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

Abstract

We investigate the ground-state phase diagram of the frustrated transverse field Ising (TFI) model on the checkerboard lattice (CL), which consists of Néel, collinear, quantum paramagnet and plaquette-valence bond solid (VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network (TTN) ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region (J2 = J1), we observe a second order phase transition from plaquette-VBS state to paramagnet phase at the critical magnetic field, Γc = 0.28, with the associated critical exponents ν = 1 and γ ≃ 0.4, which are obtained within the finite size scaling analysis on different lattice sizes N = 4 × 4, 6 × 6, 8 × 8. The stability of plaquette-VBS phase at low magnetic fields is examined by spin-spin correlation function, which verifies the presence of plaquette-VBS at J2 = J1 and rules out the existence of a Néel phase. In addition, our numerical results suggest that the transition from Néel (for J2 < J1) to plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of TFI on CL to be the same model on J1 − J2 square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point J2 = J1 on CL is mapped to the emergent string-VBS phase on SL at J2 = 0.5J1.

Frustrated transverse-field Ising model

“…The QMC calcul ation [7] also focusses on finite-T behaviour in the presence of both transverse and longitudinal fields. Sadrzadeh et al [12] presents a ground-state phase diagram which is very similar to figure 6 of the present paper, allowing for a factor of 2 from the different expression of the Hamiltonian. As mentioned in section 1, that work argues for a VBS state at J 2 /J 1 = 0.5.…”

“…In another recent study of this model [11], using an approximate analytic approach including multispin cluster fluctuations, evidence was obtained for the existence of a valence-bondsolid (VBS) state at exactly J 2 /J 1 = 0.5. The same group has used a tensor network method to investigate the TFIM on the checkerboard lattice, where they find [12] an extended VBS phase. A mapping to the J 1 − J 2 square lattice then also suggests a VBS state in the latter case.…”

A lattice model for condensation in Levin-Wen systems

“…Here, s( ), t( ) denote the source and target of the edge . This Hamiltonian is just the 2D transverse-field Ising model [40], so tuning t from 0 to 1 drives the system through a well-studied quantum phase transition [17,45]. Of course, when K V and K and V remain constant, the overall story is more complicated.…”

Nonlinear response of the Kitaev honeycomb lattice model in a weak magnetic field

“…In a quantum spin liquid (QSL) [1][2][3][4][5][6][7][8][9][10][11] , quantum fluctuations overcome the conventional magnetic orders even at very low temperature. Accordingly, the description of QSL state falls beyond the framework of the traditional description of magnetic phases.…”