Emergent symmetry and conserved current at a one-dimensional incarnation of deconfined quantum critical point

The deconfined quantum critical point (DQCP) was originally proposed as a continuous transition between two spontaneous symmetry breaking phases in 2D spin-1/2 systems. While great efforts have been spent on the DQCP for 2D systems, both theoretically and numerically, ambiguities among the nature of the transition are still not completely clarified. Here we shift the focus to a recently proposed 1D incarnation of DQCP in a spin-1/2 chain. By solving it with the variational matrix product state in the thermodynamic limit, a continuous transition between a valence-bond solid phase and a ferromagnetic phase is discovered. The scaling dimensions of various operators are calculated and compared with those from field theoretical description. At the critical point, two emergent $\mathrm{O}\left(2\right)$ symmetries are revealed, and the associated conserved current operators with exact integer scaling dimensions are determined with scrutiny. Our findings provide the low-dimensional analog of DQCP where unbiased numerical results are in perfect agreement with the controlled field theoretical predictions and have extended the realm of the unconventional phase transition as well as its identification with the advanced numerical methodology.

Figure 1

Schematic phase diagram for the model in Eq. (1). When $J_z<J_c$ the system is inside a VBS phase with translational symmetry breaking and when $J_z>J_c$ the system is inside a FM phase with ${\mathbb{Z}}_2^x$ on-site symmetry breaking. At $J_c$, an incarnation of DQCP emerges. In the VBS phase there exists an exactly solvable point at $J_z=1$, where the ground state is a product state of dimerized neighbor spins (orange point).

Figure 2

(a) The first and second derivatives of the ground state energy $E^{\left(1\right)}=\frac{\partial \langle H\rangle }{\partial J_z}$ and $E^{\left(2\right)}=\frac{{\partial }^2\langle H\rangle }{\partial J_z^2})$ with respect to $J_z$. (b) $z$-FM order parameter $n_3$ and VBS order $n_4$ as a function of $J_z$ calculated from $D=600$ MPS.

Figure 3

(a) The correlation length $\xi$ for different $J_z$ across the critical point. A divergence at $J_c=1.4645$ manifests. (b) The correlation length $\xi$ from different $D$ MPS wave function in log-log plot. Only at $J_c$, the $\xi \left(D\right)$ is a power-law function.

Figure 4

The entanglement entropy $S$ as a function of correlation length $\xi$ at the DQPC $J_z=J_c$. The entanglement entropy at odd bond(A bond) and even bond(B bond) in the MPS are both used for the fitting, with the fitting form $S=\frac{c}{6}log\left(\xi \right)$ and $c=0.99$ is obtained.

Figure 5

The log-log plot for (a) the $z$-FM order parameter $n_3$ and (b) the VBS order parameter $n_4$ as a function of correlation length $\xi$. The fittings at $J_c$ give rise to the critical exponents ${\mathrm{\Delta }}_3$ and ${\mathrm{\Delta }}_4$ (for $n_4$ the power law fitting is done at a lightly different point $J=1.46375)$.

Figure 6

Data collapse of the $z$-FM order $n_3$ and the VBS order $n_4$ with correlation length $\xi$, according to Eq. (10). $\delta =J_z-J_c$ is the driving parameter with respect to the critical point. Critical exponents $\mathrm{\Delta }$ and $\nu$ can be obtained from the collapse.

Figure 7

(a)–(d) The correlation functions defined in Eq. (7) calculated from MPSs of various $D$ at the critical point. (e)–(h) Data collapse of the correlations functions with different correlation lengths $\xi$. With this approach, data beyond the correlation length can also be utilized. In each channel, the exponent $\eta$ can be extracted based on Eq. (10).

Figure 8

(a) The subleading component of the correlation function $G_x$ corresponds to $G_x^{\left(\pi \right)}$. (b) The subleading component of the correlation function $G_y$ corresponds to $G_y^{\left(0\right)}$. Their exponents $\eta$ can be extracted by power-law fitting.

Figure 9

The correlation function of the conserved currents. (a) and (b) The correlation function of the emergent conserved current $J_x^{\varphi }$ or $J_{\tau }^{\theta }$ before and after rescaling with correlation length $\xi$. (c) The Fourier components of the correlation function $G_z\left(r\right)$. (d) The log-log plot of the subleading component $G_z^{\left(\pi \right)}$.