering across the Fermi surface with energy transfer $\omega =0$. The interpocket scattering $\mathrm{\Gamma }\leftrightarrow M$ and $M\leftrightarrow M$ across the SFS (Fig. 3F) leads to a large joint density of states with momentum transfer $\mathbf{k}=M$ that results in peaks at the $M$ points (Fig. 3G), consistent with our DMRG calculations of $S_{tr}\left(\mathbf{k}\right)$ in the intermediate phase (Fig. 3E). Other large circular features in $\mathcal{F}\left(\mathbf{k}\right)$ arise from similar scattering processes: for example, a spinon with momentum $\mathbf{q}$ around the $\mathrm{\Gamma }$ pocket can scatter to all $\mathbf{q}+\mathbf{k}$ points within the pockets at the $M$ points. The combination of these large circular features leads to the three-petal patterns also found in $S_{tr}\left(\mathbf{k}\right)$ at the $K/K\prime$ points (Fig. 3E). In summary, our proposed SFS is able to qualitatively capture even subtle features of the $S_{tr}\left(\mathbf{k}\right)$ in the intermediate phase, providing strong evidence for a QSL with an SFS shown in Fig. 3F.

We next turn to the effect of a magnetic field on the spin structure factor. Fig. 4 shows $S_{tot}\left(\mathbf{k}\right)$ for $C_1$ cut and $C_2$ cut at various field strengths $H$. Note that, for zero fields gapless and $H\lesssim 0.2$ gapped QSL phases, there is a broad feature that is consistent with short-ranged correlations established by an exponential decay of spin-spin correlations in real space (Fig. 1C). Our result is also consistent with previous calculations of peaks in the dynamical structure factor $S_{tot}\left(\mathbf{k},\omega \right)$ arising from plaquette flux and Majorana excitations (33) that, on integrating $S_{tot}\left(\mathbf{k}\right)={\int }_0^{\infty }d\omega S_{tot}\left(\mathbf{k},\omega \right)$, give the broad feature in Fig. 4.

Fig. 4.

$S_{tot}\left(\mathbf{k}\right)$ for various field strengths along (A) cut $C_1$ and (B) cut $C_2$ (defined in Fig. 3D). The peaks at $M_e$ points are most robust for the intermediate phases $H=0.26,0.30$, and 0.33. The peak/hump-like feature close to $K$ points monotonically decreases with increasing $H$.

The intermediate phase with long-range correlations shows robust peaks at $M_e$ (Fig. 4A). Additionally, Fig. 4B demonstrates that the plateau at momentum point $K$ monotonically decreases with an increasing magnetic field. On the other hand, $S_{tot}\left(\mathbf{k}=M_e\right)$ increases with the field, reaching its maximum value in the intermediate phase and decreasing again on transitioning into PPM phase. All peaks get suppressed in the PPM phase, as expected, because of shorter-ranged spin-spin correlations (Fig. 1D). Overall, the combined results of Figs. 3 and 4 show that the dominant peaks of the spin structure factor $S\left(\mathbf{k}\right)$ at the $M$ points exist only in the intermediate gapless phase.

Discussion

Several conflicting values of the central charge $c$ have been reported for the intermediate phase, where arguments based on the central charge were used to deduce the SFS (33–35). However, it seems to be very difficult to obtain a reliable value for $c$ unambiguously in a gapless phase using DMRG. Ref. 34 proposes c = 1, 0 using L₂ = 3, 4, respectively; ref. 33 finds a $c=4$ using $L_2=5$; and ref. 35 calculates c = 1, 2 using L₂ = 2, 3, respectively, using DMRG. Based on the central charge arguments, the SFS was argued to have pockets around the $K/K\prime$ and $\mathrm{\Gamma }$ points of the first BZ (34, 35). This clearly differs from our proposal of the SFS with pockets around $\mathrm{\Gamma }$ and $M$ points; our results are also consistent with the exact diagonalization results of the dynamical structure factor on small clusters (30). We further emphasize that the total $S\left(\mathbf{k}\right)$ must respect the symmetries within the extended BZ because of two atoms in the unit cell at different locations for a honeycomb lattice (Fig. 3A), a point that seems to have been ignored in the literature.

We note that the $S\left(\mathbf{k}\right)$ surfaces shown in Fig. 3 D and E cannot result from the tight-binding model with uniform hopping on a honeycomb lattice (the graphene problem). This is because any scattering $K\leftrightarrow K\prime$ will only lead to $S\left(\mathbf{k}\right)$ with peaks at the $K$ and $K\prime$ points, which are not present in our DMRG calculations of $S\left(\mathbf{k}\right)$ in the intermediate phase. To have peaks at the $M$ point in $S\left(\mathbf{k}\right)$, as we have found in our calculations, there must be pockets around the $M$ points of the SFS. Note that all of the $6\hspace{0.17em}M$ points in the BZ are related by $C_3$ rotation and momentum translation symmetries and therefore, must be identical. Such a SFS with $M\leftrightarrow M$ scattering will produce a peak with momentum transfer around the $M$ point in $S\left(\mathbf{k}\right)$. Moreover, the size of pockets on the SFS is picked such that more subtle three petal-shaped patterns around the $K/K\prime$ points (Fig. 3E) are also captured in $\mathcal{F}\left(\mathbf{k}\right)$ (Fig. 3G). In the graphene problem, the Dirac points at the $K/K\prime$ are protected by inversion and time-reversal symmetry. However, the magnetic field breaks time-reversal symmetry, allowing for the possibility of shifting the pocket positions away from the $K/K\prime$ points. Additionally, we expect the largest scattering contribution to occur between a particle and hole pocket. In summary, we propose an SFS with two bands, creating Fermi pockets at the $\mathrm{\Gamma }$ point and $M$ with opposite particle and hole character (Fig. 3F).

The procedure described above for the reconstruction of the SFS relies on the assumption that the $\omega -$integrated $S_{tot}\left(\mathbf{k}\right)\simeq S_{tot}\left(\mathbf{k},\omega =0\right)$ is well described by the dynamical correlations at low energies, also used previously to propose an SFS in 1T-${\mathrm{T}\mathrm{a}\mathrm{S}}_2$ (46). In the Kitaev model (at zero field), indeed most of the weight in $S_{tot}\left(\mathbf{k},\omega \right)$ is concentrated at small $\omega$ (4). We test this assumption in the new QSL phase by calculating the numerically challenging dynamical spin structure factor (48, 49) at low energies $S_{tot}\left(\mathbf{k},\omega =0\right)$ on $8\times 3$ unit cell (48 sites) clusters. We find that the dynamical spin structure factor also shows peaks at the $M$ points for $\omega =0$ (SI Appendix), justifying our method to reconstruct the SFS.

We expect the emergent neutral fermions that form a Fermi surface in the gapless QSL phase to show quantum oscillations in a magnetic field, similar to observations of quantum oscillations in ${\mathrm{S}\mathrm{m}\mathrm{B}}_6$, a topological Kondo insulator (50–52). Going forward, it would also be useful to dope the different classes of Mott insulators that could harbor QSL ground states to explore the emergent superconducting phases (53).

Conclusion

Our most significant finding is that of an intermediate gapless QSL with an SFS sandwiched between the well-known gapped nonabelian Kitaev spin liquid at low magnetic fields and a partially polarized phase at high magnetic fields, studied here for a field along the [111] direction. The two quantum-phase transitions are revealed by kinks in the magnetization and peaks in the susceptibility. The gapless nature of the intermediate phase $0.2\lesssim H\lesssim 0.35$ is indicated by the slow power law decay of spin-spin correlations as opposed to an exponential decay in the other two phases. The temperature dependence of the thermodynamic quantities, specific heat and entropy, also corroborates the gapless nature of this intermediate phase.

Finally, using large-scale DMRG and detailed analysis along many cuts of the spin structure factor $S\left(\mathbf{k}\right)$ in momentum space, we propose an SFS in the intermediate phase with pockets at the $\mathrm{\Gamma }$ and $M$ points with opposite particle hole character. We expect our findings to encourage the search for QSLs in materials with Kitaev interactions via spectroscopy and thermodynamic measurements.