Phase diagram of the three-dimensional axial next-nearest-neighbor Ising model

The three-dimensional axial next-nearest-neighbor Ising model is studied by a modified tensor product variational approach. A global phase diagram is constructed with numerous commensurate and incommensurate magnetic phases. The devil’s stairs behavior for the model is confirmed. The wavelength of the spin modulated phases increases to infinity at the boundary with the ferromagnetic phase. Widths of the commensurate phases are considerably narrower than those calculated by mean-field approximations.

Figure 1

The layer-to-layer transfer matrix $\mathcal{T}[\sigma \mid \overline{\sigma }]$ (left) illustrated in the case for $L=5$ and the local Boltzmann weight $W_{i,j}^{\mathrm{B}}\{\sigma \mid \overline{\sigma }\}$ (right).

Figure 2

Graphical representation of the local variational weight $V_{i,j}\left\{\sigma \right\}$ (left) used to construct the trial function $\Psi$ (right) in the particular case for $L=5$.

Figure 3

The global phase diagram of the 3D ANNNI model obtained by the TPVA. The Lifshitz point $P_{\mathrm{L}}$ is denoted by the black circle. The dotted lines enclose extremely narrow commensurate phases.

Figure 4

The five selected lines $A_1{B}_1$, $A_2{B}_2$, $A_3{B}_3$, $P_{\mathrm{L}}{B}_0$, and $A_4{B}_4$ in the region of modulated phases. The points $C_1$ and $C_2$ are marked by the white triangles

Figure 5

The spontaneous magnetization $⟨{\sigma }_i⟩$ vs the position $i$ on the lattice in the $x$ direction calculated at $C_1$ (the upper graph) and at $C_2$ (the lower one) with the lattice size $L=401$. In order to show the data in detail, we plot $i=120,\dots ,170$.

Figure 6

The spontaneous magnetization obtained at $C_2$ for the lattice size $401\times \infty \times \infty$. The upper, middle, and lower graphs display $⟨{\sigma }_i⟩$ for the three different boundary conditions.

Figure 7

The divergence of $\lambda$ at the boundary with the ferromagnetic phase. The black circles, squares, and triangles, respectively, correspond to the selected points on the lines $A_1{B}_1$, $A_2{B}_2$, and $A_3{B}_3$. The inset shows behavior of $\lambda$ around the commensurate phase ⟨3⟩.

Figure 8

The behavior of the wavelength $\lambda$ on the line $P_{\mathrm{L}}{B}_0$. Several commensurate phases with the integer value of $\lambda$ are labeled. The inset shows the area around the commensurate phase ⟨5⟩.

Figure 9

The calculation of $\lambda$ on the line $A_4{B}_4$ at the temperature $k_{\mathrm{B}}T∕J_1=1.5$. The dashed rectangle borders an area shown in Fig. 10. The inset illustrates the behavior of the corresponding wave vector $q$.

Figure 10

The resulting devil’s stairs with $\lambda ⩾6$ observed on the line $A_4{B}_4$.

Figure 11

Comparison of the numerical results at $k_{\mathrm{B}}T∕J_1=1.5$ (the black squares) with the low-temperature series expansions represented by the dashed stairslike curve (in Ref. 6).

Figure 12

Typical convergence process of the free energy $\mathcal{F}$ for the lattice size of $1001\times \infty \times \infty$. The inset shows a detailed view near the vicinity of the free energy minimum up to the eleventh sweep (11RL).