Exact solution of an integrable anisotropic $J_1-J_2$ spin chain model

TL;DR

This paper constructs and solves an integrable anisotropic $J_1-J_2$ spin chain model, revealing unique excitation structures and effects of interactions, and offers a method to develop new integrable models with next-nearest-neighbour couplings.

Contribution

It presents the exact solution of a new integrable anisotropic spin chain with complex interactions, including scalar chirality, and analyzes its ground state and excitations.

Findings

Spinon excitation has a triple arched structure.

Elementary excitations are gapless for real anisotropy parameter.

Next-nearest-neighbour and chiral interactions increase the excitation gap.

Abstract

An integrable anisotropic Heisenberg spin chain with nearest-neighbour couplings, next-nearest-neighbour couplings and scalar chirality terms is constructed. After proving the integrability, we obtain the exact solution of the system. The ground state and the elementary excitations are also studied. It is shown that the spinon excitation of the present model possesses a novel triple arched structure. The elementary excitation is gapless if the anisotropic parameter $\eta$ is real while the elementary excitation has an enhanced gap by the next-nearest-neighbour and chiral three-spin interactions if the anisotropic parameter $\eta$ is imaginary. The method of this paper provides a general way to construct new integrable models with next-nearest-neighbour interactions.

Tables

Table 1. Table 1: Numerical solutions of the BAEs ( III ) for real η 𝜂 \eta case, where η = 1 𝜂 1 \eta=1 , b = 1 𝑏 1 b=1 , 2 ​ N = 4 2 𝑁 4 2N=4 , n 𝑛 n indicates the number of the energy levels and E n subscript 𝐸 𝑛 E_{n} is the corresponding energy. The energy E n subscript 𝐸 𝑛 E_{n} calculated from the Bethe roots is exactly the same as that from the exact diagonalization of the Hamiltonian ( 1 ).

$u_1$	$u_2$	$E_n$	$n$
$-2.0080+0.0000i$	$2.0080+0.0000i$	$-100.4304$	$1$
$2.6286+0.0000i$	$---$	$-20.0748$	$2$
$-2.0253-3.1416i$	$2.0253+0.0000i$	$-20.0748$	$2$
$0.0000+0.0000i$	$---$	$5.0260$	$3$
$0.0000-3.1416i$	$0.0000+0.0000i$	$17.9135$	$4$
$0.0000-1.3032i$	$0.0000+1.3032i$	$18.1853$	$5$
$---$	$---$	$22.2360$	$6$
$0.0000-3.1416i$	$---$	$35.1235$	$7$
$-2.0777-3.1416i$	$2.0777-3.1416i$	$60.0091$	$8$

Table 2. Table 2: Numerical solutions of the BAEs ( III ) for imaginary η 𝜂 \eta case, where γ = 1 𝛾 1 \gamma=1 , a = 1 𝑎 1 a=1 and 2 ​ N = 4 2 𝑁 4 2N=4 .

$u_1$	$u_2$	$E_n$	$n$
$-1.9566+0.0000i$	$1.9566+0.0000i$	$-12.1765$	$1$
$-3.1416+0.0000i$	$0.0000+0.0000i$	$-4.3247$	$2$
$-1.8439+0.0000i$	$---$	$-1.8476$	$3$
$-1.5708-0.9497i$	$-1.5708+0.9497i$	$-1.8476$	$3$
$-3.1416+0.0000i$	$---$	$0.1830$	$4$
$-3.1416-1.1002i$	$-3.1416+1.1002i$	$1.1932$	$5$
$0.0000-1.3426i$	$0.0000+1.3426i$	$2.9633$	$6$
$0.0000+0.0000i$	$---$	$3.5122$	$7$
$---$	$---$	$8.0199$	$8$

Equations

$$H$$

$${\vec{\sigma }}_{2N+j}={\vec{\sigma }}_j,j=1,\cdots ,2N,$$

$$H$$

$$R_{0,j}(u)$$

$$\text{mbox}Initialcondition:R_{0,j}(0)=P_{0,j},$$

$$\text{mbox}Unitaryrelation:R_{0,j}(u)R_{j,0}(-u)=\varphi (u)\times \mathrm{id},$$

$$\text{mbox}Crossingrelation:R_{0,j}(u)=-{\sigma }_0^y{R}_{0,j}^{t_0}(-u-\eta ){\sigma }_0^y,$$

$$\text{mbox}PT-symmetry:R_{0,j}(u)=R_{j,0}(u)=R_{0,j}^{t_0{t}_j}(u),$$

$$R_{j,k}(u)=P_{0,j}{R}_{0,k}(u)P_{0,j}.$$

$$R_{1,2}(u_1-u_2)R_{1,3}(u_1-u_3)R_{2,3}(u_2-u_3)$$

$$=R_{2,3}(u_2-u_3)R_{1,3}(u_1-u_3)R_{1,2}(u_1-u_2).$$

$$T_0(u)$$

$${\hat{T}}_0(u)$$

$$R_{1,2}(u-v)T_1(u)T_2(v)=T_2(v)T_1(u)R_{1,2}(u-v).$$

$$t(u)=t{r}_0{T}_0(u),\hat{t}(u)=t{r}_0{\hat{T}}_0(u).$$

$$t(u)=\hat{t}(-u-\eta ),\hat{t}(u)=t(-u-\eta ).$$

$$[t(u),t(v)]=[\hat{t}(u),\hat{t}(v)]=[t(u),\hat{t}(u)]=0.$$

$$H$$

$$T_{0}(u)=\left(\begin{array}[]{cc}A(u)&B(u)\\ C(u)&D(u)\\ \end{array}\right).$$

$$\displaystyle|0\rangle=\left(\begin{array}[]{c}1\\ 0\\ \end{array}\right)_{1}\otimes\cdots\otimes\left(\begin{array}[]{c}1\\ 0\\ \end{array}\right)_{2N}.$$

$$A(u)|0⟩=a(u)|0⟩,B(u)|0⟩\ne 0,$$

$$C(u)|0⟩=0,D(u)|0⟩=d(u)|0⟩,$$

$$a(u)=\frac{{\sin }^N(u+a+\eta ){\sin }^N(u-a+\eta )}{{\sin }^{2N}\eta },$$

$$d(u)=\frac{{\sin }^N(u+a){\sin }^N(u-a)}{{\sin }^{2N}\eta }.$$

$$|{\lambda }_1,\cdots ,{\lambda }_M⟩=\prod _{j=1}^{M}B({\lambda }_j)|0⟩,$$

$$[A(u),A(v)]=[B(u),B(v)]=[C(u),C(v)]=[D(u),D(v)]=0,$$

$$A(u)B(v)=\frac{\sin (u-v-\eta )}{\sin (u-v)}B(v)A(u)+\frac{\sin \eta }{\sin (u-v)}B(u)A(v),$$

$$D(u)B(v)=\frac{\sin (u-v+\eta )}{\sin (u-v)}B(v)D(u)-\frac{\sin \eta }{\sin (u-v)}B(u)D(v),$$

$$[B(u),C(v)]=\frac{\sin \eta }{\sin (u-v)}[D(v)A(u)-D(u)A(v)].$$

$$t(u)=A(u)+D(u).$$

$$t(u)|{\lambda }_1,\cdots ,{\lambda }_M⟩$$

$$\Lambda (u)=a(u)\prod _{j=1}^M\frac{\sin (u-{\lambda }_j-\eta )}{\sin (u-{\lambda }_j)}+d(u)\prod _{j=1}^M\frac{\sin (u-{\lambda }_j+\eta )}{\sin (u-{\lambda }_j)},$$

$$\displaystyle\!\!\Lambda_{j}(u)=\frac{\sin\eta}{\sin(u-\lambda_{j})}\bigg{\{}a(\lambda_{j})\prod_{l\neq j}^{M}\frac{\sin(\lambda_{j}-\lambda_{l}-\eta)}{\sin(\lambda_{j}-\lambda_{l})}-d(\lambda_{j})$$

$$\displaystyle\quad\quad\quad\times\prod_{l\neq j}^{M}\frac{\sin(\lambda_{j}-\lambda_{l}+\eta)}{\sin(\lambda_{j}-\lambda_{l})}\bigg{\}}.$$

$${\left[\frac{\sin ({\lambda }_j+a+\eta )\sin ({\lambda }_j-a+\eta )}{\sin ({\lambda }_j+a)\sin ({\lambda }_j-a)}\right]}^N=\prod _{l\ne j}^M\frac{\sin ({\lambda }_j-{\lambda }_l+\eta )}{\sin ({\lambda }_j-{\lambda }_l-\eta )},$$

$$j=1,\cdots ,M.$$

$${\left[\frac{\sinh \frac{1}{2}(u_j-2b-i\eta )\sinh \frac{1}{2}(u_j+2b-i\eta )}{\sinh \frac{1}{2}(u_j-2b+i\eta )\sinh \frac{1}{2}(u_j+2b+i\eta )}\right]}^N$$

Full text

Quantum spin chain; Bethe Ansatz; Yang-Baxter equation

††preprint: APS/123-QED

I Introduction

It is well known that the Heisenberg model has played an important role to account for magnetism in condensed matters. An interesting fact is that this model in one-dimension is exactly solvable Bethe (1931). Based on Bethe’s exact solution, the ground state energy Hulthén (1938), the low-lying excitation spectrum des Cloizeaux and Pearson (1962) and the magnetization at zero temperature Griffiths (1964) had been studied extensively. This exact solution provided a benchmark to understand a variety of physical phenomena in low-dimensions such as the Luttinger liquid behavior and the fractional excitations. In addition, this model also becomes a typical model in developing new theoretical methods to approach general quantum integrable systems Sklyanin et al. (1979); Korepin et al. (1993); Takahashi (1999); Wang et al. (2015).

Besides the Heisenberg model with nearest-neighbor (NN) exchanges, its generalization with next-nearest-neighbor (NNN) interactions, known as the $J_{1}-J_{2}$ model, also attracted a lot of attentions Zeng and Parkinson (1995); White and Affleck (1996); Eggert (1996); Shastry and Sutherland (1981); Okamoto and Nomura (1992); Jafari and Langari (2006); Djoufack et al. (2016). The $J_{1}-J_{2}$ model is interesting because there exists a topological phase transition at the point of $J_{2}/J_{1}=0.241$ Bursill et al. (1995); Jafari and Langari (2007). At the Majumdar-Ghosh point, $J_{2}/J_{1}=0.5$, the model Hamiltonian degenerates into a projector operator and the ground state can be obtained exactly Majumdar and Ghosh (1969). The ground state is two-fold degenerated and can be expressed by the direct product of spin singlets, supposed the number of site of the system is even. Another interesting development is that Frahm proposed an integrable $J_{1}-J_{2}$ model containing chiral three-spin interactions Frahm (1992). The extra scalar chirality terms are introduced for ensuring the integrability. Later, Frahm and Rödenbeck studied the properties of the chiral spin liquid state in the system Frahm and Rödenbeck (1997). Wen, Wilczek and Zee Wen et al. (1989) and Baskaran Baskaran (1989) proposed that the expectation value of the spin chirality operator can be used as the order parameter for chiral spin liquids Kalmeyer and Laughlin (1987). Recently, the models with chirality terms have attracted renewed interest in the context of quantum spin liquids Popkov and Zvyagin (1993); Gorohovsky et al. (2015); Chen et al. (2017).

In this paper, we propose a systemic method to construct new integrable models with the NNN and the scalar chirality terms interactions. We use the anisotropic XXZ quantum spin chain as an example to show the validity of the method. Before that, we first introduce the main result that we construct an integrable anisotropic $J_{1}-J_{2}$ spin chain with the Hamiltonian

[TABLE]

where $\vec{\sigma}_{j}\equiv(\sigma^{x}_{j},\,\sigma^{y}_{j},\,\sigma^{z}_{j})$ are the Pauli matrices at site $j$, $a$ and $\eta$ are the generic constants describing the coupling strengths, and the periodic boundary condition

[TABLE]

is imposed. The first two terms describe an anisotropic NN interaction, the third term is an isotropic NNN interaction (i.e., the J2 term) and the last one corresponds to an anisotropic chiral three-spin interaction. We shall show that the anisotropic $J_{1}-J_{2}$ spin chain with the Hamiltonian (1) is integrable and can be exactly solved by the Bethe ansatz.

Some remarks are in order. (i) The hermitian of the Hamiltonian (1) requires that $a$ must be real if $\eta$ is imaginary (gapped regime), and $a$ must be imaginary if $\eta$ is real (gapless regime). (ii) The NN interactions are anisotropic while the NNN interactions are isotropic. (iii) The anisotropic scalar chirality terms are added to ensure the integrability of the system. (iv) The coupling strengths in the NNN terms and those in the scalar chirality terms are not independent but related by the parameters $a$ and $\eta$. (v) The model degenerates into the conventional XXZ spin chain at the points of $a=n\pi$ with integer $n$. (vi) After parameterizing $a=\bar{a}\,\epsilon$, $\eta=\epsilon$ and then taking the limit of $\epsilon\rightarrow 0$, our Hamiltonian (1) becomes

[TABLE]

The resulting Hamiltonian describe an integrable isotropic $J_{1}-J_{2}$ spin chain model, which was studied previously by Frahm et al Frahm (1992); Frahm and Rödenbeck (1997).

The paper is organized as follows. The model is constructed and the integrability is proved in section II. The exact energy spectrum and the Bethe Ansatz equations are derived in section III. The ground state energy and spinon elementary excitation for real $\eta$ are given in section IV and the corresponding results with imaginary $\eta$ are given in section V. The results of the non-hermitian case are discussed in section VI and section VII is attributed to the concluding remarks.

II Model and Integrability

Throughout, ${V}$ denotes a two-dimensional linear space and let $\{|m\rangle,m=0,1\}$ be an orthogonal basis of it. We shall adopt the standard notations: for any matrix $A\in{\rm End}({V})$, $A_{j}$ is an embedding operator in the tensor space ${V}\otimes{V}\otimes\cdots$, which acts as $A$ on the $j$-th space and as identity on the other factor spaces. For $B\in{\rm End}({V}\otimes{V})$, $B_{ij}$ is an embedding operator of $B$ in the tensor space, which acts as identity on the factor spaces except for the $i$-th and $j$-th ones.

Let us introduce the $R$-matrix $R_{0,j}(u)\in{\rm End}({V}_{0}\otimes{V}_{j})$

[TABLE]

where $u$ is the spectral parameter. The $R$-matrix (8) satisfies the following relations

[TABLE]

where $\phi(u)=-\sin(u+\eta)\sin(u-\eta)/\sin^{2}\eta$, $t_{0}$ (or $t_{j}$) denotes the transposition in the space ${V}_{0}$ (or ${V}_{j}$) and $P_{0,j}$ is the permutation operator possessing the property

[TABLE]

The $R$-matrix satisfies the Yang-Baxter equation (YBE)

[TABLE]

We define the monodromy matrices Sklyanin (1988); Wang et al. (2015) as

[TABLE]

where $V_{0}$ is the auxiliary space, $V_{1}\otimes V_{2}\otimes\cdots\otimes V_{2N}$ is the physical or quantum space, $2N$ is the number of sites and $a$ is the inhomogeneous parameter. From the YBE (11), one can prove that the monodromy matrix $T(u)$ satisfies the Yang-Baxter relation

[TABLE]

The transfer matrices are the trace of monodromy matrices in the auxiliary space

[TABLE]

Using the crossing symmetry in Eq.(9), we obtain the relations between transfer matrices $t(u)$ and $\hat{t}(u)$

[TABLE]

From the Yang-Baxter relation (13) and Eq.(15), one can prove that the transfer matrices $t(u)$ [or $\hat{t}(u)$] with different spectral parameters commute with each other. Meanwhile, the transfer matrices $t(u)$ and $\hat{t}(u)$ also commute with each other

[TABLE]

Therefore, both $t(u)$ and $\hat{t}(u)$ serve as the generating functions of all the conserved quantities of the system. We note that the transfer matrices $t(u)$ and $\hat{t}(u)$ can be diagonalized simultaneously.

The model Hamiltonian (1) can be constructed as (for details, see Appendix A)

[TABLE]

From the construction (17) and the commutation relation (16) of generating functions $t(u)$ and $\hat{t}(u)$, we conclude that the quantum spin chain (1) with the periodic boundary condition is integrable.

III Exact solution

Based on the integrability discussed in the previous section, the Hamiltonian (1) can be solved exactly via the algebraic Bethe Ansatz Korepin et al. (1993). The matrix form of monodromy matrix $T_{0}(u)$ in the auxiliary space is

[TABLE]

where $A(u)$, $B(u)$, $C(u)$ and $D(u)$ are the operators acting in the quantum space. We denote the all spins aligning up state as the vacuum state $|0\rangle$

[TABLE]

The matrix elements of the monodromy matrix $T_{0}(u)$ acting on the vacuum state gives

[TABLE]

where

[TABLE]

From Eq.(24), we know that the operator $B(u)$ can be regarded as the creation operator of all the eigenstates of the system. Assume the eigenstates take the form

[TABLE]

where $M$ is the number of flipped spins and $\{\lambda_{j}\}$ are the Bethe roots. From the Yang-Baxter relation (13), we obtain the commutative relations among the elements of monodromy matrix as

[TABLE]

From the definition (14), the transfer matrix $t(u)$ is

[TABLE]

Acting the transfer matrix $t(u)$ on the Bethe state (25) and with the help of the commutation relations (26), we have

[TABLE]

where

[TABLE]

The first term in Eq.(28) corresponds to the eigenvalue term, while the last terms in Eq.(28) are the unwanted ones. The state (25) becomes an eigenstate (or the Bethe state) of the transfer matrix provided that the parameters $\{\lambda_{j}|j=1,\cdots,M\}$ satisfy the Bethe Ansatz equations (BAEs)

[TABLE]

For convenience, we put $\lambda_{j}=iu_{j}/2-\eta/2$ and $a=ib$ for real $\eta$ and $\lambda_{j}=u_{j}/2-\eta/2$ for imaginary $\eta=i\gamma$ . The BAEs become

[TABLE]

for real $\eta$, and

[TABLE]

for imaginary $\eta=i\gamma$. From Eqs.(15), (17) and (29) we obtain the eigenvalue of the Hamiltonian (1) in terms of the Bethe roots as

[TABLE]

where $\eta$ is real and $\{u_{j}\}$ should satisfy the BAEs (III), or

[TABLE]

where $\eta=i\gamma$ is imaginary and $\{u_{j}\}$ should satisfy the BAEs (III).

Next, we check above results numerically. Numerical solutions of the BAEs and exact diagonalization of the Hamiltonian (1) are performed for the case of $2N=4$ and randomly choosing of model parameters. The results are listed in Table 1 for real $\eta$ and Table 2 for imaginary $\eta$. We note that the eigenvalues obtained by solving the BAEs are exactly the same as those obtained by the exact diagonalization of the Hamiltonian (1). The energies of the system are degenerated and there are only 8 separated energy level. Therefore, the expression (III) or (III) gives the complete spectrum of the system.

IV Ground state and elementary excitations for real $\eta$

In this section we study the ground state and elementary excitations of the system. First, we consider the real $\eta$ case. Taking the logarithm of BAEs (III), we have

[TABLE]

where

[TABLE]

Here the quantum number $\{I_{j}\}$ are certain integers (or half odd integers) if $M$ is odd (or even). For convenience, we define the counting function

[TABLE]

Obviously, $Z(u_{j})=I_{j}/2N$ corresponds to the Eq.(IV). In the thermodynamic limit, $N\rightarrow\infty$, $M\rightarrow\infty$ and $N/M$ finite, taking the derivative of Eq.(37) with respect to $u$, we obtain

[TABLE]

where

[TABLE]

$\rho(x)$ and $\rho^{h}(x)$ are the densities of particles and holes, respectively.

IV.1 Ground state

From the analysis of Eq.(IV), we know that at the ground state, $M=N$ which is half of the number of sites. Meanwhile, all the Bethe roots constrained by Eq.(IV) are real and the corresponding quantum numbers are

[TABLE]

From Eq.(III), we learn that each real Bethe root $u_{j}$ contributes a negative energy. At the ground state, the Bethe roots should fill the whole real axis and leave no hole, i.e., $\rho^{h}(u)=0$. This means that the density of particles $\rho_{g}(u)$ at the ground state satisfies

[TABLE]

Let us define the following Fourier transformation

[TABLE]

Without losing generality, we consider the case $\eta\in(0,\pi)$. Taking the Fourier transformation of Eq.(41), we obtain

[TABLE]

with $\delta_{n}\equiv\frac{n\eta}{2\pi}-\lfloor\frac{n\eta}{2\pi}\rfloor$ denoting the fractional part of $\frac{n\eta}{2\pi}$. Thus the solution of Eq.(41) is

[TABLE]

The Bethe root distribution at the ground state is shown in Fig.1. The magnetization at the ground state is

[TABLE]

indicating a singlet ground state.

The energy density at the ground state reads

[TABLE]

IV.2 Spinon Excitations

Now we consider the elementary excitations. The simplest excitation is the case of one less spin flipped, i.e., $M=N-1$. Such a configuration is described by putting two holes in the Fermi sea. Meanwhile, all the Bethe roots constrained by Eq.(IV) are real and the corresponding quantum numbers are

[TABLE]

where $0<r<s<N$. The positions of holes are denoted as $u^{h}_{r}$ and $u^{h}_{s}$. In this case all $N-1$ quasi-momentum are real numbers and the total momentum is $\Delta K=\pi(r+s)/N$. In the thermodynamic limit, the momentum of this excitation is calculated as

[TABLE]

The density of holes is

[TABLE]

The corresponding Bethe root density becomes $\rho_{e}(u)=\rho_{g}(u)+\delta\rho(u)$ Takahashi (1999). The density $\rho_{e}(u)$ will deviate from $\rho_{g}(u)$ by $\delta\rho(u)$ because of the presence of the two holes. From Eqs.(38) and (49), we obtain

[TABLE]

After some calculations, we have

[TABLE]

The excitation energy is

[TABLE]

where

[TABLE]

We see that the energy of such an excitation is the summation of the energies of two holes. Here the two holes together carry spin-1, and each of them carries spin-$\frac{1}{2}$. These excitations are usually called spinons Faddeev and Takhtajan (1981), a typical fractional excitation in the one-dimensional quantum systems.

The dispersion relation of the spinon excitations can be derived from equations (48) and (52). The numerical results are shown in Fig.2. From it, we see that the spinon excitation is gapless which can be reached by putting the holes at the points [math] or $\pm\pi$. Meanwhile, if $b$ is very small, the excitation spectrum is quite similar to that of the conventional XXZ model Takahashi (1999). With the increasing of $b$, the excitation spectrum turns to the triple arched structure. Unlike the conventional $J_{1}-J_{2}$ model, there is no dimerization in the present model for any real $b$. If $b$ takes some imaginary values, dimerization indeed occurs as hinted from the solution of the BAE’s. However, in such a case the Hamiltonian is non-hermitian.

V Ground state and elementary excitations for imaginary $\eta$

In this section we study the ground state and elementary excitations of the system for imaginary $\eta=i\gamma$. Without losing generality, we assume $\gamma>0$. Similarly, let us introduce

[TABLE]

The Fourier transformation of $a_{n}(x)$ is

[TABLE]

Note that $\omega$ takes values of integers. From the energy expression (III), we know that at the ground state the Bethe roots still take real values and fill the region $(-\pi,\pi]$. Using the similar procedure mentioned above, we find the density of Bethe roots at the ground state satisfies

[TABLE]

Using Fourier transformation we have

[TABLE]

The Bethe root distribution at the ground state is shown in Fig.3. Interestingly, we find $\rho_{g}(\pm\pi)=\rho_{g}(0)$ if $a=\pi/4+k\pi/2$, and $k$ is an arbitrary integer. Please see the lower one in Fig.3.

The total magnetization at the ground state is still zero. The energy density at the ground state reads

[TABLE]

In the thermodynamic limit, the momentum of spinion excitation is calculated as

[TABLE]

After some calculations similar with above real $\eta$ case, we obtain the excitation energy

[TABLE]

where

[TABLE]

The dispersion relation of the spinon excitations can be derived from equations (59) and (60). The numerical results are shown in Fig.4. From it, we see that if $a$ is very small, the excitation spectrum is quite similar to that of the conventional XXZ model Takahashi (1999). With the increasing of $a$, the excitation spectrum turns to the triple arched structure. Similar with real $\eta$ case, there is also no dimerization in the present model for any imaginary $\eta$ and real $a$.

From the excitation spectrum in Fig.4, we also find that the elementary excitations possess a finite gap. Now we determine the values of the gap. Without losing generality, we assume $a\in[0,\pi]$. We should consider the positions of holes first. From Fig.3, we know that there are two minimal points located at $u=0$ and $u=\pi$. As we mentioned before, $\rho_{g}(\pi)=\rho_{g}(0)$ if $a=\pi/4$ or $a=3\pi/4$. By detailed analysis, we conclude that $\rho_{g}(\pi)<\rho_{g}(0)$ if $0<a<\pi/4$ or $3\pi/4<a<\pi$ and $\rho_{g}(\pi)>\rho_{g}(0)$ if $\pi/4<a<3\pi/4$. Thus we put the holes at the point of $\pi$ if $a\in[0,\pi/4]\cup[3\pi/4,\pi]$, and put the holes at the point of [math] if $a\in(\pi/4,3\pi/4)$. We note that only in the thermodynamic limit, two holes can be put at the same position. After some calculations, we obtain the energy gap of the model (1) in these intervals as

[TABLE]

if $a\in[0,\pi/4]\cup[3\pi/4,\pi]$, and

[TABLE]

if $a\in(\pi/4,3\pi/4)$.

The gap is shown in Fig.5. From it, we see that the gap is enhanced by the NNN and chiral three-spin interactions. At the points of $a=0$ and $a=\pi$, the gap takes its minimum, which is the same as that of the conventional XXZ spin chain, because that the model (1) degenerates into the conventional XXZ spin chain at these points. At the point of $a=\frac{\pi}{2}$, the gap also takes its minimum. In this case the model (1) degenerates into the XXZ spin chain only with NN interaction where the couplings along the $x$ and $y$ directions are negative. At the points of $a=\frac{\pi}{4}$ and $a=\frac{3\pi}{4}$, the gap arrives at its maximal value. This is because at the these points, the coupling strengths of NNN and chiral three-spin interactions reach their maximum and the NN couplings along the $x$ and $y$ directions are zero. Besides, the gap also has the property

[TABLE]

which means that the gap is symmetric with respect to the points of $\frac{\pi}{4}$, $\frac{\pi}{2}$ and $\frac{3\pi}{4}$. This symmetry is different from that of the Hamiltonian, where the Hamiltonian is symmetric only with respect to $\pi$, i.e.,

[TABLE]

VI Non-hermitian case

In this section, we consider the case that both $a$ and $\eta$ are real, which implies that the Hamiltonian (1) is non-hermitian. By the analysis of possible couplings in the Hamiltonian (1), we restrict the values of $a$ in the interval $[0,\pi]$. It is easy to check that following identity holds

[TABLE]

This means that the values of $\eta$ can also be restricted in the interval $(0,\pi)$. Then the parameters $a\in[0,\pi]$ and $\eta\in(0,\pi)$ can describe all the coupling strengths.

Using the direct diagonalization method, up to $2N=12$, we find that all the eigenvalues of the Hamiltonian (1) are real if $a$ takes the values in some intervals. After detailed calculation, the intervals are determined as $a\in[0,\eta/2]\cup[(\pi-\eta)/2,(\pi+\eta)/2]\cup[\pi-\eta/2,\pi]$. We note that if $\eta>\pi/2$, these intervals are connected with each other and the parameter $a$ fills the whole interval $[0,\pi]$, which means that with an arbitrary $a$, all the eigenvalues of the model (1) are real.

The BAEs (30) are true for real $a$ and real $\eta$. Put $\lambda_{j}=iu_{j}/2-\eta/2$ and the BAEs become

[TABLE]

From Eqs.(17) and (29) we obtain the eigenvalue of the Hamiltonian (1) in terms of the Bethe roots as

[TABLE]

Solving the BAEs (VI) with $2N=6$ and substituting the values of Bethe roots into (VI), we find that the eigenvalues calculated from the Bethe roots are exactly the same as those obtained from the exact diagonalization of the Hamiltonian (1). The energy spectrum is complete. Meanwhile, the intervals that all the eigenvalues are real keep unchanged.

VII Conclusion

In this paper, we propose a new integrable anisotropic $J_{1}-J_{2}$ spin chain model with extra scalar chirality terms. By means of the Bethe Ansatz method, we obtain the exact solution of the system. The ground state and the novel structure of the elementary excitation spectrum are obtained. We find that the elementary excitation is gapless if the anisotropic parameter is real while the elementary excitation has a gap if the anisotropic parameter is imaginary. Moreover, it is shown that the spinon excitation spectrum of the model possesses a novel triple arched structure. The method of this paper can be used to construct other new integrable models with next-nearest-neighbour couplings.

Acknowledgments

We would like to thank Prof. Y. Wang for his valuable discussions and continuous encouragements. The financial supports from the National Program for Basic Research of MOST (Grant Nos. 2016YFA0300600 and 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11434013, 11425522, 11547045, 11774397, 11775178 and 11775177), the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2017KCT-12, 2017ZDJC-32), Australian Research Council (Grant No. DP 190101529), the Strategic Priority Research Program of the Chinese Academy of Sciences and the Double First-Class University Construction Project of Northwest University are gratefully acknowledged.

Appendix: Derivation of the Hamiltonian

Using the initial condition (9) of the $R$-matrix $R(u)$ given by (8), we can evaluate the values of the transfer matrix $\hat{t}(u)$ at some points: $u=\pm a$:

[TABLE]

Taking the derivative of the transfer matrix $t(u)$ with respect to $u$ at the point of $u=a$, we have

[TABLE]

where $R^{\prime}_{i,j}(u)=\frac{\partial}{\partial u}\,R_{i,j}(u)$. Similarly we can calculate the derivative of $t(u)$ at the point of $u=-a$

[TABLE]

Substituting the above relations (A1)-(Appendix: Derivation of the Hamiltonian) into the expression (17), we obtain

[TABLE]

The derivative of the $R$-matrix (8) reads

[TABLE]

The commutative relation between the permutation operators is

[TABLE]

Substituting Eqs.(8), (A5) and (A6) into (A4) and after some tedious calculations, we arrive at the form of the Hamiltonian (1).