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Magnetic Field-Induced Intermediate Quantum Spin Liquid with a Spinon Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface Niravkumar D. Patela and Nandini Trivedia,1 aDepartment of Physics, The Ohio State University, Columbus, OH 43210 Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved April 26, 2019 (received for review December 15, 2018) The Kitaev model with an applied magnetic field in the H k [111] In this article, we theoretically address the following question: direction shows two transitions: from a nonabelian gapped quan- what is the fate of the Kitaev QSL with increasing magnetic field tum spin liquid (QSL) to a gapless QSL at Hc1 ' 0:2K and a second (Eq. 1) beyond the perturbative limit? Previous studies, using a transition at a higher field Hc2 ' 0:35K to a gapped partially variety of numerical methods (30–33), have pushed the Kitaev polarized phase, where K is the strength of the Kitaev exchange model solution to larger magnetic fields outside the perturbative interaction. We identify the intermediate phase to be a gap- regime. At high magnetic fields, one would expect a polarized less U(1) QSL and determine the spin structure function S(k) and phase. What is surprising is the discovery of an intermediate S the Fermi surface F (k) of the gapless spinons using the density phase sandwiched between the gapped QSL at low fields and the matrix renormalization group (DMRG) method for large honey- polarized phase at high fields when a uniform magnetic field is comb clusters. Further calculations of static spin-spin correlations, applied along the [111] direction (Fig. 1C). The main question magnetization, spin susceptibility, and finite temperature-specific that we focus on is as follows: what is the nature of this inter- heat and entropy corroborate the gapped and gapless nature of mediate phase? A study based on exact diagonalization on small the different field-dependent phases. In the intermediate phase, clusters claims to get an intermediate QSL phase with a stable the spin-spin correlations decay as a power law with distance, spinon Fermi surface (SFS) (30). An independent analysis based indicative of a gapless phase. on the central charge and symmetry arguments also deduce an SFS (34, 35). In this article, we present exact calculations of the quantum spin liquid j fractionalization j Kitaev model j spinon spin structure factor S(k) on large systems and reconstruct the PHYSICS S SFS F (k) in the intermediate phase, details of which are distinct from other recent proposals. These are the primary significant he inspired exact solution of the Kitaev model on a two- results of this work. Tdimensional honeycomb lattice with bond-dependent spin exchange interaction (1) (Eq. 1) has emerged as a paradigmatic Model and Method model for quantum spin liquids (QSLs). For equal bond interac- We consider the Kitaev model on a honeycomb lattice in the K tions ( ), the ground state of the Kitaev model is known to be presence of an external magnetic field along the [111] direction. a topologically nontrivial gapless QSL with fractionalized excita- The model Hamiltonian is defined as tions. The multiple ground-state degeneracy reflects the topology of the lattice manifold (twofold degeneracy on a cylinder and X X γ γ X HK = K S S − H · Si , [1] fourfold degeneracy on a torus). The promise of using fraction- i j γ = x,y,z hij i i alized excitations for applications in robust quantum computing γ (1–5) has led to considerable excitement and activity in the field where γ = x, y, z is the nearest neighbor bond index of each site from diverse directions, all of the way from fundamental devel- and S γ is the γ projection of spin-1=2 degrees of freedom on opments to applications. On breaking time-reversal symmetry each site. The first term is the Kitaev model where K = 1 eV, (for example, by applying a magnetic field), the exact solvabil- and the second term represents Zeeman field with a uniform ity 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). Significance To realize the exotic properties of the Kitaev model, there has been keen interest to discover Kitaev physics in materials. To this In a quantum spin liquid, the spins remain disordered down to end, Mott insulators with magnetic frustration and large spin- zero temperature, and yet, it displays topological order that orbit coupling have been proposed (6), leading to the discovery is stable against local perturbations. The Kitaev model with of α − RuCl3 and A2IrO3 (A = Na, Li). These are candidate anisotropic interactions on the bonds of a honeycomb lattice QSL materials with the desired honeycomb geometry. Although is a paradigmatic model for a quantum spin liquid. We explore given additional interactions beyond the Kitaev interaction in the effects of a magnetic field and discover an intermediate materials, there is still controversy about the regimes where gapless spin liquid sandwiched between the known gapped Kitaev physics can be accessed (7–12). Additionally, triangular Kitaev spin liquid and a polarized phase. We show that the and Kagome lattice with spin frustration has also been proposed gapless spin liquid harbors fractionalized neutral fermionic as a candidate for a QSL (13–19). In this context, organics, such excitations, dubbed spinons, that remarkably form a Fermi as κ-(BEDT-TTF)2Cu2 (CN)3 (20) and EtMe3Sb[Pd(dmit)2]2 surface in a charge insulator. (21), the transition metal dichalcogenides 1T-TaS2 (19), and a large family of rare earth dichalcogenides AReX2 (A = alkali Author contributions: N.T. designed research; N.D.P. performed research; and N.D.P. and or monovalent ions, Re = rare earth, X = O, S, Se) (22) have N.T. wrote the paper.y been explored. Remarkably, recent NMR and thermal Hall con- The authors declare no conflict of interest.y ductivity experiments on α − RuCl3 demonstrate that one can This article is a PNAS Direct Submission.y drive the magnetically ordered phase into a QSL phase using an Published under the PNAS license.y external magnetic field (23, 24). To this end, various numerical 1 To whom correspondence may be addressed. Email: [email protected] tools were used to explore the possible QSL phases in the mod- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. els relevant to candidate honeycomb materials with an external 1073/pnas.1821406116/-/DCSupplemental.y magnetic field (25–29). www.pnas.org/cgi/doi/10.1073/pnas.1821406116 PNAS Latest Articles j 1 of 5 Downloaded by guest on September 25, 2021 in the gapped nonabelian QSL phase (H = 0:15), the C (R) decays exponentially, consistent with a finite gap to flux excita- tions (4, 30, 33, 45). In the PPM phase, the behavior of C (R) 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 (R) / 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 (R), where m is weakly dependent on the field in the intermediate phase. We provide further evidence of the gapless nature of excita- tions in Fig. 2 through the temperature-dependent specific heat Cv (T ) and the thermodynamic entropy S(T ). We capture the characteristic two-peak structure of Cv and two hump structures in the entropy at zero field. As expected, S saturates to ln(2) at large temperatures for all field values. The two-peak/hump fea- ture is also found in the gapped QSL phase (H = 0:15), where Fig. 1. (A) Spin susceptibility (χ) for L = 4, 16 and (B) different measures 1 low-temperature peak/hump shifts to even lower temperatures of magnetization as a function of magnetic field strength (H) for L1 = 16. The vertical dashed lines show critical magnetic fields where phase transi- compared with H = 0. The behavior of the low-temperature tions occur. (C) Corresponding phase diagram that shows transitions from S(T ) and Cv (T ) is consistent with a gapped spectrum both gapless QSL to gapped QSL to an intermediate gapless QSL with an SFS into at low and high fields. However in the intermediate phase, the a PPM phase (H = 0 to large H, respectively). (D) Average spin-spin correla- peak at low T =K is suppressed, leading to an algebraic behavior tions between sites at a fixed distance of R. The dashed black line represents that is distinct compared with both gapped phases. In fact, the the power law decay with power −1.5 as a guide to the eye. All results were low-temperature Cv at H = 0:33 shows linear behavior in tem- obtained using DMRG at zero temperature. perature (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 magnetic field H = jH j(^ex +e ^y +e ^z ) (in units of K ) applied in an exact dependence of Cv on temperature, and therefore, we the [111] direction, i.e., perpendicular to the two-dimensional only discuss the qualitative trends. The analysis of the statistical honeycomb lattice. This model is solved using density matrix error in calculations of Cv and S is also shown in SI Appendix renormalization group (DMRG) and Lanczos on a lattice with along with Cv at low temperatures.
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