Baxter-Wu model in a transverse magnetic field

We investigate the low-energy properties as well as quantum and thermal phase transitions of the Baxter-Wu model in a transverse magnetic field. Our study relies on stochastic series expansion quantum Monte Carlo and on series expansions about the low- and high-field limits at zero temperature using the quantum finite-lattice method on the triangular lattice. The phase boundary consists of a second-order critical line in the four-state Potts model universality class starting from the pure Baxter-Wu limit meeting a first-order line connected to the zero-temperature transition point ($h\approx 2.4$, $T=0$). Both lines merge at a tricritical point approximately located at ($h\approx 2.3J$, $T\approx J$).

Figure 1

Illustration of the triangular lattice $\Lambda$. Quantum spins $\frac{1}{2}$ are located at the vertices of the lattice. Additionally, we sketch the elementary spin-flip excitation above the ferromagnetic ground state of the BW model with unit cell $|\uparrow \uparrow \uparrow \rangle$. This excitation can be created by the single operator ${\sigma }_i^x$ acting on site $i$ which flips the eigenvalues of the six face operators sharing this site. Alternatively, the spin-flip excitation can also be seen as six excitations centered on triangles which form the dual honeycomb lattice $\tilde{\Lambda }$.

Figure 2

Ground-state energy per site, $e_0$, of the BWTF model as a function of the transverse field $h$ obtained by QMC (open symbols) and pCUT (filled symbols). QMC data is obtained for fixed inverse temperature $\beta J=16$ and for different linear sizes $L$. For $L=3$, a dashed line indicates the $T=0$ exact result. The vertical dotted lines (respectively the vertical dashed lines) correspond to the location of the first-order phase transition $h/J\simeq 2.39$ ($h/J\simeq 2.405$) deduced from the different QMC observables (from the crossing point of the two high-order series expansions [19]). Bottom inset: Zoom on the region close to the phase transition. Top inset: Equilibration effects for the energy curve of a $L=18$ QMC sample, as a function of inverse temperature $\beta$. In the latter case, we do not plot error bars when simulations are not converged.

Figure 3

(a) Magnetization per site along field $\langle {\sigma }^x\rangle$ vs transverse field $h$ at low temperature $\beta J=16$ for various sizes obtained by QMC and pCUT results. Inset: zoom on the transition region where a sharp jump is forming. Note that lines are guide to the eyes except for $L=3$, which displays the exact $T=0$ results from ED.

Figure 4

(a) Sublattice magnetization density square $\langle m_s^2\rangle$ vs transverse field $h$ at low-temperature $\beta J=16$ for various sizes. Inset: zoom on the transition region where a sharp jump is forming. (b) Binder cumulant $U_L$ vs transverse field $h$ at low-temperature $\beta J=16$ for various sizes. Note that lines are guide to the eyes except for $L=3$, which displays the exact $T=0$ results from ED.

Figure 5

${\chi }_E=-{\partial }^2{e}_0/\partial h^2$ vs transverse field $h$ at low-temperature $\beta J=16$ for various sizes. Inset: maximum of this quantity vs $L$ showing a divergence (the dotted line displays the expected $L^2$ divergence for comparison).

Figure 6

Specific heat $C_v$ vs temperature $T$ for various sizes. The three panels correspond to $h=0$, $h=2J$, and $h=2.35J$ respectively. For $h=0$ (classical case), the exact $T_c$ is indicated by the dotted vertical line.

Figure 7

Collapse analysis using the same data as in Fig. 6. Using the known results for BW model and the obtained $T_c$'s, we plot $C_v/L^{\alpha /\nu }$ vs $(T-T_c\left(h\right))L^{1/\nu }$ with $\alpha =\nu =2/3$.

Figure 8

(a) Specific heat maximum vs length $L$ for various $h$. (b) Dip size of $V_L$ vs $1/L$ for various $h/J$.

Figure 9

Energy Binder cumulant $V_L$ vs $T$ for various sizes and transverse fields. A dip in this quantity signals a phase transition (see text).

Figure 10

Tentative data collapse of the $V_L$ dip times $L$ vs $(T-T_c)L^{1/\nu }$ using the critical exponents of the BW model. A rather good collapse can be obtained for $h=0$ and $2J$, but not for $h=2.35J$.

Figure 11

Order parameter squared $\langle m_s^2\rangle$ as a function for temperature $T/J$ for different system sizes, for different field values.

Figure 12

Scaling of the order parameter squared $\langle m_s^2\rangle$ times $L^{2\beta /\nu }$ vs $(T-T_c)L^{1/\nu }$ with the values $\beta /\nu =1/8$ and $\nu =2/3$ showing an excellent data collapse (two first panels for $h=0$ and $h=2J$), and no collapse (right panel) for $h=2.35J$.

Figure 13

(a) Order parameter Binder cumulant $U_L$ vs $T$ for various sizes and transverse field $h$. (b) Zoom close to the transition temperature. For $h=0$, the exact $T_c$ is indicated as a dashed line.

Figure 14

Scaling of the order parameter Binder cumulant $U_L$ vs $(T-T_c)L^{1/\nu }$ using the known exponents of the BW model.

Figure 15

Order parameter histograms on $L=36$ cluster for (a) $h=0$, $T/J=2.274$; (b) $h/J=2$, $T/J=1.53$; and (c) $h/J=2.35$, $T/J=0.86$.

Figure 16

Energy histograms on $L=36$ cluster for (a) $h=0$, $T/J=2.274$; (b) $h/J=2$, $T/J=1.53$; and (c) $h/J=2.35$, $T/J=0.86$. In each case, we have split data depending whether $m_s^2$ is smaller or greater than 0.25.

Figure 17

Phase diagram of the BWTF vs $(h,T)$. Full (respectively dashed) lines are used for second- (respectively first-) order phase transitions. Both lines meet at a putative tricritical point close to $h^{*}\simeq 2.3J$ and $T^{*}\simeq J$.