Quantum transitions of the XY model with long-range interactions on the inhomogenous periodic chain
The isotropic XY model in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of segments with different exchange interactions and magnetic moments, is exactly solved by introducing the integral gaussian transformation and the generalized Jordan-Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of th order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at , for the induced magnetization and isothermal susceptibility as function of the transverse field.
pacs:05.70.Fhsep 05.70.Jksep 75.10.Jmsep 75.10.Pq
The study of the critical quantum behavior of systems(1), which is induced by quantum fluctuations, has been object of great interest in recent years. This critical behavior, which controls the properties of the systems at very low temperature, is present in different systems. In particular, for magnetic systems, they have been responsible for unusual properties observed in low dimensional magnetic materials (2); (3). Therefore, the study of the critical behavior of spin systems in low dimension, particularly the exactly soluble, is of great importance for understanding the properties of these materials.
Among these models, the one-dimensional XY model introduced by Lieb, Schultz and Mattis (4), despite being almost fifty year old, is still the best one to describe exactly magnetic quantum transitions. The rather rich quantum critical behavior presented by the model can be seen in a recent work by the authors (5) and in the references therein, where they study the anisotropic model on the inhomogeneous periodic lattice. The study of the model on the inhomogeneous open lattice has also been recently addressed by Feldman (6).
The isotropic model on the inhomogeneous lattice has also been studied by the authors (7), where a detailed study of the static and dynamic critical properties is presented. Although the model has been applied mostly in the study of the quantum critical behavior of magnetic systems (2); (3); (8); (9); (10), more recently it has also been applied in the study of quantum entanglement, which plays an essential role in the quantum computation. These applications can be found in the recent work by Amico et al.(11) and in the references therein. In particular, in a recent work on quantum communication in a spin system, Avellino et al.(12) have studied the strong effect of long-range interaction on the fidelity of transmission of quantum information.
Besides the importance of the long-range interaction in the transmission of quantum information in spin chains, its presence can induce classical critical behavior in these systems which is essential for the study of the classical/quantum crossover.
The one-dimensional XY model is among the models which present this behavior, provided a uniform long-range interaction is considered. In particular, for the isotropic one, in the presence of a homogeneous long-range interaction along the transverse field direction, it can still be solved exactly, and its solution has been obtained by the authors (13); (14). Besides the appearance of classical critical behavior, the most important features presented by the homogeneous model have been the existence of quantum bicritical points and, particularly, the existence of first order quantum transitions. Another relevant result obtained in the study of the homogeneous case was the exact determination of the classical/quantum crossover for first and second order phase transitions, and the verification of the scaling relations proposed Continentino and Ferreira (15) for first order quantum transitions.
As pointed out by Pfleiderer (16), quantum first order transitions can be driven by different mechanisms, and to look at this new quantum critical behavior, in an exactly soluble problem, has been the one of the motivations to analyze the isotropic model on the inhomogeneous periodic chain with long-range interaction. Besides this point, the possibility of having multiple first and second order quantum transitions and multicritical points of higher order have also been important motivations for considering the long-range interaction in the inhomogeneous model.
Therefore, in this paper we will extend our previous results (7); (13); (14) by looking at an extended version of the models previously considered. Although the model presents classical and quantum critical behavior, we will restrict our analysis to the quantum one.
In Section 2 we present the exact solution of the model and obtain the functional of the Helmholtz free energy at arbitrary temperatures. The quantum critical behavior is discussed in Section 3, and it is determined from the equation of state at . Explicit results for the quantum phase diagrams, induced magnetization and isothermal susceptibility for systems with different sizes of the unit cell are also presented. Finally in Section 4 we summarize the main results of the paper.
Ii The model and the functional of the free energy
We consider the isotropic XY model on the inhomogeneous periodic chain with cells, sites per cell, and lattice parameter , in a transverse field, with long-range interactions among the spin components in the direction. The unit cell of the inhomogeneous lattice is shown in Fig. 1, and the Hamiltonian of the model in its more general form is given by
where the parameters are the exchange coupling between nearest-neighbours, the magnetic moments, the external field, the uniform long-range interaction among components, and where we have assumed periodic boundary conditions. If we introduce the ladder operators
and the Jordan-Wigner transformations
where and are fermion annihilation and creation operators, we can write the Hamiltonian as(17)
with given by
Introducing periodic and anti-periodic boundary conditions on for and , respectively, the wave-vectors in the Fourier transforms
are given by for periodic condition and , for anti-periodic condition (20), with , and and can be written in the form
Although and do not commute, it can be shown that in the thermodynamic limit all the static properties of the system can be obtained in terms of or (17); (18); (19). Therefore, by considering periodic boundary conditions on we can identify and , can be written in the form
The partition function is then given by
Since the long-range interaction term commutes with the Hamiltonian, we can introduce the gaussian transformation
in the previous expression, so that the partition function can be rewritten in an integral representation as
where and the effective Hamiltonian is given by
By introducing the canonical transformations
and by imposing the condition
this leads, for the coefficients to the equation
where is given by
and the satisfy the orthogonality relations,
and Therefore the effective Hamiltonian can be written in the diagonal form
where the spectrum of is determined from the determinantal equation
and the operators are given in terms of and by the expression
In the thermodynamic limit, the partition function can be evaluated by Laplace´s method by imposing the condition that where is given by
and is equal to
which can be written in terms of the average induced magnetization , defined as
in the form
Finally, we can obtain from the previous results the functional of the Helmholtz free energy per lattice site which is given by
The equation of state is obtained numerically from this functional by imposing the conditions
The numerical solution is more easily obtained for where there are analytical solutions for eqs.(20) and (25) (7) and in the cases where we have uniform magnetic moments, namely, In this situation, the term commutes with the Hamiltonian, and consequently the effect of the field is to shift the spectrum. This means that the solution of eq.(25) can be written as
where is the energy of the excitations of the model for zero transverse field and long-range interaction, and we identify the expression as an effective field (14).
Iii The quantum critical behavior
In the limit the functional of the Helmholtz free energy per lattice site, eq.(30), can be explicitly written as
where is obtained by imposing the condition and we have considered the lattice spacing and uniform magnetic moments
Under these conditions the average induced magnetization, , given in eq.(29), can also be written in the form
The quantum phase diagram is determined from the equation of state, which is obtained numerically from the previous equations by considering the conditions shown in eq.(31), and in particular we can determine the magnetization as a function of the transverse field. As it is well known, the second order phase transitions are determined by imposing the limit and it can be shown numerically, for arbitrary unit cell sizes, that the critical field varies linearly with the long-range range interaction and that these phase transitions, as in the homogenous model (14), occur for only.
The first order phase transitions are determined by imposing the additional condition
where are the magnetization plateaus, which are identical to the ones in the model without the long-range interaction (7). This is due to the fact that, for uniform magnetic moments, the long-range interaction does preserve the azimutal symmetry of the model and consequently the magnetization plateaus satisfy the quantization condition (21)
which does only depend on the symmetry of the Hamiltonian.
As in the homogenous case (14), the first order phase transitions occur for only, and in this case, besides the bicritical points where the second order lines meet the first order ones (13), there are triple points which correspond to the point where three first order lines meet.
From the magnetization as a function of the field we can obtain the isothermal susceptibility, which is given by
The main results are shown in Figs. 2 to 11. In Fig. 2 we present the phase diagram for and as expected there are two second order lines for which end up at bicritical points for . For , there are two first order lines which collapse into a single one at a triple point. In Fig. 3(a) we present the magnetization as function of the field, and as it can be seen the different critical behaviors are explicitly shown depending on the sign of the long-range interaction. It should be noted that as we approach the triple point the two first order phase transitions collapse into a single one, which corresponds in this case to a jump in the induced magnetization from zero to one half.
As in the homogeneous model(14), there is a universal curve to which all magnetization data collapse, independently of the order of the transition. This is presented in Fig. 3(b), where we show the magnetization as a function of the effective field
In Fig.4 we present the quantum phase diagram for As in the previous case, for we have second order phase transitions and for we have first order phase transitions. There are three second order lines and two first order lines meet at a unique triple point. The magnetization is presented in Fig.5(a) for different values of , which characterize the different behaviors, and in Fig 5(b) we present the collapse of the magnetization when plotted as a function of the effective field
It should be noted that for a different set of parameters we could have two triple points, and in this case we will have a single first order line beyond the critical value which is associated to the second triple point. This situation can be seen in the phase diagram shown in Fig 6, for , where it is explicitly shown that we can have three triple points. Since we have no suppression of a phase transition, there are four second order transition lines. As it has been shown for (7), an adequate choice of the exchange parameters can suppress a phase transition. For , this condition corresponds to and the phase diagram for this case is presented in Fig. 7. As it can be verified in this figure, we have three second order lines, instead of four, and just one triple point.
The magnetization associated to the parameters defined in Fig. 6 is presented in Fig. 8. As in the previous cases, in Fig 8(a) we have the magnetization as a function of the field for different values of , and in Fig.8(b), the universal curve for the magnetization as a function of the effective field In Fig. 9, we present the phase diagram for Since is odd, there is no suppression of any transition for (7), and, as expected, there are five second order transition lines. The associated magnetization as a function of the field is presented in Fig 10(a) and the magnetization universal curve is shown in Fig. 10(b).
Finally, in Fig. 11, we present the isothermal susceptibility, for and different values of As in the homogeneous model, the isothermal susceptibility, at the second order phase transitions, diverges for only, and its multiple phase transitions have the same critical exponents of the homogeneous model and consequently belong to same universality class.
In this work we have considered the isotropic XY model with an uniform long-range interaction along the direction, in a periodic inhomogenous lattice with cells and sites per cell. The exact solution of the model was formally obtained at arbitrary temperatures and distribution of magnetic moments and exchange constants. Explicit equations have been obtained for the functional of the Helmholtz free energy from which the equation of state can be determined numerically.
The analysis of the critical behavior has been restricted to the quantum phase transitions and we have shown, as in the homogeneous model, that the system presents first order phase transitions when the long-range interaction is ferromagnetic, and second order phase transitions when the long-range interaction is antiferromagnetic. The model also presents multiple first and second order phase transitions and the number of critical lines is equal to for odd, and less than for even, provided the exchange constants in the plane satisfy a special relation.
The second order critical lines meet the first order ones at which are bicritical points and there are multiple triple points, where three first order critical lines meet, depending on the unit cell size. The critical exponents have been obtained numerically, and it has been shown, as expected, that the model belongs to the same universality class of the homogeneous one.
Finally, we would like to point out that is of paramount importance the presence of multiple first order quantum transitions and triple points, which we have shown exactly to exist in the model, since we believe that this behavior is related to the different mechanisms from which the first order phase transitions are driven. These mecanisms have been thoroughly discusssed by Pfleiderer(16), by analyzing quantum critical behavior obtained for different materials.
The authors would like thank the Brazilian agencies CNPq and Capes for partial financial support.
- preprint: Preprint
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