Phase diagram of J1-J2 transverse field Ising model on the checkerboard lattice: a plaquette-operator approach

Abstract

We study the effect of quantum fluctuations by means of a transverse magnetic field (Γ) on the antiferromagnetic J1 − J2 Ising model on the checkerboard lattice, the two dimensional version of the pyrochlore lattice. The zero-temperature phase diagram of the model has been obtained by employing a plaquette operator approach (POA). The plaquette operator formalism bosonizes the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a plaquette ordered solid (RPS) state, which breaks lattice translational symmetry, in addition to a unique collinear phase for J2 > J1. The bosonic excitation gap vanishes at the critical points to the Néel (J2 < J1) and collinear (J2 > J1) ordered phases, which defines the critical phase boundaries. At the homogeneous coupling (J2 = J1) and its close neighborhood, the (canted) RPS state, established from an-harmonic fluctuations, lasts for low fields, Γ/J1 0.3, which is followed by a transition to the quantum paramagnet (polarized) phase at high fields. The transition from RPS state to the Néel phase is either a deconfined quantum phase transition or a first order one, however a continuous transition occurs between RPS and collinear phases.

Quantum phase diagram of the two-dimensional transverse-field Ising model: Unconstrained tree tensor network and mapping analysis

“…Utilizing finite-size scaling analysis on N = 4 × 4, 6 × 6 and 8 × 8 lattices, we obtain the associated critical exponents to be ν ≃ 1 and γ ≃ 0.44. We did not observe a signature of a canted Néel phase predicted by the Monte-Carlo study [25], which is in agreement with previous results based on cluster operator approach [22]. In addition, we found the nature and associated critical exponents of the quantum phase transitions from the plaquette-VBS phase to the adjacent Néel and collinear antiferromagnetic phases and also to the quantum paramagnetic phase of high fields, summarized in table-II.…”

“…Before presenting the results, let us mention that the interesting and controversial part of TFI model on the CL is in the low magnetic field limit around the highly frustrated coupling J 2 = J 1 . This clarifies the reason that we concentrate on this region, while the other parts of the phase diagram are known by other methods without doubt [22,26]. To obtain an accurate phase diagram for J 1 −J 2 TFI model on the CL via TTN approach, we compute the first and second derivatives of the ground state energy by TTN simulation in two distinct directions on the phase diagram.…”

“…In addition, we support the plaquette-VBS nature of the ground state at low fields by calculating the ground state expectation value of resonating plaquette operator (Ô) [22,25]. This operator is defined aŝ…”

“…We implement the mapping established in the previous section and apply it to the GS phase diagram of TFI model on CL-which has been obtained by COA, [22]to get the GS phase diagram of J 1 − J 2 TFI model on SL. To this end, we insert the location of critical boundaries of the CL phase diagram in Eqs.…”

Emergence of string valence-bond-solid state in the frustrated J1−J2 transverse field Ising model on the square lattice

“…30,31 to obtain the effective theory. We associate a boson to each of the three eigenstates |u of each dimer (u = 1, 2, 3).…”