The Model and order parameters

The model we study is the staggered dimer spin 1/2 model on the square lattice, and will be referred as the J-J’ model form now on. It is a quantum Heisenberg model with two kinds of nearest-neighbor exchange couplings, and its Hamiltonian is

H = J X

<ij>

S~i· ~Sj+ J⁰ X

<ij>⁰

S~i· ~Sj , (3.1)

where the summations over < ij > and < ij >⁰ represent sums over the nearest-neighbor bonds as shown in Fig.1. Each square lattice consists of three J bonds (the thin ones) and one J⁰ bond (the thick one).

We fix the J bonds to be antiferromagnetic, i.e., J > 0, and consider −∞ < J⁰/J < ∞ as the parameter of this model. For the antiferromagnetic J⁰-bond regime, i.e., J⁰/J > 0, this model has been studied by using the perturbation theory [155], exact diagonalization (ED) [156], the coupled cluster method (CCM) [156], iPEPS [94] and quantum Monte Carlo [157, 158]. In this regime, there exists a quantum phase transition by tuning J⁰ with a critical point at J⁰/J ≈ 2.51 [155], separating the classical N´eel ordered phase and a finite-gap disordered phase [157, 158].

J’

J _G

G

G

G

Figure 3.1: The J-J’ model on the 2D square lattice with two different nearest neighboring bond couplings J and J’ (thin and thick, respectively). The red dash square is a 2 × 2 unit cell.

We are more interested in the regime of the ferromagnetic J⁰-bond, i.e., J⁰/J < 0 because the plaquettes are now frustrated. The quantum Monte Carlo method cannot be used for the frustrated system because of the sign problem. Instead, some mean-field based approximation methods such as the renormalized spin wave theory (RSWT) [159], exact diagonalization [156] and coupled cluster method (CCM) [156] are used to study this regime. These studies found a phase transition at J⁰/J = −1/3 separating the N´eel phase from a helical phase for classical spins, and the maximal frustration occurs around J⁰≈ 1 region. However, due the frustrations we suspect that a quantum spin liquid phase with topological order may appear around the region of maximal frustration.

Based on the solved ground state from Iterative Projection method, we can calculate

some quantities to characterize the phase diagrams. For the symmetry breaking phase we can evaluate the order parameters which are the vacuum expectation value (vev) of some physical operators. However, when we insert a physical operator at some particular site such that the translational invariance is lost, one needs some efficient method to contract the exponentially large number of bounds. Here we will adopt the TRG method [92, 96]

to do that. The basic idea of TRG is to renormalize the tensors of the TPS by keeping the relevant entanglement when coarse graining. In this way, we can reduce the size of the system while retaining the essential quantum correlations of the original ground state. Finally the whole system will be reduced to a unit cell with the renormalized tensor, we can then evaluate the vev’s faithfully enough within the renormalized unit cell.

Besides, there are alternative ways to implement polynomially efficient evaluation of the vev. One interesting method is to contract the tensors in the 1D scheme vertically and horizontally [160], and the other one is to use the Monte Carlo sampling to enhance the rate of contracting the tensors [161].

3.2.1 Order parameters for N´eel and dimer phases

For the J-J’ model considered here, we will evaluate the N´eel order parameter i.e., given by M_s^z = _M¹ PN

i=1(−1)ⁱhψ_g|S_i^z|ψ_gi to characterize the N´eel ordered phase. We also evaluate the spin-spin correlations and the dimerization [162, 163, 164]:

D_x = |h ~S_i,j · ~S_i+1,ji − h~S_i+1,j· ~S_i+2,ji| (3.2) and

D_y = |h ~S_i,j · ~S_i+1,ji − h~S_i,j · ~S_i,j+1i| (3.3) to characterize the dimmer strength of the disordered “dimer phase”. Note that the Hamiltonian of the model does not possess the dimerized order by construction. Besides, we also use TRG to evaluate the ground-state energy per site as _N^E = hψ_g|H|ψ_gi/N . 3.2.2 Characterizing topological phase by degeneracy of singular value

spec-trum

Another quantity in characterizing the topological phase is the degeneracy of the entangle-ment spectrum. It has been used to characterize the topological order for some quantum Hall states [53], and recently to characterize some symmetry-protected topological or-dered Haldane phase in spin 1 chain [56]. The entanglement spectrum is refereed to the

spectrum of the Schmidt values for the bi-partition of the whole system. By definition, these Schmidt values are the square roots of the eigenvalues of the reduced density matrix of either of the two partitions. Basically, the degeneracy of the entanglement spectrum implies the spins at different sites are almost in the maximally entangled state, hence it implies that the ground state has topological order. In 1D, the entanglement spectrum is the same as the singular value spectrum of the matrix product state, and its power in characterizing the topological order is fully demonstrated [56, 147, 149, 150]. However, for general 2D spin system there is no consensus on the power of entanglement spectrum in characterizing the topological order, see [154] for recent study on this issue ¹. Despite that, one may expect that it still works in 2D since the topological order is closely re-lated to the long range entanglement characterized by the degeneracy of the entanglement spectrum.

Unlike the 1D case, the 2D entanglement spectrum is not the same as the singular value spectrum of the 1-site tensor in TPS though these twos should be related, especially for translationally invariant states. Numerical task in evaluating the 2D entanglement spectrum is almost the same as for the topological entanglement entropy, and thus difficult.

Instead, it is far more easier and straightforward to evaluate singular value spectrum for TPS by merging the tensors of neighboring sites along x- or y-direction and then doing SVD. In this way, we can obtain four kinds of the singular value spectrum, which are associated with the four different links out of a site. For example, in the J-J’ model there are three J -bonds and one J⁰-bond, and they may have different singular spectrum. This can be done by just using Iterative Projection method without further invoking TRG.

Besides, from singular value spectrum one can straightforwardly evaluate the bipartite entanglement measure per length as well as the single-site von Neumann entropy (1-tangle). Both can be used to characterize the quantum critical point [165].