$1/f{}^{\alpha }$ noise and generalized diffusion in random Heisenberg spin systems

We study the “flux-noise” spectrum of random-bond quantum Heisenberg spin systems using a real-space renormalization group (RSRG) procedure that accounts for both the renormalization of the system Hamiltonian and of a generic probe that measures the noise. For spin chains, we find that the dynamical structure factor $S_q\left(f\right)$, at finite wave vector $q$, exhibits a power-law behavior both at high and low frequencies $f$, with exponents that are connected to one another and to an anomalous dynamical exponent through relations that differ at $T=0$ and $T=\infty$. The low-frequency power-law behavior of the structure factor is inherited by any generic probe with a finite bandwidth and is of the form $1/f^{\alpha }$ with $0.5<\alpha <1$. An analytical calculation of the structure factor, assuming a limiting distribution of the RG flow parameters (spin size, length, bond strength) confirms numerical findings. More generally, we demonstrate that this form of the structure factor, at high temperatures, is a manifestation of anomalous diffusion which directly follows from a generalized spin-diffusion propagator. We also argue that $1/f$-noise is intimately connected to many-body-localization at finite temperatures. In two dimensions, the RG procedure is less reliable; however, it becomes convergent for quasi-one-dimensional geometries where we find that one-dimensional $1/f^{\alpha }$ behavior is recovered at low frequencies; the latter configurations are likely representative of paramagnetic spin networks that produce $1/f^{\alpha }$ noise in SQUIDs.

Figure 1

(a) Phase diagram of the strongly disordered Heisenberg spin chain (with $1/\left|J\right|$ distribution of initial couplings in a range $\left|J\right|\in [1,e^D])$ extrapolated to finite temperatures using RSRG results at zero and infinite temperatures; the color scheme encodes the variation of the dynamical exponent $z$ as a function of the bias ${\eta }_i$ [initial proportion of F/AF (−1/1) bonds in chain]. (b) The low-frequency noise power law $\alpha$ and (c) the dynamical critical exponent $z$ are plotted against ${\eta }_i$ at $T=0$ (data-points and error bars) and $T=\infty$ (flat line with dashed lines indicating error), for disorder strength $D=3$ (see main text). At $T=0$, the purely F system harbors heavily damped spin wave excitations whose localization length diverges as $1/\omega$ in the zero-frequency limit. The purely AF system, which is known to flow to a infinite-randomness fixed point, exhibits $1/\omega$ noise. Intermediate biases (nonhatched region) flow to strong-randomness fixed points accompanied by $1/{\omega }^{\alpha }$ noise spectra. The hatched region is inaccessible to RSRG. At high temperatures $\left(T\gg \omega \right)$, the $1/{\omega }^{\alpha }$ noise spectrum is generic to strongly disordered Heisenberg chains and can be interpreted in the context of generalized diffusion. The crossover from zero-temperature to infinite-temperature behavior occurs at $T/\omega \sim 2^{z_0}$, where $z_0$ is the dynamical exponent found for the zero temperature system.

Figure 2

(a) Two-spin dynamics. Two spins $A$ and $B$ strongly coupled to one another revolve around their net moment ${\vec{S}}_{+}={\vec{S}}_A+{\vec{S}}_B$ with a large frequency $J|{\vec{S}}_{+}|$, while ${\vec{S}}_{+}$ moves slowly owing to weak interactions $J_A^{\prime }$ and $J_B^{\prime }$. ${\vec{S}}_{-}={\vec{S}}_A-{\vec{S}}_B$ is effectively decoupled from the slow dynamics of ${\vec{S}}_{+}$ and results in the “noise” evaluated at this RG step. (b) Demonstration of RG rules. The strongly coupled spin-pair are decimated to form an effective spin (with quantized net moment $s_{+})$, and Heisenberg couplings are renormalized from $J_{A\left(B\right)}$ to $J_{A^{\prime }\left(B^{\prime }\right)}$. The effective spin couples to the probe with amplitude $g_{+}$. If a singlet is formed, both spins are integrated out, providing an effective interaction $J^{\prime }$ between the neighboring spins. (c) Scaling behavior at $T=\infty$ (at $T=0$, the bonds, composed of long-range singlets scale as the cluster size). Size of clusters $n$, their spin size $s$ and the time-scale of their dynamics $1/\omega$ are connected through the anomalous dynamical exponent $1/\beta$. Harmonic probes (illustrated in blue) measure a low-(high-)frequency noise-exponent $\alpha$ or ${\alpha }^{\prime }$ in the regimes where the probe length $1/q$ is smaller (greater) than the clusters whose dynamics they probe.

Figure 3

RG-determined spectrum (red) compared with exact diagonalization determined spectrum (blue) for a particular 12-site random spin chain with initial $1/\left|J\right|$ distribution, disorder strength $D=3$, and equal mix of F/AF bonds.

Figure 4

Scaled dynamical structure factor (a) ${\tilde{S}}_q^0\left(\tilde{\omega }\right)=x^{1/2-1/\beta }{S}_q^0\left(\omega x^{1/\beta }\right)$ at $T=0$, (b) ${\tilde{S}}_q^{\infty }\left(\tilde{\omega }\right)=x^{-1/\beta }{S}_q^{\infty }\left(\omega x^{1/\beta }\right)$ at $T=\infty$ for $q\in 2\pi /\left[40\right(\text{yellow}),80(\text{cyan}),120(\text{green}),400(\text{blue}),\text{and}800(\text{red}\left)\right]$, and $x=(2\pi /40)/q$; the results shown correspond to a system of an equal proportion of F/AF bonds, and disorder strength $D=3$; (c) scaling relations at $T=0$ and $T=\infty$ between the power laws $\alpha$ and ${\alpha }^{\prime }$ determining the high- and low-frequency dependence of $S_q\left(\omega \right)$ as given in Eq. (1). $\alpha$ and ${\alpha }^{\prime }$ were plotted for $D=3$ and ${\eta }_i=[1,0.75,0.5,0.25,0,-0.25,-0.5,-0.75]$ for $T=0$, while, at $T=\infty$, the results are independent of ${\eta }_i$, and the data-points correspond to $D=[1,3,9]$. Decreasing $D$ and decreasing ${\eta }_i$ correspond to smaller values of $\alpha$. At $T=\infty ,{\alpha }^{\prime }\approx 1+2(1-\alpha )$, while at $T=0,{\alpha }^{\prime }\approx 1+3(1-\alpha )$; these relations are plotted as dashed lines. The optimal scaling collapse in (a) (at $T=0)$ and (b) (at $T=\infty )$ was found for $\beta =0.40$ and $\beta =0.36$ respectively. These values are in accordance with analytically determined relation to the low-frequency noise exponent $\alpha$, i.e., $\beta =2(1-\alpha )$ at $T=0$ and $\beta =1-\alpha$ at $T=\infty$.

Figure 5

(a) The $T=\infty$ dynamical structure factor $S_q^{\infty }\left(\omega \right)$ is plotted as a function of frequency for various $q=2\pi /l$ $\left(l\in [40,800]\right)$, system size $L=15000$. (b) $S_q^{\infty }\left(\omega \right)\sim 1/q$ for $\omega \ll q^{1/\beta }$ is evident from the scaling collapse at low frequencies. (c)$S_q^{\infty }\left(\omega \right)\sim q^2$ for $\omega \gg q^{1/\beta }$ is evident from the scaling collapse at high frequencies.

Figure 6

(a) The $T=0$ dynamical structure factor $S_q^0\left(\omega \right)$ is plotted as a function of frequency for various $q=2\pi /l$ $\left(l\in [40,800]\right)$, system size $L=15000$. $S_q^0\left(\omega \right)\sim \text{const.}$ for $\omega \ll q^{1/\beta }$ is evident from the collapse at low frequencies (without any scaling). (b) $S_q^0\left(\omega \right)\sim q^2$ for $\omega \gg q^{1/\beta }$ is evident from the scaling collapse at high frequencies.

Figure 7

The structure factors $S_q^{\infty }\left(\omega \right)$ $(T=\infty$, blue) and $S_q^0\left(\omega \right)$ $(T=0$, red) are plotted simultaneously, for initial bias ${\eta }_i=0$, disorder strength $D=3$, system size $L=15000$, and $q=2\pi /160$. The powers ${\alpha }^{\prime }$ and $\alpha$ are also indicated for the two cases with the corresponding subscripts. They approximately satisfy the respective scaling relations for zero and infinite temperature cases.

Figure 8

The average (a) bias, (b) noise frequency, and (c) “fidelity” is plotted against the RG step for a system of size $L=10000$. Initial bias ${\eta }_i=0.5$, temperature $T=0$, disorder strength $D=1$.

Figure 9

The fidelity is plotted as a function of RG steps at $T=\infty$ for a system of size $L=15000$ and initial disorder strength $D=1$.

Figure 10

Distribution of the gap $\mathrm{\Delta }$ after 98% of the spins were eliminated. The system has Initial bias ${\eta }_i=0.5$, temperature $T=0$, disorder strength $D=1$. The noise power law $\alpha$ and the power law of the distribution of the gap are not the same.

Figure 11

The power $\gamma$ (red) of the gap distribution $P_{\mathrm{\Delta }}\left(\mathrm{\Delta }\right)\sim 1/{\mathrm{\Delta }}^{\gamma }$ and the final bias ${\eta }_f$ (blue) are plotted as a function of the initial bias ${\eta }_i$ of a $1/\left|J\right|$ distribution with disorder strength $D=3$.

Figure 12

The distribution function of lowest gapped excitation energies, $P_L\left(\epsilon \right)$ is obtained for two cases corresponding to different initial AF/F composition of the spin chain: (a) ${\eta }_i=0.5$, (b) ${\eta }_i=0$, and system sizes $L=250$ (yellow), $L=500$ (orange), $L=1000$ (blue). Solid red line is the fit according to the Fréchet distribution. See Appendix pp7 for more details.

Figure 13

The distribution $D\left(n\right)$ of the cluster sizes $n$ at various (different colors) stages of the RG for $L=15000$ spin chain at $T=0$, and initial bias ${\eta }_i=0.5$. The features of the result are not dependent on the specifics of the system parameters. The important facet here is that $D\left(n\right)$ has appreciable weight at all stages of the RG in the $n\to 0$, while it decays beyond a certain scale $n_0$, which depends on the energy scale at which the RG was terminated. This justifies the form of the distribution we use in the main text to calculate the $T=0$ finite-$q$ susceptibility of the spin chain.

Figure 14

Scatter plot of the logarithm of the couplings ${log}_{10}\left(J\right)$ of effective spins of size $s$ remaining after the RG procedure has reached convergence operating on a $L=5000$ spin chain at $T=\infty$. The maximum coupling is seen to decrease with the spin size $s$.

Figure 15

Various features of the simulations on two-dimensional system of size $50\times 50$ are shown. (a) The noise spectrum for low frequencies has a power law tail $S_q\left(\omega \right)\sim 1/\sqrt{\omega }$ $[q=(\pi ,\pi )$ here]. (b) The distribution of gaps is fairly broad. (c) The fidelity $f$ does not rise over the course of RG. (d) Approximate scaling collapse using the generalized-diffusion ansatz using exponent $\beta =0.55\pm 0.05$ for wave vectors $q=(2\pi n/L,2\pi n/L)$ with $n\in \left[2\right(\text{blue}),4(\text{red}),6(\text{magenta}),8(\text{cyan}),10(\text{black}),\text{and}12(\text{green}\left)\right]$. and $L=50$. The lower (upper) frequency power laws are found to be $\alpha \approx 0.5\pm 0.05$ and ${\alpha }^{\prime }\approx 1.75\pm 0.05$; it is unclear whether there is data for a sufficiently large range of frequencies to extract the high-frequency power law reliably, especially since it exhibits a drift to higher values as we go to smaller wave vectors.

Figure 16

The energy spectrum of a small $3\times 3$ lattice spin system (disorder strength $D=3)$ is calculated using exact diagonalization (blue) and compared with the RG generated spectrum (red).

Figure 17

(a) The finite-$q$ structure factor $S_q\left(\omega \right)$ $[q=(2\pi /25,\pi \left)\right]$ as a function of frequency; (b) the fidelity $f$ and (c) the noise frequency are plotted as a function of the RG steps for a two-dimensional strip of spins of lattice size $500\times 6$, with $D=3$ at $T=\infty$. There are $\sim 50$ clusters remaining when these are of the size of the width of the spin network; these remaining clusters behave as a one-dimensional network of effective spins and produce the anomalous $1/{\omega }^{\alpha }$ noise. Here we see that about 50 (red dashed lines) spins remain when the maximum frequency $\omega \sim 2\times {10}^{-2}$ and this approximately agrees with the frequency below which $1/{\omega }^{\alpha }$ noise behavior is observed. Above this frequency, a $\sim 1/\sqrt{\omega }$ tail is observed as in the case of the square network of spins.

Figure 18

(a) The inverse participation ratio is finite [seen by scaling with system sizes $L=500$ (yellow), $L=1000$ (blue), and $L=2000$ (red) ] and scales with frequency as $\text{IPR}\left(\omega \right)\sim \omega$ (dashed line fit); (b) spin-wave peak seen in the structure factor $S_q\left(\omega \right)$; (c) the peak occurs at frequency ${\omega }_c\left(q\right)\sim q^2$ (red line fit corresponds to $q^{1.95})$; (d) the width (calculated as inverse height of the peak) scales as $\mathrm{\Gamma }\left(q\right)\sim q^2$ (red line fit corresponds to $q^{2.2})$.

Figure 19

An example localized spin-wave wave-function for the disordered purely F chain of size $L=500$.

Figure 20

(a) The inverse participation ratio decreases to zero in the thermodynamic limit [seen by scaling with system sizes $20\times 20$ (yellow), $30\times 30$ (blue) and $40\times 40$ (red)]; (b) spin-wave peak seen in the structure factor $S_q\left(\omega \right)$; (c) the peak occurs at frequency ${\omega }_c\left(q\right)\sim q^2$ (red line fit corresponds to $q^{2.1})$; (d) the width (calculated as inverse height of the peak) scales as $\mathrm{\Gamma }\left(q\right)\sim q^4$ (red line fit corresponds to $q^{3.95})$.

Figure 21

Infinite-temperature $S_q^{\infty }\left(\omega \right)$ for classical spins of magnitude $\left|\vec{S}\right|=1/2$ is plotted (blue) and compared with the quantum result (red) at infinite temperature. $L=15000,D=3$, and ${\eta }_i=0$.

Figure 22

Comparison of the structure factor obtained for classical spin chains using direct integration of Bloch-equations and RSRG rules. The direct simulation is computationally hard to perform over very long time scales, but the agreement is very reasonable for the simulated frequencies.