Theory of extreme correlations using canonical Fermions and path integrals

Theory of extreme correlations using canonical Fermions and path integrals

B Sriram Shastry Physics Department, University of California, Santa Cruz, Ca 95064
25 February 2014

The  -  model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson-Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitean quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further a transparent physical interpretation of the previously introduced auxiliary Greens functions and the “caparison factor” is obtained.

The low energy electron spectral function in this theory, with a strong intrinsic asymmetry, is summarized in terms of a few expansion coefficients. These include an important emergent energy scale that shrinks to zero on approaching the insulating state, thereby making it difficult to access the underlying very low energy Fermi liquid behavior. The scaled low frequency ECFL spectral function, related simply to the Fano line shape, has a peculiar energy dependence unlike that of a Lorentzian. The resulting energy dispersion obtained by maximization is a hybrid of a massive and a massless Dirac spectrum , where the vanishing of , a momentum type variable, locates the kink momentum. Therefore the quasiparticle velocity interpolates between over a width on the two sides of , implying a kink there that resembles a prominent low energy feature seen in angle resolved photoemission spectra (ARPES) of cuprate materials. We also propose novel ways of analyzing the ARPES data to isolate the predicted asymmetry between particle and hole excitations.


I Introduction

The intensely studied  -  model is often regarded as the effective low energy Hamiltonian for describing several observed phenomena in cuprate superconductors Anderson (). Here the limit is presupposed, and hence the Hilbert space is restricted to a maximum of single occupancy at each site, i.e. Gutzwiller projectedGutzwiller (). A few words on the choice of the  -  model are relevant here. The implied infinite limit eliminates high energy ( scale) electronic states, known as the upper Hubbard band states. The residual low energy ( meV scale) excitations are probed by sensitive spectroscopies and transport phenomena, making the  -  model suitable for our task. At reasonably high , say comparable to the band width in a Hubbard model, this elimination of the upper Hubbard band must already occur in part. Therefore the limit must be regarded as a useful mathematical idealization of the very strong, or extreme correlation phenomenon. The resulting Gutzwiller projected electron operators, denoted by Hubbard’s convenient notation of operators Hubbard (), are rendered non canonical. The non-canonical nature of the electrons precludes the Wick’s theorem underlying the Feynman diagram approach, whereby leading to the fundamental difficulty of the  -  model, namely the impossibility of a straightforward Feynman type perturbative expansion. This situation leads to enormous calculational difficulties, so that systematic and controlled analytical calculations with this model have been very difficult.

In a series of recent papers ECFL (); Monster (); Hansen-Shastry (); Asymmetry (); Anatomy (); DMFT-ECFL (); Large-D (), we have shown that it is possible to overcome some of these difficulties by using alternate methods based on Schwinger’s treatment of field theory with time dependent potentials. This idea yields exact equations of motion for the electron Greens function. These equations have the nature of functional differential equations, and provide a powerful launching pad for various approximations. The specific approximation pursued is a systematic expansions in a parameter related to double occupancy. Using this we have presented an analytical theory of the normal state of the  -  model termed the extremely correlated Fermi liquid (ECFL) theory. An interesting feature of the theory is that we find a non-Dysonian representation of the projected electron Greens function. This is a significant structural departure from the usual field theories, and arises in a most natural fashion. The Greens function is determined by a pair of self energies, denoted by and , instead of the standard Dyson self energy (see Eq. (21) below). The latter can be reconstructed from the pair by a simple inversion. Starting with rather simple pairs of self energies, it is found that non trivial complexity is introduced into the Dyson self energy through this inversion process. Explicit self consistent calculations in parameter have been carried out to so far, and yield reliable results for electron densities . The detailed dynamical results of the ECFL theory have been benchmarked against independent theories in overlapping domains; e.g. against high temperature series results in Ref. (High-T, ). The ECFL theory has been shown to have a momentum independent Dyson self energy in the limit of infinite dimensions Ref. (Large-D, ). This enables benchmarking against the dynamical mean field theory (DMFT) in Ref. (DMFT-ECFL, ). Importantly, the results from the ECFL theory for the spectral function compare well with a large Hubbard model solved by the DMFT method, and not just infinite . The ECFL theory has also been benchmarked in Ref. (AIM, ) against the exact solution of the asymmetric Anderson impurity model, obtained from the numerical renormalization group study of Krishnamurthy, Wilson and Wilkins Ref. (KWW, ). In addition, a detailed comparison between the data on cuprate superconductors at optimal filling and the theoretical photoemission spectral lines of the ECFL theory has been carried out in Ref. (Gweon-Shastry, ) and Ref. (Kazue-Gweon, ), where excellent agreement is found. In all cases studied, the comparisons with ECFL are good, and seem to indicate the utility of this approach.

The ECFL formalism could initially seem somewhat unfamiliar, in view of its reliance on the analysis of the Schwinger equations of motion. This analysis was originally used to derive the main constituents of the theory, namely the auxiliary Greens function and the two self energies (detailed below). This type of analysis is somewhat removed from the toolkit of “standard” many body physics courses, and hence might obstruct a ready visualization of these objects. One goal of the present work is to show that these results are (A) minimal, i.e. having least redundancy, and (B) available more transparently. The latter follows from an important and novel hat removal theorem, leading to a compact mapping of the Hubbard operators to canonical Fermions. The mapping is given in Eq. (1) and described further in Section (III.2), leading to a path integral formulation (Section (VII)). It is possible that such a simplified presentation could lead to improved strategies for devising approximate methods, especially close to the insulating state.

This method rests on an exact replacement rule for the Hubbard operators in terms of the canonical Fermi operators


This replacement rule is shown to be exact when “right-operating” on states which satisfy the Gutzwiller constraint. This replacement is similar in spirit to the Dyson-Maleev representation Ref. (Dyson, ), Ref. (Maleev, ), where spin operators are expressed in terms of canonical Bosonic operators. With the advantage of this representation, most steps in the ECFL theory, such as the factorization of the Greens function into an auxiliary Greens function, the two self energies and the caparison function (see Eqs ( 18, 19, 21)) becomes very intuitive.

The analogy can be pushed further to establish a parallel between the parameter of the ECFL theory, and the small parameter of the Dyson Maleev Dyson (); Maleev () theory, namely the inverse spin . Finally we are able to make contact with the illuminating work of Harris, Kumar, Halperin and Hohenberg Ref. (HKHH, ). In a detailed work these authors computed the Greens function of the spins for two sublattice antiferromagnet using the Dyson-Maleev scheme and extracted the lifetime of the magnons of the theory. We find that their calculation contains the precise Bosonic counterparts of the auxiliary Greens function and the second self energy defining the “caparison function” of the ECFL theory (see Eqs ( 18, 19, 21)). Unlike the spin problem with variable number of excitations, the  -  model has a fixed number of particles. Hence there are significant new elements in the ECFL theory involving the imposition of the Luttinger Ward volume theorem, as discussed later.

A few comments on the canonical description of the equations of motion are appropriate. The general problem is to represent a time evolution of a state of the  -  model


where the primed states and operators are in the  -  model Hilbert space defined with the three allowed states at each site as usual (see Sec  (II.1) for details). The operators may be thought of as the exponential of the  -  Hamiltonian: written in terms of the projected operators. Since the algebra of the projected electrons is very inconvenient, one seeks a reframing of the problem into a canonical space. This involves mapping the states, the Hamiltonian and all other operators of the original theory, into the unconstrained Hilbert space of two Fermions at each site. This canonical space is of course described by the usual Fermi operators and their adjoints. This gives us an enlarged space with four states per site, with one redundant state corresponding to double occupancy, eliminated using Gutzwiller projection. There are various possibilities for doing this elimination leading to the different theories in literature. This includes the popular slave Boson or slave Fermion technique slave (); Read (); lee (), where additional degrees of freedom, over and above the already enlarged 4 dimensional local state space, are introduced and finally eliminated as best as possible. This handling of the redundancy leads to gauge theories for the  -  model that are reviewed in Ref. (lee, ).

In the enlarged state space let us block diagonalize the state space into physical and unphysical states and write the projection operator as


where is the identity operator in the physical space. The relevant operators in the theory e.g. the Hamiltonian, the creation operators or the destruction operators, are now written in terms of the canonical Fermions:


The next goal of the construction is to ensure that a state resulting from the application of a sequence of operators on a projected state remains in the projected space, i.e.


and . If this condition is not ensured, the projector has to be introduced at all intermediate time slices, thus making the calculations intractable. A sufficiency condition for this is the commutation for all . The slave Boson- Fermion technique uses the conservation of the Gutzwiller constraint by writing a suitable version of the Hamiltonian. This enables the use of a time independent Lagrange multiplier, as demonstrated in the work of Read and Newns Ref. (Read, ). In Sec (III.1), we display a compact Hermitean representation that also achieves this, without however adding further states (beyond the four states) into the problem.

While the commutation condition is sufficient, it is not necessary, and a much less restrictive condition can be found. We note that if the operators have a vanishing then the product in Eq. (5) remains in the physical sector with


The property of a commuting projection operator , requires that as well as , whereas the vanishing property of the unphysical components noted in Eq. (6) requires only . Then as well as are quite arbitrary. With this property, all the operators in Eq. (4) are block triangular


In more formal terms the sufficiency condition with least constraints that leads to Eq. (6) (via the block triangularity Eq. (7)) is


This condition is also expressible as ; a conditional vanishing of the commutator, when right operating on projected states. This observation provides some intuition for why Eq. (8) is sufficient in the present context. In view of the block triangular operators in Eq. (7), the adjoint property, namely of representing conjugate operators by their matrix Hermitean conjugates, is lost in this representation. This is seen clearly in Eq. (1), where the first two operators are mutual adjoints in the defining representation, but not so in the canonical basis. In general this situation is expected to lead to non Hermitean Hamiltonians. The non Hermitean representation in Eq. (64) and Sec (III.2) implements this idea and therefore leads to the most efficient canonical theory. We show that it exactly matches the minimal theory, found from the minimal description of the  -  model in terms of the Hubbard operators and the Schwinger equations of motion. It is notable that the Gutzwiller projection operator does not appear explicitly in the equations of motion, although it does play a crucial role in the canonical theory, and is at the root of its difficulty.

The plan of the paper is as follows. In Sec (II.1, II.2, II.3) we review the Schwinger equations of motion for the  -  model, and the ingredients of the recent method developed for a systematic expansion in a parameter . In Sec (II.4) we summarize the general form of the Greens function at low frequencies near the Fermi surface, and obtain the prototypical spectral function of the theory. We summarize in Sec (II.5) a kink in the electronic dispersion that arises from the theory, and seems to be closely related to that seen in many photoemission experiments. We also present simple but important ideas for analyzing photoemission data, with a view to isolating important feature of asymmetry predicted by the ECFL theory.

In Sec (III) we formulate the “best possible” representation of the Hubbard operators in terms of canonical Fermions, as discussed above. Sec (III.1) summarizes the well known representation and Sec (III.2) implements the block triangular idea to obtain a non Hermitean method with least redundancy. Sec (III.3, III.4) give further details of the Hamiltonian in this representation and the proof of the antiperiodic temporal boundary conditions necessary for defining the new framework.

In Sec (IV), the above non Hermitean representation is used to analyze the nature of the Greens function of projected electrons. Quite remarkably this process also yields the Greens function as a convolution of an auxiliary Greens function and a caparison function, in complete parallel to that obtained from the Schwinger method employed in Sec ( II.2, II.3). In Sec (V) we generalize the above representation to define Fermions where the Gutzwiller projection is only partial, and becomes full at . The equations of motion from these Fermions are shown to be those obtained in the expansion of Sec (II.3).

In Sec (VI) we display a close analogy between the non Hermitean representation of the Gutzwiller projected electrons and the well known Dyson-Maleev representation of spin operators in terms of canonical Bosons. This connection also provides further meaning of the small parameter in the Fermion theory, as a parallel of the expansion parameter of the Dyson Maleev theory. A connection with the work of Harris, Kumar, Halperin and Hohenberg (HKHH) HKHH () is noted, who invented a method for computing the lifetime of spin waves in antiferromagnets, with considerable overlap with our representation of the Greens function with two self energies.

In Sec (VII), we cast the canonical theory in terms of Fermionic path integrals, and show how the exact Schwinger equations of motion can be obtained directly from this representation, thereby validating all the links in the argument. The subtle role of the Gutzwiller projection operator is explored, it does not appear explicitly in the equations of motion and yet plays an important role in the theory. In Sec (VIII) we summarize the main points of the paper.

In Appendix ( A) we summarize the derivation of the minimal equations of motion from the Schwinger viewpoint. In Appendices (B, C, D) we provide the details of the coherent state path integrals and the implementation of the Gutzwiller projection. In Appendix (E) we provide a more detailed interpretation of the caparison function in terms of a change of variable of the source fields.

Ii Summary of the ECFL theory for the  -  model

Step(I) Step(II) Step(III) Step(IV) Step(V) Step(VI) Step(VII)
Green’s function in terms of Hubbard operators Exact Schwinger equations of motion for . Product expression into canonical part and adaptive spectral weight (caparison) part . Exact equations for and . Introduction of interpolating parameter connecting the Fermi gas to the extreme correlation limit. Shift invariance requires second chemical potential . Same sum rule for both GreenÕs functions so that Fermi surface volume is conserved. Successive orders in expansion satisfying shift invariance for practical calculations.
Table 1: A flowchart of the ECFL theory as developed in Ref. (ECFL, ) and Ref. (Monster, ). See Sections (II.2, II.3) for a detailed description.

ii.1 The  -  model preliminaries

The well studied  -  model is a two component Fermi system on a lattice, defined on the restricted subspace of three local states, obtained by excluding all doubly occupied configurations. The allowed states are with , and the double occupancy state is removed by the (Gutzwiller) projection operator. These Gutzwiller projected electron operators are denoted, in the convenient notation due to Hubbard, as . Its Hamiltonian is expressed in terms of the operators so that the single occupancy constraint is explicit. Summing over repeated spin indices we write


In computing the Green’s functions we add two kinds of Schwinger sources to the Hamiltonian; the anticommuting Grassman pair coupling to electron creation and destruction operators, and the commuting potential , coupling to the charge as well as spin density. These sources serve to generate compact Schwinger equations of motion (EOM), and are set to zero at the end. Explicitly we write


and all time dependences are as in . The generating functional of Green’s functions of the  -  model is


it reduces to the standard partition function on turning off the indicated source terms. The Green’s functions for positive times , are defined through the Martin-Schwinger prescription MS (); angleaverage ():


The functional conveniently yields the Green’s functions upon taking functional derivatives with respect to the sources, e.g.


where the sources are turned off at then end. We note that , the number of particles per site, is determined from the number sum rule:


and the chemical potential is fixed by this constraint.

ii.2 The Schwinger equations of motion

The detailed theory of the  -  model developed so far ECFL (); Monster () uses the Schwinger equations of motion. Since these equations play a fundamental role in the theory, we summarize next the equations of motion and their extension, obtained by introducing a parameter . We relegate to Appendix  (A) the derivation of the “minimal theory” equations. In the minimal theory, the most compact set of Schwinger equations are established, and some redundant terms from Ref. (ECFL, ) are omitted. This minimal version of the theory is important for the purposes of the present paper, since our goal in this paper is to recover these from a canonical formalism.

As the Schwinger school hasMS (); Kadanoff-Baym (); KM () emphasized, a field theory is rigorously determined by its equations of motion plus the boundary conditions. We can also establish alternate descriptions such as path integrals formulations, from the requirement that they reproduce these equations of motion- we present an example of this approach in Section (VII.2). In terms of the original description of the  -  model involving the Hubbard operators, the Schwinger equation of motion is a partial differential equation in time and also a functional differential equation involving the derivatives with respect to a Bosonic source:

where is the noninteracting Green’s function Eq. (LABEL:gnon), is a functional derivative operator Eq. (132), is the local Green’s function obtained from as (see Eq. (137)) and is the band hopping times Eq. (133); further details can be found in the Appendix (A). This equation has been written down in Ref. (ECFL, ) and Ref. (Monster, ): Antiperiodic boundary conditions with respect to both times (as in Eqs. (75) and (76)), and the number sum-rule Eq. (14) together with the equation of motion Eq. (LABEL:Minimal-eq), specify the theory completely.

ii.3 The expansion, the shift identities and second chemical potential

The idea of introducing a parameter into the EOM Eq. (LABEL:Minimal-eq) becomes quite natural when we observe the Schwinger EOM for the Hubbard model closely. These can be written schematically as . By scaling the interaction , with a parameter (), the interacting theory is connected continuously to the Fermi gas by tuning from to . The standard perturbative expansion can be organized by counting the various powers of , setting at the end before evaluating the expressionscomment-perturb (). Below in Section  (V) we provide a more microscopic argument for introducing the parameter in the Hubbard operators directly, this method leads back to the equations found here.

In the corresponding equation for the  -  model  (LABEL:Minimal-eq), we observe that the Green’s function differs from that for the free Fermi gas through two terms on the left hand side, exactly as in the Hubbard model, but also through one term on the right hand side. Scaling these three terms by , we rewrite  (LABEL:Minimal-eq) schematically as:


The strategy of the perturbative expansion method is to build up the solution of this equation at through a suitable expansion in , starting from the free Fermi limit . Thus corresponds to the admixture of a finite fraction of double occupancy that vanishes at . Insights from sum rules, the skeleton graph expansion and the physics of the Hubbard sub bands has played a major role in formulating a systematic expansion described in detail in Ref. (ECFL, ) and Ref. (Monster, ).

Within this approach it is also necessary to add a term to the Hamiltonian, and a corresponding term to the EOM, so that the and in Eq. (16) are suitably redefined. Here is an extra Hubbard interaction type parameter that is determined by a sum rule as explained below. At such a term makes no difference since the double occupancy is excluded. This parameter also enables us to enforce a simple but crucial symmetry of the  -  model- the shift invariance, noted in Ref. (Monster, ). This invariance arises from the twofold function of the hopping in the  -  model when expressed in terms of the canonical operators, of providing hopping as well as the four Fermion (interaction) terms. Therefore in an exact treatment, adding a constant times the identity matrix to the hopping matrix: , shifts the center of gravity band innocuously. In approximate implementations it has the unphysical effect of also adding to the interaction (i.e. four Fermion type) terms. Such a change must therefore be compensated by an adjustable parameter that can soak up this additive constant. Indeed provides precisely this type of a parameter. It also plays the role of a second chemical potential (Ref. (Monster, )) to fix the number of Fermions in the auxiliary Green’s function through , while the thermodynamical chemical potential (residing in the non interacting ), is fixed by the number sum rule Eq. (14)). Enforcing this shift invariance to each order in the expansion plays an important “watchdog” role on the expansion, in addition to other standard constraints such as the Ward identities.

To summarize some key points of the expansion, we first decompose the Greens function into the space time convolution of an auxiliary Greens function and a caparison function as:


With this the operator in equation  (16) acts on the two factors of Eq. (17), and breaks into two equations upon using the ansatz that has a canonical structure . The expansion Ref. (Monster, ) is then an iteration scheme that proceeds by an expansion of the caparison function and () in powers of . Dyson’s skeleton graph idea is implemented by keeping the auxiliary intact ( i.e. unexpanded in ), while all other variables are expanded in powers of and , thereby obtaining self consistent equations for and the vertex functions. Successive levels of approximation are obtained by retaining increasing powers of . At each approximation level, we set before actually evaluating the expressions, and implement the antiperiodic boundary conditions  (75),  (76), and the number sum-rule (Eq. (14)).

Elaborating on the representation Eq. (17) of , we note that the term on the right hand side of  (16) is due to the non canonical anticommutator of the projected Fermi operators. As noted in Ref. (ECFL, ), this term contains the essential difficulty of the  -  problem, having no parallel in the (canonical) Hubbard type models. After turning off the sources, in the momentum-frequency space we can further introducing two self energies , and with


where the constant in Eq. (18) is fixed by the condition that vanishes at infinite frequency. The auxiliary Greens function satisfies a second sumrule analogous to Eq. (14), written in the Fourier domain:


Clearly the same sumrule holds for . Eq. (17) can now be written explicitly in the non-Dysonian form proposed in Ref. (ECFL, ) and Ref. (Anatomy, )


As argued in ECFL (); Monster (); Asymmetry (); DMFT-ECFL (), simple Fermi liquid type self energies and can, in the combination above, lead to highly asymmetric (in frequency) Dyson self energies from the structure of Eq. (21), thus providing a considerable tactical advantage in describing extreme correlations. We further discuss the physical meaning of this decomposition and the twin self energies in Section (IV). Table  (1) provides an overview of the various steps in the construction of the theory.

ii.4 and the low energy spectral function in ECFL theory

We summarize here the low temperature low energy theory near the Fermi surface that follows from the general structure of Eq. (21) in terms of a small number of parameters, upon assuming that the two self energies have a Fermi liquid behavior at low energies. In the limit of large dimensions, a similar exercise gives a very interesting spectral function that matches the exact solution of the Hubbard model found from the dynamical mean field theory (DMFT) Ref. (DMFT-ECFL, ). The presentation below generalizes that to include a momentum dependence that is absent in high dimensions, and is supplemented by a discussion of the behavior of the various coefficients as the density of electrons approaches unity, or equivalently the hole density .

The Dyson self energy can be inferred from a simple inversion, and has a strong set of corrections to the Fermi liquid theory that we delineate here. We assume here a Fermi liquid type state that survives the limit of small hold density . In reality at very small several other broken symmetry states would compete and presumably win over the liquid state, so that this Fermi liquid state would be metastable. It characteristics are of interest and hence we proceed to describe these.

We study Eq. (21) by analytically continuing and write


Let us define as the normal deviation from the Fermi surface i.e. , and the frequently occurring Fermi liquid function


We carry out a low frequency expansion as follows:


where is the constant term at the Fermi surface, and a similar expansion for so that


where is the bare Fermi velocity. The expansion coefficients above are in principle functions of the location of on the Fermi surface, and have suitable dimensions to ensure that is dimensionless and is an energy. The dimensionless velocity renormalization constants and capture the momentum dependence normal to the Fermi surface, arising from the two respective self energies. The Greens function near the Fermi surface can now be written as


where is the net quasiparticle renormalization constant. The spectral function can be computed from in the ECFL form of a Fermi liquid function times a caparison function as follows:


where the (Fermi liquid) width function (or decay rate)


with an extra phenomenological parameter required to describe elastic scattering Gweon-Shastry () in impure systems. The caparison function is


where we introduced an important (emergent) low energy scale combining the other parameters:


and the dimensionless momentum dependence coefficient


A cutoff is implicit in Eq. (29), so that the function is assumed to be zero at large positive frequencies as discussed in Ref. (ECFL, ). The five final parameters defining the spectral function Eq. (27) are . For fitting experimental data, it may be best to think of them as adjustable parameters that determine the line shapes, their asymmetries and also features in the spectral dispersions. In addition the parameter is needed to describe impurities that are not contained in the microscopic theory. In the early fit Gweon-Shastry () the total number of free parameters is even smaller-just two instead of five. The corrections to the Landau Fermi liquid theory are encapsulated in the caparison factor, which contains a correction term that is odd in frequency and seems to be ultimately responsible for the asymmetric appearance of the line shapes Gweon-Shastry (); Asymmetry ().

For reference we note that in the limit of high dimensions DMFT-ECFL (), the coefficient of the momentum dependent term vanishes in Eq. (27), while the earlier fits to experiments in Gweon-Shastry (), it is non zero, and in modified fits Kazue-Gweon () its magnitude is varied to get a good description of the constant energy cuts of the data.

It is useful to consider the approach to the Mott insulating limit, where the parameters behave in a specific fashion to satisfy the expected behavior. We consider the limit of density , and a frequency scale , where the above expression Eq. (27) may be expected to work. For reference, it is useful to note that in this limiting case, the widely used Gutzwiller-Brinkman-Rice theory Gutzwiller (); BR () gives the quasiparticle propagator as:


where vanishes linearly with as . This leads to a delta function spectral weight . In contrast Eq. (27) provides the spectral function at non zero and .

As in Eq. (24) we expect that the constant , in order to reach the Mott insulating limit continuously. This implies that in this regime, and this drives the various other coefficients as well. We summarize the expected behavior of the above five coefficients


by using an overline for denoting a non vanishing limit of the stated variable DMFT-ECFL (); 2DMFT (). The scaling of the velocity constants is guided by the results in high dimensions, and ensure that the dispersing quasiparticles have a vanishing bandwidth as we approach the insulator- as emphasized by Brinkman and Rice BR (). From this we find that the ECFL spectral function Eq. (27) satisfies a simple homogeneity (i.e. scaling) relation valid in the low energy regime for a scale parameter :


where the dependence on the temperature and hole density are made explicit. The momentum variable does not scale with due to the assumed behavior of the ’s. The scaling holds for , and generalizes to a non zero values if we scale . This scaling relation describes a Fermi liquid including significant corrections to Fermi liquid theory through the caparison function. It rests upon the specific behavior for the coefficients as the density varies near the insulating state, unlike other generalized scaling relations that have been proposed in literature Ref. (senthil, ) for non Fermi liquid states. If set with say , then the ratio and we infer


relating the low hole density system to an overdoped (i.e. high hole density) system at a high effective temperature. This relation provides basic intuition for why the  -  model, near the insulating limit behaves almost like a classical liquid, unless one fine tunes parameters very close to the limit.

ii.5 Electronic origin of the low energy kink and further tests of dynamical asymmetry

In this section we summarize the origin of the important low energy kink feature of the dispersion relation obtained in the ECFL theory. Since a similar feature is seen in the experiments on angle resolved photoemission studies (ARPES) of various groups Johnson (); Lanzara (); Campuzzano (); Gweon-Shastry (), it is worth clarifying the purely electronic origin of this feature within the ECFL theory. A higher (binding) energy kink is also seen and is well understood in terms of the behavior of the self energy over a greater range Anatomy (); DMFT-ECFL (), and is not pursued here. Rather we focus on the low energy kink seen around eV in several compounds Johnson (); Lanzara (); Campuzzano (); Gweon-Shastry (), and finds a natural interpretation within ECFL.

We also present a few experimentally testable features relating to dynamical asymmetry, i.e. the asymmetric in correction to the Fermi liquid theory contained in ECFL, arising from the caparison function in Eq. (27).

Let us assume that at low enough frequency relative to so that we may treat as a constant. We may then bring Eq. (27) to an interesting form studied in Ref. (Anatomy, ) by defining variables


so that the spectral function reduces to the standard form occurring in the ECFL theory:


with . This expression is valid for small enough Anatomy (); ECFL (), and can be viewed as the (weighted) sum of the real and imaginary parts of a simple damped oscillator with a scaled susceptibility . It is interesting to note that the scaled spectral function Eq. (37) can be related to the (scaled) Fano line shape


This spectrum is often considered with the Fano parameter , it is highlighted by a vanishing at negative energies , representing the destructive interference of a scattering amplitude with a background term arising from a continuum of states. However we can flip the sign of and by choosing , we can relate these through

For the purpose of representing ARPES spectral functions, the scaled spectral function Eq. (37) gains an advantage over the Fano line shape Eq. (38) by the absence of a background at large . In relating them via Eq. (LABEL:connect-Fano), the background term in the Fano process is killed, while its interference with the peak is retained.

Unlike the simple Lorentzian obtained at , the energy variable enters the numerator as well as the denominator in both Eq. (37) and the Fano shape. This feature gives rise to the characteristic skew to the ECFL spectrum. The spectral function can be maximized with respect to the frequency at a fixed , yielding the energy distribution curve (EDC) dispersion , or with respect to at a fixed frequency , giving the momentum distribution curve (MDC) dispersion . Let us introduce the convenient variables


giving the ratio of the two velocity factors. The ratio in the limit of high dimensions DMFT-ECFL (). In the simplified ECFL analysis in ECFL (); Gweon-Shastry (), we find due to the suppression of relative to by a quasiparticle renormalization factor . We see below that the magnitude and sign of play a significant role in determining the location of the kink, and its observability in ARPES respectively. We also introduce a (linear in ) energy variable:


In terms of these, the two dispersions are obtained as


Simplifying the notation, both energy dispersions are of the form , i.e. the hybrid of a massless and a massive Dirac spectrum. As varies from to , the energy crosses over from to , thus exhibiting a knee or a kink near , with its sharpness determined by the “mass term”. The mass term in the MDC spectrum depends on the ratio , and this generally leads to a smaller magnitude. Upon turning off the decay rate , both the EDC and MDC spectra reduce to the expected spectrum , arising from the pole of the auxiliary Greens function in Eq. (21). These expressions illustrate an unusual feature of this theory: the two dispersions are influenced by the emergent energy scale , as well as the width Eq. (28).

The above dispersions exhibit an interesting kink feature at in both spectra. The condition locates the kink momentum as


it corresponds to occupied momenta provided , we will confine to this case below. For the other case , a kink would arise in the unoccupied side, for this reason we do not pursue it here. For , the two dispersions asymptotically become and . Hence these spectra exhibit a change in velocity (i.e. slope) around of magnitude for the EDC and the usually larger for the MDC spectrum. The change in slope of the spectrum occurs over a range , thus becoming sharper as decreases.

The value of the EDC energy at the kink is found by substituting and gives


The MDC spectrum shows a kink for at the same momentum Eq. (44), with energy


this feature is sharper than in the EDC spectrum since the effective damping is smaller.

When , the MDC energy is real only for , where the (negative) momentum

For beyond the cut off, the root becomes complex implying the loss of a clear peak in the MDC spectrum. Thus the spectrum “fades” before reaching the kink momentum Eq. (44). Therefore in this case, the kink is less than ideal, unlike the EDC kink or the MDC kink for , which should be visible on both sides of the kink momentum. From Eq. (33) we may extract the hole density dependence of all the kink parameters, while , determining the kink width, is given in Eq. (28).

We observe in Fig. (1) that the kink becomes sharp when decreases. The MDC curves display a sharper kink than the EDC curves, this is easy to understand since the effective damping is smaller in this case, and also the net change in velocity across the kink is greater, as discussed above. From Eq. (28) we see various parameters that control , in case of laser ARPES, it is argued Gweon-Shastry () that is small so we expect to see sharper kinks in this setup. Further, as drops below , the d-wave superconductor has gapless excitations along the nodal direction , and the quasiparticles seen in this case are sharper. Theoretical considerations Shastry-SC () show that in the superconducting state, a reduction in the available gapless states responsible for the linewidth implies a reduction of and hence to a sharper kink.

Figure 1: A kink feature in the MDC dispersion relation from Eq. (42) and in the inset from the EDC dispersion Eq. (43) with parameters eV, and three values of in eV from top to bottom. The kink is more pronounced in the MDC curve as discussed in text.

We next discuss the feature of dynamical asymmetry in the spectra. It is also important to note that the ECFL spectral function Eq. (27) has an unusual correction to the standard Fermi liquid part, embodied in the caparison function . This function is odd in frequency, thus disturbing the particle hole symmetry of the Fermi liquid part, and it grows in importance as we approach the insulating state since as indicated in Eq. (33). It is also interesting that the spectral line shape in the calculation of Anderson and Casey Ref. (Anderson-Casey, ) (AC) as well as Doniach and Sunjic Ref. (Doniach-Sunjic, ) (DS) also have such odd in corrections to the Fermi liquid part. In fact the AC result may be viewed as the vanishing of the scale so that the ground state is non Fermi liquid like. At finite and the AC and DS theories are parallel with the ECFL line shapes regarding the asymmetry as remarked in Ref. (Asymmetry, ), and we wish to make a few comments about the experimental tests for such an asymmetry, going beyond standard measures such as the skewness factor.

DSDoniach-Sunjic () make the interesting point that the asymmetry is best isolated by looking at the inverse of the spectral function in a plot of


where is the peak location in the EDC. With this plot, a Fermi liquid yields two coincident straight lines above and below , whereas an asymmetric contribution, as in Eq. (27) or the DS lineshape Doniach-Sunjic (), would split into two distinct non linear curves, from below and above . The inversion of the spectral function is an interesting device, since it refocuses attention on the asymmetric parts. For very similar reasons Ref. (ECFL, ) (Fig. 1 inset) also advocates plotting the inverse of the spectral function. On the other hand an untrained examination of the EDC curves invariably focuses on the close proximity of the peaks of , these are arguably the least interesting part of the asymmetry story!

In fact armed with the explicit knowledge of the spectral function of the ECFL theory in Eq. (27), we can aim to do better in establishing the asymmetry and in determining the various parameters. We first redefine the frequency by subtracting off the EDC peak value


so that the spectral peak occurs at . The inverse spectral function can be computed as a function of and reads:

where the peak value of the spectral function at is :


We next construct the object from Eq. (LABEL:ratio) by subtracting unity and cross multiplying:


This variable is designed to be a independent constant in a simple Fermi liquid with a Lorentzian line shape (i.e. Eq. (27) without the caparison function ). Here has dimensions of the square of energy, and when plotted against in the small range surrounding zero i.e. it exhibits a linearly decreasing behavior with within the ECFL spectral function Eq. (27)


Note that this function is flat for the usual Fermi liquid state without asymmetric corrections, since in this case . If found in data, this linear in behavior is the distinctive aspect of the asymmetric lineshapes. We can then read off various physical quantities once the curve of versus is obtained. For this purpose we need the intercept and the slope near the origin . Clearly the function will deviate from a straight line sufficiently far from , and it will also be contaminated with background terms as well as noise. However, with high quality data this procedure could be useful in inverting the data to fit simple functional forms, and to make decisive tests of the predictions of the theories containing asymmetry, namely the DS and AC theories as well as ECFL.

Iii Exact formulation in terms of a canonical Fermions

We will next rewrite this in canonical Fermi representation in an enlarged Hilbert space where double occupancy is permitted, and the singly occupied states form a subspace. We regard the physical subspace of states as those that satisfy the condition of single occupancy, i.e. with the double occupancy operator is given by:


and and denote the canonical Fermionic destruction and creation operators. The unphysical states contain one or more doubly occupied states. In terms of these, the Gutzwiller projector over all sites is written as:


This projection operator can be introduced into a partition function to deal with unphysical states, as we show below.

The next goal (see Table 1) is to write the most efficient representation in the enlarged space of the  -  model Green’s functions, in terms of the canonical operators and the projection operator. As pointed out in the Introduction, we note that pairs of operators that are mutual adjoints in the  -  model (e.g. ), are allowed to be represented by operators that violate this adjoint property. The main result of this section is that this possibility leads to the most compact canonical theory; we term it the non-Hermitean theory. However we first warmup with a short summary of the more obvious Hermitean theory, which sets the stage for the main result.

iii.1 A Hermitean canonical representation with redundancy

Projected Fermi operators distinguished by the hats can be written in a familiar construction chats ()


where , and , with the property that these conserve the number of doubly occupied sites locally:


and therefore also globally i.e. with in place of . It implies that any Hamiltonian written in terms of these operators with hats commutes with the individual as well as the global , and thus conserves the local symmetry of the model. Therefore acting within the physical subspace of states,  (55) provide a faithful realization of the operators as and , and clearly satisfies the mutual adjoint property. We are also interested in the product of two ’s in order to represent the kinetic energy term of the effective Hamiltonian below. The optimal choice is seen to be


While the choice


is also a faithful representation, it contains an extra term , over and above  (57), which is redundant since  (57) already commutes with  (54).

Using  (57) we write a canonical expression for the Hamiltonian




we call this as the symmetrized kinetic energy in view of its obvious symmetry under the exchange , and write with the spin and number operators written in terms of ’s and