Quantum criticality in Ising chains with random hyperuniform couplings

We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\alpha$ is tuned. For $\alpha =0$, one recovers the familiar infinite-randomness critical point. For $0<\alpha <1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\alpha =0$ are absent at any $\alpha >0$. When $\alpha >1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.

Figure 1

Upper panel: simple model with strongly hyperuniform ($\alpha =2$) bond randomness. Each site is displaced by an independent random amount ${\xi }_i$ from its equilibrium position; this leads to bonds $J_i$ that are evidently strongly hyperuniform, in the sense that the variance of ${\sum }_{i=1}^l{J}_i$ tends to a constant independent of $l$. Lower panel: phase diagram of the random hyperuniform TFIM. Away from criticality, all excitations are localized for any randomness; however, the universality class of the critical point changes as the hyperuniformity parameter $\alpha$ is varied. A line of infinite-randomness critical points, with continuously varying exponents, terminates at a multicritical point, beyond which disorder is dangerously irrelevant (the critical Ising insulator).

Figure 2

Critical length scales and dynamics in the strongly hyperuniform Ising model. Left: the length-energy scaling of two relevant scales at the Ising critical point: the clean correlation length ${\xi }_{\mathrm{clean}}\sim 1/\left|E\right|$ and the localization length ${\xi }_{\mathrm{loc}}\sim 1/{\left|E\right|}^{\alpha }$. When $\alpha >1$ the localization length diverges faster, so it appears asymptotically larger than the correlation length in the low-energy critical properties. The critical properties are then controlled by the clean Ising correlation length. Right: expansion of an initially localized wave packet along the CII line at $\alpha =2$. The bulk of the wave packet is localized (corresponding to high-energy components of the wave packet), but there is a rapidly attenuating component that propagates at the light cone, depositing weight as it goes.

Figure 3

Weakly hyperuniform systems. Critical exponents at the Ising transition vs hyperuniformity parameter $\alpha$, extracted from the strong-disorder renormalization group. Upper panel shows the length-time scaling $logt\sim l^{\psi }$, with the analytically exact results for $\alpha =0$ [8] in agreement with the numerical results for $0<\alpha \le 1$. Lower panel plots the average order parameter scaling dimension $\left[〈{\tau }_i^x{\tau }_{i+l}^x〉\right]\sim l^{-2{\mathrm{\Delta }}_{\sigma }}$, extracted from the correlations, and the scaling of the magnetic moment ${\mu }_l\sim l^{\psi \varphi }$. The relation ${\mathrm{\Delta }}_{\sigma }=1-\psi \varphi$ is seen to hold.

Figure 4

Zero-mode wave-function profiles $|c_i|$ [see Eq. (5)] in a representative sample in the weakly (left) and strongly (right) hyperuniform regimes.

Figure 5

Density of states for strongly hyperuniform systems. At low energies the density of states tends to a finite value as in the clean case. The hyperuniform modulation gives rise to a subleading correction in the form of a nonanalytic cusp. The density of states was computed using the recursive method of Ref. [58] for parameters $L={10}^7, s=\frac{3}{16}$, and $n=40$ disorder realizations.

Figure 6

Top: spreading of a wave packet at the critical point for $\alpha =2$. A well-defined ballistic front exists and moves out with the clean critical velocity; however, the weight at the front attenuates with time: its height shrinks as $1/t$ (dashed line) and it broadens as $\sqrt{t}$ (Supplemental Material, Appendix A [73]). Bottom: regimes of $\{{\gamma }_{i+r}\left(t\right),{\gamma }_i\left(0\right)\}$. The autocorrelator is small outside the light cone, grows to $r^{-2/\alpha }$ when $r\sim t$, then saturates at later times to a value $\{{\gamma }_{i+r}\left(\infty \right),{\gamma }_i\left(0\right)\}=1/r^{1+1/\alpha }$. Other correlation functions can exhibit multiples $p=1,2,\cdots$ of this basic power law.

Figure 7

Minimum in the inverse localization length $1/\xi \left(E\right)$: disorder averaged $1/\xi$ is plotted versus $E$. At criticality the localization length $\xi$ is monotonically increasing with decreasing excitation energy $E$. When one detunes away from criticality by $\delta$, the localization length is minimized at an energy $E=O\left(\delta \right)$. Data for system sizes $L=5000, \alpha =1.5, s=\frac{3}{16}$ error bars smaller than plot points.

Figure 8

Density of states for weakly hyperuniform systems. At low energies the integrated density of states scales as $N{\left(E\right)}^{(\alpha -1)/2}=c_{1}logE+c_0$ (for constants $c_0,c_1$), generalizing the familiar $N\left(E\right)\sim 1/{log}^{2}E$ of the iid ($\alpha =0$) case. Mean values of $N\left(E\right)$ are calculated by disorder averaging; statistical error is shown by error bars. Parameters: $L={10}^7, s=\frac{3}{16}$.

Figure 9

Susceptibility of a spin at the edge of the chain vs distance from the transition $\delta$, for the cases $\alpha =0$ (independent random couplings), $\alpha =0.5$ (weakly hyperuniform), and $\alpha =1.5$ (strongly hyperuniform). Evidently, even for weakly hyperuniform couplings, the susceptibility approaches linear behavior with field even very close to the transition, suggesting the absence of a Griffiths phase. Parameters: $L=2000, s=\frac{3}{16}$.

Figure 10

Strong-disorder RG exponents. Upper panel: flow of typical coupling scale $\left[\zeta \right],\left[\beta \right]$ vs length scale $l$ under SDRG for various values of $\alpha$. The power law $\left[\zeta \right],\left[\beta \right]\sim l^{\psi }$ is exhibited with exponent $\psi =(1-\alpha )/2$. Lower panel: average correlator $C_{xx}\left(r\right)=\langle {\tau }_i^x{\tau }_{i+r}^x\rangle$ as a function of spacing. The decay is algebraic, and fits to the form $C_{xx}\left(r\right)\sim 1/r^{2(1-\psi \varphi )}$ where the typical moment at scale $l$ scales as $\mu \sim l^{\psi \varphi }$ (scaling of moments is shown in the Supplemental Material, Appendix C [73]). Parameters: $L={10}^8, s=\frac{3}{16}$.

Figure 11

Typical (upper) and mean (lower) order-parameter correlation functions $〈{\tau }_i^x{\tau }_{i+r}^x〉$ as a function of distance for different $\alpha$ (legend inset). The behavior of the typical correlation function is consistent with a power law in the strongly hyperuniform case ($\alpha >1$) and with a stretched exponential in the weakly hyperuniform case ($0<\alpha <1$). The mean correlator decays with the clean Ising exponent in the strongly hyperuniform case, but clearly faster in the weakly hyperuniform case. In the weakly hyperuniform case we do not see a clean power law at large scales; it seems that our data here are still dominated by typical rather than mean behavior. Parameters: $L=2000$ periodic chain, $s=\frac{3}{16}.$

Figure 12

Histograms of the order-parameter correlation function $〈{\tau }_i^x{\tau }_{i+r}^x〉$ for exponentially spaced values of $r$, for weakly hyperuniform (upper) and strongly hyperuniform (lower) systems. In the former case, the histograms broaden strongly, while in the latter case they do not broaden. Parameters: same as Fig. 11.

Figure 13

Entanglement entropy $S_{\ell }$ of the region $1...\ell$ as a function of $\ell /L$. The predicted growth is $S_{\ell }\sim (c/6)log\left(\ell \right)$ (solid black line) with the coefficient $c$ set by (16). The collapse of data for different $L$ (legend inset) onto the predicted line for $\ell \ll L$ validates the conjectured form (16). The dashed black lines exhibit the expected growth of the uncorrelated ($\alpha =0$) and clean Ising ($\alpha =1$) cases for reference. The panels correspond to different values of $\alpha$ (values inset), with further data given in the Supplemental Material [73].