Quantum spin liquid at finite temperature: Proximate dynamics and persistent typicality

Quantum spin liquids are long-range entangled states of matter with emergent gauge fields and fractionalized excitations. While candidate materials, such as the Kitaev honeycomb ruthenate $\alpha \text{−}{\mathrm{RuCl}}_3$, show magnetic order at low temperatures $T$, here we demonstrate numerically a dynamical crossover from magnonlike behavior at low $T$ and frequencies $\omega$ to long-lived fractionalized fermionic quasiparticles at higher $T$ and $\omega$. This crossover is akin to the presence of spinon continua in quasi-1D spin chains. It is further shown to go hand in hand with persistent typicality down to very low $T$. This aspect, which has also been observed in the spin-1/2 kagome Heisenberg antiferromagnet, is a signature of proximate spin liquidity and emergent gauge degrees of freedom more generally, and can be the basis for the numerical study of many finite-$T$ properties of putative spin liquids.

Figure 1

Dynamical crossover near a Kitaev QSL (schematic). Magnon excitations, characteristic of the long-range magnetic order at low $T$ and $\omega$, give way to long-lived fractionalized quasiparticles (here Majorana fermions) at higher $T$ and $\omega$. This crossover is accompanied by a remarkable persistence of typicality to very low $T$ near the QSL.

Figure 2

Persistence of typicality near the Kitaev QSL, I. Logarithm of contributions to the partition function ${\zeta }_r$ [Eq. (3)] from 320 random states for three points inside the Néel state, stabilized by a Heisenberg coupling $J$: (a) $\theta =0$, (b) $\theta =0.4\pi$, and (c) $\theta =0.49\pi [J=cos\theta ,K=sin\theta ]$. All data refer to the symmetric 24-site cluster of Appendix pp2.

Figure 3

Persistence of typicality near the Kitaev QSL, II. (a), (b) Real-time evolution of the correlator $f_r\left(t\right)$ [Eq. (3)] for 320 random $\left(T=\infty \right)$ states $|r\rangle$, for the $JK$ model at $\theta =0.47\pi$, with $T=\left|K\right|$ (a) and $0.2\left|K\right|$ (b). (c) $T$-dependence of the maximum standard deviation in $\langle \mathcal{R}\left(t\right)\mathcal{R}\left(0\right)\rangle ={\sum }_r{d}_r{f}_r\left(t\right)/{\sum }_r{d}_r{\zeta }_r$ [see Eq. (3)] for various values of $\theta$. All data refer to the symmetric 24-site cluster of Appendix pp2.

Figure 4

Low-energy spectrum of the $JK$ model on the 24-site cluster. Energies are measured from the ground-state energy $E_0\left(\theta \right)$. The nonlinear horizontal axis is used to better highlight what happens in the narrow regions around the Kitaev QSL points $\theta =\pi /2$ and $3\pi /2$. The various symbols correspond to the irreducible representations of the symmetry exploited here. The latter includes (i) the 12 translations, with the symbols $\mathrm{\Gamma },K^{*}$, and $M^{*}$ corresponding, respectively, to zero momentum, the corners of the first Brillouin zone (BZ), and the midpoints of the BZ edges; (ii) real-space inversion through the middle of the hexagons, with the letters e and o standing for the even and the odd sectors, respectively; and (iii) spin inversion (global rotation around the $x$ axis in spin-space), with the symbols Sze and Szo standing for the even and odd sectors, respectively. The numbers in parentheses give the degeneracy of each level. The spectrum is obtained by the standard Lanczos method. Specifically, we show the 100 lowest energy states in each sector. Among these, only the lowest five are converged to the requested high precision (here relative precision in the ninth digit). The rest give a good representation with precision that lowers as we go up in energy.

Figure 5

Raman scattering intensity in the ‘xy’ polarization channel for various phases proximate to the Kitaev QSL. (a) Ideal AF Kitaev point, (b) $JK$ model with $\theta =0.47\pi$ (Néel phase), (c) $JK$ model with $\theta =0.45\pi$ (Néel phase), (d) $JK$ model with $\theta =0.53\pi$ (zigzag phase), (e) $JK$ model with $\theta =1.43\pi$ (FM phase), (f) $K\mathrm{\Gamma }$ model with $K=-1$ and $\mathrm{\Gamma }=0.2$. Dashed lines show the analytical $T=0$ result at the Kitaev QSL point. [46] $T$ and $\omega$ are in units of $\sqrt{K^2+J^2}=1$ (a)–(e) or $\left|K\right|$ (f). Results are obtained for the symmetric 24-site cluster of Appendix pp2, by propagating 320 random $\left(T=\infty \right)$ states in real and imaginary time.

Figure 6

Fermionic character of the Raman response. (a), (b) $T$ dependence of the integrated Raman intensities $n_{\text{tot}}={\int }_0^{\infty }d\omega I\left(\omega \right)$ (black circles), $n_L={\int }_0^{{\omega }_1}d\omega I\left(\omega \right)$ (blue triangles), and $n_H={\int }_{{\omega }_2}^{\infty }d\omega I\left(\omega \right)$ (red diamonds), where ${\omega }_1=0.25,{\omega }_2=0.5$ (as in Ref. [47]) for the $JK$ model with $\theta =0.49\pi$ (a) and $0.46\pi$ (b). Solid lines are fits to Eqs. (5), with ${\varepsilon }_L^{*}=0.42$ and ${\varepsilon }_H^{*}=0.58$ (as in Ref. [47]) with $a_L,b_L,a_H$ and $b_H$ determined by a least squared fitting procedure. (c) The standard deviations of the fits, ${\sigma }_L=\sqrt{{\sum }_{i=1}^p{(n_L\left(T_i\right)-y_L\left(T_i\right))}^2/p}$, and similarly for ${\sigma }_H$ [where $y_L$ and $y_H$ are defined in Eqs. (5)], as we depart from the AF Kitaev point in the JK model parametrized as $K=sin\theta ,J=cos\theta$.

Figure 7

Expectation value of the energy per site, ${\langle \mathcal{H}\rangle }_{r,\beta }/N$, in the normalized state $|r,\beta \rangle /\sqrt{{\zeta }_r}$ [see definition in Eq. (A5)] for a number of initial random states $|r\rangle$ (ten in each irreducible representation; in total, 240 states for the 16-site cluster, 320 for the 24-site cluster, and 200 for the 32-site cluster; for the latter, we only show data from the sectors that are even under spin inversion). Panels (a) and (b) correspond to the AF Heisenberg and the AF Kitaev point, respectively.

Figure 8

Evolution of the quantity $ln\left({\zeta }_r\right)/N$, where ${\zeta }_r=\langle r,\beta |r,\beta \rangle$, with inverse temperature for a number of initial random states $|r\rangle$ (same as in Fig. 7). Panels (a) and (b) correspond to the AF Heisenberg and the AF Kitaev point, respectively.

Figure 9

Finite-size clusters used in our simulations. All clusters have periodic boundary conditions, and $N=16$ (left top), 24 (left bottom and middle), or 32 sites (right). The first has the full point-group symmetry of the model.

Figure 10

Phase boundaries in the $JK$ model. Ground-state expectation value of Kitaev's hexagonal plaquette operator [7] $W=2^6{S}_1^x{S}_2^y{S}_3^z{S}_4^x{S}_5^y{S}_6^z$ (site labeling shown in the inset) in the $JK$ model, with $K=sin\theta ,J=cos\theta$. Note the nonlinear horizontal axis.

Figure 11

Results for the LTLM method. Raman response in the $xy$ channel at (a) the Kitaev QSL point, and (b) $\theta =0.485\pi$ (within the Néel phase). Standard error due to the sampling involved in Eq. (D2) is shown as error bars. Dashed grey lines mark the shift of the main peak with $T$. Peaks are broadened with a Lorentzian of width $\eta =0.2$.