Universal thermodynamics in the Kitaev fractional liquid

In the Kitaev honeycomb model, the quantum spin fractionalizes into itinerant Majorana fermion and gauge flux upon cooling, leading to rich experimental ramifications at finite temperature and an upsurge of research interest. However, accurate modelings of the Kitaev materials by including realistic couplings beyond the pure Kitaev model constitute a major challenge for the community. With recently developed exponential tensor-network approach, we clear the pathway and perform finite-temperature simulations of the extended Kitaev model with additional interactions common in real materials. At intermediate temperature, we find an emergent Curie law of magnetic susceptibility and a stripy spin-structure factor characterizing the robust Kitaev fractional liquid. With this insight, we revisit the susceptibility measurements of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and $\alpha \text{−}{\mathrm{RuCl}}_3$ and find evidence of ferromagnetic Kitaev coupling and finite-temperature fractionalization. Bridging the gap between theories and experiments with the state-of-the-art tensor-network simulations, our findings provide guidance for future experimental exploration of the spin liquids in Kitaev materials by thermodynamic measurements and spin-resolved structure factor probes.

Figure 1

Cylindrical honeycomb lattice of size YC $4\times 6\times 2$ illustrated with the snakelike quasi-1D mapping path (underlying gray lines), adopted in the XTRG calculations. Three types of Kitaev bonds $K_x$, $K_y$, and $K_z$ are marked in blue, green, and red, respectively. The ${\mathbb{Z}}_2$ flux $W_P={\sigma }_1^z{\sigma }_2^x{\sigma }_3^y{\sigma }_4^z{\sigma }_5^x{\sigma }_6^y=+1$ (hexagon in white) or $-1$ (purple), with ${\sigma }_i^{\gamma }=2S_i^{\gamma }$, can be pair flipped by applying a magnetic field $h\parallel \left[001\right]$ on the spin-space cubic axis. Typical nearest and next-nearest bonds $d_x$, $d_z$, $d_{x+y}$, and $d_{x+z}$ are indicated by the dashed lines.

Figure 2

Spin correlations $\langle S_i^z{S}_j^z\rangle$ with (a) $h=0$ and (b) $h=0.045$ on bond $\langle i,j\rangle =d_x$, $d_z$, $d_{x+y}$, and $d_{x+z}$ as depicted in Fig. 1, where the NN spin correlations are established at $T_H\simeq 0.373$ and converge to the ground-state (GS) DMRG results (closed symbols) at low temperature. In (b) the mean magnetization squared ${\langle M\rangle }^2$ has been subtracted in the correlation data (see the text). (c) Thermal average of flux $W_P$ and (d) thermal entropy $S$ results under various fields. As temperature decreases, the flux increases monotonically for $h\lesssim h_c$ and builds up rapidly at around $T_L\simeq 0.017$, while it becomes nonmonotonic for $h\gtrsim h_c$ and starts to decrease below a certain low temperature after the initial stage of increasing. (d) Entropy with $h\lesssim h_c$ exhibits a stable $S\simeq \frac{1}{2}ln2$ quasiplateau, above which the curves almost coincide; for $h\gtrsim h_c$, the entropy leaves a shoulderlike structure at intermediate temperature.

Figure 3

Contour plot of (a) thermal entropy $S/ln2$ and (b) specific heat of the Kitaev model under magnetic fields $h$, where two crossover temperature scales and an intermediate Kitaev fractional liquid (KFL) regime are evident. The $T_H$ and $T_L$ ($T_L^{\prime }$) determined from (b) coincide with results from Ref. [33] (the white dots, available for $T\gtrsim 0.05$ for $h\ne 0$). (c) Flux $W_P$ and (d) magnetization $M$ are shown vs various $h$ and $T$. Three finite-$T$ ($T=0.07$) spin-structure factors at $h=0$, 0.03, and 0.06 are presented in the inset of (c) (whose intensity color bar ranges from 0.2 to 0.5). A structure peak emerges at the center ($\mathrm{\Gamma }$ point) of the extended Brillouin zone due to the induced FM moment $M$ at $h>0$, together with a stripy background. (d) The field-induced FM phase is clearly indicated and the black dots represent the spins parallel to the $\left[001\right]$ field in the inset.

Figure 4

Specific heat $C_V$ under various magnetic fields $h$. The linear-$T$ dependence of $C_V$ at intermediate temperature is evident as indicated by the black dashed line, which also underlines the quasiplateau in the $C_V/T$ curves plotted in the inset.

Figure 5

(a) Uniform magnetic susceptibility $\chi$ under various fields $h$, with $1/\chi$ plotted in the inset. The Curie-Weiss fittings in the high- and intermediate-$T$ regimes are indicated by the red and blue dashed lines, respectively. The high-$T$ behavior can be fitted well by $1/(4T-J)$, and immediately below $T\simeq T_H$ there emerges a second Curie behavior. (b) Zero-field susceptibility of the anisotropic model with $K_x=K_y\ne K_z$, where the emergent Curie behaviors are observed in a wider temperature range. Also presented are the corresponding (c) $1/\chi$ fittings and (d) spin correlations on the $d_z$ bond.

Figure 6

Phase diagram of (a) the Kitaev-Heisenberg and (b) the Kitaev-$\mathrm{\Gamma }$ model obtained by entropy $S/ln2$, where the white diamonds indicate the crossover temperature determined from the double-peaked specific heat. The insets show the $T=0.07$ static structure factors of (a) $J=0.12$ and (b) $\mathrm{\Gamma }=0.06$ in the extended Brillouin zone. The XTRG results of the correlations $\langle S_i^z{S}_j^z\rangle$ along four NN bonds with (c) $J=0.08$ and 0.16 and (d) $\mathrm{\Gamma }=0.03$ and 0.07 all converge at the lowest temperature to the GS values calculated by the DMRG method. For the Kitaev-Heisenberg case with $J=0.08$ and the Kitaev-$\mathrm{\Gamma }$ case, $d_z$ correlations arise at around $T_H$ and stay predominate across the whole temperature range; for the $J=0.16$ case, all correlations increase abruptly at the low-temperature scale, indicating the onset of SAF order. (e) and (f) Flux $W_P$ with various $J$ or $\mathrm{\Gamma }$ and (g) and (h) magnetic susceptibility $\chi$ and the Curie-Weiss fittings for two models. The $\chi$ data are computed under a small uniform field $h=0.002$.

Figure 7

Specific heat $C_V$ vs $T$ for the (a) Kitaev-Heisenberg and (b) Kitaev-$\mathrm{\Gamma }$ models with various coupling strengths. The black dashed lines indicate the linear $T$ dependence of $C_V$ at intermediate temperature and the corresponding $C_V/T$ data are shown in the insets. With Heisenberg interactions in (a), the $C_V/T$ plateau is apparent for $J\lesssim 0.08$ and leaves a shoulderlike structure until $J_c\simeq 0.12$; for the $\mathrm{\Gamma }$ terms this plateau and its remnant signature are clear for $\mathrm{\Gamma }\lesssim 0.04$. (c) and (d) Thermal entropy results of both models. Around the stable high-$T$ scale $T_H$, the entropy curves almost collapse for various $J$ and $\mathrm{\Gamma }$ strengths and consequently leave a quasiplateau with $S\simeq \frac{1}{2}ln2$.

Figure 8

Experimental ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ [14] and $\alpha \text{−}{\mathrm{RuCl}}_3$ susceptibility data [57] and their Curie-Weiss fittings at both (a) high and (b) intermediate temperatures under fields $H\parallel ab$ (orange circles) and $H\parallel c$ (blue squares), with inverse susceptibility plotted in the insets. (c) Fitted Curie constants ($C$ for high and $C^{\prime }$ for intermediate $T$) and the Curie temperatures (corresponding $\theta$ and ${\theta }^{\prime }$), for both the $c$ and $ab$ directions, are listed in the table.

Figure 9

Static spin-structure factors $S^{\gamma }\left(q\right)$ calculated from the equal-time spin-correlation functions [see Eq. (2)] at an intermediate temperature $T\simeq 0.15$: (a)–(d) pure Kitaev model results of three spin-structure factors $S^{x,y,z}\left(q\right)$ as well as their sum $S\left(q\right)$ and (e)–(h) results of the Kitaev-Heisenberg model with $J=0.14$.

Figure 10

(a) Relative errors $\delta f/f$ between the ED and XTRG results of a isotropic pure Kitaev model on the $N=12$ (YC4) cluster, with retained bond dimensions $D=200$, 400 and 600. (b) Corresponding specific-heat results of the pure Kitaev model (blue), the Kitaev-Heisenberg model with $J=0.2$ (red), and the Kitaev-$\mathrm{\Gamma }$ model with $\mathrm{\Gamma }=1$ (yellow). Comparisons between the large-scale XTRG and QMC simulations [27] are presented for (c) the thermal entropy $S/ln2$ (orange) and flux $W_P$ (blue), (d) specific heat $C_V$, and (e) spin correlations $\langle S_i^z{S}_j^z\rangle$. We also compare the XTRG susceptibility results to the QMC data on the $L=20$ torus [29], where the high- and intermediate-$T$ fittings are also performed on the latter (dashed lines). We show in (c)–(f) both the YC $4\times 6\times 2$ results (solid curves) and the YC4 data (dotted curves) obtained by subtracting length $L=6$ by $L=4$ results. The XTRG results of susceptibility are computed under the $\left[001\right]$ field, which agree excellently with the QMC results in (f) above the low temperature $T_L$, although the latter is along the $\left[111\right]$ direction.

Figure 11

High- and intermediate-$T$ spin correlations of the extended Kitaev model, along the (a), (c), and (e) zigzag and (b), (d), and (f) armchair paths, with the reference site marked in red and the hexagon edge length as the unit of distance. Note that in (a) and (b) we have subtracted the (field-induced) mean magnetization squared ${\langle M\rangle }^2$ from the spin correlations.

Figure 12

Bipartite entanglement $S_E$ cut at the center of the MPO representation of density matrices for (a) the Kitaev model with fields, (b) the Kitaev-Heisenberg model, and (c) the Kitaev-$\mathrm{\Gamma }$ model. At intermediate temperature, $S_E$ roughly scales as $\sim 2/3ln\beta$, which characterizes the required computational resource of the simulations.

Figure 13

(a) Illustration of the magnetic fields $h$ along $\left[001\right]$ and $[-101]$ (parallel to the honeycomb $ab$ plane). Also shown are (b) the magnetic susceptibility $\chi$ and (c) its inverse $1/\chi$, calculated with a small field $h=0.002$. (d) Susceptibility results along various field directions, obtained by ED calculations on the $N=12$ cluster.

Figure 14

(a) Uniform susceptibility $\chi$ of the AF Kitaev model under various fields $h$. The black dashed line indicates the high-$T$ Curie-Weiss fitting, and at intermediate temperature no emergent Curie behavior is found, which is also evident as shown in the $1/\chi$ plot (inset). Since the AF and FM Kitaev models are exactly equivalent under zero fields, here the two temperature scales $T_L$ and $T_H$ coincide with those in Fig. 5. (b) Staggered susceptibility ${\chi }_S$ of AF Kitaev model exhibiting a universal Curie behavior in the intermediate-$T$ regime (see also $1/{\chi }_S$ in the inset). In fact, the ${\chi }_S$ data of the AF Kitaev model are in quantitative agreement with the uniform susceptibility $\chi$ of the FM Kitaev model.

Figure 15

Inverse susceptibility $1/\chi$ of (a) the Kitaev-Heisenberg and (b) Kitaev-$\mathrm{\Gamma }$ models with various $J$ or $\mathrm{\Gamma }$ interactions. The linear fittings are shown in both the high-$T$ (the red dashed lines) and intermediate-$T$ regimes, with the same fitting parameters as those in Figs. 6 and 6.

Figure 16

Landscape of specific-heat data for the (a) Kitaev-Heisenberg and (b) Kitaev-$\mathrm{\Gamma }$ models. The corresponding gauge fluxes $W_P$ are contour plotted in (c) and (d) for the two models.