Quantum critical dynamics in the two-dimensional transverse Ising model

In the vicinity of the quantum critical point (QCP), thermodynamic properties diverge toward zero temperature governed by universal exponents. Although this fact is well known, how it is reflected in quantum dynamics has not been addressed. The QCP of the transverse Ising model on a triangular lattice is an ideal platform to test the issue, since it has an experimental realization, the dielectrics being realized in an organic dimer Mott insulator, $\kappa \text{−}{\mathrm{ET}}_{2}X$, where a quantum electric dipole represents the Ising degrees of freedom. We track the Glauber-type dynamics of the model by constructing a kinetic protocol based on the quantum Monte Carlo method. The dynamical susceptibility takes the form of the Debye function and shows a significant peak narrowing in approaching a QCP due to the divergence of the relaxation timescale. It explains the anomaly of dielectric constants observed in the organic materials, indicating that the material is very near the ferroelectric QCP. We disclose how the dynamical and other critical exponents develop near QCP beyond the simple field theory.

Figure 1

(a) Two-dimensional conducting plane of $\kappa$-(${\mathrm{ET})}_{2}X$. In the insulating phase the electric dipole (right panel) is defined on each dimer as arrows that could point in two different directions depending on the location of the charges. As the charge hops between two molecules, the dipole fluctuates quantum mechanically by $\mathrm{\Gamma }$. (b) Schematic description of the transverse Ising (TRI) model on an anisotropic triangular lattice. Arrows indicate the Ising degrees of freedom. Phase diagram (right panel) near QCP for the material parameters of $\kappa$-(${\mathrm{ET})}_{2}X$ extracted from our results (see Fig. 4). The broken yellow line is the phenomenologically discussed crossover line of the critical region, whereas the region marked with a red dotted line is the one obtained from our calculation where the static and dynamical properties, ${\chi }_0$ and $\tau$, take enough large values.

Figure 2

Results of QMC for the TRI model on an anisotropic triangular lattice given in units of $\mathrm{\Gamma }=1$. (a) Schematic illustration of a set of world lines describing the partition function of the TRI model given on the place of $\mathrm{space}(i$) and imaginary $\mathrm{time}(\tau$), which are periodic about $\tau =[0:\beta ]$. Kinks (cross symbols) including old (black) and new (blue) ones separate the world lines into segments and the highlighted/plain segments carry ${\sigma }^z=1/-1$. (b) Phase diagram on the plane of $J, J^{\prime }$, and $k_{B}T$. The shaded region corresponds to the ordered phase (ferroelectric order of dipoles). The material parameters of $\kappa$-($\mathrm{ET}){}_{2}X$ (Appendix pp1) fall near the blue point $(J\sim 0.1,J^{\prime }\sim 0.5)$ in the diagram at $k_{B}T=0$. The right panel shows the cross section of the phase diagram at $J=0.1$ at low $k_{B}T$ in the vicinity of QCP. The case of square lattice ($J^{\prime }=0$) in green cross section is given in Appendix pp3. (c) Relaxation function $\mathrm{\Psi }(q=0,t)$ obtained by the QMC calculation at $L=64$ and $J^{\prime }=0.47,J=0.1$ for several choices of $k_{B}T$. (d) ${\tau }_L$ and ${\chi }_{0;L}$ extracted from the relaxation function at several $L$, plotted as functions of $k_{B}T$. The envelope lines (broken lines) $\tau =c_1{\left(k_{B}T\right)}^{-z},{\chi }_0=c_2{\left(k_{B}T\right)}^{-\frac{\gamma }{\nu }}$ are their thermodynamic limit for fitted $z=2.11$ and $\gamma /\nu =2.10$. The solid line represents the same function using the 3D critical exponents $z=2.02$ and $\gamma /\nu =1.966$ with $c_1=4.34,c_2=4.61$, which is almost the same as the case of square lattice (Appendix pp3, Fig. 7). (e) Dynamical finite-size scaling analysis. Correlation time ${\tau }_{\mathrm{int}}$ is obtained for a series of $L=8,16,32$, and 64 for $k_{B}T/\mathrm{\Gamma }=0.5,0.25,0.125,0.0625,0.03125,0.0078125$, with $\mathrm{\Gamma }/k_{B}T=2L$. The data collapse to a single scaling function $\varphi$.

Figure 3

Dynamical susceptibility $\chi (q=0,\omega )$ obtained by the kinetic TRI protocol as functions of $\omega$ and $k_{B}T$. (a) Three-dimensional plot showing a set of Debye functions about $\omega$ for different choices of $k_{B}T$. We choose $(J^{\prime },J)=(0.47,0.1)$ that exhibits QCP in Fig. 2. Inset shows the temperature dependence of $\tau$ and ${\chi }_0$ used for this plot and are extracted from the relaxation function. Small arrows indicate the peak positions, $T_m\left(\omega \right)$. (b) $\omega$ dependence of $\chi (q=0,\omega )$ for several choices of $k_{B}T$. The half-width of the $q=0$ peak gives ${\tau }^{-1}$ and its peak height gives ${\chi }_0$. (c, d) Temperature dependence of $\chi (q=0,\omega )$ for several choices of $\omega$, ranging over $5-30\times {10}^{-5}$ and $2-10\times {10}^{-3}$. (e) $\omega$ dependence of $T_m$ for the QCP data [in panels (c, d)] and for slightly off QCP, $(J^{\prime },J)=(0.48,0.1)$. Solid and broken lines are $\propto {\omega }^{1/2.03}$ fitted by the QCP data and ${\omega }^{1/z},z=2.095$, respectively.

Figure 4

Density map of (a) static susceptibility ${\chi }_0$ and (b) the relaxation time $\tau$ extracted from the envelope of the relaxation function, plotted as functions of $k_{B}T$ for the TRI model on a triangular lattice taking $\mathrm{\Gamma }=1$ as a unit.

Figure 5

Schematic description of the models of $\kappa \text{−}{\mathrm{ET}}_{2}X$. (a) Mapping of the extended Hubbard model based on molecules (circles) to the TRI model based on dimers (ovals). The indices on four independent bonds are those of $t_i,V_i, i=1\sim 4$. (b) The 16 basis of the extended Hubbard model on a single dimer, where $\uparrow$ and $\downarrow$ indicate the electrons of up and down spins, respectively. The four different configurations with one electron per dimer form the low-energy local Hilbert space at large $V_1,U$. When the spin degrees of freedom are neglected at the leading order of perturbation, they are reduced to 2. (c) Configuration of electrons on adjacent two dimers, where the arrows indicate the corresponding pseudospin configuration. The second panel has energy $V_2$ and others zero, which yields the Ising interactions between pseudospins, see text.

Figure 6

(a) Phase diagram of the transverse Ising model in Eq. (1) at $\mathrm{\Gamma }=1$ on the triangular lattice. The cross sections of the $J-J^{\prime }-k_{B}T$ diagram [small square corresponds to the parameter region of Fig. 2 right panel] for several fixed $J^{\prime }$ on the plane of $J$ and $k_{B}T$ and for fixed $J=0.1–1.0$ on the plane of $J^{\prime }$ and $k_{B}T$ are shown in two panels. The data points that mark the transition $T_c$ are determined by the crossing of the Binder ratio $g$ for $k_{B}T\lesssim 0.5$ (deep colored symbols) and as the peak position of the specific heat of $L=16$ results at $k_{B}T\gtrsim 0.8$ (light colored symbols). (b) Binder parameter calculated for $L=8,16,32,64$ at $J=0.1$ as a function of $J^{\prime }$ around the phase transition point $J_c^{\prime }\sim 0.47$ at two low temperatures. The right inset shows the Binder crossing as a function of $k_{B}T$ that almost coincides with the crossing point of the middle panel, $T_c=0.125, J^{\prime }\sim 0.4796$. (c) Specific heat $c/k_B$ of $J^{\prime }=0.8, J=0.1$ with $L=16$.

Figure 7

Results of QMC calculations for the square lattice, $J^{\prime }=0$. (a) Static susceptibility ${\chi }_{0;L}$ and (b) relaxation time ${\tau }_L$ extracted from the relaxation function $\mathrm{\Psi }(q=0,t)$ at $L=8,16,32,64$, plotted as functions of $k_{B}T$ for $J=0.32840$ and 0.33547 at QCP and slightly off QCP. Error bars are smaller than the symbols. The solid and broken lines follow $\tau =c_1{\left(k_{B}T\right)}^{-z}$ and ${\chi }_0=c_2{\left(k_{B}T\right)}^{-\gamma /\nu }$. The red solid lines follow the critical exponent $(z,\nu ,\gamma )=(2.02,0.629,1.2379)$ and (2.183,1,1.75) for $J^{\prime }=0.32840$ (top) and 0.33547 (bottom) of the 3D and 2D Ising universality class, respectively, where the former (top panel) yields $c_1=4.225,c_2=4.623$, which is similar to the case of the anisotropic triangular lattice [Fig. 2]. Broken lines are the fitted envelope functions describing the thermodynamic limit with an exponent of $(z,\gamma /\nu )=(2.04,1.87)$ and (2.11,1.74) in the top and bottom panels, respectively. (c, d) Density map of the static susceptibility ${\chi }_0$ and the relaxation time $\tau$ obtained from the Monte Carlo calculation at $J^{\prime }=0$ (square lattice), the latter from the kinetic TRI protocol.

Figure 8

Dynamic susceptibility, $\chi (q,\omega )$, as a function of $k_{B}T$ for three different choices of $J$ an $J^{\prime }$ with $\mathrm{\Gamma }=1$. (a) Square lattice at QCP, $J=0.3284, J^{\prime }=0$; (b) square lattice off QCP, $J=0.3354, J^{\prime }=0$, with $k_B{T}_c=0.1206$; and (c) triangular lattice at QCP, $J=0.1, J^{\prime }=0.48$, with $k_B{T}_c=0.1163$. Panels (b) and (c) follow the 2D Ising universality class.

Figure 9

Dielectric constant ${\epsilon }^{\prime }\left(\omega \right)$ in units of ${\epsilon }_0$ of ($\mathrm{a})\kappa \text{−}{\mathrm{ET}}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ and ($\mathrm{b})\kappa \text{−}{\mathrm{ET}}_2\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{CN}\right)}_2\right]\mathrm{Cl}$, reported in Refs. [5, 6]. Data is provided by the courtesy of Sasaki and Lunkenheimer. Solid lines are the Lorentzian fit [Eq. (10)], and the value of ${\tau }_{\mathrm{lab}}$ for (a) is given in the inset as functions of $(T-T_c)$ with $T_c=6\mathrm{K}$. The solid/broken lines in the inset show the function ${(T-T_c)}^{-z\nu }$ with $z=2.18$ (2D Ising universality) and 3, respectively.