Quantum phase transitions of the extended  isotropic XY model with long-range interactions

Quantum phase transitions of the extended  isotropic XY model with long-range interactions

F. G. Ribeiro Departamento de Física, Universidade Federal do Piauí, Campus Ministro Petrônio Portela,64049-550, Teresina, Piauí, Brazil.    J. P. de Lima Departamento de Física, Universidade Federal do Piauí, Campus Ministro Petrônio Portela,64049-550, Teresina, Piauí, Brazil.    L. L. Gonçalves Departamento de Engenharia Metalúrgica e de Materiais, Universidade Federal do Ceará, Campus do Pici, Bloco 714, 60455-760, Fortaleza, Ceará, Brazil lindberg@fisica.ufc.br   

The one-dimensional extended isotropic XY model (s=1/2) in a transverse field with uniform long-range interactions among the z components of the spin is considered. The model is exactly solved by introducing the gaussian and Jordan-Wigner transformations, which map it in a non-interacting fermion system. The partition function can be determined in closed form at arbitrary temperature and for arbitrary multiplicity of the multiple spin interaction. From this result all relevant thermodynamic functions are obtained and, due to the long-range interactions, the model can present classical and quantum transitions of first- and second-order. The study of its critical behavior is restricted for the quantum transitions, which are induced by the transverse field at The phase diagram is explicitly obtained for multiplicities and as a function of the interaction parameters, and, in these cases, the critical behavior of the model is studied in detail. Explicit results are also presented for the induced magnetization and isothermal susceptibility , and a detailed analysis is also carried out for the static longitudinal and transversal correlation functions. The different phases presented by the model can be characterized by the spatial decay of the these correlations, and from these results some of these can be classified as quantum spin liquid phases. The static critical exponents and the dynamic one, have also been determined, and it is shown that, besides inducing first order phase transition, the long-range interaction also changes the universality class the model.

One-dimensional extended XY-model; long-range interaction; quantum phase transitions; static properties; quantum spin liquids.
05.70.Fhsep 05.70.Jksep 75.10.Jmsep 75.10.Pq
preprint: Preprint


I Introduction

The study of the quantum phase transitions(QPTs) has been the object of many theoretical and experimental investigationsSchadev 2000 (), since they play an essential role in understanding the low temperature properties of the materialsColeman 2005 (). In particular, for low dimension magnetic materials, the study of quantum spin chains has provided much insight in this directionSchadev 2008 (). Motivated by this, in the last years, others tools have been used to investigate QPTs in spin chains, such as, quantum discordRaoul 2008 (), geometric phasesCarollo 2005 (); Zhu 2006 () and quantum fidelityZanardi 2006 (), where the last two concepts were unified in the approach of the geometric tensorsVenuti 2007 (). All these tools have only one purpose, that is, to point out the existence of the quantum critical behavior. On the other hand, the QPTs are also directly related to quantum entanglement, which has been object of study in quantum computationAmico 2006 (). Therefore, the study of spin systems is of great importance for understanding the behavior of the materials in low temperature regime.

Among the quantum spin models, the one-dimensional XY model introduced by Lieb et al.Lieb 1961 (), with its generalized classes, have attracted much interest in the last decades. In particular, the models with multiple spin (see Refs.Titvinidze 2003 (); Krokhmalskii2008 (), and references therein) and long-range interactions(see Refs. Lenilson 2005 (); Bouchett 2008 (), and references therein) have been object of intensive investigations. As pointed out by DerzhkoDerzhko 2009 () and in the references therein, quantum spin systems with multiple spin interactions work as effective spin models for the standard Hubbard model under certain conditions. On the other hand, models with long-range interactions are important to understand the process of quantum information in spin chains as well as for the study of the classical and/or quantum crossover, as it was explained by de Lima and Gonçalves.de Lima 2008 (). Another importance of the long-range interaction is that it can induce first-order QPTs, which play an important role in the quantum critical behaviourPfleiderer 2005 ().

Therefore, in this paper, we will study the effect of the long-range interaction on the quantum critical behaviour of the extended one-dimensional XY-model with multiple spin interactionsSuzuki 1971 (). The solution of the model can be obtained exactly, at arbitrary temperatures in a one-dimensional and the main purpose of the paper is to study the quantum critical behaviour of the model.

We analyze these two kind of interactions, where we will show the role of each one, that is, we will show that the long-range interactions induce first and second order QPTs and that their presence change the universality class of the model, as we will also verify the scaling relations proposed by Continentino and FerreiraContinentino 2004 () for first-order QPTs. While for the multisite interaction among spins, we will show the presence of many different kinds of quantum spin liquid phases, which after some time returned to be a relevant topic in the present researchLee 2008 ().

In Sec. II we introduce the model and obtain its exact solution by means of  Jordan-Wigner fermionization and the integral  Gaussian transformation, and present its basic results, such as, the spectrum of the energy, magnetization at arbitrary temperatures. The quantum phase diagrams, as function of the interaction parameters, are presented and discussed in Sec. III, and in Sec. IV we study the scaling behaviour of the longitudinal and transversal correlations functions on the different phases presented by the model. In Sec. V we evaluate the critical exponents, and discuss the change in the universality class due the presence of the long-range interactions. Finally, in Sec. VI we summarize the main results.

Ii The Model and Basic Results

We consider the one-dimensional isotropic extended XY model () with uniform long-range interactions among the components of the spins,  whose Hamiltonian is explicitly given by


where the parameters , are the exchange coupling between nearest neighbors, the uniform long-range interaction among the components, is the multiplicity of the multiple spin interaction, and is the number of sites of the lattice.

Introducting the Jordan-Wigner transformation


where and are fermion operators, the Hamiltonian can be written in the well known decomposed form(siskens 1974 () also references therein)






with given by


As it is also well known, the Hamiltonian can be diagonalized by imposing  antiperiodic (for and periodic (for boundary conditions, and, in the thermodynamic limit, the static properties can be described by siskens 1974 (); capel 1977 (); goncalves 1977 (). Therefore, since we are interested in the determination of the static properties, we will identify the Hamiltonian of the system with By taking into account that the long-range interaction term commutes with the Hamiltonian, the partition can be written in the form


where and is the temperature. Introducting the Gaussian transformationamit1984 ()


the partition function can be written in the integral representation as


where and

Introducing the canonical transformation


with   Eq. (10)  can be written in the form






In the thermodynamic limit, , we use the Laplace’s methodmurray 1984 () to evaluate the partition function, and can be written in the form




with satisfying the conditions


and explicitly given by


Therefore, from Eq.(15) we can write the Helmholtz free energy as


Taking into account that the magnetization per site can be written in the form

by using Eq.(18), we can express in terms of in the form


and from this result it follows that the functional of the Helmholtz free energy per site is given by




From Eq. (21), we can  determine the equation of state  imposing  the stability conditions


which leads to the result


Iii Quantum Critical Behavior

In the limit , the functional of the Helmholtz free energy per site[Eq. (21)] is given by




and the equation of state, given by Eq. (25), can be written in the form


The quantum phase diagram of the model, for second order phase transitions and arbitrary , can be obtained from the previous expression by imposing the divergence of the isothermal susceptibility, and for first-order phase transitions, by using the equation of state, Eq. (25), and by imposing the condition


where and are the values of the induced magnetization at the transition.

Following Titvinidze and JaparidzeTitvinidze 2003 (), by introducing the unitary transformation


and the time reversal one


we can show that the Hamiltonian is invariant under the transformation and This means that only the signs of and are relevant in determining the critical behaviour of the system, and the appearance of multiple phases are result of the competition between these interactions which induce frustration in the system. In particular, they can induce the so-called quantum spin liquid phasesLee 2008 (), as in the case of the model without long-range interactionTitvinidze 2003 (). Therefore, without loss of generality, we will consider in all results presented only.

From Eqs. (26-28), we can show that, for arbitrary the system presents second order quantum transitions for , and first-order quantum transitions as in the previously studied XY-models with similar long-range interactionsLenilson 2005 ().

Although the Eqs. (26-28) allow us to obtain the phase diagrams for arbitrary , we will only consider the cases and since they present the main features and, in these cases, analytical expressions can be obtained for the critical surfaces and critical lines associated to the second order quantum transitions.

For the case which has also been studied by Titvinidze and JaparadzeTitvinidze 2003 ()and Krokhmalski et al. Krokhmalskii2008 () for the model without long-range interactions, we show in Figs. 1 and 2 the critical field as a function of in the regions and respectively,  for different values which are projections of the global phase diagram. It is worth mentioning that in this case the results for can be  obtained from the results for by introducing the transformations and as can be verified in the results shown in Figs. 1 and 2.

In Figs. 3 and 4 we present the magnetization and the isothermal susceptibility, respectively, and from their behavior we can conclude that for the system undergoes second order transitions, and first-order transitions for

As can it be seen in Figs. 1 and 2, the number of transitions of first and second order depends on and it can also be shown that these transitions correspond to three phases for and to four phases for Following Titvinidze and JapararidzeTitvinidze 2003 (), we can classify the intermediate phases, which are limited by the two saturate ferromagnetic phases, as quantum spin liquid phasesLee 2008 (). As will be shown later, these spin liquid phases will be characterized by the spatial decay of the transversal static correlation function and by the modification of the oscillatory modulation of longitudinal static correlation function .

The global phase diagram for is shown in Figs. 5 and 6, for and respectively. For the critical surfaces can be obtained explicitly and, for where there are four critical surfaces, which are given by the equations




and for there are four critical surfaces given by




The critical lines, shown in Figs. 1 and 2, can be obtained from Eqs. 32-36 for and and from Eqs. 37-41 for and .

It should be noted that for the critical surfaces meet at a bicritical lineLenilson 2005 () given by


For where the phase transitions are of first-order, critical surfaces are obtained numerically from the solution of the system


where and are given by


are the values of the magnetization at the transition, with  for , and with for and is given by Eq. (26). As in the case of second order transitions, the critical lines shown in the Fig. 1 are determined from the previous systems by considering and .

For there are four critical surfaces and three of them, for meet at a critical line which is determined by the following system of equations


where is obtained from the Eq. (26), and with and This is a triple line which meets the bicritical line at

A second triple line can be determined by imposing and in the system


where and are given by Eqs. (26) and (45), which begins at the point and which has been obtained Gonçalves et al. Lenilson 2005 ()

For the special case , we can find the critical surfaces by using the same procedure used in the case . In this case, due to many intersections of the critical surfaces, the global phase diagram becomes too complicated, as we will show below. Therefore, we will present some projections of the global diagram which contain the main characteristics of this diagram and are shown in Figs. 7-10.

In this case, the fermion excitation spectrum, obtained from Eq. (27), is given by


and from this result we can  determine the equations of the critical surfaces for and which are




Identically, we can show that for and , the critical surfaces are




In this case there are three bicritical lines which are given by


For as for the case , the first order transition surfaces can be determined numerically and the triple lines are determined by following the same procedure adopted for

In Fig. 7 the phase diagram is shown for the positive region where there are four critical surfaces which are given by the Eqs (49-53). These critical surfaces meet at a bicritical line, which is given by Eq. 62.

The phase diagrams for the negative region and for different values of are shown in Figs.8-10. In this case there are six critical surfaces, given by Eqs 54-59, which meet at bicritical lines given by the Eqs.(64-63). As we can see in Figs.8-10, the system presents in this case identical critical behavior to the one obtained for the case , as far as the critical behavior is concerned. However, it is worth mentioning the appearance of quadruple point at and which  is shown in Fig. 9 and is not present in the case . The behavior of the functional of the Helmholtz free energy at this point is presented in Fig. 11.

We have also analysed the cases and , for , where the model presents new quantum spin liquid phases. Although in these cases there are no first-order transitions, we have restricted these analyses to the model without the long-range interaction, since the main purpose was to study appearance of new quantum spin liquid phases, which is mainly controlled by the multiple short-range interaction.

In Fig. 12 we show the phase diagram for the case where the critical lines are given by


As it can be seen, there are six quantum spin liquid phases which, as we will show later, can be classified  in three different classes as far as the critical behavior is concerned.

The phase diagram for is shown in the Fig. 13, where the critical lines are given by


where with