Extended Ashkin-Teller transition in two coupled frustrated Haldane chains

Abstract

We report an extremely rich ground state phase diagram of two spin-1 Haldane chains frustrated with a three-site exchange and coupled by the antiferromagnetic Heisenberg interaction on a zig-zag ladder. A particular feature of the phase diagram is the extended quantum phase transition in the Ashkin-Teller universality class that separates the plaquette phase, which spontaneously breaks translation symmetry, and the uniform disordered phase. The former is connected to the Haldane phase, stabilized by large inter-chain coupling, via the topological Gaussian transition. Upon decreasing the inter-chain interactions, this intermediate disorder phase vanishes, giving place to a dimerized phase separated from the plaquette phase on one side via a non-magnetic Ising transition and from the Haldane phase on the other side by a topological weak first-order transition. Finally, in the limit of two decoupled chains, we recover a quantum critical point that corresponds to two copies of the Wess-Zumino-Witten $\mathrm{SU(2)}_2$ criticality with a total central charge $c=3$.

Figures (14)

• Figure 1: Sketch of a zig-zag ladder with interaction terms appearing in Eq.\ref{['eq:Hamiltonian']} marked: nearest-neighbor coupling $J_1$, next-nearest-neighbor coupling $J_2$, and the three body term $J_{3,\mathrm{leg}}$ acting on the legs of the ladder.
• Figure 2: (a) Phase diagram of the spin-1 $J_1-J_2-J_{3,\mathrm{leg}}$ chain defined in Eq. \ref{['eq:Hamiltonian']}. Each phase has a simple valence-bond-solid (VBS) representation sketched on the diagram. A topologically trivial disordered phase (green) is separated on one side from the $\mathbb{Z}_4$ ordered phase (blue) by an Ashkin-Teller transition (red circles), and on the other side from the topological Haldane phase (magenta) by a Gaussian transition. Both transitions are characterized by the central charge $c=1$. The Haldane phase vanishes for $J_1 \lesssim 2$, and the transition between the $\mathbb{Z}_4$ ordered and Haldane phase becomes first order (grey circles) up to the point where intermediate $\mathbb{Z}_2$ ordered dimerized phase (red) appears. This is clearly visible in panel (b) that zooms in on $J_1$ small. The $\mathbb{Z}_2$ and $\mathbb{Z}_4$ ordered phases are separated by a continuous Ising transition. Haldane phase is separated from the dimerized and NNN-Haldane phases by a pair of first order transitions. At $J_1=0$ these transitions merge together with the Ising critical line into a multi-critical point that corresponds to two copies of WZW SU(2)$_2$ critical theory, separating the leg-dimerized phase from two uncoupled Haldane chains.
• Figure 5: Evidence for a Gaussian transition between the topological Haldane phase and the topologically trivial uniform phase. (a) Entanglement spectrum as a function of $J_{3,\mathrm{leg}}$ for a chain of $N=200$ sites with open boundaries. We show the 10 lowest levels across the Haldane, uniform and the $\mathbb{Z}_4$ ordered phase. The data points (black dots) are slightly shifted horizontally so the degeneracy of the levels are clearly visible. Red and blue lines indicate the estimated location of the Gaussian and Ashkin-Teller transitions correspondingly. Bi-partition is made (a) across an even bond between sites $N/2$ and $N/2+1$ and (b) across an odd bond between sites $N/2+1$ and $N/2+2$. Data in (a)-(c) are shown for $J_1=50$. Note that the multiplicity of the spectrum in the uniform phases is unaltered, in contrast to those for the $\mathbb{Z}_4$ ordered phase. (c) Scaling of the reduced entanglement entropy with the logarithm of the conformal distance $d_N(j)$. Colored dots denote different chain lengths. Data of all chain lengths but $N=140$ are shifted vertically for visual clarity. The central charge $c$ is extracted by fitting the data to Eq.\ref{['eq:entanglement_entropy']}; its numerical value approaches $c=1$ of the Gaussian transition upon approaching the thermodynamic limit. (d)-(f) Sketches of bi-parition of the $\mathbb{Z}_4$ ordered phase across (d)-(e) even and (f) odd bonds resulting in (d)-(e) unique and (f) three-fold degenerate lowest state in the entanglement spectrum.
• Figure 6: Numerical evidence for an Ashkin-Teller transition between the uniform phase and the $\mathbb{Z}_4$ ordered phase. Presented example is for $J_1=20$. Symbols show DMRG data, lines state for the fits. (a) Finite-size scaling of the mid-chain leg dimerization as defined in Eq.\ref{['eq:dimerization_order_parameter']} in a log-log plot. We associate a critical point with a separatrix at $J_{3,\mathrm{leg}}^c \approx 6.29$; its slope corresponds to a scaling dimension $d \approx 0.130$ that is in excellent agreement with the universal $d=1/8$ of the Ashkin-Teller critical theory. Concave and convex curves are shifted vertically for visual clarity. (b) Scaling of the reduced entanglement entropy with conformal distance at the critical point $J_{3,\mathrm{leg}}^c$ identified in (a). We only show data (dots) for bi-partitions made across even bonds. Curves for $N>440$ are shifted for visual clarity. By fitting the data with the Eq.\ref{['eq:entanglement_entropy']} we extract the central charge $c$ that is in excellent agreement with $c=1$. (c) Inverse of the correlation length for three system sizes. Lines mark the fit with $\xi \propto |J_{3,\mathrm{leg}}-J_{3,\mathrm{leg}}^c|^{-\nu}$; numerically extracted values of $\nu$ fits well the interval $2/3 \leq \nu \leq 1$ of Ashkin-Teller critical theory. Data with $\xi > N/10$, or those far away from the critical point, are excluded from the fit (pale symbols). (d) Mid-chain dimerization on the legs near the critical point. Curves show a fit with $D^\mathrm{leg}_{N/2} \propto (J_{3,\mathrm{leg}}-J_{3,\mathrm{leg}}^c)^{\beta}$. Numerically extracted critical exponent $\beta$ matches well the Ashkin-Teller interval $1/12 \leq \beta \leq 1/8$. Inset: Same data but in a log-log scale.
• Figure 7: Proof of the non-magnetic nature of the Ashkin-Teller transition. (a) - (b) Sketches of the domain walls between the plaquette phase and the uniform disordered phase. We depict them for both possibilities of the disordered phase that are illustrated in Fig.\ref{['fig: Uniform phase']}. (c) Finite size scaling of the singlet-triplet gap $E_{S^z_\mathrm{tot}=1} - E_{S^z_\mathrm{tot}=0}$ (red diamonds) and the singlet-quintuplet $E_{S^z_\mathrm{tot}=2} - E_{S^z_\mathrm{tot}=0}$ gap (blue squares) at the critical point $J_{3,\mathrm{leg}}^c=6.29$ for $J_1=20$ in the thermodynamic limit (see finite size scaling of the order parameter in Fig.\ref{['fig: Ashkin-Teller']}). Both energy gaps show no tendency to close. (d) Singlet-triplet gap as a function of $J_{3,\mathrm{leg}}$ for two different chain lengths. Vertical dashed purple line shows the location of the critical point. Lack of a dip in the energy at the transition indicates a finite gap.