Mixed-order transition in the antiferromagnetic quantum Ising chain in a field

The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field ($h$) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\Gamma }\to \infty , \mathrm{\Gamma }$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\Gamma }>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\Gamma }$ the transition is of mixed order.

Figure 1

The zero-temperature phase diagram of the Ising chain with antiferromagnetic coupling $J$, in transverse ($\mathrm{\Gamma }$) and longitudinal ($h$) magnetic fields calculated by the DMRG method. In the absence of longitudinal field, $h=0$, the transition between a quantum antiferromagnetic (AFM) phase and a quantum paramagnetic (PM) phase (indicated by thick green line) is controlled by the fixed point of the transverse Ising model (TIM) at $(J/\mathrm{\Gamma }=1,h/\mathrm{\Gamma }=0)$. For finite value of $h>0$ the AFM phase survives and at the other side of the phase boundary there is a quantum ferromagnetic (FM) phase. In the classical limit, $\mathrm{\Gamma }=0$, thus $J/\mathrm{\Gamma }\to \infty$ and $h/\mathrm{\Gamma }\to \infty$ there is a classical endpoint (CEP) at $(h/J=2)$ having a first-order transition.

Figure 2

Illustration of the block renormalization for $n=3$.

Figure 3

(a) The $J$ dependence of the entanglement entropy at $h=40.0$ and $\mathrm{\Gamma }=1.5$ for different lengths of periodic chains. The position of the maximum is shifted by $\sim lnL/L^2$ from the phase transition point; see in (b). The value at the maximum scales as $\approx 0.17lnL$; see in (c).

Figure 4

The correlation length, extracted from the entropy through Eq. (11) as a function of the difference from the transition point $J<J_c$ in log-log scale for different values of $h$ at $\mathrm{\Gamma }=1.5$. In the inset the local slope of the curves, approaching the critical exponent $-\nu$ is shown. For the case $h=0$ the slope is taken from the analytical result in Ref. [23].

Figure 5

Correlation function (a) at $\mathrm{\Gamma }=1.5, h=0$ and (b) at $\mathrm{\Gamma }=1.5, h=0.4$ as a function of the coupling for different lengths. In the insets the scaling plots $u=C(J,L)L^{1/4}$ vs $(J-J_c)L$ are shown.

Figure 6

(a) Correlation function at $h=0.4$ (denoted by $\circ$), $h=0.8$ (denoted by $\square$), and 1.2 (denoted by $\diamond$) for $\mathrm{\Gamma }=1.5$ as a function of the antiferromagnetic coupling, $J>0$, close to the transition point, calculated by DMRG at different lengths: $L=256\circ , L=512 \circ , L=1024 \circ$. The expected form of the correlation function in the thermodynamic limit is indicated by green lines, as defined in Eq. (13); its endpoint is shown by an $*$. The limiting values $C^{-}\left(h\right)$ as a function of $1/h$ are plotted in (b). In (c) a scaling plot of $u=C(J,L)L^{1/4}$ vs $(J-J_c)L$ is shown for $h=0.4$; here points for a length $L=128$ are indicated by $\circ$. The point indicated by an arrow is the critical endpoint, which separates the critical and the linear regions; see text.

Figure 7

End-to-end $C_{ee}$ (denoted by $\circ$) and end-to-bulk $C_{eb}$ correlation functions (denoted by $\square$) for $\mathrm{\Gamma }=1.5$ and $h=0.4$ as a function of $J$ for different lengths of the open chain. $L=32\circ , L=64 \circ , L=128 \circ , L=256 \circ , L=512 \circ$, and $L=1024 \circ$. (Inset) Scaling of the finite-size correction to the minimum value of $[C_{ee}(J_c,L)+0.193]$ (denoted by $\square$) and the radius of curvature at the minimum $R\left(J_c\right)$ (denoted by $\circ$), as a function of $L$ in log-log scale; see text.

Figure 8

Illustration of the coupling dependence of the spin-spin correlation function in the bulk ($C$) and between end spins ($C_{ee}$) at $h=0$ (main panel) and at $h>0$ (inset).