Magnetization jump in one dimensional J − Q2 model with anisotropic exchange

We investigate the adiabatic magnetization process of the one-dimensional J − Q 2 model with XXZ anisotropy g in an external magnetic field h by using density matrix renormalization group (DMRG) method. According to the characteristic of the magnetization curves, we draw a magnetization phase diagram consisting of four phases. For a fixed nonzero pair coupling Q, (i) when g < −1, the ground state is always ferromagnetic in spite of h; (ii) when g > −1 but still small, the whole magnetization curve is continuous and smooth; (iii) if further increasing g, there is a macroscopic magnetization jump from partially- to fully-polarized state; (iv) for a sufficiently large g, the magnetization jump is from non- to fully-polarized state. By examining the energy per magnon and the correlation function, we find that the origin of the magnetization jump is the condensation of magnons and the formation of magnetic domains. We also demonstrate that while the experienced states are Heisenberg-like without long-range order, all the jumped-over states have antiferromagnetic or Néel long-range orders, or their mixing.

where H J = −J ∑ i P i,i+1 and H Q = −Q ∑ i P i,i+1 P i+2,i+3 . For the system described by the Hamiltonian in Eq. (S1), with size L and periodic boundary conditions (PBC), the energy of this zero-magnon state is L.
The one-magnon excited state with momentum k can be defined as By acting the Hamiltonian on the state, we get for J term, and for Q term. Notice that H is diagonal in |k⟩ basis, we can easily obtain the energy dispersion of the system with one-magnon As in the main text, define N magnon excitation energy asẼ(N ) = E(N ) − E(0). The one-magnon excitation energy isẼ (S7) * Electronic address: luohg@lzu.edu.cn HereẼ (1) can also be considered as the energy of a free magnon.
For two-magnon state, we choose the basis with a total momentum k and a relative distance d defined as Acting the Hamiltonian on the basis set, for the J term, we obtain: For the Q term, there are Then the ground state energy E 2 (k) of the two-magnon state with momentum k can be obtained by numerically diagonalizing the (L − 1) × (L − 1) Hamiltonian matrix in the basis set |d, k⟩. In Fig. S1 we show the dispersion E k (2) − E(0) of the two-magnon exited states for several different parameters. As one can see, for these examples the two-magnon ground state always has k = 0. Actually, we have carefully checked the dispersion for all the parameters we concern in this work, and the minimum E k (2) − E(0) for each point in the parameter space always has zero momentum. Therefore, the Hamiltonian matrixH (2) in the two-magnon basis can be simplified asH In order to see the finite size effect in the few magnon limit, we plotẼ(2) − 2Ẽ(1) for g = 0.5 as a function of Q for different system sizes, as shown in Fig. S2. All these curves have a precise cross atẼ(2) − 2Ẽ(1) = 0 and a critical Q c (g = 0.5) = 0.386145 even for L = 8, which is the minimum system size to include all the information of the effective Hamiltonian described by Eq. (S15). Therefore, the finite size effect in the few magnon limit is negligible.

II. THE ASYMPTOTIC BEHAVIOR
From Fig.3(c) in the main text we can see that the phase boundary between the N-MJ and PF-MJ phases can be exactly determined by comparing the energy of one-and two-magnon excitations. The phase boundary obtained in the few magnon limit perfectly agrees with the numerical results by DMRG.
We also notice the asymptotic behavior of this curve when the pair coupling Q is extremely large. In the limit of Q → ∞, the one-magnon excitation energy is here we have ignored the infinite small terms proportional to 1/Q. Similarly, ignoring the O(1/Q) terms, the twomagnon excitation is described by the matrix We can numerically obtain the critical anisotropy g c satisfyingẼ(2) −Ẽ(1) = 0. For the ground state wave function (|G⟩ 2 = ∑ L−1 d=1 α d |d⟩) at the critical point, we find that i)α 2 d is a constant number, ii) α d+1 = −α d , for d and d + 1 in range [3, L − 3]. Thus, the critical wave function can be assumed as where |d⟩ ≡ |d, k = 0⟩, η is the normalization coefficient. Applying the Hamiltonian in Eq. (S17) to the wavefunction |G⟩ 2 ,we can get a set of equations. By solving them, the critical g in the Q → ∞ limit can be obtained as /3. For any anisotropy g below this value, the magnetization curve of the system is always smooth and continuous.

III. THE EFFECTIVE HAMILTONIAN IN LARGE ANISOTROPY LIMIT
Divided by g 2 on both sides, the Hamiltonian in this study can be written as where In the limit of g → ∞, Q is finite, the O(1/g) and O(1/g 2 ) terms can be neglected . Thus the effective Hamiltonian in the large anisotropy limit is

IV. THE CORRELATION FUNCTION
In this section we discuss the different behaviors of the spin-spin correlation function in the experienced sectors and the jumped-over sectors. Fig. S3 displays the correlation function C S (r) with different parameters as examples. In the N-MJ phase (g = −0.5), independent of the magnon number N , the spin-spin correlation C S (r) is negative for all the distance r > 0, and rapidly decays to 0 as r increases. In this case, a spin has anti-ferromagnetic correlation with its environment, and is screened due to the strong quantum fluctuation. The states in the experienced sector have no long-range order (LRO).
In the PF-MJ phase (g = 0.5), the magnetization curve is continuous at some magnetization density, and then has a sharp jump. The blue triangle-line shown in Fig. S3(a) for N = 4 is the correlation function of an jumped-over state. In this state, C S (r) is positive when the distance r is small but negative when the spins are far apart from each other, which is the typical behavior of the system having two ferromagnetic domains. We also notice C S (r) has a finite value even when r = L/2, which is the largest distance possible for the finite system with system size L. In fact, C S (r) seems to converge when r is large enough. This indicates the anti-ferromagnetic (AFM) long-range order of the jumped-over states. The AFM-LRO still can be observed when N = 35, as this state is also jumped over in the magnetization process, as the blue triangle-line shown in Fig. S3(b). Further increasing the magnon number N , the correlation function C S (r) of the 56-magnon state decays rapidly to zero, similar to the situation in the N-MJ phase. The AFM-LRO disappears, and the magnetization curve in this region is continuous.
The correlation functions are more complicated in the NF-MJ phase (g = 4.0). Since the anisotropy g is sufficient large, the diagonal term of the Hamiltonian dominates and the quantum effect has been partially depressed. We can observe the strong fluctuation of C S (r) at very large distance, especially when the magnon number N is large. Nevertheless, when N is small (N = 4 and 35), besides the fluctuations, the long-range nature of domain wall is still true at large distance. For a large N = 56, which is near the zero magnetization, the correlation function shown in Fig. S3(c) exhibits a classical Néel order.