First Order Phase Transition in the Anisotropic Quantum Orbital Compass Model

We investigate the anisotropic quantum orbital compass model on an infinite square lattice by means of the infinite projected entangled-pair state algorithm. For varying values of the $J_x$ and $J_z$ coupling constants of the model, we approximate the ground state and evaluate quantities such as its expected energy and local order parameters. We also compute adiabatic continuations of the ground state, and show that several ground states with different local properties coexist at $J_x=J_z$. All our calculations are fully consistent with a first order quantum phase transition at this point, thus corroborating previous numerical evidence. Our results also suggest that tensor network algorithms are particularly fitted to characterize first order quantum phase transitions.

Figure 1

(a) Energy per link $e$ in the AQOCM on an infinite square lattice obtained by using the iPEPS algorithm with $D=2$, 3 (results for $D=4$, 5, 6 are very similar to those for $D=3$). The energy $e$ has a sharp peak with discontinuous derivative at the phase transition. Dotted lines correspond to the results from Ref. 15 up to $16\times 16$ lattices using exact diagonalization and Green’s function Monte Carlo calculations (plotted with permission). Lines linking numerical points are a guide to the eye. (b) Comparison at $s=0.5$ of the energy per bond computed with the iPEPS algorithm for $D=2,\dots ,6$ and the finite-size analysis from Ref. 15. Lines linking numerical points are a guide to the eye.

Figure 2

Expected energy per lattice link for the ground state $|{\Psi }_{\mathrm{GS}}(s)⟩$ and the adiabatically evolved states $|L(s)⟩$ and $|R(s)⟩$, as computed with the iPEPS algorithm with $D=2$.

Figure 3

Expected values of the local order parameter operators $X^{[i,j]}$ and $Z^{[i,j]}$ (in absolute value) for the ground state $|{\Psi }_{\mathrm{GS}}(s)⟩$ ($m_x$ and $m_z$) and the adiabatically evolved states $|L(s)⟩$ ($m_x^L$ and $m_z^L$) and $|R(s)⟩$ ($m_x^R$ and $m_z^R$), obtained by using the iPEPS algorithm with $D=2$, 3 (results for $D=4$, 5 are very similar to those for $D=3$). The lines correspond to the results from Ref. 16 using mean field theory after fermionization of the Hamiltonian (plotted with the author’s permission).