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.
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
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
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
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
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
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
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
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.
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