Quantum critical behavior of diluted quasi-one-dimensional Ising chains

TL;DR

This work analyzes quantum criticality in a quasi-one-dimensional transverse-field Ising magnet with quenched site dilution, modeled as a 3D anisotropic TFIM mapped to a 4D classical Ising system with columnar disorder. Using Wolff-Metropolis Monte Carlo simulations, the authors perform finite-size scaling to extract critical exponents and demonstrate activated scaling indicative of an infinite-randomness fixed point. They find ν ≈ 0.90, β/ν ≈ 1.94, and ψ ≈ 0.33, consistent with the disordered 3D TFIM universality class rather than the 1D case, despite strong anisotropy. These results imply that dilution breaks the effective one-dimensional criticality by fragmenting chains, driving the transition into a three-dimensional disordered regime with measurable experimental consequences for doped quasi-1D magnets.

Abstract

CoNb$_2$O$_6$ is a unique magnetic material. It features bulk three-dimensional magnetic order at low temperatures, but its quantum critical behavior in a magnetic field is well described by the one-dimensional transverse-field Ising universality class. This behavior is facilitated by the structural arrangement of magnetic Co$^{2+}$ ions in nearly isolated zig-zag chains. In this work, we investigate the effect of random site dilution on the critical properties of such a quasi-one-dimensional quantum Ising system. To this end, we introduce an anisotropic site-diluted three-dimensional transverse-field Ising model. We find that site dilution leads to unconventional activated scaling behavior at the quantum phase transition. Interestingly, the critical exponents of the quantum critical point are in good agreement with those of the disordered three-dimensional transverse-field Ising universality class, despite the strong spatial anisotropy. We discuss the generality our findings as well as implications for experiments.

Figures (10)

• Figure 1: Crystal Structure of CoNb$_2$O$_6$. Atom positions are taken from Ref. matproj.
• Figure 2: Equilibration test: Order parameter $m$ and energy $E$ versus Monte Carlo time (number of Monte Carlo sweeps) for a single sample of large system with optimized geometry: $L_\tau = 3080$, $L_s = 280$, $L_\perp = 7$, comparing hot and cold initial conditions. The interactions are $J_s = J_\tau = 100J_\perp$, and the dilution is $p=0.1$. The classical temperature, $T=2.22$, is close to the phase transition.
• Figure 3: Normalized Binder cumulant $g_{\text{av}}/g_{\text{av}}^{max}$ at $L_s=160$, $T = 2.21$, $p = 0.10$, $J_\perp = 0.01$ vs. system size $L_\tau$ in imaginary time direction. Error bars indicate the uncertainty in $g_{\text{av}}$ obtained from the standard deviation over disorder realizations. Dashed vertical line indicates maximum obtained via fitting with (\ref{['gavfit']}).
• Figure 4: Binder cumulant $g_{\text{av}}$ vs. $\ln(L_\tau)$ for several $L_s$ and dilution $p=0.1$. The ratio $L_s/L_\perp$ is fixed at 40. The interaction strengths are $J_s = J_\tau = 100J_\perp$. (a) $T =2.213$. (b) $T = 2.215$. (c) $T =2.217$.
• Figure 5: Binder cumulant $g_{\text{av}}$ vs. temperature $T$ for interaction strengths $J_s = J_\tau = 1 = 100J_{\perp}$ and the dilution is $p=0.1$. The sample geometry is given by $L_s = 40L_{\perp}$. The imaginary time size corresponds to the optimal shapes $L_{\tau} = L_{\tau}^\text{max}$ (values given in Table \ref{['tab:Ltau']}). Statistical errors are about a symbol size or smaller. The lines between the data points serve as visual aids only.