Rounding of a first-order quantum phase transition to a strong-coupling critical point

We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the $N$-color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme.

Figure 1

Schematic of the renormalization-group flow diagram in the disorder-coupling strength parameter space. For $\epsilon <{\epsilon }_c$ (left arrows), the critical flow approaches the usual Ising infinite-randomness critical point of Ref. 9. For $\epsilon >{\epsilon }_c$ (right arrows), we find a distinct infinite-randomness critical point with even stronger thermodynamic singularities.

Figure 2

Spectrum of the unperturbed Hamiltonian (2) as a function of the number of colors flipped with respect to the ground state $|\to ,\to ,...,\to \rangle$. As long as $T\lesssim \Omega /{\epsilon }_h$, the pseudo ground state $|{\varphi }_0^{\prime }\rangle =|\leftarrow ,\leftarrow ,...,\leftarrow \rangle$ can be neglected when computing observables (stage 1 of the RG). When $T\gtrsim \Omega /{\epsilon }_h$, $|{\varphi }_0\rangle$ and $|{\varphi }_0^{\prime }\rangle$ become effectively degenerate implying that both states need to be taken into account (stage 2 of the RG).

Figure 3

(a) Decimating a site results in a renormalized bond (characterized by $\tilde{k}$ and ${\tilde{\epsilon }}_J$) between its neighbors, and it introduces an extra binary “sector” degree of freedom represented as an Ising spin $\tilde{\zeta }=\pm 1$ in an external field $\tilde{\omega }$ [see Eq. (7)]. (b) Decimating a bond results in a renormalized site characterized by $\tilde{g}$, ${\tilde{\epsilon }}_h$ and another sector degree of freedom.

Figure 4

Phase diagram of the $N$-color quantum Ashkin-Teller model as function of $r=\ln (h_{\mathrm{typ}}/J_{\mathrm{typ}})$ and the intercolor coupling $\epsilon$ at fixed disorder strength. The critical line is located at $r=r_c=0$ ($J_{\mathrm{typ}}=h_{\mathrm{typ}}$) as expected from the duality transformation. PM and FM denote the conventional paramagnetic and ferromagnetic (Baxter) phases. OG and DG denote the ordered and disordered Griffiths phases.