Gapless metallic charge-density-wave phase
driven by strong electron correlations
We analyze the transformation from insulator to metal induced by thermal fluctuations within the Falicov-Kimball model. Using the Dynamic Mean Field Theory (DMFT) formalism on the Bethe lattice we find rigorously the temperature dependent Density of States () at half filling in the limit of high dimensions. At zero temperature () the system is ordered to form the checkerboard pattern and the has the gap at the Fermi level , which is proportional to the interaction constant . With an increase of the evolves in various ways that depend on . For the gap persists for any (then ), so the system is always an insulator. However, if , two additional subbands develop inside the gap. They become wider with increasing and at a certain -dependent temperature they join with each other at . Since above the is positive at , we interpret as the transformation temperature from insulator to metal. It appears, that approaches the order-disorder phase transition temperature when is close to or , but is substantially lower than for intermediate values of . Having calculated the temperature dependent we study thermodynamic properties of the system starting from its free energy . Then we find how the order parameter and the gap change with and we construct the phase diagram in the variables and , where we display regions of stability of four different phases: ordered insulator, ordered metal, disordered insulator and disordered metal. Finally, we use a low temperature expansion to demonstrate the existence of a nonzero DOS at a characteristic value of on a general bipartite lattice.
One of the most successful methods for describing strongly correlated electron systems is the dynamical mean-filed theory (DMFT) GeorgesKotliarKrauthRozenberg (); FreericksZlatic (). This formalism appears to be particularly useful in studying the Falicov-Kimball model (FKM) FalicovKimball (), as it enables to get analytical, or high precision numerical results, which become exact in the limit of large dimensions. Most of the findings have been obtained in the high-temperature homogeneous phase GeorgesKotliarKrauthRozenberg (), but the ordered phase was also considered in a few papers FreericksZlatic (); BrandtMielsch (); vanDongen (); vanDongenLeinung (); FreericksLemanski (); GruberMacrisRoyerFreericks (); ChenFreericksJones (); HassanKrishnamurthy (); MatveevShvaikaFreericks (). The results presented in these papers are remarkable, as they give a clear evidence that the static mean field theory is not an adequate tool for describing correlated electron systems. Indeed, physical quantities obtained using the static and dynamic mean field approach are substantially different one from another. This discrepancy is particularly clearly demonstrated by Hassan and Krishnamurthy HassanKrishnamurthy (), and by Matveev, Shvaika and Freericks MatveevShvaikaFreericks (). Both teams analyzed the spinless FKM at half filling in the ordered charge-density-wave (CDW) phase having the form of checkerboard phase. Hassan and Krisnamurthy HassanKrishnamurthy () considered the square lattice and the Bethe lattice in the limit of infinite dimension and focused mostly on spectral properties, whereas Matveev, Shvaika and Freericks MatveevShvaikaFreericks () examined the hypercubic lattice in the limit of high dimensions and they focused mainly on transport properties. It is quite interesting that even though these studies were performed on different lattices, they lead to similar spectral properties of the model. Namely, in all the cases the energy spectrum has a gap at the Fermi level at and with an increase of two additional subbands develop inside the gap in such a way, that the density of states (DOS) at the Fermi level becomes positive still in the ordered phase (above a certain temperature ), i.e. below the order-disorder transition temperature . In fact, the energy subbands developing inside the gap in the ordered checkerboard phase were already noticed by Freericks and Zlatić FreericksZlatic (). Here it is worthy to note that the Monte Carlo calculations performed on the 2D systems also give results similar to those obtained within DMFT MaskaCzajka (); ZondaFarkasovsky ().
On the other hand, the data based on the static mean field theory calculations show that the gap disappears only at FGebhard (). Indeed, according to a conventional mean field theory this gap gradually diminishes with an increase of temperature, but still persists until the CDW phase exists, i.e. until the order-disorder (O-DO) phase transition temperature is reached FGebhard (). Surprisingly, the same conclusion was also formulated by van Dongen, who studied the FKM on the Bethe lattice using a different variant of DMFT vanDongen (); vanDongenLeinung (). In fact, van Dongen derived analytical formulas on the temperature Green functions in the ordered phase, but he analyzed them only in the limiting cases of small and large coupling parameter . Since in these two limits the gap is always present in the ordered phase, he concluded that it exists for any . However this is in contradiction to the results reported in Refs. HassanKrishnamurthy (); MatveevShvaikaFreericks ().
Since the demonstration of the existence of the gapless ordered phase in Refs. HassanKrishnamurthy (); MatveevShvaikaFreericks () is quite surprising, but in the literature still practically unnoticed result, in this contribution we develop studies of the subject. Our purpose is to perform a more detailed analysis of spectral properties of the system focusing mainly on intermediate values of the parameter . Following the approach derived by van Dongen vanDongen () we perform non-perturbative calculations that allows us to reconstruct in a simple way the data obtained by Hassan and Krisnamurthy HassanKrishnamurthy () and to get analytical expressions for some characteristics of the spectrum not reported before. In addition, we calculated the electronic part of the specific heat and found that it behaves monotically around the temperature of metal-insulator (MI) transformation. Hence, we conclude that the transformation is not a phase transition in the usual sense.
Our analysis of the single electron energy spectrum of the spinless FKM is based on exact formulas for the temperature-dependent derived for the Bethe lattice within a version of the DMFT formalism derived by van Dongen vanDongen (); vanDongenLeinung (). There are two types of localized particles and in the system, whose densities and , respectively, are equal to each other and equal to , () and spinless electrons. The localized particles may correspond, for example, to two different components of an alloy. We focus on the half-filling case, when the density of electrons . Then the ground state has the checkerboard-type structure composed of two interpenetrating sublattices and , each of which is occupied only by one type of particle: the sublattice by particles and the sublattice by particles, respectively. Consequently, the density () of particles on the sublattice is equal to (), whereas the density () of particles on the sublattice is equal to ().
With an increase of temperature the densities , () diminish below 1, while , () increase above 0 and in the disordered phase all these densities are equal to . Then the quantity is equal to 1 at and equal to in the high-temperature, disordered phase, thus it is chosen to be the order parameter. It turns out that changes of cause significant changes in the . In particular, some energy states appear within the energy gap if . If it happens around the Fermi level, it corresponds to the MI transformation.
In fact, the depends explicitly on the order parameter and its temperature dependence comes out entirely from the temperature dependence of . Consequently, the order parameter and the are determined selfconsistently from the following procedure. First we determine the -dependent and from that the free energy . Next we find the temperature dependence of the order parameter from minimization of over . Then, we find the temperature dependent by inserting into . And finally we calculate the internal energy , the energy gap and the value of at the Fermi level .
The Hamiltonian we use is (see Ref. FreericksLemanski ())
where means the nearest neighbor lattice sites and , () is an annihilation(creation) operator of itinerant electrons, whereas is their particle number operator. The quantity is equal to 1/2(-1/2) for the lattice site occupied by the particle A(B), so the Coulomb-type on-site interaction between itinerant electrons and the localized particles amounts . The hopping electron amplitude we henceforth set equal to one for our energy scale.
We suppose that our results should be relevant to various experimental systems that display charge density or magnetic order such as for example , (see Ref. MatveevShvaikaFreericks () and the citations given therein) or perovskite compounds and SzymczakSzewczyk (); SzymczakSzymczak ().
In the next section we provide a detailed analysis of the as a function of and and in the section III we show the temperature dependence of the . In the section III we also discuss the relationship between the O-DO and MI transformations and present the phase diagram of the system. Then the existence of a nonzero DOS at a characteristic value of is derived within a low temperature expansion on a general bipartite lattice (Sect. IV). Finally, the last section contains some concluding remarks on our findings and a summary.
Ii Density of states (DOS)
All physical properties analyzed in this paper are derived from calculated from the Laplace transformation of the retarded Green function defined for complex with using the standard formula
In the remainder of this paper we will sometimes use simplified notations or instead of and instead of , respectively.
For the two sublattice system one has
where the corresponding system of two equations for Green functions and on the Bethe lattice reported by van Dongen vanDongen () is as follows.
At zero temperature , so the system of eqs. (4) reduces to the following simple form
and the Green functions are expressed by the analytical formulas
It comes out from (6) that the imaginary parts of and , so the , have non-zero values within the intervals and . Then the energy gap at the Fermi level is equal to . Consequently, for any non-zero the system is an insulator at zero temperature.
The situation is quite different at high temperatures, when the system is in a disordered, homogeneous state. In this case , so and the system of eqs. (4) reduces to one polynomial equation of 3rd rank (eq. (7)) on . In fact, the equation (7) was first derived and analyzed already by Hubbard in his alloy analogy paper Hubbard () (within the Hubbard-III-approximation of the Hubbard model). Then it was re-derived by Velicky et al. Velicky () and later on by van Dongen and Leinung vanDongenLeinung (). Here we rewrite it in the following form.
The equation (7) has nontrivial analytic solutions that are significantly different for small and large . Consequently, for there is no gap in the electronic energy spectrum, whereas for there is the finite gap at the Fermi level that increases with . So the system is a conductor when is smaller than the critical value , otherwise it is an insulator.
In Fig. 1 we display the in the ordered phase at (left column) and in the disordered phase (right column) for a few representative values of . It comes out that for the energy gap at the Fermi level persists in the disordered phase, then the system is an insulator. On the other hand, for the gap disappears in the high-temperature phase, so the order-disorder phase transition is accompanied by the insulator-metal transformation. However, it turns out that temperatures where these two transformations occur are usually different.
The natural question that now arises is how the evolves with temperature starting from and ending at high temperature, where the system is in the disordered phase. As we mentioned in the Introduction, preliminary studies of the for the ordered phase at finite temperatures were already reported in the review paper by Freericks and Zlatic FreericksZlatic (). Then this problem was examined by Hassan and Krishnamurthy HassanKrishnamurthy () and independently by Matveew, Shvaika and Freericks MatveevShvaikaFreericks (). In all these papers the authors calculated using the method of summation over Matsubara frequencies. Here we get similar results using a different method. Namely, we solve the system of eqs. (4) for arbitrary and then we calculate from eqs. (2) and (3). In fact, the system of eqs. (4) reduces to the polynomial equation of 5rd rank on (see eq. (8)) or (not displayed, but knowing one can find from (4)).
The coefficients , , , , , are functions of , and . Since the expressions on these coefficients are rather lengthy, we put them into Appendix A.
The resulting is displayed in Fig. 2 for and a set of values, whereas in Fig. 3 for and a set of values. By viewing Fig. 2 one can see how evolves when the system undergoes the MI transformation and by viewing Fig. 3 one can notice how the process of filling in the gap starts up when the order parameter begins to be less than one. As we already mentioned before this filling is quite surprising, as being completely different from the expectations based on the conventional mean field theory FGebhard (). Indeed, according to this theory the process of closing the gap is due to gradual increase in the width of two DOS subbands: one lying just below and the other just above the Fermi level. As a result, the upper edge of the valency band and the lower edge of the conduction band converge to each other if, and only if .
On the other hand, our results confirm findings reported in Refs. HassanKrishnamurthy (); MatveevShvaikaFreericks () that the filling of the gap occurs due to two additional subbands developing inside the gap. These additional subbands are located symmetrically with respect to the Fermi level and their initial positions depend on (for just below 1). With a decrease of the width of the subbands increases, and they merge together to form one band at certain , if .
From our calculations it was easy to obtain a simple analytical formula for DOS at the Fermi level as a function of and . Indeed, it appears that at this special point the polynomial in eq. (8) factorizes, so that the eq. (8) has the following simple form.
Then the eq. (8) can be solved analytically and the resulting DOS is as follows.
Hence it follows that inside the whole interval the system is metallic (i.e. ) not only in the disordered phase where (then ), but also in the ordered phase, if only .
Moreover, at the maximum value is attained, so the system is then metallic even for infinitesimally close to the limit , that corresponds to the fully ordered phase at .
Having the formula for DOS derived from the system of eqs. 4 we are also able to analyze the insulating phase characterized by its energy gap at the Fermi level (then obviously ). If the system is an insulator both in the disordered and ordered phase for any . On the other hand, if , then it is an insulator only for . As we already mentioned before, at , i.e. in the fully ordered phase () one has . However, it appears that is not a continuous function of at . Indeed, when and (i.e. ) we got the following analytical formula
and also the analytical expression for () (see Eq. (12)).
In Fig. 4 we display how the energy gaps change with for a set of few fixed values of . At is represented by the straight dotted line . However, for , but being infinitesimally close to 1 the function behaves non-monotonically. Starting from zero at it first increases, attains its local maximum equal to at and then goes down to 0 at . In the opposite limit of the homogeneous phase () one has for and the curve starts to rise up for according to the formula (12). The behavior of between these two limits is represented in Fig. 4 for by the dashed line.
Iii Order-disorder versus insulator-metal transition
and by minimizing over we can find the order parameter . Then, by inserting into we get . Next, from we determine the internal energy using the standard formula (14)
and the temperature dependence of two quantities characterizing the MI transformation: the energy gap and the at the Fermi level . We display as a function of in Fig. 5, where it can be seen that this function has a kink at , but no kink or any noticable anomaly at . This is why we conclude that the metal-insulator transformation at is not a phase transiton in the usual sense. On the other hand, at the system undergoes a typical order-disorder phase transition.
The temperature dependencies of , and for are displayed in Fig 6, where is the relative value of the gap. Note, that has a jump at because but, as it comes from Eq. (11), . Then, for one has and . Obviously, the energy gap , so do is positive in the insulating phase, i.e. for and is equal to zero in the metallic phase. On the other hand, is equal to zero in the insulating phase, but is positive in the metallic phase.
By viewing Fig. 6 one can see that and for , so is substantially smaller than . One can also notice that MI transformation occurs when the order parameter , so is still close to its maximum value 1. Another interesting observation is that clearly increases with temperature up to the maximum value attained at and this value is preserved for higher temperatures.
After inserting into (10) we get as a function of and . This function is quite non-trivial as it can be seen in Figs. 7 and 8. In Fig. 7 we display as a function of for a set of fixed temperatures and in Fig. 7 one can observe as a function of for a few values.
After collecting the data on and for a representative set of values we constructed the phase diagram of the system that is displayed in Fig. 9. Let us note that in this diagram the region of ordered insulator phase located below (continuous) line consists of two parts corresponding to insulating phases separated by an ordered metallic phase. This is quite unexpected finding obtained neither within the conventional mean field theory FGebhard (), nor through the exact procedure of expanding in series for large or small values vanDongen (). In fact, the finding is not inconsistent with the result obtained by Van Dongen vanDongen (), as indeed, for small and large the gap exists in the ordered phase up to . However, for intermediate values the metallic ordered phase appears below down to . What’s more, for the metallic phase can be stable down to .
The phase diagram displayed in Fig. 9 is almost identical to the one presented in Ref. HassanKrishnamurthy (). However, there is a substantial difference between the two diagrams at , where in our case the end point of the homogeneous phase is at , whereas in Ref. HassanKrishnamurthy () it lies slightly above . This is due to difference in calculation techniques used in the two cases. We were able to fix this end point at using the analytical formula (10). Then the question arises about quantum effects related to the MI transformation for this particular value of . In order to clarify this point some additional studies need to be done.
Iv Low-temperature expansion on the lattice
Now we consider a grand-canonical ensemble with the spinless Falicov-Kimball Hamiltonian (1) on a -dimensional bipartite lattice. To distinguish the following calculation from the previous one on the Bethe lattice, we introduce lattice coordinates , . With , where the absence (presence) of a heavy fermions at site is represented as classical binary number (), we can write for the Hamiltonian matrix
with the chemical potential . At half filling we have for the latter .
According to Ref. [ates05, ], the heavy particles are distributed by the thermal distribution at the inverse temperature
which we can approximate by an Ising distribution as
Adding or removing a heavy particle from the ground state (staggered configuration) appears then with the weight , where
This provides us a low temperature expansion for the density of states (cf. App. B) by adding or removing particles from the groundstate configuration:
Thus, in order we have a Dirac delta function for the DOS which is peaked at and has a weight , where the parameter can be calculated as an integrals for a given lattice with known hopping term :
The contribution to the DOS in Eq. (19) vanishes with decreasing temperature, similar to the DOS in Figs. 7, 8. With increasing temperature we must include higher order terms in which might lead to a broadening of the DOS around . These results indicate that the singular DOS around a special value of in Fig. 7, is not an artifact of the DMFT or the Bethe lattice but a general feature of the FK model on any bipartite lattice.
V Final remarks and conclusions
Here we focus on the quantitative analysis of a relationship between the degree of disorder in a correlated electron system and the transformation from insulator to metal. Using exact formulas for the temperature-dependent for the FK model on the Bethe lattice we demonstrate the effect of closing of the energy gap in the in the insulating phase (for not too large ) and then of increasing of the value at the Fermi level in the metallic phase with an increase of degree of disorder. Our results confirm and extend the findings presented in Refs. HassanKrishnamurthy (); MatveevShvaikaFreericks ().
One of the most surprising conclusions drawn from these studies is that an increase of disorder may lead to a closure of the energy gap still before the system transforms into a completely disordered phase. In view of this result, we suggest a re-examination of those experiments, in which transition temperatures and are found to be the same RajanAvignonFalicov (); SzymczakSzewczyk (); SzymczakSzymczak (). But one should keep in mind that the distinction between and can be difficult to detect in some systems, as a clear difference between these temperatures was found only in a relatively narrow range of values of the parameter . An additional difficulty is that just above the at the Fermi level is still small, as only above it begins to rise with temperature, starting from zero and reaching a maximum value at (see Fig. 7). Therefore, we expect, that one will be able to notice a difference between and only in precise enough experiments.
As we have demonstrated within a low temperature expansion, the results emerging from the DMFT calculation on the Bethe lattice might be quite general. The reason is, that the FK model, called by some authors the simplified Hubbard model MichielsenRaedt (), contains basic ingredients that are present in many other models of correlated electron system. On the other hand, properties of the relevant for these studies, such as existence of the gap in the homogeneous phase for sufficiently large and closing the gap with decreasing in the homogeneous phase but not in the ordered phase, are common for all examined lattices (hypercubic 1D, 2D, 3D and the infinite D, as well as the Bethe lattice in the infinite D limit) vanDongen (); vanDongenLeinung (); MichielsenRaedt (); MaskaCzajka (); ZondaFarkasovsky ().
Interestingly enough, there are some similarities between our phase diagram displayed in Fig. 6 and the phase diagram found for the Hubbard model with disorder ByczukVollhardt (). In fact, we cannot directly compare our results with those reported in Ref. ByczukVollhardt (), as these latter were obtained not for the FK model but for the Hubbard model, and only at zero temperature. However, in these two cases the same sort of phases appear on the phase diagram, only insulating phases survive for large and the ordered metallic phase occupies a relatively small region in the phase diagram.
Finally, let us hope that the existence of gapless checkerboard-type charge density wave phase found first for the FK model will be confirmed by studies for on the Hubbard model and other models of strongly correlated electrons.
Acknowledgements.We express our best thanks to K. Byczuk and J.K. Freericks for useful discussions on some issues raised in this paper and for critical reading of the manuscript.
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Appendix A Coefficients of the polynomial given in Eq. (8)
Appendix B Green’s function on the bipartite lattice
Then the Green’s function of the light fermions reads as an average with respect to a grand-canonical distribution of the heavy fermions
At half-filling, where , the ground state of the heavy particles on a bipartite lattice is a staggered (or generalized checkerboard) configuration. Using a sublattice representation for the hopping of the light fermions, we obtain
where the sublattice 1 (2) has the effective potential (). Here we have assumed that the hopping is only between nearest neighbors. Therefore, the hopping terms are , in the off-diagonal elements of our sublattice matrix. Now we apply a Fourier transformation on the translational invariant sublattice to get as Fourier components matrices
with the two-band dispersion . The sum over other configurations in (21) is now an expansion in powers of a weight . This implies for the Green’s function
The latter expressions mean that () removes (adds) a heavy particle at site on sublattice 1 (2). The expressions , can be easily computed by using the identity
where refers to the projection of the matrix space with nonzero . In our case is just the single site , such that this identity reads with
The elements of the Green’s function can be evaluated from their Fourier components as
Since we have a gap , the Green’s function is real in the limit . Therefore, the density of states reduces to
with the Dirac delta function .