Quantum Curie-Weiss Magnet Induced by Violation of Cluster Property ()
1. Introduction
The concept of entanglement strongly contradicts with the classical one about locality, which has been extensively studied by many researchers [1] [2] [3] [4] [5]. One reason for this active study is the possibility of applying entanglement to quantum computer [6] [7] and quantum information [8] [9] [10].
Entanglement found in many-body systems is reviewed in [11]. Its observation was discussed in [12]. Entanglement was discussed in terms of the spontaneous symmetry breaking [13] [14] [15] because the correlation must be found even at the long distance where the whole system changes entirely from the classically ordered state to the disordered one. The work [16] discussed quantum communication when the symmetry breaks spontaneously.
When the system is at the critical point, we could suppose that correlation must be found even in the far distance. By this quite long-range correlation we have to consider that the concept of the cluster property [17] or the cluster decomposition [18] is not trivial. The works [19] [20] discussed the relation between the violation of this property and confinement in QCD. Also we find active studies in quantum field theory [21] [22]. While in the macroscopic system, authors in works [23] [24] studied the cluster property in the term of the stability. Recently another type of the cluster property is discussed in [25].
In the previous paper [26] we have investigated the cluster property of spin 1/2 XXZ antiferromagnet on the square lattice. For this antiferromagnet, the ground state realizes semi-classical Neel order [27], in other words, spontaneous symmetry breaking (SSB) [17] [28] of U(1) symmetry. This semi-classical order has been confirmed by spin wave theory [29] and the quantum Monte Carlo method [30] [31]. The review article [32] is quite useful in this order in the spin system. Also see [33] for the experimental review.
The essential point in these studies is that SSB requires the quasi-degenerate states between which the expectation value of the local operator is not zero. The energy difference between these quasi-degenerate states decreases as the lattice size increases. Therefore, in order to determine the ground state definitely, we introduced an additional interaction that explicitly breaks the symmetry. Then we showed that the violation of the cluster property occurs in this model. The magnitude of the violation is order of
, where g is the strength of the explicit symmetry breaking interaction and N is the size of the system. We concluded that it is possible to observe this effect, though it is tinny except for the extremely small g. As for the Heisenberg model which has SU(2) symmetry, see [34].
In this paper, we propose another approach which enables us to observe the violation even when g is not so small. We consider a new spin system added to the one on the square lattice we studied in [26]. The whole Hamiltonian is
. Here
denotes the Hamiltonian which operates the states on the square lattice and
is the interaction which breaks U(1) symmetry explicitly. The newly added interaction,
, consists of spin operators both on the additional system and on the square lattice. It contains a parameter u to represent the strength of the interaction.
Applying the perturbation theory with small u, we obtain the effective Hamiltonian
for the spins in the additional system. We see that it includes Curie-Weiss model. In this model, it is known that the mean field approximation for the thermodynamic properties gives the exact results. We then find that this effective Hamiltonian
has the overall factor
. Since Curie-Weiss model has to contain the factor 1/N in order that the thermodynamical properties are well-defined, the essential factor for the system is
. We conclude, therefore, that one would be able to observe the violation when the inverse temperature
is of order of
.
Contents of this paper are as follows. In Section 2, we describe our model in some detail. The first subsection is devoted to a brief explanation of the spin 1/2 XXZ antiferromagnet on the square lattice. Also we collect the results related to the Hamiltonian
[26]. In the second subsection, we define
which describes an extended part of the model. In Section 3, using the perturbation theory, we derive the effective Hamiltonian
from
. A general discussion to derive the effective Hamiltonian is given in appendix A and the concrete form of
is calculated in appendix B. We show that the effective Hamiltonian contains Curie-Weiss model, whose Hamiltonian is the square of the sum of all spin operators on the extended sites. We also show that this Hamiltonian
contains the ferromagnet with the finite-range interaction induced by Nambu-Goldstone mode.
In Section 4, we calculate the energy and the specific heat of Curie-Weiss model. For this purpose, we use the mean field approximation, which is discussed in appendix C in detail. It should be noted that this method is absolutely reliable for the model when the system is infinitely large. In order to assure that our results are sufficiently accurate, we numerically calculate the specific heat on finite lattices.
In Section 5, we investigate the thermodynamic properties of the effective Hamiltonian
. The first subsection is to calculate the energy and the specific heat when the temperature is high. Here we employ the high temperature expansion described in appendix D. We find the effect by Nambu-Goldstone mode only in this region. In the second subsection, we calculate these thermodynamic properties at a low temperature. Here we employ the mean field approximation which is exact for the ferromagnet due to the degenerate states and reasonable for the one due to Nambu-Goldstone mode. The final section is devoted to summary and discussion.
Since many symbols are used in our paper, we list them in Table 1 for convenience.
2. Our Model
2.1. Spin System on the Square Lattice
We will consider the quantum spin system on the square lattice. On each site i (
) we have the spin operator
(
). Then we define the Hamiltonian
by
Table 1. Symbols used in our paper. The third column denotes the equation number, if any, where the symbol is defined.
(1)
Here
denotes the nearest neighbor pair on the square lattice and
is the parameter between 0 and 1. The eigen state is given by the linear combination of states
, where
(
). The vector space of the states
is denoted by
.
For this antiferromagnet, we divide the whole lattice into two kinds of sub-lattices called A sub-lattice and B sub-lattice. In order to define these sub-lattices we introduce a symbol
using integers
and
for the site
.
(2)
Then we introduce the spin operator on each sub-lattice,
(3)
In order to obtain the ground state, we introduce the symmetry breaking interaction
,
(4)
Then we have the Hamiltonian
,
(5)
It is well known that in this system there exists Nambu-Goldstone mode, which can be described successfully by spin wave theory. On the other hand, adding the explicit symmetry breaking interaction into the Hamiltonian, we have obtained the lowest energy eigen state and the excited states which are linear combinations of the degenerate states [26] [34]. This leads us to consider two kinds of excited states, which are states due to degenerate states and those from Nambu-Goldstone mode. In order to describe these excited states we will employ two kinds of Hamiltonian
and
.
Following the previous work [26], we present the Hamiltonian
which describes the excited states by the degenerate states
. They are defined by
(6)
Here
is the generator of U(1) symmetry and n is an integer. Then we define
as
(7)
Here
denotes the lowest energy with
and
is the constant which is fixed by
. The eigen state
of
is given by a linear combination of
,
(8)
Detailed expression of
is found in Appendix B.1.
Next we define
, which describes the excited states of Nambu-Goldstone mode, based on the spin wave theory.
(9)
Here
is the annihilation operator of Nambu-Goldstone mode with the wave vector
, and
denotes the ground state energy. The effect due to the symmetry breaking interaction
is included in
, which is the energy of Nambu-Goldstone mode. Detailed expression of
is given in Appendix B.2.
2.2. Extended Spin System
Let us consider a new system which consists of the spin system on the square lattice and the one on
additional sites. The state for the additional sites is represented by
(
) where
. The vector space
is spanned by these states. We will consider the spin system on the square lattice and the additional spin system. Whole vector space is
. The extended interaction
is given by
(10)
Here
is the additional site fixed by the site i as is shown in Figure 1. Note that the summation for i runs over A sub-lattice only. The whole Hamiltonian
of the system is then defined by
(11)
3. Effective Hamiltonian of the Extended Spin System
In Appendix A we have derived the effective Hamiltonian using the perturbation theory. We apply it to our model, where
is
(5) and the perturbed interaction is
(10).
Figure 1. The extended system to the spin system on the square lattice. The ordinal spin is located at the cross point of the horizontal lines and the vertical lines. The line between two nearest points denotes the interaction. The full circle shows the additional spin and the line between the full circle and the cross point shows the additional interaction.
We have two kinds of the excited states. One is the excited state
(
) that consists of the degenerate state and the other is the one-magnon state
with the wave vector
, which is Nambu-Goldstone mode. Following the discussions in the previous works [26] [34], we suppose that these excited states are independent.
We obtain the effective Hamiltonian
that operates states in
,
(12)
Here
are sum of the coefficients
due to the degenerate states and
due to Nambu-Goldstone mode.
(13)
From Appendix A we obtain
(14)
We have calculated these coefficients in Appendix B. We obtain
(15)
Here v denotes the expectation value of the spin operator in the ground state. When we consider the terms of
only and neglect those of
for
, we obtain
(16)
As for contributions by Nambu-Goldstone mode, we obtain
(17)
Here
is the modified Bessel function. When
is as large as 1020 and
, the simple expression by the modified Bessel function is reliable for
[26]. From (15), (16) and (17) the effective Hamiltonian for our model on the vector space
is given by
(18)
The first term of
has the factor
and is independent of the site. We then come to an important conclusion that this effective Hamiltonian contains modified Curie-Weiss model induced by the degenerate state. In the next section, we will discuss this model in some detail.
4. Curie-Weiss Model
Curie-Weiss model [35] [36] [37] is defined by, with the site number
,
(19)
In this model, we can exactly calculate the specific heat for the infinitely large lattice at any temperature by the mean field approximation. Since this fact is quite important we will make a numerical examination in this section. We compare the specific heat calculated by the eigen values on the large lattices with the result obtained from the mean field approximation.
In Curie-Weiss model, the partition function
with the inverse temperature
is given by
(20)
Here
, which we suppose to be even, is the lattice size and
denotes the energy eigen value with the magnitude
of the total spin.
The multiplicity
is given in the following way. First, we consider the state of k up-spins and
down-spins. For this state
, the z-component of the total spin, is given by
. For the fixed
we have the multiplicity
because we pick up k spins among
spins. Using
instead of k we have
. Since the multiplicity
is a number of possible
for each
the difference
is the number
, which is the multiplicity for the fixed
. We take account of the multiplicity of
for the fixed
, since the energy
does not depend on
. Thus we obtain the multiplicity of states
for the fixed
.
In Figure 2, we plot the specific heat calculated from the partition function
in (20) for
and 20,000. For comparison we also plot the mean field results we obtain from discussion in Appendix C, noting that
and
for Curie-Weiss model (19). When the positive solution
of the equation
exists, the specific heat is given by
(21)
Since no positive solution exists for
, we have
in this region. Note that this is the characteristic property of Curie-Weiss model. Figure 2 indicates that the result for
lattice differs from the mean field result, specially around the critical temperature. When the lattice size becomes large, however, the difference clearly decreases. For
, we find the excellent agreement between the results by both methods except for the narrow region around
. We conclude, therefore, the mean field approximation for Curie-Weiss model on large lattices is satisfyingly reliable at any temperature.
Figure 2. The specific heat of Curie-Weiss model (19) for finite size
. They are calculated by the exact partition function
in (20) for the fixed
. The red curve is calculated from the mean field approximation, which is given by (21).
5. Thermodynamical Properties of the Effective Hamiltonian
5.1. High Temperature Region
Let us study the thermodynamical properties at very small
. It is known that the high temperature expansion described in Appendix D is a powerful tool in this region. We apply the results by this method to our effective Hamiltonian
given in (18), for which
(22)
Then we have
(23)
Here
. We obtain, with
,
(24)
Similarly,
(25)
Therefore, we obtain
(26)
Note that the first term of (26) is quite small compared to the second term when
is quite large. The energy
and the specific heat per site
are then given by
(27)
In (27), we see the effect due to the first term of the effective Hamiltonian and that due to Nambu-Goldstone mode only.
5.2. Low Temperature Region
In order to calculate the thermodynamical properties at a low temperature, we employ the mean field approximation described in Appendix C to the effective Hamiltonian
. Taking the translational invariance into account and using (24), we obtain
(28)
Let us study how the specific heat
depends on the parameters u and g. Based on results in Appendix C we see that
(29)
In the case of
, we find the effect due to the degenerate states. Since
we need the condition
, which is equivalent to
, to observe this effect.
In Figure 3, we plot
for various values of
. We find that there exists the gap
at the critical temperature
. We plot
as a function of
in Figure 4, which shows that
is finite when
while it becomes infinitely large as
goes to 1. Also in Figure 5, where we show the gap
, we see that
gradually decreases as
increases and it vanishes when
is 1.
For the analytic discussion, we expand
and
by the polynomial of
. At the second order, we obtain
Figure 3. The specific heat of the effective Hamiltonian
(18) given by (29) in mean field approximation. The horizontal axis is
. When
we do not find any gap of the specific heat. If
the gap is ~3/2 as is shown in (30).
Figure 4. The critical temperature
of the effective Hamiltonian
in mean field approximation as a function of
. The vertical axis measures
. The red curve is for the case
, which is given in (30).
Figure 5. The gap of the specific heat
at the critical temperature
in (86) (
and
) as a function of
. The red curve is for the case
given in (30).
(30)
These results are also plotted in Figure 4 and Figure 5 for comparison. We see that the polynomial expansion of
is reliable for
. This suggests that the perturbation theory on
gives us the good approximation, which will be important in future study on the effective Hamiltonian.
Note that if the degenerate states are absent we should use
instead of
because the system is the ferromagnet induced by Nambu-Goldstone mode only. Measuring the specific heat, therefore, we would be able to judge if the degenerate states exist or not.
To summarize this section we present in Figure 6, a region formed by g and u, where one can observe the effect by Curie-Weiss model due to the degenerate states. The red curve in the figure gives the boundary for the validity of the perturbation theory. The black curve shows the boundary where we can observe the specific heat by this model. Therefore one can confirm the effect by the violation of the cluster property in the region between the red and the black curves.
6. Summary and Discussions
The cluster property is deeply connected with the classical concept about locality, but it is not trivial in quantum physics. In the previous papers [26] [34], we showed the violation of the cluster property (VCP) in spin 1/2 XXZ antiferromagnet and Heisenberg antiferromagnet on the square lattice. Our results indicate that the magnitude of VCP is order of
, where g is the strength of the explicit symmetry breaking interaction and N is the size of the system, which we suppose
. The observation of VCP in experiments is not easy, therefore, because of its smallness.
In this paper, we proposed an extended spin system so that we find a better chance to observe the effect by VCP. We added a new spin system to the original spin system on the square lattice. The Hamiltonian is
. Here
contains spin operators of the original system only, while
contains
Figure 6. Region of the parameters g and u, where we can observe the effect by Curie-Weiss model. For the validity of the perturbation theory we must impose condition
, which is above the red curve. We also need the condition that the critical inverse temperature is finite. The yellow (blue, green) curve shows the values of g and u where the critical inverse temperature
(104, 102).
spin operators on both systems. Applying the perturbation theory to
for a small coupling constant in
, we obtained the effective Hamiltonian
which operates only on the vector space of the additional system. Then we found that
contains Curie-Weiss model induced by the degenerate states. In order to calculate thermodynamic property of the effective Hamiltonian at a low temperature, we employed the mean field approximation, where the difference between the effect due to the degenerate states and that due to Nambu-Goldstone mode is found in the magnitude of the specific heat. Our conclusion is that it is possible to find the effect of the violation of cluster property in our extended model.
Our study in this paper is based on the effective Hamiltonian, which is derived by the perturbation theory. In order to examine the validity of the theory, we consider the Hamiltonian
on a small lattice, where the energy gap is so large that we do not need the symmetry breaking interaction
. Here let us give a brief description of the model and show the results obtained by the diagonalization on the
and
lattice (16 + 8 lattice). From (49) in Appendix A the effective Hamiltonian
reads, i and j being the site on the A sub-lattice,
(31)
Here we consider only the first excited states so that
, and
(
). Then the energy eigen values for
should be given by
(32)
For comparison, we directly diagonalize
on the 16 + 8 lattice to obtain the energy eigen values, which we denote
, for fixed values of
and
. By making the least square fitting for
by
, we can estimate the value
and
of
, which should be compared with the value of
for
. The agreement between both values is satisfactory to assure the validity of the perturbation theory.
Several comments are in order for our calculations and results.
• First let us discuss on effects of higher-order terms in the perturbation theory. On large lattices the energy gap is of order of
and
is of order of
. Then the next order term is of order of
, but the factor N should be included into the Hamiltonian for the consistency. Therefore we conclude
should be small in order to neglect higher-order terms.
• Let us consider to estimate the parameters u and g in experiments. One can estimate u by measuring the specific heat at high temperature because, as we have seen in (27), the first term
dominates compared to the second term with the factor
. In order to estimate g, on the other hand, one should measure the correlation function of the spin operator which is given in (96) in Appendix D.
• Next we discuss the qualitative difference between the effect due to the degenerate states and that due to Nambu-Goldstone mode. Since we do not see any effect due to the degenerate states at a high temperature, we need to examine the thermodynamic quantities at a low temperature. In this region, where the mean field approximation is valid, it is difficult to distinguish the effect due to the degenerate states from that due to Nambu-Goldstone mode. Therefore we have to investigate the property connected with the excited states which cannot be calculated in the mean field approximation. This subject will be studied in a future work where we investigate the effective Hamiltonian in the extended system with SU(2) symmetry.
• The last comment is about experimental realization of the proposed spin system. One idea to realize our model is following. In experiments for the spin system on the square lattice, the material contains multi layers. It will be possible to consider the material that has the sandwich structure where the magnetic layer and the quasi non-magnetic layer appear alternately. The magnetic layer realizes the spin system on the square lattice, while in the quasi non-magnetic layer we can partially add the magnetic elements such as Cu. In this additional system, the magnetic elements are sparse so that the coupling between spins on the additional system is weak. Therefore we can suppose that such material realizes
in our model.
Acknowledgements
The author specially thanks Dr. Yasuko Munehisa for constructive suggestions and fruitful discussions which are provided through the critical reading of the manuscript.
Appendix A
In this appendix we show how to derive the effective Hamiltonian by means of the perturbation theory. Here we suppose that the whole vector space V is the direct product of the vector space
and the vector space
, namely
. We also suppose that the unperturbed Hamiltonian
operates only states on
and there is no degenerate state for
on the vector space
. They are expressed by
(33)
Here
,
and
is the lowest energy state. The basis state in
is given by
, while the basis state in
is denoted by
. Then the basis state in V is given by
(34)
For the unperturbed Hamiltonian
,
(35)
We suppose that the perturbed Hamiltonian contains the products of the operator
on the vector space
and the operator
on the vector space
.
(36)
The eigen state of
is given by
(37)
The coefficient
is a polynomial function of u and contains the term of
. In order to formalize the perturbation theory we employ the variational method, where we introduce a function defined by
(38)
By the variation on the coefficients we obtain the eigen equation,
(39)
In order to calculate
we divide the Hamiltonian
to
and the perturbed interactions. For the expectation value of
we obtain, from (37),
(40)
As for the expectation value of
,
(41)
In the second order perturbation theory, we neglect the terms of
in F. Then the variation on
becomes
(42)
The variation on
is, on the other hand,
(43)
Requesting
we obtain the expression for
,
(44)
We then replace
in (42) by the above expression (44). The result is
(45)
Here we exchange the order of the summation on
and that of
and use
(46)
From
we obtain the equation for
. Since we can replace
by
in the second order of u we obtain the eigen equation on
,
(47)
Using this
we introduce the effective Hamiltonian
on
which should satisfy
(48)
Since the matrix elements of
on
apply to any state, we can express them by the operators on
. Finally we obtain the effective Hamiltonian
(49)
Appendix B
Here we calculate the inner product
in (49), where
(
) for the site i on the A sub-lattice.
Part 1
In this subsection we calculate the contributions due to the degenerate states. In [26] we obtained the eigen state
of
and the eigen energy
(
), which are given by (with
in (28) and (29) of [26])
(50)
Here
denotes the Hermite polynomial and
is the normalization factor. Note that we do not need any explicit expression for
, since any physical quantity contains the form of
.
For
we have [26]
(51)
Then we obtain
(52)
Similarly we have
(53)
Then the inner product is given by
(54)
Finally we obtain
(55)
Here we use
and
.
Part 2
Next we discuss the matrix elements due to Nambu-Goldstone mode. We employ the results calculated in the previous work [26] based on spin wave theory. Here the ground state
is
and the excited state is the one magnon state with the wave vector
, which we denote by
.
In spin wave theory, it is known that
(56)
As for the operator
we obtain [26]
(57)
Here we use the symbols defined by
(58)
The inner products are then given by
(59)
Therefore we obtain
(60)
For large
the contribution from small
dominates in the sum of
. For small
and small g we see that
(61)
Using these approximations and replacing the sum by the integration, we obtain
(62)
Here we use Bessel functions
and
.
Appendix C
The mean field approximation is based on Gibbs-Bogoliubov-Feynman inequality [38] [39], which is given by
(63)
Here
denotes the statistical average.
Let us apply this inequality to the classical Hamiltonian
. The statistical average here is defined by
(64)
By this definition we have
(65)
For
and
we then obtain
(66)
This inequality is valid for the quantum mechanics, too [38] [39].
(67)
Let us start our discussion with
(68)
Here we introduce operators,
(69)
The parameters
and
are determined later so that they maximize
. It is easy to see that, with
,
(70)
Thus, using
and
for
, we obtain
(71)
In order to find values
and
which maximize
, we examine following equations.
(72)
(73)
Since
we see from (73) that, with
,
(74)
Let us first consider the case (CASE 1) where
holds and there exists a positive solution for the equation
(75)
Note that
is necessary for this case. The maximum value of
is then
(76)
The average energy E and the specific heat per one spin
are then given by
(77)
For
, we carry out the differential on the condition.
(78)
Then we obtain
(79)
Next let us consider another case
(CASE 2), which means
. Note that
is necessary for this case. We have
(80)
The solution for
is trivial, which is
. Then we obtain
(81)
For later use, we employ the quantity m instead of
,
(82)
From (70), we see that m is
at
. Using this m the energy and the specific heat in CASE 1 are given by
(83)
Here, since
, m should satisfy the equation
(84)
In CASE 2, where
, we obtain
(85)
Finally let us examine whether the Equation (74) has a solution which is greater than h. We see that there must be a critical value of
,
, above which we can find the solution. One can easily see that
if
and
if
. Also we have to pay attention that there is no solution for any
if
. Therefore we obtain
when
. It should be noted that the energy in CASE 1 coincides with that in CASE 2 at
while the specific heat does not. Namely, using
in the limit
and
,
(86)
Especially, when h is small, we obtain the results
(87)
Appendix D
The Hamiltonian
we consider here is given in (68). The partition function
is defined by
(88)
In the high temperature expansion
is expanded by small
,
(89)
We employ the following equations
(90)
In the last equation, we exclude the case where
and
.
From the Equation (90) we see that the first order term of
vanishes.
(91)
In the second order,
(92)
Here we use
and
as well as the translational invariance. Thus we obtain
(93)
Therefore the energy and the specific heat are given by
(94)
Finally we consider a correlation function of the spin operator
which will be useful to estimate the coupling constants of the model. It is defined by
(95)
For small
, we obtain
(96)