Gapless and ordered phases in spin-1/2 Kitaev-XX-Gamma chain

In this work, we study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By a combination of Jordan-Wigner transformation and density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1, 1, 0)- and (1, -1, 0)-spin directions, whereas in the gapless phases, the low energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. Transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev-XX-Gamma chain