Ising ferromagnets and antiferromagnets in an imaginary magnetic field

We study classical Ising spin-$\frac{1}{2}$ models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent $\beta$ and the anomaly number $c$ are shown to be constant along the critical line, equal to their values $\beta =\frac{1}{8}$ and $c=\frac{1}{2}$ for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.

Figure 1

The (dimensionless) free energy per spin $\beta f$ of the 2D antiferromagnetic Ising model versus the coupling $F\le 0$ for four values of the (dimensionless) imaginary magnetic field $\beta H=i\theta /2$: zero field $\theta =0$ (solid curve), $\theta =1$ (dashed curve), $\theta =2$ (dash-dotted curve), and $\theta =3$ (dotted curve).

Figure 2

Plotted for the antiferromagnet is the value of the critical coupling $F_c$ as a function of the imaginary magnetic field $\theta \in [0,\pi ]$. The present data (open circles) are compared with those of Ref. [6] (stars) and Ref. [11] (crosses). The $F$ dependence of the second derivative of the free energy $\beta f$ with respect to $F$ is pictured in the inset for four values 0,1,2,3 of the imaginary magnetic field $\theta$; the cusp divergence of the second derivative determines the critical point $F_c\left(\theta \right)$.

Figure 3

Plotted for the antiferromagnet is the difference between sublattice magnetizations (44) $-i{m}_{AB}$ versus the coupling $F$, for imaginary fields $\theta =1,2,3$. The spontaneous magnetization for the zero magnetic field $\theta =0$ is drawn for comparison.

Figure 4

Plotted for the antiferromagnet is the linear dependence of the eighth power of the difference between sublattice magnetizations in the ordered phase on small deviations from the critical coupling $F_c\left(\theta \right)-F$ for the zero magnetic field $\theta =0$ and the values $\theta =1,2,3$ of the imaginary field. The linear form of the plots indicates the uniformity of the critical index $\beta =\frac{1}{8}$ along the line of critical points when changing the parameter $\theta \in [0,\pi ]$.

Figure 5

Plotted for the antiferromagnet is the von Neumann entropy $S$ versus the number of sites of the square lattice $N$, in logarithmic scale. The inset shows that with increasing $N$ the effective anomaly number $c_{\mathrm{eff}}\left(N\right)$ [Eq. (42)] tends to the Ising value $\frac{1}{2}$ for the zero magnetic field $\theta =0$ as well as the values $\theta =1,2,3$ of the imaginary magnetic field.

Figure 6

Plotted for the ferromagnet is the free energy $\beta f$ as a function of the ferromagnetic coupling $F>0$ for zero magnetic field $\theta =0$ and the values $\theta =1,2,3$ of the imaginary field. The inset shows the $F$ dependence of the first derivative of the free energy with respect to $F$; the cusp divergence of the derivative signals a first-order transition at the coupling $F^{*}\left(\theta \right)$. The free energy is well defined in the high-temperature region $F<F^{*}\left(\theta \right)$; the continuous plot of the free energy in the low-temperature region $F>F^{*}\left(\theta \right)$ ignores the divergence of $\beta f$ at a dense set of Yang-Lee zeros.

Figure 7

Plotted for the ferromagnet is the transition coupling $F^{*}$ as a function of the imaginary magnetic field $\theta$. The analytic 1D result (10) is shown by the dashed curve and the numerical 2D data are represented by open circles.

Figure 8

Plotted for the ferromagnet is the divergence of the magnetization $-im$ for the 2D Ising ferromagnet in an imaginary field as the coupling constant $F$ approaches the transition coupling $F^{*}\left(\theta \right)$ (vertical dotted lines) from below; the imaginary magnetic field $\theta =1$ (dashed curve) and $\theta =2$ (dash-dotted curve). The dependences of $-im$ versus $F$ yielded by the asymptotic $F\to 0$ formula (45) are depicted by the solid lines.