Effective theory and emergent symmetry in the flat bands of attractive Hubbard models
In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Neverethless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher lying bands via perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian and that the compressibility is diverging as a consequence of an emergent symmetry. This symmetry is broken by second order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensional ladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality between the Drude weight and the winding number , which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.
pacs:74.25.F-, 74.20.Fg, 74.20.-z, 67.85.Lm
In quantum mechanics particles can localize due to the destructive interference between different classical trajectories. Such localization can result in the formation of flat bands with a diverging effective mass, or equivalently zero group velocity. Historically the first example of this phenomenon is the formation of the Landau levels of a particle in the presence of a uniform magnetic field. While this localization is a purely single-particle effect, the presence of a flat band can have a profound impact on the physics of interacting many-body systems. A prime example is the fractional quantum Hall effect. Tsui et al. (1982) Beyond the physics in a strong magnetic field, flat, or nearly-flat bands are a relatively common occurrence in lattice Hamiltonians where they can be realized by engineering suitable hopping matrix elements. Lieb (1989); Mielke (1991); Motruk and Mielke (2012); Creutz (1999); Tovmasyan et al. (2013); Takayoshi et al. (2013); Sticlet et al. (2014); Biondi et al. (2015); Sun et al. (2011); Tang et al. (2011); Neupert et al. (2011) Given the vanishing group velocity, one could expect an electronic flat band system to be a particularly bad conductor. This is certainly true for the high-temperature phase. However, matters are less clear if a superconductor is formed in such a flat band. Here, we investigate this scenario with a special emphasis on the transport properties.
Why can we expect a superconductor to appear in a flat band to begin with? Let us recall that the BCS superconducting transition temperature scales as , where is the interaction strength and is the density of states at Fermi energy. Given that diverges in a flat band system, we can indeed hope for a strong superconducting instability on a flat band. Khodel and Shaginyan (1990); Volovik (1991); Khodel et al. (1994); Kopnin et al. (2011); Heikkilä et al. (2011) This now raises questions regarding the superfluid properties of such a flat-band superconductor.
That superfluid transport can occur in the limit of strictly flat bands has been known from the experiments on the exciton condensate in Quantum Hall bilayers. Kellogg et al. (2004); Tutuc et al. (2004); Eisenstein (2014) However, it has been realized recently that even flat bands that emerge from the geometry of the lattice can sustain a large superfluid current in the presence of attractive interactions. This occurs because in the flat band limit the superfluid weight is controlled by a band structure quantity called the quantum metric which is distinct from, but related to, the Chern number. Peotta and Törmä (2015)
Despite the recent progress in the study of many-body states in the flat bands of lattice Hamiltonians, there are still many open questions. For example, it has been found in the case of bipartite lattices that the ground state in the presence of an attractive Hubbard interaction is given exactly by the BCS wave function. Julku et al. (2016) However, it is quite unclear what are the properties of the normal state above the superconducting transition.
By definition the normal state is not a Fermi liquid since as the interaction is turned off the system becomes an insulator for any filling of the flat band. Indeed, the picture that the divergence of the effective mass implies the absence of transport is true for a noninteracting system.
Transport in a flat band is a consequence of either interaction or disorder. It is therefore interesting to characterize the properties both of the normal state and of the superconducting state at nonzero temperature as these can be accessed in current ultracold gas experiments. Moreover, the situation where the superconducting ground state is well-known but the normal metallic state is less understood is reminiscent of the one in high- superconductors, especially cuprates and iron-pnictides, where the enigmatic pseudogap phase is believed to be important for unveiling the pairing mechanism that gives rise to the superconducting phase. It may be possible that idealized flat band models can provide clues in this direction.
In this work we address the general problem of providing a reliable low-energy effective theory for the flat band of a multiband lattice Hamiltonian in the presence of an attractive Hubbard interaction. It is important to consider a multiband Hamiltonian since in the case of a single band Hamiltonian, i.e., a Hamiltonian defined on a simple lattice, the only flat band that can be obtained is trivial and corresponds to the limit where all the sites are decoupled (atomic limit). Furthermore, in the multiband case the flat band can have a wide variety of properties encoded in suitable invariants constructed from the Bloch/Wannier functions. An example is the Chern number , a topological invariant signaling, when nonzero, that the flat band cannot be connected adiabatically to the atomic limit () or to a band with different Chern number. Interestingly, even a topologically trivial flat band () can have a nonzero quantum metric and therefore host a superconducting state. Peotta and Törmä (2015) The subject of multiband superconductivity has recently become important with the discovery of materials such as magnesium diboride and iron pnictides superconductors. Flat band superconductivity is an exotic example of the rich variety of phenomena encountered in multiband superconductivity.
The strategy of this work is to combine the result of Ref. Peotta and Törmä, 2015 for the superfluid weight of a flat band in terms of the quantum metric with the general approach of Ref. Huber and Altman, 2010 where flat bands are studied by projecting the interaction Hamiltonian on the Wannier functions of the flat band. This latter approach has the advantage of providing a simple low-energy effective Hamiltonian which is often accurate in predicting the properties of the ground state.
We are able to prove the useful result that, in the wide class of Hubbard models considered here, the BCS wave function is the exact ground state of the projected interaction Hamiltonian under a simple condition on the Wannier functions. Concomitantly, the compressibility in the partially filled flat band is diverging. In fact we show how both these properties are the manifestation of an emerging symmetry, which is due to the band flatness.
Guided by these rigorous results, we are lead to approximate the projected interaction Hamiltonian by an effective ferromagnetic Heisenberg model. The drawback is that the Wannier functions are defined up to a unitary transformation, and therefore any approximation performed on the projected interaction Hamiltonian depends on the specific choice of the Wannier functions. Using the gauge invariant result of Ref. Peotta and Törmä, 2015, we show that there is a preferred choice for the Wannier functions, which coincides with the maximally localized Wannier functions in the Marzari-Vanderbilt sense.
As a concrete example of our general results, we consider an Hubbard model defined on a one-dimensional ladder, the Creutz ladder, which is graphically defined in Fig. 1. The band structure of the Creutz ladder consists of two perfectly flat bands. In the case of the attractive Creutz-Hubbard model the resulting effective ferromagnetic Heisenberg model takes the form of the integrable chain.
We find also that in order to account for a finite compressibility it is necessary to include the effect of interband transitions resulting in higher order terms in the effective Hamiltonian. We provide an analytic result for the compressibility up to second order in the ratio between interaction and band gap in the case of the Creutz-Hubbard model. This result is tested against Density Matrix Renormalization Group (DMRG) simulations and we find an excellent agreement.
Finally, we prove an important bound between Drude weight and the one-dimensional winding number, which extends the result for the superfluid weight valid in two dimensionsPeotta and Törmä (2015). It is shown using the Creutz-Hubbard model that the inequality is in fact optimal.
The paper is organized as follows. In Section II we introduce the model, the basic notations, and the perturbative Schrieffer-Wolff (SW) transformation for a generic Hubbard model. In Section III we derive the result that the BCS wave function is exact in the isolated flat band limit and provide the generators of the emergent symmetry. In Section IV we derive the ferromagnetic Heisenberg model by dropping the pair-breaking terms in the projected interaction Hamiltonian. We show that this mapping is in fact exact for the Creutz-Hubbard model. In Section V we discuss how the result of Ref. Peotta and Törmä, 2015 for the superfluid weight of a flat band can be recovered from the Heisenberg model. This is done by introducing an overlap functional for Wannier functions, whose relation with the usual Marzari-Vanderbilt functional is analyzed in detail. The proof of various results relating the two functionals is detailed in Appendix A. In Section VI we derive the second order corrections to the effective Hamiltonian using the SW transformation in the case of the Creutz-Hubbard model. In Section VII we use the result of the previous section to derive an analytic result for the compressibility of the Creutz-Hubbard model which is then compared to DMRG simulations. In Section VIII the inequality between superfluid weight and winding number is proved. Our results and the future perspectives are discussed in the last Section.
Ii Effective Hamiltonian from the SW transformation
ii.1 The model
Here we introduce the class of flat band models with attractive Hubbard interactions defined on a -dimensional lattice that are considered in this work. We shall also give some useful definitions and consider an one-dimensional example, the Creutz-Hubbard model.
The kinetic part of the tight-binding Hamiltonian is given by
where, as usual, and are the creation and the annihilation operators, respectively, of a fermion with spin at unit cell and sublattice . According to Eq. (1) the number of spin- and spin- particles are separately conserved. In order to preserve the time-reversal symmetry and favor the occurrence of Cooper pairs in the spirit of Anderson’s theorems, Anderson (1959, 1984) we take the hopping matrix such that the kinetic Hamiltonian is time-reversal invariant, namely , where the star denotes the complex conjugate. Moreover, we consider models in which the kinetic Hamiltonian has an isolated flat Bloch band separated from the other bands by a finite energy gap . Without loss of generality, we concentrate on the case where the flat band is the lowest lying band to simplify the notation.
The attractive Hubbard interaction term has the form
where is the fermionic number operator and we assume that .
Occasionally we comment on the corresponding repulsive Hubbard model, where by “corresponding”we mean that the spin- and spin- kinetic Hamiltonians are equal () and equal to the spin- kinetic Hamiltonian of the attractive model. Thus the corresponding Hubbard model possess full spin rotational symmetry. We will see that there is a general duality between flat band superconductivity in the attractive model and flat band ferromagnetism in the corresponding repulsive model.
For the discussion of the interacting problem below, the Wannier states form a convenient local orthonormal basis. Huber and Altman (2010); Tovmasyan et al. (2013) The Wannier function of the flat band centered at the unit cell is constructed from the Bloch functions of the flat band according to Wannier (1937); Marzari et al. (2012)
where denotes the volume of the unit cell and is the lattice vector corresponding to unit cell .
It is a well known fact that the Wannier functions are not uniquely defined because of the gauge freedom of the Bloch functions. In fact, we can change the Bloch functions by , where is an arbitrary real function of quasi-momentum . Below, when necessary, we discuss the consequences of this gauge freedom in details.
As an example we shall consider the Creutz ladder, a one-dimensional model of two cross-linked chains, depicted in Fig. 1. The hopping matrix elements for the spin- fermions are also given in Fig. 1. The band structure of this model is extremely simple consisting in two perfectly flat bands at (), which allows to evaluate the interband effects analytically. Moreover, the Wannier functions of the Creutz model for both the upper and lower flat band can be chosen to be perfectly localized on only two adjacent rungs, also shown in Fig. 1. The lower band Wannier function for the spin- fermions correspond to the following periodic and analytic Bloch functions
where is the lattice spacing. Note that, from the time-reversal invariance of the kinetic Hamiltonian, we have or, equivalently, .
ii.2 The effective Hamiltonian
In this subsection we give the framework which we use to construct an effective low-energy theory for the class of Hubbard models introduced above. We are interested mainly in the attractive case, but this framework can be used in the case of repulsive Hubbard interactions as well.
A simple low-energy theory can be constructed by projecting the interaction term into the subspace where only the flat band states are occupied. This is done in practice by truncating the expansion of field operators in terms of Wannier orbitals of all bands to retain only the orbitals corresponding to the chosen flat band, Huber and Altman (2010) thereby restricting the Hilbert space to the flat band subspace. However, one expects interband effects which become more relevant with increasing interaction strength . To include these effects, we use the perturbative SW transformation that allows to take them into account by means of a low-energy effective Hamiltonian, which involves only the degrees of freedom of the lowest flat band. Bravyi et al. (2011)
Let us define the field operators projected in the flat band as
where is the Wannier function of the flat band centered at unit cell and is the annihilation operator corresponding to this Wannier orbital. Hereafter we drop the spin index of the Wannier and Bloch functions and refer to the ones for the spin-, that is, . By virtue of the orthonormality of the Wannier functions, namely , fermionic operators satisfy the canonical anticommutation relations. It is important to note that the projected field operators satisfy modified anticommutation relations
where we have introduced the lower band projector defined by . All other commutation relations are trivial.
Despite the gauge freedom in the definition of the Wannier functions, the projected field operators given in Eq. (5) are gauge independent, i.e., they are the same for any choice of the Wannier function. Indeed, an equivalent way of defining the projected operators is via the lower band projector introduced above according to and . From the last relations the gauge invariance of the projected operators becomes explicit, as the projectors are gauge independent.
From the projected operator given in Eq. (5) we can define the projected number operator as . Accordingly, the projected interaction Hamiltonian becomes . This low-energy effective Hamiltonian neglects all interband effects and it describes the system quite well if the minimum band gap between the flat band and other bands is much bigger then the interaction strength . If this condition is not satisfied, one needs to include the effects from other bands.
Let us denote the second-quantized projection operator that projects on the subspace of the Hilbert-Fock space where only the states of the flat band have nonzero occupancy by , where is the projector relative to spin . Then the following properties are immediate from the definitions
where is the field operator projected into the complement of the flat band subspace, namely . The operator can be expressed by the Wannier orbitals of the higher bands similarly to (5). The operators commute with the flat band field operators and with the operators with opposite spin. It is then easy to verify that , namely the projected interaction operator is obtained by sandwiching with . We call the complementary projector. Using the notation of Ref. Bravyi et al., 2011, we define superoperators and acting on a generic operator as
The superoperator extracts the diagonal part of the operator in the argument, while the off-diagonal one. Therefore, . Let us define another superoperator
where the labels run over the eigenstates of the noninteracting Hamiltonian . Since the matrix element is nonzero only if belongs to the flat band subspace and to the complementary subspace, or vice versa, the above sum is well defined. It follows that .
With the help of the above introduced superoperators, the effective Hamiltonian up to second order in an expansion in the coupling constant reads Bravyi et al. (2011)
The zero order term is the projected kinetic Hamiltonian which is a trivial constant for a flat band, while the first order term is simply the projected interaction term . Below we will use this general result for the Creutz-Hubbard model.
Iii Exactness of the BCS wave function and emergent symmetry
In this Section we prove that the ground state of the projected attractive interaction for arbitrary filling the flat band is given exactly by the BCS wave function. This statement is analogous to the well known fact that the completely polarized ferromagnetic state is the ground state for a half-filled flat band in a repulsive Hubbard model if the flat band is the lowest lying oneMielke and Tasaki (1993). An important difference is that in the former case we known the ground state only of the projected Hamiltonian , while the latter is a statement regarding the full Hamiltonian .
This result generalizes the one relative to bipartite lattices. In a bipartite lattice it is possible to relate a repulsive Hubbard model to an attractive Hubbard model by a particle-hole transformation. Using this transformation and the Lieb theoremLieb (1989), it was shown in Ref. Julku et al., 2016 that the completely polarized ferromagnetic state maps into the BCS wave function, which is then the exact ground state of the attractive Hubbard model for arbitrary filling of the flat band. Below we show that the BCS wave function is the exact ground state in a more general setting.
First, let us define the operator which creates a Cooper pair in a plane wave state with zero quasi-momentum in the flat band
The above introduced operator creates a fermion with quasi-momentum in the flat Bloch band. Note that the pair creation operator takes the above simple form only in the flat band limit. Generally, it is given by , where defines the Cooper pair wave function. In the flat band limit the BCS coherence factors and are independent of the quasi-momentum and only in this case the second equality in Eq. (11) is valid. Here is the filling of the flat band with the number of unit cells and the number of fermions. Hence, the BCS wave function in the grand canonical ensemble can be written in the following equivalent forms
where is the vacuum containing no fermions. In expanding the exponential in Eq. (12) we have made use of Fermi statistics, which imply .
It is straightforward to check that commutes with the projected spin operator , i.e., . From this commutation relation one has that . Therefore, the BCS wave function is a zero eigenvector of the positive semidefinite operator . Note that, as a consequence of the modified anticommutation relations (6), , while the usual number operators satisfy . Using this, one can expand the square
The last term in the above equation is precisely the attractive Hubbard interaction, while the first term is in general a nontrivial orbital-resolved potential. Consider now the case in which the flat band Bloch/Wannier functions have the same weight on the orbitals were they are nonzero, which means
in a certain subset of orbitals and for . Here is the number of orbitals on which the flat band states have nonvanishing weight and, equivalently, on which the Cooper pair wave function is uniformly delocalized. We call the Eq. (14) the uniform pairing condition. Then the operator reduces to
with the projected particle number operator. In this case differs from the projected attractive Hubbard interaction by a trivial term proportional to the particle number operator. Following the usual argument Mielke and Tasaki (1993) we conclude that the BCS wave function is the ground state of if the condition in Eq. (14) is satisfied. The ground state energy is
with the flat band energy. It is important to note that this result is only valid asymptotically for small , contrary to the repulsive case. The reason is that the pair creation operator commutes only with the projected spin operator and it does not commute with the full spin operator . In fact one can show that .
From the Cooper pair creation operator , we can construct the generators of the emergent symmetry of the projected interaction Hamiltonian which also includes the usual particle number conservation symmetry. The operators , and form the generators of . Using the commutator , it is straightforward to check that the three generators given above satisfy the commutation relations of the algebra.
A consequence of this symmetry is that the compressibility is diverging; as it also follows from Eq. (16) and the definition of the inverse compressibility . Moreover, since is a zero eigenstate of , one obtains the relation between density and double occupancy (expectation values are here taken on ). Both the results for the energy and the relation between density and double occupancy have been verified with DMRG for the Creutz-Hubbard model ().
We mention also that, as a consequence of the uniform pairing condition (14), the expectation values of the projected density and of the pairing order parameters are constant as a function of and vanish for . In particular, the condition of constant has been used in Ref. Peotta and Törmä, 2015 to derive the relation between superfluid weight and quantum metric.
Iv Spin chain form of the effective Hamiltonian
As a next step, we show that it is possible to drop a large amount of terms in the projected interaction Hamiltonian at the same time preserving all of the properties mentioned above, namely the BCS ground state and the emergent symmetry. The result of this truncation is an effective ferromagnetic Heisenberg model, where the components of the pseudospin are the creation, annihilation and occupation operator of a Cooper pair in a Wannier function. The effective spin model is computationally much easier to deal with and offers an intuitive model of a flat band superconductor as a ferromagnet.
The effective Hamiltonian obtained by the projection technique with Wannier functions is generally given by
where the coefficients can be written in terms of Wannier functions. In the case of attractive interactions it is energetically more favorable that fermions with opposite spins form a Cooper pair. Hence, we truncate the projected Hamiltonian to the subspace defined by the conditions for every . In other words, only the terms in the expansion (17) that preserve and act nontrivially in this subspace are retained. After this truncation, the only remaining degree of freedom is the presence or absence of a Cooper pair in a given Wannier state. This is encoded in the components of the pseudospin defined by
Note that and the generators of the symmetry given in the previous section correspond to , and , where and .
Using Eq. (18) we can map the projected Hamiltonian (after the truncation) into an effective spin model which is an isotropic ferromagnetic Heisenberg model given by
with couplings defined as
The physical content of the spin Hamiltonian (19) is that the effective spins located at unit cells and interact through the density overlap of Wannier functions centered at unit cells and , substantiating the intuition that the density overlap of Wannier functions is responsible for the superconducting order. The ground state of the effective spin Hamiltonian is a product state where all spins are aligned to the same direction. In the language of the attractive Hubbard model this corresponds to the BCS wave function (12) where the order parameter is given by the expectation value . Moreover, the spin Hamiltonian is manifestly invariant.
The main drawback of the spin Hamiltonian (19) is that it is not gauge invariant. The above truncation depends on the gauge choice for the Wannier states, namely on the definition of the operators . Whereas the BCS ground state and the symmetry is correctly reproduced in any gauge, the choice of gauge affects in a substantial way the low-energy spectrum of the effective Hamiltonian. We show in the next section that there is a preferred gauge choice, the maximally localized Wannier functions in the Marzari-Vanderbilt sense. Marzari and Vanderbilt (1997) In this basis the spin Hamiltonian is the best possible approximation of in the sense that the gauge invariant result for the superfluid density obtained in Ref. Peotta and Törmä, 2015 is recovered.
The subtle point of the gauge noninvariance of the spin Hamiltonian is very well illustrated in the Creutz-Hubbard model. As we mentioned before, the Wannier functions of the Creutz model can be chosen to be perfectly localized on a plaquette, see Fig. 1. These plaquette states are in fact the maximally localized Wannier functions in the Marzari-Vanderbilt sense. If is expanded in annihilation and creation operators of these plaquette states, pair-breaking and pair-creation terms subject to truncation are absent, and the mapping from the projected interaction Hamiltonian to is exact. However, this is not the case for arbitrary choices of the Wannier functions. Utilizing the maximally localized lower band Wannier functions of the Creutz model, we obtain the projected Hamiltonian given by
where . Indeed, we see that the above Hamiltonian does not mix the subspaces with different number of pairs, and hence for the Creutz model the above described truncation is not an approximation. Using Eqs. (18), we can map the projected Hamiltonian (21) into the ferromagnetic chain with Hamiltonian
It is worth mentioning that the spin Hamiltonian (19) describes also the completely polarized ferromagnetic state of the corresponding repulsive Hubbard model with half-filled lowest flat band. As discussed in Section II.1 in the case of the repulsive Hubbard model the kinetic Hamiltonian is spin-isotropic. In this case it is energetically more favorable that all the Wannier states are only singly occupied, i.e., we consider the subspace defined by . Hence, the only remaining degree of freedom is the spin and the effective spin of Eq. (19) coincides with the true spin of the fermions. In fact the attractive and repulsive flat band models are exactly related by a particle-hole transformation up to first order in , while this mapping is broken by interband transitions. The effective spin Hamiltonian (19) expresses in a concise way the duality between the ferromagnetic and BCS ground states in a flat band.
V Superfluid weight from spin chain
The gauge noninvariance of the spin Hamiltonian (19) manifests in the fact that the superfluid weight evaluated from it depends on the choice of the Wannier functions. On the other hand, in Ref. Peotta and Törmä, 2015 a gauge invariant result for the superfluid weight of a flat band in the presence of an attractive Hubbard interaction has been derived using mean-field BCS theory. The solution of this inconsistency is that, within the spin Hamiltonian approximation, there exists a preferred gauge choice for the Wannier functions which allows to obtain a result for the superfluid weight as close as possible to the gauge invariant result of Ref. Peotta and Törmä, 2015. Specifically, in the following we prove the inequality
valid in arbitrary dimension . Here the Quantum Geometric Tensor (QGT) is defined by
where are the flat band Bloch functions. Moreover, we show that the gauge noninvariant quantity attains a global minimum if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. In this preferred gauge the superfluid weight tensors calculated using the two different methods coincide in one dimension, while for the spin Hamiltonian generally overestimates the superfluid weight as shown by Eq. (23).
Our argument is based on the fact that the superfluid weight calculated from the spin Hamiltonian is proportional to a functional, called , that measures the degree of overlap between Wannier functions. This functional is similar to the Marzari-Vanderbilt functional, called , which measures the spread of the Wannier functions. Marzari and Vanderbilt (1997) The above results follow from some general relations between these two functionals whose detailed proof is given in the Appendix A. Our proof relies on some assumptions: the first is the uniform pairing condition (14) which is also assumed in Ref. Peotta and Törmä, 2015 to derive Eq. (24). Since we have proved that the BCS wave function is exact in this case, the use of mean-field BCS theory and thus the result of Eq. (24) are justified. The second assumption is the existence of a gauge in which the Bloch functions are periodic and analytic functions of quasi-momentum, which is equivalent to requiring that the flat band has zero Chern number(s). Brouder et al. (2007); Panati (2007); Panati and Pisante (2013); Monaco et al. () The final assumption is Eq. (60) which is discussed in Appendix A. All conditions are verified in the case of the Creutz ladder.
The overlap functional is introduced by considering the state with finite uniform superfluid current given approximately, for small phase-winding wavevector , by the ansatz
It is easy to verify that and . Therefore, the energy change due to the finite phase-winding with wavevector is given by
Here we have defined the overlap functional for Wannier functions which reads
We call it the overlap functional since, from the second line of the above equation, it is apparent that it can be expressed as the density overlap between the Wannier functions located at unit cells and , given by , summed over all possible relative position vectors .
We also introduce the Marzari-Vanderbilt localization functional which differs from the overlap functional only by an additional summation over the orbitals
Properly speaking, the Marzari-Vanderbilt localization functional is given by . Indeed, one has
Upon replacing the summations with integrals in the last equation, the usual definition of the Marzari-Vanderbilt localization functional in the continuum case Marzari et al. (2012); Marzari and Vanderbilt (1997) is recovered. In this sense both and are “generating functionals” whose expansion in the wavevector generates various moments of the Wannier function density distribution. However, we use the same name for both (29) and (30) since the object we are referring to should be clear from the context. The same applies to the overlap functional.
The main technical results of this section are some general relations between the two functionals which are proved in the Appendix A under the conditions stated above. Let us first introduce the maximally localized Wannier functions in the Marzari-Vanderbilt sense, denoted by , that are the global minimizers of the Marzari-Vanderbilt functional, namely, they satisfy
for all Wannier functions obtained from by a gauge transformation. The first result is that the maximally localized Wannier functions in the Marzari-Vanderbilt sense are in fact global minimizers of the overlap functional as well; in other words,
The second result is the equality, up to the constant factor defined in Eq. (14), of the second derivatives of the two functionals calculated on the maximally localized Wannier functions
where the first equality is the general definition of superfluid weight and the second equality holds only if the coefficients in the spin Hamiltonian are calculated using maximally localized Wannier functions. For any other gauge one obtains the inequality (23) as an immediate consequence of Eqs. (27), (32) and (34).
The crucial point of Eq. (34) is that the superfluid weight obtained from the spin Hamiltonian has been related to the Marzari-Vanderbilt functional which is well-known to be the sum of a gauge invariant term and a gauge noninvariant one. Marzari and Vanderbilt (1997); Marzari et al. (2012) Not by chance, the result of Eq. (24) can be recovered from the gauge invariant term of the Marzari-Vanderbilt functional. In fact one has where the gauge invariant term is expressed in term of the QGT as
Using Eqs. (34) and (35) one recovers Eq. (24) from the gauge invariant part only, which is a positive semidefinite matrix. Also the gauge noninvariant part is positive semidefinite, which means that the spin Hamiltonian overestimates the superfluid weight of the original Hubbard model if . However, in one dimension if maximally localized Wannier functions are used. In general this is not true in due to the noncommutativity of the components of the projected position operator. Marzari and Vanderbilt (1997) Also it can be shown that is not vanishing in general when the Berry curvature is nonzero.
A concrete example of the above general results is provided by the Creutz ladder. Indeed, it is easy to verify that the plaquette states are maximally localized Wannier functions in the Marzari-Vanderbilt sense, and in this preferred gauge the number of Cooper pairs is a conserved quantity as shown by Eq. (21). As a consequence there is no approximation in going from the projected interaction Hamiltonian to the spin Hamiltonian and the result for the superfluid weight coincides with Eq. (24). On the other hand, if we had chosen to perform the truncation in any other gauge, we would have obtained a superfluid weight which is strictly larger than the correct one as it can be seen from Eq. (23) and the results of Appendix A.
We now summarize the results of this section and comment on their significance. The first result is that the approximation underlying the spin effective Hamiltonian (19) is justified at least in one dimension, since the result of Eq. (24) is reproduced exactly. This means that the spin Hamiltonian is not only able to capture the correct ground state, but also low-lying excited states. In higher dimensions this approximation is justified only if is significantly smaller than in some sense. In particular, we anticipate that effects due to the Berry curvature, which are absent in one dimension, may play an important role and the low-energy effective Hamiltonian may differ substantially from Eq. (19). This is reflected in that we have purposefully avoided the case of nonzero Chern number which is the result of a nonzero average Berry curvature.
As a second important result, we believe we have provided a good understanding of the reason why the quantum metric (the real part of the QGT) enters in the result of Eq. (24) for the superfluid weight of a flat band. The physical mechanism for transport in a flat band is the correlated hopping of Cooper pairs induced by the Hubbard interaction between Wannier functions with finite density overlap. This picture is nicely captured by the effective Hamiltonian (19). We emphasize, though, that the gauge invariant part of the Marzari-Vanderbilt functional measures the spread of the Wannier functions and not their overlap. As we have proved, under some conditions the two distinct functionals measuring spread and overlap of Wannier functions, respectively, are equivalent, but this may not be true in general. Better insight into this matter can be gained by studying specific models which do not satisfy the conditions required for the validity of the relations (32) and (33) between the overlap and spread functionals.
Finally, we have provided a clean example where it is necessary to employ a specific preferred gauge for the Wannier functions in order to obtain a good approximation for the low-energy Hamiltonian. Very often the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used to derive simplified model Hamiltonians from electronic structure calculations. This is a generally accepted heuristic prescription, but we have shown here that there are some rigorous underlying constrains behind it, in our case the minimization of an observable quantity such as the superfluid weight.
Vi Second order corrections to the effective Hamiltonian and finite compressibility
Contrary to the repulsive case for which the completely polarized ferromagnetic state is the exact ground state for any value of if the half-filled flat band is the lowest band, in the attractive case the BCS wave function is the exact ground state of the effective Hamiltonian (10) only up to the first order term. As we shall see now the second order term in the expansion leads to a breaking of the emergent symmetry. It is convenient to study the breaking of the emergent symmetry in the specific case of the Creutz model which is particularly simple. However, this is a general fact, since this symmetry is not present in the generic model under consideration.
The off-diagonal part, as defined in Eq. (8), of the interaction term can be written as
where if and viceversa. The operators introduced above correspond to virtual processes where one particle with spin or two particles with opposite spins are excited to the upper band, respectively. Then the second order term in the expansion of the effective Hamiltonian (10) reads
In this last result we have made crucial use of the fact that the upper band of the Creutz ladder is also flat. For a generic lattice Hamiltonian with flat bands the computation of the second order term is more involved.
In order to compute the expansion of the products in terms of Wannier operators the following identity is useful
Recall that we impose time-reversal symmetry in the attractive Hubbard model which implies . Using Eq. (40) we obtain