Low-temperature transition of 2d random-bond Ising model and quantum infinite randomness

Abstract

At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-driven ferromagnet-to-paramagnet transition controlled by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix -- the Hamiltonian of a noninteracting quantum problem -- this RG procedure corresponds to an iterative diagonalization of the quantum Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size: $\log \varepsilon_{\it min}^{-1} \sim L^ψ$. The tunneling exponent $ψ$ is equal to the spin stiffness exponent $θ_c$ characterizing the zero-temperature fixed point.

Figures (8)

• Figure 1: Qualitative phase diagram of the 2d random-bond Ising model, tuned by temperature $T$ and the inverse of the mean $\mu$ of the distribution of couplings, showing ferromagnet (FM), paramagnet (PM), and $T=0$ spin glass (SG) phases. Arrows are RG flows, with colors corresponding to destination phases. Black arrows are critical flows. The thick black line indicating the low-$T$ FM-SG transition below the multicritical point, and particularly the $T=0$ fixed point which controls it, are our focus here.
• Figure 2: Classical and quantum representations. (a) A spin configuration, and the corresponding set of unsatisfied strings (dashed) on the dual lattice that cross unsatisfied bonds and terminate on frustrated plaquettes (crosses). Antiferromagnetic bonds are shown in red. (b) Four unit cells, corresponding to four Ising spins, of the lattice on which the antisymmetric pure-imaginary ${\bf H}$ lives. The four-site "city" $1\{R,U,D,L\}$ surrounds spin $1$, etc. Elements of ${\bf H}$ are shown, with orientation dictating sign: for example, $H_{1D,1R}=-H_{1R,1D}=i$ (all intra-city elements are $i$ in the direction of arrows), and $H_{1U,2D}=-H_{2D,1U}=i\tanh(\beta J_{12})$.
• Figure 3: Building up the ground state by adding in frustration. Frustrated plaquettes are labeled by letters. Dashed lines cross strings of unsatisfied bonds that connect frustrated plaquettes. $\zeta_{pq}$ is the cost of directly matching plaquettes $p$ and $q$ in the ground state, calculated as the sum of the magnitudes of the $J$'s along the least costly path between the plaquettes. In this example, the first two RG steps, $r_1$ and $r_2$, are formed by joining up plaquettes $\{ a,b\}$ and $\{ c,d\}$ along the blue and green paths respectively, indicated in the upper left. When frustrated plaquettes $e$ and $f$ are added, various options are shown for the path (in red) along which the added matching flips the candidate ground state, with new configurations of unsatisfied bonds crossed by the black dashed lines. Of the candidates shown, the red path with the smallest value of $r_3$ is the one that will be selected in the ground state at this stage of the RG.
• Figure 4: Scaling of the optimized defect energy $r_{\it max}$ with $L$. In the spin glass ($\mu=0$), $r_{\it max}$ grows more slowly than $\log L$. In the ferromagnet ($\mu=1.5$), $r_{\it max}$ grows as $\log L$. (Blue and red dashed lines are $\sim\log L$.) At the critical point ($\mu=1.0307$), $r_{\it max}$ grows as $L^{0.16}$ with a constant correction to scaling (dashed black line fit to the gray sequence of points). Error bars are smaller than marker sizes. The inset shows, at criticality, the scaled distribution of $r_{\it max}$ minus its mean, for several $L$ between $64$ and $512$. Averages are over at least $3\times 10^4$ disorder realizations.
• Figure 5: Left: the dual Kasteleyn lattice and, in red, a perfect matching on it, which maps to the displayed spin configuration. Each four-site clique (Kasteleyn city) corresponds to a plaquette. The grey regions, containing spins that are overturned relative to the uniform state, are surrounded by domain walls which are closed polygons on the dual lattice that also define dimer-hosting intercity bonds. Right: the primal Ising lattice (solid lines). Green bonds are antiferromagnetic, and their positions define the gauge invariant locations of the frustrated plaquettes (crosses). The same state is represented by strings of unsatisfied bonds (dashed) that in the ground state constitute a minimal-weight perfect matching of the frustrated plaquettes.