Interface roughening in the 3-D Ising model with tensor networks

Abstract

Interfaces in three-dimensional many-body systems can exhibit rich phenomena beyond the corresponding bulk properties. In particular, they can fluctuate and give rise to massless low energy degrees of freedom even in the presence of a gapped bulk. In this work, we present the first tensor-network study of the paradigmatic interface roughening transition of the 3-D Ising model using highly asymmetric lattices that are infinite in the $(xy)$ direction and finite in $z$. By reducing the problem to an effective 2-D tensor network, we study how truncating the $z$ direction reshapes the physics of the interface. For a truncation based on open boundary conditions, we demonstrate that varying the interface width gives rise to either a $\mathbb{Z}_2$ symmetry breaking transition (for odd $L_z$) or a smooth crossover(for even $L_z$). For antiperiodic boundary conditions, we obtain an effective $\mathbb{Z}_q$ clock model description with $q=2L_z$ that exhibits an intermediate Luttinger liquid phase with an emergent $\U(1)$ symmetry.

Figures (6)

• Figure 1: Plane-to-plane transfer matrix. The tensor is defined in terms of the six-legged $\delta$ tensor [Eq. \ref{['eq:delta']}] and the square root of the $t$ matrix [Eq. \ref{['eq:t_matrix']}] encoding the elementary Boltzmann weights.
• Figure 2: The interface partition function. We start from $L_z$ layers of the plane-to-plane transfer matrix, and trace with matrices $Y$ that define the boundary conditions.
• Figure 3: Numerical results for open boundary conditions. (a) Boundary MPS correlation length, (b) central charge from TNR simulations, and (c)total magnetization as a function of temperature for different values of $L_z$ (for even $L_z$ the magnetization is always zero within machine precision);(d) magnetization profiles for a well chosen points from boundary MPS. In all these simulations the boundary MPS bond dimension is $\chi=64$. The dashed line is the Monte Carlo estimate for the roughening transition temperature Hasenbusch1997.
• Figure 4: Clock model interpretation. The six local interface configurations on a single column of the 3-D Ising model with $L_z=3$ and antiperiodic boundary conditions. The antiperiodic bond is denoted by antiferromagnetic interaction $(-J)$, the other two bonds have a ferromagnetic interaction $(+J)$. The dashed line denotes the location of the interface: either two parallel spins on the antiperiodic bond, or two antiparallel spins on a normal bond. In the middle, we show the identification of these six configurations with a clock variable $\theta=2\pi k /(2L_z)$.
• Figure 5: Numerical results for antiperiodic boundary conditions. (a) Correlation length from boundary MPS calculations at $\chi=64$, (b) the scaling of the MPS entanglement entropy as a function of the correlation length at $T=2.5$ for three different values of $L_z$, fitted with the form $S_\chi\propto c/6\log\xi_\chi$ (for all three fits we find $c=1$ within one percent error). The temperature dependence of the central charge and Luttinger parameter computed from TNR for (c), $L_z=1$(cross) and 2(diamond), (d)$, L_z=3$ and (e)$\,L_z=4.$ The vertical dashed lines correspond to $T_c^{(1)}$ and $T_c^{(2)}$. $T_c^{(1)}$ corresponds to the roughening transition. In the TLL regime concerned, the lowest excitation corresponds to $W_1=:e^{i\theta}:$, whose scaling dimension is $\Delta_{W_1}=1/4K$. We therefore extract the Luttinger parameter $K$ from $\Delta_{W_1}$. The Luttinger parameters (red circles) take the universal values at the BKT transition points.