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

Abstract

We investigate the ground state nature of the transverse field Ising model on the J1 − J2 square lattice at the highly frustrated point J2/J1 = 0.5. At zero field, the model has an exponentially large degenerate classical ground state, which can be affected by quantum fluctuations for non-zero 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 multi-spin cluster type fluctuations above a non-magnetic 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 stringtype 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 non-magnetic string-VBS phase is gapped and breaks lattice rotational symmetry with only two-fold degeneracy, which bears a continuous quantum phase transition at Γ/J1 ∼ = 0.50 to the quantum paramagnet phase of high fields. The critical behavior is characterized by ν ∼ = 1.0 and γ ∼ = 0.33 exponents.

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

“…An important outcome of our mapping is to clarify the VBS nature of the intermediate phase of square-lattice phase diagram at low fields around the highly frustrated point J 2 = 0.5J 1 . In fact, we showed that the plaquette-VBS phase of the checkerboard lattice is mapped to the string-VBS phase of sqaure lattice at the highly frustrated point J 2 = 0.5J 1 , completely in agreement with the previous results of J 1 − J 2 TFI model on square lattice by cluster operator approach, which describes such VBS ordering [29]. Briefly, we claim that the low-energy effective theory of J 1 − J 2 TFI model on checkerboard is given by the same model on square lattice with renormalized parameters.…”

“…This is in agreement with our earlier results in Ref. [29], which states that the highly degenerate classical ground state of J 1 − J 2 TFI model on SL at J 2 = 0.5J 1 goes to a unique string-VBS phase, when taking into account quantum fluctuations. This justifies the mapping procedure introduced here.…”

“…For instance, the critical point Γ c /J 1 = 0.3 at J 2 = J 1 on CL is mapped to Γ c /J 1 = 0.32 at J 2 = 0.5J 1 on SL. This result is consistent with the result Γ c /J 1 = 0.51 obtained from TTN and COA data on the square lattice [29]. Moreover, Fig.…”

“…In other words, we claim that the low-energy effective theory of frustrated TFI on CL is given by frustrated TFI on SL. This mapping suggests a string-VBS order at the highly frustrated regime of SL, which is in agreement with the results of COA [29]. It is worth mentioning that the TFI model could represent the large easy-axis anisotropic limit of the antiferromagnetic J 1 − J 2 Heisenberg model, where the true nature of a non-magnetic (VBS) phase is still under debate on SL [32][33][34][35][36][37][38][39] .…”

“…On the other hand, the J 1 −J 2 TFI model on the square lattice shows an emergent string-VBS phase at the fully frustrated point J 2 = 0.5J 1 [29,30]. It can be expressed that quantum fluctuations by means of transverse field, lift the classical degeneracy toward a doubly degenerate VBS states along the horizontal or vertical directions of the square lattice called string-VBS phase.…”