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PHYSICS 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 = |H |(ˆex +e ˆy +e ˆz ) (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. Finally, with increasing H L1 × L2 number of unit cells and with two sites per unit cell across Hc2 ' 0.35 into the PPM phase (H = 0.45), the two peaks labeled as A or B (the total number of sites equals 2L1L2) in Cv merge, and similarly, only one hump feature is visible in (see Fig. 3A). All DMRG simulations are performed with fixed the entropy. L1 = 16 and cylindrical boundary conditions using up to 1,200 We also analyze the low-temperature missing part of the DMRG states per block, ensuring the truncation error ≤ 5 × entropy (S ) resulting from finite size effects (shown in SI −6 low 10 for L2 ≤ 5 (≤ 10 sites in a2 direction) within the two- Appendix). The H = 0 and gapped phases have a quadratic site grand canonical DMRG (36–43). Additionally, we also use increase in Slow with temperature. On the contrary, the inter- finite-temperature Lanczos method (FTLM) for calculations of mediate phase shows almost a linear behavior of Slow at low specific heat and thermodynamic entropy at finite temperatures temperatures. This is consistent with our earlier arguments of T (in units of K ) on small clusters (39, 44). SI Appendix presents gapless nature of the intermediate phase. To summarize, we definitions of all observables and convergence analysis of DMRG observe the following features in the intermediate phase: (i) a and FTLM calculations. slow power law decay of the C (R) (Fig. 1D) that is indicative of a gapless phase and (ii) a Cv (T ) consistent with linear behavior at Results low T indicating a presence of itinerant emergent fermions. All We begin with the magnetic and thermodynamic properties of of this evidence taken together provides strong evidence for an the model. Fig. 1 shows the spin susceptibility (χ = ∂Stot/∂H ) intermediate phase that is indeed gapless. Additionally, recently and various measures of magnetization as a function of field calculated T = 0 spin dynamics on small systems also show a strength H . The clear two-peak structure of χ demonstrates continuum in energy, suggesting that the intermediate phase is the presence of two-phase transitions at finite field strengths. The corresponding magnetization also shows kinks at the crit- ical fields H ' 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 AB spectrum and fourfold topological ground-state degeneracy that is associated with a symmetry-protected topological phase (1). At finite fields H . 0.2, the ground state is twofold degener- ate 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 & 0.35, we find a par- tially polarized magnetic (PPM) phase, where the magnetization monotonically increases toward its saturation value (M111 = 1.5) in the trivially field-polarized product state. Remarkably, the two-peak structure of χ indicates that an intermediate phase is sandwiched between the gapped Kitaev QSL and polarized phases (Fig. 1C). Fig. 2. Specific heat Cv (T) and thermodynamic entropy S(T) as a function What are the properties of the intermediate phase? We study of temperature T. Shown results are using a 3 × 3 unit cell (18 sites) cluster the decay of spin-spin correlations C (R) shown as a function solved using FTLM averaged over 50 different random runs (39, 44) for each of the distance R between the two spins in Fig. 1D. As expected, fixed magnetic field strength H (key label).

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1821406116 Patel and Trivedi Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 ean h nesbatc correlations. intersublattice the retains the to 3 the shift at Fig. ply peaks comparing dominant the fact, that In observed around circle. peaks a and patterns forming petal-shaped three forming points 3E Fig. M BZ. factor first structure the spin within symmetries trace the from of factor Apart locations structure 3D. Fig. spin in total circles the red high-intensity as curve. visible green tinctly 3C, Fig. in atc vectors lattice L and Γ the of cuts fixed at ihitniy h ignllnsaeseilcuts special are lines in diagonal labeled also The are intensity. BZ extended high and BZ the of points symmetry in cuts momentum crystal the along shown 3. Fig. BZ. first the vs. BZ between extended distinguish the to of e subscript points use high-symmetry shall We respectively, 3B. BZs, Fig. extended in labeled and first the in points high-symmetry a indeed ae n Trivedi and Patel at peaks robust reproduces SFS proposed The sublattices S short, In 3E). (Fig. BZ first the using only S considered be 3 (Fig. may scheme BZ therefore, extended the and in orbitals), sidered (unlike cell that unit a Note Appendix ). (SI sublattices the correlations intrasublattice and inter- (total) while correlations, (trace) intrasublattice of calculation correla- For spin-spin The 3. intrasublattice tions. Fig. and inter- in of (BZ) transform zone (A Fourier cell Brillouin unit the per sites of a two cuts on various case, along our before. tor in discussed as even SFS that an stable have likely have may is to we it lattice, shown honeycomb Therefore, recently 47). was (46, lattice SFS triangular a on phase B AC 2 tr tr ,weega ice at circles gray where (k), = on n ek rudthe around peaks and point nFg 3 Fig. In oepoeti osblt,w aclt h pnsrcuefac- structure spin the calculate we possibility, this explore To Γ = (Γ) onso h rtB,wt otpasaon the around peaks soft with BZ, first the of points ,4 6 ,1 ie nthe in sites 10 8, (6, 5 4, 3, C 3 h pnsrcuefco n eue F nteitreit phase. intermediate the in SFS deduced and factor structure spin The astruhtwo through pass H U S A tr and 0 = (1) S C k (Γ tr 1 = .Teei upeso fitniyaon the around intensity of suppression is There (k)]. .3K e and B A als S 3) nfc,sc als pnliquid spin gapless such fact, In (30). QSL gapless M but ), cteigsetu ftepooe F acltdusing calculated SFS proposed the of spectrum Scattering (G) model. Kitaev the in as equivalent are (2π and e onswe osdrn the considering when points , eso usadcnoro the of contour and cuts show we D, IAppendix [SI phase intermediate the in −2π/ B S tot M r oae ndfeetpstoswithin positions different on located are and M √ (Γ) S onsrpeetlrepas (F peaks. large represent points a )and 3) tot 2 e ieto) hw eefor here shown direction), ons hwn outpasvisible peaks robust showing points, =S 6 B (k) ,w en tutr atrusing factor structure define we ), S M tot tot k S e talo the of all at 2 (Γ eas nlz h peak the analyze also we (k), tr = ons h usbetween cuts The points. (k) e 0 4π/ (0, B where ), ihnteetne Za fixed at BZ extended the within M hw ag ek tthe at peaks large shows spromduigonly using performed is M onso h Zadcrua atrsaon the around patterns circular and BZ the of points S ,wiethe while B–D), tot √ S C onsin points S ) hr ice,surs n rage ersn usotie using obtained cuts represent triangles and squares, circles, where 3), D 1 tot (k) tr , C Γ M (k) (k) and 2 S and , and usoe both over sums e tot L onsi dis- is points utb con- be must htrespects that (k) 2 tcnbe can it E, rpsdSFS Proposed ) B. = S Γ C K tr e 3 D .Teui etr are vectors unit The 5. htalso that (k) endin defined ee to refer hw otu ltof plot contour a shows and S S tot tr sim- has (k) (k) K C Γ 2 0 A(k, H (E C. = rps h orsodn SFS corresponding the propose erae iha nraigmgei ed nteohrhand, other the On field. point magnetic momentum increasing an at with plateau decreases the that strates at peaks robust integrating on that, (33) S excitations space Majorana factor and real flux structure calculations plaquette dynamical in previous the with in correlations consistent peaks also of spin-spin is result Our of by 1C). established (Fig. decay correlations exponential short-ranged with an consistent is that strengths and less field shows various 4 at Fig. factor. structure shown SFS an with QSL 3F a Fig. for in evidence the strong qualitatively of providing to features phase, able is subtle the SFS even proposed to capture our leads summary, in In features 3E). found (Fig. circular also large patterns these three-petal of combination The all to momentum ter with features spinon circular a large Other 3E). of in (Fig. calculations phase DMRG intermediate our the with consistent 3G), (Fig. transfer 3F momentum (Fig. SFS the ω q 3G, k, Fig. in function scattering the rlfnto,dfie by defined function, tral and A tot + ω nteitreit hs.Teect r enduigtereciprocal the using defined are cuts These phase. intermediate the in 0.3 0 = hw h oecm atc ihclnrclbudr odtoswith conditions boundary cylindrical with lattice honeycomb the shows ω h nemdaepaewt ogrnecreain shows correlations long-range with phase intermediate The ie h nomto rvddby provided information the Given enx unt h feto antcfil ntespin the on field magnetic a of effect the to turn next We h eklctos(e ons fteitaultiesrcuefactor structure intrasublattice the of points) (red locations peak The ) F = (k) = k htpcsu h oetmtransfer momentum the up picks that 0), = M (k) 0) h nepce scattering interpocket The . E D cteigars h em ufc iheeg transfer energy with surface Fermi the across scattering a = 1 onso h Z(i.3F (Fig. BZ the of points = ∫ H S rs rmsmlrsatrn rcse:frexample, for processes: scattering similar from arise δ . 0 tot ∞ (ω 0 )and 1) (0, q . ,wt lerpeetn o nest n e representing red and intensity low representing blue with (k), − d + 0. ω  k F S Γ 2 S )wt okt around pockets with (k)) tot ons iia oteDR eut in results DMRG the to similar points, ed oalrejitdniyo ttswith states of density joint large a to leads ) M onswti h okt tthe at pockets the within points apdQLpae,teei ra feature broad a is there phases, QSL gapped (k, a e 2 k = .Adtoal,Fg 4B Fig. Additionally, 4A). (Fig. ω = (0.5, ietebodfauei i.4. Fig. in feature broad the give ), M A(k, H √ htrslsi ek tthe at peaks in results that q ) h pnsrcuefactor structure spin The 3/2). oeta,frzr ed gap- fields zero for that, Note . rudthe around ω S F = ) tot  (k) F F S L Γ (k) S (k) ∼ 2 δ (k) .Uigtesio spec- spinon the Using ). 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PHYSICS A B in S(k). Moreover, the size of pockets on the SFS is picked such that more subtle three petal-shaped patterns around the K /K 0 points (Fig. 3E) are also captured in F(k) (Fig. 3G). In the graphene problem, the Dirac points at the K /K 0 are pro- tected 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 0 points. Additionally, we expect the largest scattering contribu- tion to occur between a particle and hole pocket. In summary, we propose an SFS with two bands, creating Fermi pockets at the Γ point and M with opposite particle and hole character (Fig. 3F). Fig. 4. Stot (k) for various field strengths along (A) cut C1 and (B) cut C2 The procedure described above for the reconstruction of the (defined in Fig. 3D). The peaks at Me points are most robust for the inter- ω− S (k) ' mediate phases H = 0.26, 0.30, and 0.33. The peak/hump-like feature close SFS relies on the assumption that the integrated tot to K points monotonically decreases with increasing H. Stot(k, ω = 0) is well described by the dynamical correlations at low energies, also used previously to propose an SFS in 1T-TaS2 (46). In the Kitaev model (at zero field), indeed most of the

Stot(k = Me ) increases with the field, reaching its maximum weight in Stot(k, ω) is concentrated at small ω (4). We test this value in the intermediate phase and decreasing again on transi- assumption in the new QSL phase by calculating the numeri- tioning into PPM phase. All peaks get suppressed in the PPM cally challenging dynamical spin structure factor (48, 49) at low phase, as expected, because of shorter-ranged spin-spin cor- energies Stot(k, ω = 0) on 8 × 3 unit cell (48 sites) clusters. We relations (Fig. 1D). Overall, the combined results of Figs. 3 find that the dynamical spin structure factor also shows peaks at and 4 show that the dominant peaks of the spin structure the M points for ω = 0 (SI Appendix), justifying our method to factor S(k) at the M points exist only in the intermediate reconstruct the SFS. gapless phase. We expect the emergent neutral fermions that form a Fermi surface in the gapless QSL phase to show quantum oscillations Discussion in a magnetic field, similar to observations of quantum oscil- Several conflicting values of the central charge c have been lations in SmB6, a topological Kondo insulator (50–52). Going reported for the intermediate phase, where arguments based forward, it would also be useful to dope the different classes of on the central charge were used to deduce the SFS (33–35). Mott insulators that could harbor QSL ground states to explore However, it seems to be very difficult to obtain a reliable value the emergent superconducting phases (53). for c unambiguously in a gapless phase using DMRG. Ref. 34 Conclusion proposes c = 1, 0 using L2 = 3, 4, respectively; ref. 33 finds a Our most significant finding is that of an intermediate gapless c = 4 using L2 = 5; and ref. 35 calculates c = 1, 2 using L2 = 2, 3, respectively, using DMRG. Based on the central charge QSL with an SFS sandwiched between the well-known gapped nonabelian Kitaev spin liquid at low magnetic fields and a par- arguments, the SFS was argued to have pockets around the 0 tially polarized phase at high magnetic fields, studied here for K /K and Γ points of the first BZ (34, 35). This clearly dif- a field along the [111] direction. The two quantum-phase tran- fers from our proposal of the SFS with pockets around Γ and sitions are revealed by kinks in the magnetization and peaks in M points; our results are also consistent with the exact diag- the susceptibility. The gapless nature of the intermediate phase onalization results of the dynamical structure factor on small 0.2 H 0.35 is indicated by the slow power law decay of spin- clusters (30). We further emphasize that the total S(k) must . . spin correlations as opposed to an exponential decay in the respect the symmetries within the extended BZ because of two other two phases. The temperature dependence of the thermo- atoms in the unit cell at different locations for a honeycomb lat- dynamic quantities, specific heat and entropy, also corroborates tice (Fig. 3A), a point that seems to have been ignored in the the gapless nature of this intermediate phase. literature. Finally, using large-scale DMRG and detailed analysis along We note that the S(k) surfaces shown in Fig. 3 D and E many cuts of the spin structure factor S(k) in momentum space, cannot result from the tight-binding model with uniform hop- we propose an SFS in the intermediate phase with pockets at ping on a honeycomb lattice (the graphene problem). This is the Γ and M points with opposite particle hole character. We because any scattering K ↔ K 0 will only lead to S(k) with expect our findings to encourage the search for QSLs in materi- peaks at the K and K 0 points, which are not present in our als with Kitaev interactions via spectroscopy and thermodynamic S(k) DMRG calculations of in the intermediate phase. To have measurements. peaks at the M point in S(k), as we have found in our cal- culations, there must be pockets around the M points of the ACKNOWLEDGMENTS. We thank Kyungmin Lee and David C. Ronquillo for SFS. Note that all of the 6 M points in the BZ are related by their helpful comments and discussions. We also thank Gonzalo Alvarez C for help with DMRG++ open source code developed at the Oak Ridge 3 rotation and momentum translation symmetries and there- National Laboratory. N.D.P. and N.T. acknowledge support from Department fore, must be identical. Such a SFS with M ↔ M scattering will of Energy Grant DE-FG02-07ER46423. Computations were performed using produce a peak with momentum transfer around the M point Unity cluster at The Ohio State University and the Ohio supercomputer.

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4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1821406116 Patel and Trivedi Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 1 .Io,A ymd,S agw,M aua .Kt,Qatmsi iudi the in liquid spin Quantum Kato, R. an Tamura, in M. state Maegawa, liquid S. Spin Oyamada, Saito, A. G. Itou, Maesato, T. M. 21. Kanoda, K. Miyagawa, K. Shimizu, Y. 20. 1T-TaS Lee, A. P. Law, T. the K. of transport 19. and Dynamics Sachdev, S. Xu, C. Qi, Y. 18. 0 .Hce,S rbt mrec fafil-rvnU1 pnlqi nteKitaev the quantum in magnetic-field-driven liquid of Signatures spin Trivedi, U(1) N. Vengal, field-driven A. Ronquillo, a C. of D. Emergence 31. Trebst, S. Hickey, Kitaev C. the on 30. interactions the exchange off-diagonal in of Impact liquid Fujimoto, S. spin Takikawa, D. quantum 29. Field-induced Jiang, field-revealed C. the H. of Devereaux, Theory P. T. Kee, Y. Jiang, H. F. Sørensen, Y. S. 28. E. Catuneanu, A. 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