Universal crossover from ground-state to excited-state quantum criticality

We study the nonequilibrium properties of a nonergodic random quantum chain in which highly excited eigenstates exhibit critical properties usually associated with quantum critical ground states. The ground state and excited states of this system belong to different universality classes, characterized by infinite-randomness quantum critical behavior. Using strong-disorder renormalization group techniques, we show that the crossover between the zero and finite energy density regimes is universal. We analytically derive a flow equation describing the unitary dynamics of this isolated system at finite energy density from which we obtain universal scaling functions along the crossover.

Figure 1

Decimation rules for the disordered Fibonacci anyon chain. We decimate $-J_i{P}_i^A$ and set $J_L=J_{i-1}$ and $J_R=J_{i+1}$. Left: Strong bond in the chain. Right: The Hamiltonian after the decimation. The triplet (singlet) sector decimates 1 (2) spin(s) and renormalizes the Hamiltonian.

Figure 2

Schematic tree of many-body eigenstates generated by RSRG-X. Every eigenstate corresponds to a branch in this tree of choices between singlet and triplet channels at each decimation. The ground-state branch (blue path) is chosen by minimizing the energy at each RG step, whereas typical many-body eigenstates at $T=\infty$ (red path) can be obtained by performing a random walk on the tree.

Figure 3

Excited-state RG flow for the random Fibonacci chain. At zero temperature, the system is described by two infinite-randomness random singlet fixed points depending on the ratio of ferromagnetic bonds in the chain, with different $\psi$ exponents characterizing the scaling between length and energy $\mathrm{\Gamma }=-lnE\sim L^{\psi }$. Considering the system at finite energy density or temperature is a relevant perturbation in the renormalization group sense, and the system flows to a stable quantum critical glass (QCG) fixed point. In this paper, we analyze the universal finite-temperature crossover between the $T=0$ and $T=\infty$ fixed points.

Figure 4

Simplified RSRG-X tree with four eigenstates (branches) illustrating the difference between Monte Carlo and local-node finite-temperature samplings (see text).

Figure 5

Local-node probability versus energy of ${10}^3$ random eigenstates from a disordered Fibonacci chain with ${10}^3$ spins with the initial disorder strength $W=2$. Probabilities are calculated from the local-node probabilities with $\beta$ equals 1 (blue) and 0 (red). The splitting in the $\beta =0$ case is due to the finite-size effect (i.e., goes away when taking number of spins to infinity), which is completely removed when using the exact finite-size single fusion probability, $F_{N-2}/F_N$, instead of ${\mathrm{\Pi }}_s$. The fitted linear curve has a slope $\approx 0.99$. Left inset: Plot of local-node sampling probabilities at $\beta =0$ but with a wrong ${\mathrm{\Pi }}_s$ (0.1 larger than the correct one) showing a nonphysical probability distribution. Right inset: Plot of local-node sampling probabilities at $\beta =1$ sampled over the full spectrum.

Figure 6

Plot of the “error” in effective partition function versus the inverse disorder strength. This “error” is computed as the averaged value of standard deviations of effective partition functions associated with random eigenstates. We sample over ${10}^3$ eigenstates in each disorder realization and 500 disorder realizations for each $W$ with ${10}^3$ spins. Error bars are always smaller than the size of the symbols.

Figure 7

Crossover length scales $L$ versus $\frac{1}{ln\beta }$. At small but finite temperature, the short length scale physics is dominated by the ground-state fixed-point phase. After passing through the crossover region, the physics is eventually dominated by the infinite-temperature infinite-randomness fixed point. The crossover boundaries are determined by computing at which length scale ${\mathrm{\Pi }}_s(\beta ;\mathrm{\Gamma })$ approaches or departs from its UV (ultraviolet)/IR (infrared) value within one percent, using Eq. (7) to convert $\mathrm{\Gamma }$ into a distance $L$.

Figure 8

Universal scaling function $f\left(x\right)$ given by Eq. (9) that controls the typical decay of correlation functions. Data points are obtained by evaluating (7), (8) numerically for various temperatures. The dashed black curve corresponds to Eq. (10).

Figure 9

A system with $N+1$ anyons on a 2D surface with total topological charge $S$.

Figure 10

Basis vector of the Fibonacci anyon Hilbert space of $N+1$ anyons $S_1,\cdots ,S_{N+1}$, labeling “legs” of the basis. Dots are drawn to provide a connection with dots in Fig. 9. Each vertex satisfies the admissibility condition (fusion rule).

Figure 11

(a) Quantum dimension of the Fibonacci anyon system. (b) Useful identities including the “no tadpole” rule. (c) Nontrivial “F moves” in the Fibonacci anyon system.

Figure 12

Projection operators. Left: Definition of $P_i^A$ which projects spins at $i$ and $i+1$ onto the singlet (trivial anyon). Right: Definition of $P_i^F$ which projects spins at $i$ and $i+1$ onto the triplet ($\tau$ anyon).

Figure 13

Change in logarithmic bond strength upon changing the RG scale from $\mathrm{\Gamma }$ to $\mathrm{\Gamma }+d\mathrm{\Gamma }$. 0 denotes the strongest bond at $\mathrm{\Gamma }$ which is decimated away at $\mathrm{\Gamma }+d\mathrm{\Gamma }$. Left: Bond strengths at $\mathrm{\Gamma }$. Right: Bond strengths at $\mathrm{\Gamma }+d\mathrm{\Gamma }$. Due to the change in the energy scale, logarithmic variables are shifted by $d\mathrm{\Gamma }$. Diagrams at middle and bottom show how the nearby bonds change as a result of a triplet (ferromagnetic) and a singlet (antiferromagnetic) decimation.