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Journal of Physics Conference Series

Phase diagram of the frustrated J 1 − J 2 transverse field Ising model on the square lattice

M. Sadrzadeh, A. Langari

Published Mar 1, 2018• 3 citations
Publisher

Abstract

We study the zero-temperature phase diagram of transverse field Ising model on the J 1 − J 2 square lattice. In zero magnetic field, the model has a classical Néel phase for J 2 / J 1 < 0.5 and an antiferromagnetic collinear phase for J 2 / J 1 > 0.5. We incorporate harmonic fluctuations by using linear spin wave theory (LSWT) with single spin flip excitations above a magnetic order background and obtain the phase diagram of the model in this approximation. We find that harmonic quantum fluctuations of LSWT fail to lift the large degeneracy at J 2 / J 1 = 0.5 and exhibit some inconsistent regions on the phase diagram. However, we show that anharmonic fluctuations of cluster operator approach (COA) resolve the inconsistency of the LSWT, which reveals a string-valence bond solid ordered phase for the highly frustrated region.

DOI
10.1088/1742-6596/969/1/012114

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We study the zero-temperature phase diagram of transverse field Ising model on the J 1 − J 2 square lattice. In zero magnetic field, the model has a classical Néel phase for J 2 / J 1 < 0.5 and an antiferromagnetic collinear phase for J 2 / J 1 > 0.5. We incorporate harmonic fluctuations by using linear spin wave theory (LSWT) with single spin flip excitations above a magnetic order background and obtain the phase diagram of the model in this approximation. We find that harmonic quantum fluctuations of LSWT fail to lift the large degeneracy at J 2 / J 1 = 0.5 and exhibit some inconsistent regions on the phase diagram. However, we show that anharmonic fluctuations of cluster operator approach (COA) resolve the inconsistency of the LSWT, which reveals a string-valence bond solid ordered phase for the highly frustrated region.
We investigate the ground-state nature of the transverse field Ising model on the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice at the highly frustrated point J2/J1=0.5{J}_{2}/{J}_{1}=0.5J2​/J1​=0.5. At zero field, the model has an exponentially large degenerate classical ground state, which can be affected by quantum fluctuations for nonzero field toward a unique quantum ground state. We consider two types of quantum fluctuations, harmonic ones by using linear spin-wave theory (LSWT) with single-spin-flip excitations above a long-range magnetically ordered background and anharmonic fluctuations, by employing a cluster-operator approach (COA) with multispin cluster-type fluctuations above a nonmagnetic cluster-ordered background. Our findings reveal that the harmonic fluctuations of LSWT fail to lift the extensive degeneracy as well as signaling a violation of the Hellmann-Feynman theorem. However, the string-type anharmonic fluctuations of COA are able to lift the degeneracy toward a string valence-bond-solid (VBS) state, which is obtained from an effective theory consistent with the Hellmann-Feynman theorem as well. Our results are further confirmed by implementing numerical tree tensor network simulation. The emergent nonmagnetic string VBS phase is gapped and breaks lattice rotational symmetry with only twofold degeneracy, which bears a continuous quantum phase transition at $\mathrm{\ensuremath{\Gamma}}/{J}_{1}\ensuremath{\cong}0.50$ to the quantum paramagnet phase of high fields. The critical behavior is characterized by $\ensuremath{\nu}\ensuremath{\cong}1.0$ and $\ensuremath{\gamma}\ensuremath{\cong}0.33$ exponents.
We study the effect of quantum fluctuations by means of a transverse magnetic field (Γ\GammaΓ) on the antiferromagnetic J1−J2J_1-J_2J1​−J2​ Ising model on the checkerboard lattice, the two dimensional version of pyrochlore lattice. A plaquette operator formalism has been implemented to bosonise 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 state, which breaks lattice rotational symmetry for J2>J1J_2>J_1J2​>J1​ (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el (J2J1J_2 J_1J2​J1​) phases, respectively. At the homogeneous coupling (J1=J2J_1=J_2J1​=J2​) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields, Γ/J1≲0.4\Gamma/J_1\lesssim 0.4Γ/J1​≲0.4, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point (J2=0.5J1J_2=0.5 J_1J2​=0.5J1​) of the antiferromagnetic J1−J2J_1-J_2J1​−J2​ Ising square lattice.
We investigate the zero-temperature phase diagram of the fully frustrated transverse-field Ising model on the square lattice both in the classical limit and in the presence of quantum fluctuations. At the classical level (the limit of infinite spin SSS), we find that upon decreasing the transverse field $\ensuremath{\Gamma}$ this model exhibits a phase transition from the fully polarized state into an eightfold degenerate translational symmetry breaking state. This phase can be identified to correspond to plaquette order in the dimer language and remains the lowest-energy state in the entire range of fields below the critical one, ${\ensuremath{\Gamma}}_{c}$. The eightfold degenerate solution that corresponds to columnar order in the dimer language is a saddle point of the classical energy. It is degenerate with the plaquette solution at $\ensuremath{\Gamma}=0$ and is only slightly higher in energy in the whole interval $0<\ensuremath{\Gamma}<{\ensuremath{\Gamma}}_{c}$. The effect of quantum fluctuations is investigated in the context of a large S expansion both for the plaquette and columnar structures. For this purpose we employ an approximate method allowing to estimate from above the fluctuation-induced correction to the energy of a configuration which at the classical level is a saddle point of the energy, not a local minimum, and find that the harmonic quantum fluctuations show a clear tendency to overcome the energy difference between the two states. For relatively high fields, the transition from the plaquette to the columnar state takes place at values of SSS large enough to be in the domain of validity of the harmonic approximation.
Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully frustrated transverse-field Ising model on the honeycomb lattice using classical and semiclassical approximations. We show that, in addition to the fully polarized phase at a large field, the classical model possesses a multitude of phases that break the translational symmetry which, in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by quantum fluctuations has been investigated in the context of linear spin-wave theory. The extrapolation of the semiclassical energies suggests that the plaquette phase extends down to zero field for spin 1/2, in agreement with the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ phase of the quantum dimer model on the triangular lattice with only kinetic energy.
We investigate the ground-state phase diagram of the frustrated transverse-field Ising (TFI) model on the checkerboard lattice (CL), which consists of N\'eel, 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 ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region (J2=J1)({J}_{2}={J}_{1})(J2​=J1​), we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field ${\mathrm{\ensuremath{\Gamma}}}_{c}=0.28$ with the associated critical exponents $\ensuremath{\nu}=1$ and $\ensuremath{\gamma}\ensuremath{\simeq}0.4$, which are obtained within the finite-size scaling analysis on different lattice sizes $N=4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6,8\ifmmode\times\else\texttimes\fi{}8$. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at J2=J1{J}_{2}={J}_{1}J2​=J1​ and rules out the existence of a N\'eel phase. In addition, our numerical results suggest that the transition from the N\'eel (for J2<J1){J}_{2}<{J}_{1})J2​<J1​) to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point J2=J1{J}_{2}={J}_{1}J2​=J1​ on the CL is mapped to the emergent string-VBS phase on the SL at J2=0.5J1{J}_{2}=0.5{J}_{1}J2​=0.5J1​.
The ground-state properties of the S=1/2S=1/2S=1/2 transverse-field Ising model on the checkerboard lattice are studied using linear spin-wave theory. We consider the general case of different couplings between nearest neighbors (J1{J}_{1}J1​) and next-to-nearest neighbors (J2{J}_{2}J2​). In zero field, the system displays a large degeneracy of the ground state, which is exponential in the system size (for J1=J2{J}_{1}={J}_{2}J1​=J2​) or in the system's linear dimensions (for J2>J1{J}_{2}>{J}_{1}J2​>J1​). Quantum fluctuations induced by a transverse field are found to be unable to lift this degeneracy in favor of a classically ordered state at the harmonic level. This remarkable fact suggests that a quantum-disordered ground state can be instead promoted when nonlinear fluctuations are accounted for, in agreement with existing results for the isotropic case J1=J2{J}_{1}={J}_{2}J1​=J2​. Moreover, spin-wave theory shows sizable regions of instability, which are further candidates for quantum-disordered behavior.