Ground state of a spin-1/2 Heisenberg-Ising two-leg ladder with XYZ intra-rung coupling

The quantum spin-1/2 two-leg ladder with an anisotropic XYZ Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation. The particular case of the considered model with X-X intra-rung interaction resembles a quantum compass ladder with additional frustrating diagonal Ising interactions. Using an appropriately chosen unitary transformation, the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may subsequently find the ground state quite rigorously. We obtain a ground-state phase diagram and analyze the interplay of the competition between several factors: the XYZ anisotropy in the Heisenberg intra-rung coupling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction. The investigated model shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases, two different disordered quantum paramagnetic phases, as well as discontinuous or continuous quantum phase transitions between those phases.


Introduction
Quantum spin ladders with frustrated interactions are intensively studied during the last few decades, since they exhibit a rather complex ground-state behaviour to be reflected in extraordinarily rich lowtemperature thermodynamics as well [1,2]. Quite recently, a number of exact solutions have been obtained for several particular examples of quantum spin-1 2 two-leg ladders [3][4][5][6][7]. The railroad ladder considered by Lai and Montrunich [6] has a quite specific configuration of inter-spin interactions, namely, the staggering of X -X and Y -Y couplings along the legs is supplemented by the uniform Z -Z coupling present along the rungs. An exact solution of this specific quantum spin ladder has been found by adopting the method originally developed by Kitaev [8], which proved a striking spin-liquid ground state in this quantum spin ladder. On the other hand, the railroad ladder with the uniform Z -Z interaction along the legs and the uniform X -X interaction along the rungs has been rigorously solved by Brzezicki and Oleś [3][4][5]. To a certain extent, this exactly solved quantum spin ladder can be regarded as an one-dimensional analogue of the quantum compass model on a square lattice, which describes the orbital ordering in transition-metal compounds [9].
In this work, we will examine a more general model of the quantum spin-1 2 two-leg ladder, which includes the fully anisotropic X Y Z -Heisenberg coupling between spins from the same rung and two different Ising (Z -Z ) interactions between spins from neighbouring rungs considered along the legs and across the diagonals, respectively. Nevertheless, it should be mentioned that the investigated quantum spin ladder extends our previous exact calculations for the spin-1 [3][4][5] also represents a very special limiting case of the investigated model system. The main goal of the present paper is to examine the simultaneous effect of two different kinds of frustration: the geometric frustration caused by the antiferromagnetic interaction between spins from an elementary triangle plaquette and the competition between X -X and Y -Y intra-rung interactions with both Z -Z inter-rung interactions. The outline of the paper is as follows. In section 2, we define the model and show how to get the ground state by a rigorous calculation based on the appropriate unitary transformation. The groundstate phase diagram of the spin-1 2 X Z -Ising and X Y -Ising ladders is explored in section 3. Finally, some conclusions are drawn in section 4.

Model and solution
Consider the quantum spin-1 2 Heisenberg-Ising ladder with an anisotropic X Y Z intra-rung coupling and two different Ising-type couplings, which involve Z -Z spin-spin interactions along the legs and across the diagonals of a two-leg ladder (see figure 1): Here, s α j ,i denote three spatial components α = x, y, z of the spin-1 2 operator, the former subscript j = 1, 2 determines the number of a leg and the latter subscript enumerates the lattice position in a particular leg.
Apparently, the interaction terms J x 1 , J y 1 , J z 1 account for the quite anisotropic X Y Z -Heisenberg coupling between two spins belonging to the same rung, while the interaction terms J 2 and J 3 take into consideration the Ising-type interactions between the nearest-neighbor spins along the legs and across the diagonals of the two-leg ladder. It should be pointed out that the z-component of the total spin S z i = s z 1,i + s z 2,i Heisenberg-Ising ladder with X X Z intra-rung interaction. After the unitary transformation [7] one may rewrite the Hamiltonian (2.1) into the following pseudospin representation: which shows the symmetry of the model in a more explicit way. It is quite obvious that only z-components of spin operators from the first leg are present in the Hamiltonian (2.3), which means thats z 1,i are good quantum numbers. By contrast, different spatial components of spin operators from the second leg are still involved in the Hamiltonian (2.3) and thus, they still represent quantum spins with regard to the presence of two non-commuting partss x 2,i ands z 2,i of each spin operator. Altogether, the Hamiltonian (2.3) can be identified as the Ising chain of composite spins in a transverse field, whereas the values of the effective interaction and the effective transverse field locally depend on a particular choice of eigenvalues of the classical Ising spinss z 1,i . Following [7], one may also establish the following correspondence between new and initial states: To get the partition function one has to diagonalize the Hamiltonian (2.3) for all particular configurations ofs z 1,i and sum up all contributions in the trace of statistical operator. However, it is quite evident from the transformed Hamiltonian (2.3) that the chain decomposes into two independent parts whenever two neighboring spinss z 1,i ands z 1,i+1 have opposite orientation (i.e., take on different eigenvalues). In this respect, the composite chain is divided into a set of finite chains of different sizes for any chosen configuration ofs z 1,i . Generally, this problem seems to represent a quite formidable task, but the ground state of the investigated model can be found quite rigorously using the same arguments as given in [7]. Since the ground-state energy of two finite but isolated spin- 1 2 Ising chains in a transverse field is always higher than the ground-state energy of one unique spin- 1 2 Ising chain in a transverse field obtained by joining both independent finite chains, the ground state of the model under investigation should accordingly correspond only to the uniform configuration of alls z 1,i . Therefore, one may single out only two different uniform configurations with alls z 1,i = 1 2 or alls z 1,i = − 1 2 from which the ground state of the Heisenber-Ising ladder can be derived. The effective Hamiltonian (2.3) for the two uniform configurations acquires the following form: The ground state energies per site of both effective Hamiltonians can be exactly calculated using the Jordan-Wigner fermionization [10,11]: is the complete elliptic integral of the second kind.
Both Hamiltonians H + and H − imply a precise mapping correspondence between the spin-1 2 Heisenberg-Ising two-leg ladder and the spin-1 2 quantum Ising chain in a transverse field, which can be, however, characterized by different values of the effective interaction and transverse field. Bearing this in mind, one should expect quantum phase transitions of two different types. The first kind of zerotemperature phase transitions may correspond to a continuous (second-order) quantum phase transition inherent to the transverse Ising chain, which arises for one particular ratio between the effective interaction and transverse field. Beside this, there may also occur discontinuous (first-order) quantum phase transitions whenever a crossing of the lowest-energy levels inherent to both effective Hamiltonians (2.5) takes place. In the following two sections, we will illustrate all the aforementioned features of quantum phase transitions on ground-state phase diagrams of two particular cases of the model under consideration.

Ground state of X Z -Ising and X Y -Ising ladders
In this section, we will consider two particular cases of the investigated model (2.1) by switching off either the y-or z-component of X Y Z -Heisenberg coupling (i.e., either J y 1 = 0 or J z 1 = 0). It is noteworthy that the two aforementioned particular cases represent a direct extension of the quantum compass ladder [3][4][5] to which the considered model reduces when neglecting the z-component of the Heisenberg coupling (J z 1 = 0), one of the two transverse components of the Heisenberg coupling (i.e., either J x 1 = 0 or J y 1 = 0) and the frustrating Ising interaction across the diagonals (J 3 = 0). Furthermore, the problem of two-dimensional quantum compass model is quite complex and the exact solution for this model has not been found yet.
Let us first consider all possible phases that may appear in the ground state of the model under investigation. Each uniform configuration ofs z 1,i corresponds to the transverse Ising chain, which has three possible ground-state phases. The ground-state phases for alls z It is worthwhile to remark that the ground-state phases belonging to this effective model were thoroughly analyzed in our preceding paper [7] and let us, therefore, give here just their definition for the sake of easy reference: • Quantum paramagnetic (QPM1) state for 1 , which undergoes the obvious quantum reduction of magnetization.
• Néel state for 1 2 (J x 1 + J y 1 ) < J 2 − J 3 : the nearest-neighbor spins both along the legs and rungs exhibit predominantly antiferromagnetic ordering. The dependence of staggered magnetization as the relevant order parameter is quite analogous to the previous case The most fundamental difference between the ground states of the Heisenberg-Ising ladder with the X X Z -and X Y Z -Heisenberg intra-rung interaction can be found in the phases arising from the uniform configuration with alls z 1,i = 1 2 . While in the former model with the X X Z intra-rung interaction, all ground-state phases are classical in their character [7], the emergent ground-state phases of the latter model with the more anisotropic X Y Z intra-rung coupling display significant quantum features. One may indeed identify the following three quantum ground states for a particular case e + 0 < e − 0 with all s z 1,i = 1 2 : • Quantum paramagnetic (QPM2) state for 1 2 |J x 1 − J y 1 | > |J 2 + J 3 |: the equivalent transverse Ising chain H + (2.5) is in the gapped disordered state with no spontaneous magnetization 〈s x i 〉 = 0 and

13601-4
X Y Z -Heisenberg-Ising two-leg ladder non-zero magnetization 〈s z i 〉 0 induced by the effective transverse field. For the initial Heisenberg-Ising ladder, one consequently gets the ground state with the prevailing dimer state |φ i 1,− 〉 on the rungs.
• Stripe Rung (SR) state for 1 2 |J x 1 − J y 1 | < J 2 + J 3 : the equivalent transverse Ising chain exhibits a spontaneous antiferromagnetic ordering with 〈s x i 〉 = (−1) i m x 0. Due to relationships (2.2), one obtains for the Heisenberg-Ising ladder 〈s z 1,i 〉 = −〈s z 1,i+1 〉 = 〈s z 2,i 〉 = −〈s z 2,i+1 〉 0. Thus, the Heisenberg-Ising ladder shows an antiferromagnetic order along the legs and ferromagnetic order along the rungs. The staggered magnetization as the relevant order parameter in this phase is non-zero and it exhibits evident quantum reduction of the magnetization given by: Altogether, it could be concluded that the X Y anisotropy in the Heisenberg intra-rung coupling is responsible for quantum features of otherwise classical SR and FM states and, moreover, it may also lead to the appearance of a new disordered phase QPM2. Two paramagnetic phases QPM1 and QPM2 have quite similar features: they are both disordered states with the energy gap in their excitation spectrum and, consequently, their pair spin-spin correlation functions decay exponentially. Both quantum paramagnetic phases can be distinguished by the square of zth component of the total spin (S z i ) 2 on i th rung, which is equal to (S z i ) 2 = 0(1) in QPM1 (QPM2).
Now, let us pay our attention to the ground-state phase diagram established for the particular case of X Z -Ising ladder as depicted in figure 2 by considering J y 1 = 0. Assuming the X Z intra-rung interaction, one gets a striking competition between the X -X interaction along the rungs and the Z -Z interaction along the legs, while the additional Z -Z interaction along rungs acts generally against the X -X interaction. It should be also mentioned that one may recover some known examples from the ground-state phase diagram of the X Z -Ising ladder presented in figure 2. In fact, figure 2 (a) shows the particular limiting case of a quantum compass ladder with an additional diagonal frustrating Ising interaction. Let us follow the known results of a simple quantum compass ladder [3] to be obtained from our model by disregarding the frustrating Ising interaction J 3 = 0. Both Hamiltonians H + and H − (2.5) become identical under this special condition and, consequently, the ground state of the model is always two-fold degenerate due to the equality e − 0 = e + 0 . The investigated model is either in the SL or FM state for 1 either in QPM1 or QPM2 state for 1 2 J x 1 > |J 2 |, either in SR or Néel state for 1 2 J x 1 < J 2 . Note that the quantum phase transition from the disordered to the long-range ordered state takes place at It is quite evident that the diagonal interaction J 3 removes the two-fold degeneracy of the ground state The relevant ground-state behavior can be supported by the dependencies of respective order parameters as displayed in figure 3 (a). If there is no frustrating diagonal interaction, the model may stay  in the disordered QPM1 or the ordered SL and Néel phases. The corresponding nearest-neighbor correlation function along the legs shows a continuous change with a weak singularity at the quantum critical points indicated by filled circles in figure 3 (b). The curve for another particular case J 3 = 0.25 looks similar except that the diagonal interaction of this strength leads to a direct phase transition between two disordered quantum paramagnetic states QPM1 and QPM2. This unusual transition can be recognized from the relevant dependence of the nearest-neighbor correlation function, which sustains a jump at this special critical point. It is interesting to note that the further increase of a frustrating Ising interaction J 3 demolishes both disordered phases QPM1 and QPM2. Thus, one may also detect the quantum phase transition between two ordered SL and Néel phases, whereas the order parameters do not reach zero continuously in this particular case. In figure 4, the ground-state phase diagram of the X Y -Ising ladder is depicted by considering another particular case with J z 1 = 0. The effect of the Y -Y intra-rung interaction has some similarities with the one of Z -Z intra-rung interaction, although the origin is completely different. The ground-state energy of H − is generally lowered with respect to that of H + , because J Similarly to the case with the Z -Z intra-rung interaction, the Y -Y intra-rung interaction also induces the presence of the rung singlet-dimer state along a special line J 2 = J 3 .

Conclusions
In the present paper, the effect of the most general X Y Z anisotropy in the intra-rung interaction on the ground state of the spin-1 2 Heisenberg-Ising two-leg ladder was investigated in detail. It has been shown that the most general kind of anisotropy, which breaks the rotational symmetry of the Heisenberg interaction, may lead to the appearance of new quantum phases in the ground-state phase diagram. We have also considered the special case of quantum compass ladder with an additional frustrated diagonal interaction and showed that the singlet-dimer phase cannot appear in this particular case. The order parameters and the nearest-neighbor correlation function were calculated and analyzed in detail in the ground state. It has been demonstrated that the relevant behavior of the correlation function can help us to reveal the quantum phase transition between two different disordered quantum paramagnetic phases.