Strong-randomness infinite-coupling phase in a random quantum spin chain

We study the ground-state phase diagram of the Ashkin-Teller random quantum spin chain by means of a generalization of the strong-disorder renormalization group. In addition to the conventional paramagnetic and ferromagnetic (Baxter) phases, we find a partially ordered phase characterized by strong randomness and infinite coupling between the colors. This unusual phase acts, at the same time, as a Griffiths phase for two distinct quantum phase transitions, both of which are of infinite-randomness type. We also investigate the quantum multicritical point that separates the two-phase and three-phase regions, and we discuss generalizations of our results to higher dimensions and other systems.

Figure 1

Schematic ground-state phase diagram of the random quantum Ashkin-Teller chain. For $\epsilon <1$, the paramagnetic and ferromagnetic phases are connected by a direct continuous quantum phase transition. For $\epsilon >1$, they are separated by a partially ordered “product” phase characterized by strong randomness and renormalization group flow towards infinite coupling. The flow is indicated by arrows on the principal axis, $\delta =0$ and $\epsilon =1$, of the multicritical point (MCP).

Figure 2

SDRG steps (a) and (b) decimate a spin variable ${\eta }_i$ but merge two product variables, ${\sigma }_i$ and ${\sigma }_{i+1}$, into a new cluster.

Figure 3

Schematic of the Griffiths dynamical exponent $z$ as a function of $\delta$ for fixed $\epsilon >1$. Rare large magnetization clusters lead to Griffiths singularities associated with the exponent $z_m$ (green dashed line) while the singularities stemming from rare polarization clusters are associated with $z_p$ (red dash-dotted line). Thermal quantities are dominated by the larger of the two exponents $z=max(z_m,z_p)$ which features nonmonotonous behavior (thick solid line).

Figure 4

Average value $\langle ln\epsilon \rangle$ (solid lines) and standard deviation ${\sigma }_{ln\epsilon }$ (dashed lines) of the coupling strength $ln\epsilon$ versus the SDRG length scale $ln\ell$ for different values of the tuning parameter $ln{\epsilon }_{\max }=641,...,646$. The inset shows $ln|\langle ln\epsilon \rangle |$ and $ln{\sigma }_{ln\epsilon }$ for selected curves, giving a multicritical value of $ln{\epsilon }_c=643.75$.

Figure 5

Distance $ln|\langle ln\epsilon \rangle -ln{\epsilon }_c|$ from the multicritical point versus the SDRG length scale $ln\ell$ for different values of the tuning parameter $ln{\epsilon }_{\max }=640.5,...,646$.

Figure 6

${\langle ln\left(\epsilon \right)\rangle }^{1/2}$ vs $\Gamma$ for four different systems on the self-duality line $h_{\mathrm{typ}}=J_{\mathrm{typ}}$ close to the multicritical point ($w=8$ and ${\epsilon }_I=1.001$, 1.002, 1.005, and 1.01). Each curve stems from a single long chain of $2.5\times {10}^7$ sites. The solid lines are fits of the data in the range $\langle ln\epsilon \rangle <0.5$ to $\langle ln\epsilon \rangle =C{(\Gamma -{\Gamma }_0)}^{{\varphi }_{1/2}}$ with ${\varphi }_{1/2}\approx 1.823$; see Eq. (B5).