Topological transition between competing orders in quantum spin chains

We study quantum phase transitions between competing orders in one-dimensional spin systems. We focus on systems that can be mapped to a dual-field double sine-Gordon model as a bosonized effective field theory. This model contains two pinning potential terms of dual fields that stabilize competing orders and allows different types of quantum phase transition to happen between two ordered phases. At the transition point, elementary excitations change from the topological soliton of one of the dual fields to that of the other, thus it can be characterized as a topological transition. We compute the dynamical susceptibilities and the entanglement entropy, which gives us access to the central charge, of the system using a numerical technique of infinite time-evolving block decimation and characterize the universality class of the transition as well as the nature of the order in each phase. The possible realizations of such transitions in experimental systems both for condensed matter and cold atomic gases are also discussed.

Figure 1

Staggered magnetization curves for $m_{\mathrm{N}}^x$ and $m_{\mathrm{N}}^z$ in the XXZ model with (a) staggered $x$ field ($\mathrm{\Delta }=1.9$) and (b) XY anisotropy ($\mathrm{\Delta }=1.6$). The saturation value of $m_{\mathrm{N}}^{x\left(z\right)}$ is normalized to 1.

Figure 2

(a) Log-log plot of $m_{\mathrm{N}}^z$ as a function of $h_{x,\mathrm{c}}-h_x$. (b) Semi-log plot of entanglement entropy for a finite interval $S_{\mathrm{EE}}$ as a function of the size of the interval $l$ at $h_x=h_{x,\mathrm{c}}$. $M$ is the bond dimension of MPS (see the Appendix).

Figure 3

Plot of (a) ${\left(m_{\mathrm{N}}^z\right)}^8$ and (b) half-infinite entanglement entropy $S_{\mathrm{half}}$ as a function of $h_x$.

Figure 4

Dynamical susceptibility (a) ${\chi }^{xx}(q=\pi )$ and (b) ${\chi }^{zz}(q=\pi )$ for the XXZ model ($\mathrm{\Delta }=1.9$) with staggered $x$ field. The dominant low energy excitation in the low (high) $h_x$ phase corresponds to ${\chi }^{xx}$ (${\chi }^{zz}$). We see that ${\chi }^{zz}$ diverges at the transition point $h_x/J\simeq 0.071$ while ${\chi }^{xx}$ does not.

Figure 5

Dynamical susceptibility (a) ${\chi }^{xx}(q=\pi )$ and (b) ${\chi }^{zz}(q=\pi )$ for the XXZ model ($\mathrm{\Delta }=1.6$) with XY anisotropy. Both ${\chi }^{xx}$ and ${\chi }^{zz}$ diverge at the transition point $D_{xy}/J=0.6$.

Figure 6

Dynamical susceptibility (a) ${\chi }^{xx}(q=0)$ and (b) ${\chi }^{zz}(q=0)$ for the XXZ model ($\mathrm{\Delta }=1.9$) with staggered $x$ field and (c) ${\chi }^{xx}(q=0)$ and (d) ${\chi }^{zz}(q=0)$ for the XXZ model ($\mathrm{\Delta }=1.6$) with XY anisotropy.

Figure 7

The dependence of iTEBD calculations (a) on the truncation dimension $M$ with fixed $T/J^{-1}=80$ and (b) on the temporal interval $T$ with fixed $M=60$. The results of $\mathrm{Im}{\chi }^{xx}(q=\pi ,\omega )$ for the model (5) are shown with $\mathrm{\Delta }=1.9$ and $h_x/J=0.02$.