Extremely Correlated Quantum Liquids

TL;DR

This paper develops a theoretical framework for extremely correlated electron liquids, extending Fermi liquid theory, and introduces novel equations and transformations to analyze their unique physical properties.

Contribution

It generalizes Fermi liquid theory to strongly correlated systems using Schwinger source fields and matrix scale transformations, providing a new formalism for such quantum liquids.

Findings

Derived an exact Dyson equation for quasiparticle Green's functions.

Established Ward identities for systematic approximations.

Presented preliminary calculations demonstrating the formalism's application.

Abstract

We formulate the theory of an extremely correlated electron liquid, generalizing the standard Fermi liquid. This quantum liquid has specific signatures in various physical properties, such as the Fermi surface volume and the narrowing of electronic bands by spin and density correlation functions. We use Schwinger's source field idea to generate equations for the Greens function for the Hubbard operators. A local (matrix) scale transformation in the time domain to a quasiparticle Greens function, is found to be optimal. This transformation allows us to generate vertex functions that are guaranteed to reduce to the bare values for high frequencies, i.e. are ``asymptotically free''. The quasiparticles are fractionally charged objects, and we find an exact Schwinger Dyson equation for their Greens function. We find a hierarchy of equations for the vertex functions, and further we obtain Ward identities so that systematic approximations are feasible. An expansion in terms of the density of holes measured from the Mott Hubbard insulating state follows from the nature of the theory. A systematic presentation of the formalism is followed by some preliminary explicit calculations.

Equations

$$H=-\sum _{i,j}{t}_{i,j}{\hat{c}}_{{\vec{r}}_j,\sigma }^{†}{\hat{c}}_{{\vec{r}}_i,\sigma }+\frac{1}{2}\sum _{i,j}{J}_{i,j}\{{\vec{S}}_{{\vec{r}}_i}\cdot {\vec{S}}_{{\vec{r}}_j}-\frac{1}{4}{n}_{{\vec{r}}_i}{n}_{{\vec{r}}_j}+n_{{\vec{r}}_i}\}-\mu \sum _i{n}_{{\vec{r}}_i},$$

$$X_i^{{\sigma }_1{\sigma }_2}=|{\sigma }_1⟩⟨{\sigma }_2|,X_i^{0{\sigma }_1}=|0⟩⟨{\sigma }_1|,X_i^{{\sigma }_{1}0}=|{\sigma }_1⟩⟨0|,$$

$$X_i^{00}=\text{mathds}1-\sum _{\sigma }{X}_i^{\sigma \sigma }.$$

$$\{X_i^{0{\sigma }_1},X_j^{{\sigma }_{2}0}\}$$

$$[X_i^{0{\sigma }_3},X_j^{{\sigma }_1{\sigma }_2}]={\delta }_{ij}{\delta }_{{\sigma }_3{\sigma }_1}{X}_i^{0{\sigma }_2}$$

$$X_i^{ab}{X}_i^{cd}={\delta }_{bc}{X}_i^{ad}$$

$$X_i^{{\sigma }_a,0}={\hat{c}}_{{\vec{r}}_i,{\sigma }_a}^{†}$$

$$X_i^{0,{\sigma }_a}={\hat{c}}_{{\vec{r}}_i,{\sigma }_a}$$

$$X_i^{{\sigma }_a,{\sigma }_b}$$

$$H=-\sum _{i,j,\sigma }{t}_{ij}{X}_i^{\sigma 0}{X}_j^{0\sigma }-\mu \sum _{i,\sigma }{X}_i^{\sigma \sigma }+\frac{1}{2}\sum _{i,j}{J}_{ij}\{{\vec{S}}_i.{\vec{S}}_j-\frac{1}{4}{n}_i{n}_j\}+\frac{1}{2}\sum _{i,j,\sigma }{J}_{ij}{X}_i^{\sigma \sigma }.$$

$$V=\frac{1}{4}\sum _{ij}{J}_{ij}\{P_{ij}-X_i^{{\sigma }_1{\sigma }_1}{X}_j^{{\sigma }_2{\sigma }_2}\}=\frac{1}{4}\sum _{ij}{J}_{ij}\{X_i^{{\sigma }_1{\sigma }_2}{X}_j^{{\sigma }_2{\sigma }_1}-X_i^{{\sigma }_1{\sigma }_1}{X}_j^{{\sigma }_2{\sigma }_2}\},$$

$${\mathcal{G}}_{{\sigma }_1{\sigma }_2}[1,2]=-⟨T(X_1^{0{\sigma }_1}({\tau }_1)X_2^{{\sigma }_{2}0}({\tau }_2))⟩,$$

$$⟨Q⟩=\frac{Tr{e}^{-\beta H}Q}{Tr{e}^{-\beta H}}.$$

$$Q(\tau )\equiv e^{\tau H}Q{e}^{-\tau H}.$$

$$\mathcal{A}=\sum _{j,{\sigma }_1,{\sigma }_2}{\int }_0^{\beta }d\tau {\mathcal{V}}_j^{{\sigma }_1{\sigma }_2}(\tau )X_j^{{\sigma }_1{\sigma }_2}(\tau ),$$

$${\mathcal{G}}_{{\sigma }_1{\sigma }_2}[1,2]=-\frac{Tr\left[e^{-\beta H}T(e^{-\mathcal{A}}{X}_1^{0{\sigma }_1}({\tau }_1)X_2^{{\sigma }_{2}0}({\tau }_2))\right]}{Tr\left[e^{-\beta H}T(e^{-\mathcal{A}})\right]}.$$

$${\mathcal{G}}_{{\sigma }_1{\sigma }_2}[r_1{0}^{-},r_{2}0]=-{\mathcal{G}}_{{\sigma }_1{\sigma }_2}[r_1\beta ,r_{2}0],$$

$$⟨⟨Q({\tau }_1,{\tau }_2,..)⟩⟩=\frac{Tr\left[e^{-\beta H}T(e^{-\mathcal{A}}Q({\tau }_1,{\tau }_2,..)\right]}{Tr\left[e^{-\beta H}T(e^{-\mathcal{A}})\right]},$$

$${\mathcal{G}}_{{\sigma }_i{\sigma }_f}[i,f]=-⟨⟨X_i^{0{\sigma }_i}{X}_f^{{\sigma }_{f}0}⟩⟩.$$

$$\frac{\delta }{\delta {\mathcal{V}}_j^{{\sigma }_1{\sigma }_2}({\tau }_1)}⟨⟨Q({\tau }_2)⟩⟩=⟨⟨Q({\tau }_2)⟩⟩⟨⟨X_j^{{\sigma }_1{\sigma }_2}({\tau }_1)⟩⟩-⟨⟨X_j^{{\sigma }_1{\sigma }_2}({\tau }_1)Q({\tau }_2)⟩⟩$$

$$[X_i^{0{\sigma }_i},H]=-\sum _j{t}_{ij}\left[{\delta }_{{\sigma }_i{\sigma }_j}-{\sigma }_i{\sigma }_j{X}_i^{{\bar{\sigma }}_i{\bar{\sigma }}_j}\right]X_j^{0{\sigma }_j}-\mu X_i^{0{\sigma }_i}+\frac{1}{2}\sum _j{J}_{ij}\left[{\delta }_{{\sigma }_i{\sigma }_j}-{\sigma }_i{\sigma }_j{X}_j^{{\bar{\sigma }}_i{\bar{\sigma }}_j}\right]X_i^{0{\sigma }_j}.$$

$${\partial }_{{\tau }_i}T\left(e^{-\mathcal{A}}{X}_i^{a,b}({\tau }_i)\right)=-T\left(e^{-\mathcal{A}}[X_i^{a,b}({\tau }_i),H]\right)+{\mathcal{V}}_i^{{\sigma }_1{\sigma }_2}({\tau }_i)T\left(e^{-\mathcal{A}}[X_i^{{\sigma }_1{\sigma }_2}({\tau }_i),X_i^{a,b}({\tau }_i)]\right).$$

$${\partial }_{{\tau }_i}{\mathcal{G}}_{{\sigma }_i{\sigma }_f}[i,f]$$

$$\delta [i,j]$$

$$J[i,j]$$

$$(M^k{)}_{{\sigma }_1,{\sigma }_2}={\sigma }_1{\sigma }_2{M}_{{\bar{\sigma }}_2,{\bar{\sigma }}_1}.$$

$$M\cdot M^k$$

$$(M{)}^{-1}$$

$$\Delta [i]$$

$${\Delta }_{{\sigma }_1{\sigma }_2}[i]$$

$${\Delta }_{{\sigma }_1{\sigma }_2}^{-1}[i]$$

$$D_{{\sigma }_1,{\sigma }_2}[r]={\sigma }_1{\sigma }_2\frac{\delta }{\delta {\mathcal{V}}_r^{{\bar{\sigma }}_1{\bar{\sigma }}_2}}.$$

$$({\partial }_{{\tau }_i}-\mu ){\mathcal{G}}_{{\sigma }_i{\sigma }_f}[i,f]$$

Full text

1. I Introduction

2. II The bare and quasiparticle Greens functions of the  $t$-$J$  model

3. II.1 Basic framework using the Hubbard Operators

4. II.2 Calculation of the Greens function

5. II.3 Calculation of the inverse Greens function

6. III Schwinger Dyson Equation and the Limit of vanishing sources

7. IV Solution of the local Greens function

8. V Calculation of the Vertices

9. V.0.1 $\Delta[i]$ and its derivatives

10. V.0.2 $\mu[i,k]$ and its derivatives

11. V.0.3 $\nu[i,k]$ and its derivatives

12. VI Electrical Conductivity, Current conservation and Ward identities

13. VII General properties of the Greens Function and Successive Approximations

14. VII.1 Atomic Limit

15. VII.2 First approximation

16. VII.3 Hartree Approximation (HA)

17. VII.4 Hartree Fock Approximation (HFA)

18. VII.5 Non Linear Hartree (Fock) Approximation (NLH(F)A)

19. VII.6 Frequency dependent self energy

20. VIII Anisotropic d-wave Pairing and Superconducting instability of the ECQL

21. IX Physical Interpretation of the Quasiparticles

22. X Conclusions and Summary

23. XI Acknowledgments

24. A Identities involving the source derivatives.

25. A.1 The $\nu$ matrix.

26. A.2 The $\Theta$ matrix.

27. B The $\Phi_{i}$ matrix.

28. C Rotation Invariance and the Nozières relations.

29. D The Hubbard model atomic limit and FS volume

30. E The different limits of vanishing source terms

31. F Various Susceptibilities and their relationships and Sum-rules.

32. F.0.1 Susceptibility sum rules

33. G Detailed Vertices

34. H Fourier transform convention.

I Introduction

We present a theory of an extremely correlated quantum liquid (ECQL) in this report. We believe it to be both necessary and useful to make a distinction between the class of systems studied here, and the more commonly addressed strongly correlated electron systems (SCES). The latter deals with models such as the Hubbard model for transition metals or copper oxide systems, and Kondo or the periodic Anderson models as germane to heavy Fermi systems, and also refers occasionally to the  $t$-$J$  class of systems. We use the term ECQL exclusively for systems such as the  $t$-$J$  modelearly_tJ_1 , having much stronger- even extreme correlations built into them. The origin of the difference is the infinite Coulomb repulsion at each site, where double occupancy is prohibited, not just discouraged as in the Hubbard model. The result is that we must deal with a genuinely different problem, both physically and mathematically; here the fundamental operators are no longer canonical Fermions but rather Hubbard operators. As we describe below, a systematic theory of the ECQL can be built, in parallel to the Fermi liquid (FL) theory for the more conventional interacting electron systems. The ECQL is described below as a coupled spin and charge liquid that breaks no symmetry, while it accommodates the extreme correlations of the  $t$-$J$  class of models. Its Greens functions and vertex functions can be defined in a specific way that is different, and fundamentally more complex than in weak coupling (FL) models. In this formalism, the instabilities of the ECQL in charge, spin and superconducting channels can be studied systematically. The extremely correlated quantum liquid described here, is a strong coupling entity and in contrast to the Fermi liquid, it cannot be related in an adiabatic fashion to a free fermi gas qhafm . Although the $t$-$J$ model is obtainable from a non interacting Fermi gas by turning on the $J$ parameter, one is also obliged to turn on the Hubbard U term all the way to $U=\infty$. Adiabaticity is lost in the passage to the infinite $U$ limit; Appendix D illustrates this breakdown of adiabaticity within the context of the atomic limit of the Hubbard model, and outlines the general argument for the invalidity of the Luttinger Ward theorem in the extreme correlation limit. The ECQL is expected to describe the Physics for a sufficiently large $U$, and not just $U=\infty$.

The theory of strongly correlated electronic systems, as opposed to the extremely correlated matter studied here, has received considerable attention in recent times. A notable success has been the idea of large dimensionality where the dynamical mean field theorydmft-1 ; dmft-2 leads to a controlled set of calculations that are useful as well as predictive. The  $t$-$J$  model Eq(1) has also been studiedlee , but perhaps not so widely as the Hubbard model. The origin of the  $t$-$J$  model is the subject of several studies summarized in ReftJ_model . We emphasize that it is not merely a descendent of the the Hubbard model (upon using a large $U$ expansion), but rather has an independent origin via the down folding of multiband systems. Thus $t$ and $J$ may be viewed as independent parameters, rather than being fixed through $J=4t^{2}/U$ as in superexchange theory. The possibility of an expansion of the physical quantities in terms of Mott Hubbard holes, i.e. the departure from half filling, has been anticipated in 1986 early_tJ_1 , but a systematic procedure remained undiscovered. Early studies early_tJ_1 ; early_tJ_2 , have struggled with the technical difficulties of the non canonical nature of the Fermions. The proposal of Andersonpwa in 1987, that the High Tc systems are described by this model, has led to a revived and wider interest in this model. Alternate techniques such as the auxiliary field method have been employed to deal with the constraintsslave_operators ; lee , but give divergent views of the outcome of extreme correlations. The present study is motivated by this conundrum, as well as by the possibility of its applicability to certain systems found in nature. The predicted behaviour of the ECQL has certain specific signatures that distinguish it from the FL, so that it should be relatively straightforward to verify its applicability to a given system.

The plan of the paper is as follows. In Section II we present the calculation of the Greens function of the  $t$-$J$  model, along with the background definitions of the Hubbard operators and the Schwinger framework. In Section III, the exact Schwinger Dyson equation is given after the sources are turned off. In Section IV, we solve for the local Green’s function. In Section V we outline the computation of the vertices, with detailed results given in Appendix G. In Section VI the Ward identities are developed and current vertices defined. In Section VII, we initiate the programme of successive approximations to the Greens function and list several consistent schemes. In Section VIII we present a calculation of the d-wave superconducting instability of the ECQL, using one of the consistent schemes for the Greens functions. In Section IX we present a detailed picture of the quasiparticles which are argued to be fractionally charged. The argument for the changed Fermi surface volume is given further in Appendix D. We summarize the results in Section X.

Appendix A and its subsections A.1, A.2 and Appendix B provide the details of the functional derivatives and the functions that arise in that process. Appendix C gives the details of the Nozières relations that play an important role in giving us rotation invariance of the theory. Appendix E provides the details of different zero source limits that are available. Appendix F contains details of the connection between the quasiparticle susceptibilities and the physical particle susceptibilities as well as some important sum rules that set the scales for the susceptibilities. Appendix G gives all the vertices computed after throwing out higher order vertices, i.e. $\frac{\delta\Gamma}{\delta V}$. Finally Appendix H gives the conventions used for various Fourier transforms.

II The bare and quasiparticle Greens functions of the  $t$-$J$  model

We study the  $t$-$J$  model given by

[TABLE]

where $\hat{c}^{\dagger}_{\vec{r}_{i},\sigma}$ creates a electron at the site $i$ which is Gutzwiller projectedGutz to the subspace of single occupancy, and the other symbols have their usual meaning. The Gutzwiller projection creates the major technical challenge in this problem. Projected electrons no longer satisfy canonical anti-commutation relations and the power of Wick’s theoremwick is lost, therefore a Dyson equationdyson with straightforward expansion in terms of a free Greens function is not possible. Lost too are the appealing Feynman diagrams that encapsulate standard FL theory. The projected electrons satisfy instead, a set of graded Lie algebraic commutation relations. The latter are compactly expressed in the notation of Hubbard in terms of the $X$ operators that are summarized in Section II.1. Section II.1 contains the details of the definitions of the Greens functions, and of the equations of motion calculation, using the powerful technique invented by Schwinger, Martin and their schoolschwinger_school ; kadanoff_baym ; rajagopal . We find that despite the non canonical nature of the Hubbard operators, we are able to obtain equations that are at the same level of complexity as those of canonical Fermions, with some unavoidable embellishmentsremark_1 . We therefore conclude that in the case of projected electrons, the Schwinger-Dyson expansionrivers is yet possibleremark_2 . This work studies the resulting equations and their consequences.

II.1 Basic framework using the Hubbard Operators

Let us define the Hubbard operators acting upon each site as projected Fermi operators

[TABLE]

in terms of the three possible states at any site $|0\rangle,|\uparrow\rangle,|\downarrow\rangle$. The doubly occupied site is forbidden, and these operators do not connect forbidden states with the allowed ones. Thus an important statement of completeness at any site is the relation

[TABLE]

The first member $X_{{i}}^{\sigma_{1}\sigma_{2}}$ is Bosonic while the other two are Fermionic with respect to their commutation relations at different sites. The fundamental anticommutator for the Hubbard operators at *different sites * is most conveniently expressed as

[TABLE]

Also note the basic commutator

[TABLE]

The Hubbard algebra at a given site is defined by

[TABLE]

In brief we may visualize these Hubbard operators in terms of familiar (canonical) electronic operators $c^{\dagger}_{\vec{r}_{i},\sigma_{a}}$ creating an electron at site $i$ with spin $\sigma_{a}$, via the non-linear constructs

[TABLE]

where $\bar{\sigma}_{a}=-\sigma_{a}$ here and throughout this paper. The factors of $(1-n_{\vec{r}_{i}\ \bar{\sigma}_{a}})$ get rid of doubly occupied sites. Although one can work with the $c$’s and project doubly occupied states out, it is optimal to work with the $X$ operators. As Hubbard pointed out, the manifold of lower Hubbard band states defined by the reduced Hilbert space excluding double occupation, is mapped into itself under the action of $X$’s.

The Hamiltonian is expressed in terms of the $X$ operators by

[TABLE]

The last (trivial) term is a shift of the chemical potential, and is added to make the equations more compact. The third ($J$ dependent) term is rewritten as

[TABLE]

where $P$ permutes the spin indices.

II.2 Calculation of the Greens function

The Greens function is defined as

[TABLE]

where $T$ is the time ordering symbol, and for an arbitrary operator $Q$ we define

[TABLE]

Thus, ${\cal{G}}$ is a $2\times 2$ matrix in the spin space. The time dependence is given by

[TABLE]

In further work we need to add a source termkadanoff_baym via the operator ${\cal A}$

[TABLE]

with the *same * $\tau$ dependence of the operators as in Eq(12), and an arbitrary function of time ${\cal V}_{j}^{\sigma_{1}\sigma_{2}}(\tau)$ at every site. The Greens functions are now no longer time translation invariant, and are defined as

[TABLE]

This version of $G$ satisfies the Kubo Martin Schwinger boundary conditionskadanoff_baym as may be readily verified

[TABLE]

so that one may perform a Fourier series with odd integer Matsubara frequencies only. We will use the convention that the Greens function is written as the difference of frequencies after setting the source ${\cal A}=0$, whereas for ${\cal A}\neq 0$ we will display a function of two separate times.

More generally for any variable we define a modified expectation

[TABLE]

with a compact notation that includes the time ordering and the exponential factor automatically. Thus

[TABLE]

From this the variation of the Greens function can be found from functional differentiation as

[TABLE]

and we note the important commutator:

[TABLE]

We note the similarity in form between the $t$ and $J$ terms above, one can be transformed into the other by flipping the spatial indices $i,j$ on the $X$ operators; this symmetry persists in the following equations as well. From this point we will use an Einstein type convention, we sum over all internal repeated indices, while leaving the external indices fixed. By summing over an index, a spatial sum over the lattice and integration over imaginary time $0\leq\tau_{j}\leq\beta$ is implied. The external indices (both space-time and spin) are recognizable since they appear in the LHS of all the equations.

Let us compute the time derivative of the G. For this we need the derivative

[TABLE]

This follows from the definition of the time ordering and the form of ${\cal A}$. Using this we find:

[TABLE]

We further use the abbreviations,

[TABLE]

We next introduce a useful and convenient notion of “k-conjugation” of any matrix $M$, denoted by $M^{k}$, such that

[TABLE]

This is the time reversal operation, but confined to the spin space indices and excludes transforming the space-time indices. The k-conjugation of a matrix in the spin space produces the transpose of its cofactor matrix, and thus the inverse of any matrix is proportional to its conjugate, as in Eq(24). More explicitly we see that

[TABLE]

where $\mathds{1}$ is the identity matrix in the $2\times 2$ dimensional spin space.

We need an object $\Delta[i]$ that plays the role of a dynamical Gutzwiller factor in this theory. It is defined by

[TABLE]

where the second and third lines follow from Eq(23). We also define a matrix functional derivative operator

[TABLE]

In terms of these, we find the equation of motion (EOM)

[TABLE]

We employ a useful relation with an arbitrary operator ${\cal Q}$:

[TABLE]

to rewrite Eq(26) in component form as

[TABLE]

This may finally be written compactly in matrix form as

[TABLE]

where we used the definitions

[TABLE]

The space-time indices are displayed but the spin indices are hidden in the above matrix structure.

We next perform a scale (or local gauge) transformation with a space time spin dependent factor described below. This scale transformation is a key step in our work, and it is important to appreciate its motivation. If we work with the EOM Eq(29), the resulting vertex, i.e. schematically the object $-\frac{\delta}{\delta V}{\cal{G}}^{-1}$, turns out to have pathological “overhangs”. By this we mean that the vertex will contain not only ${\cal{G}}$, but also ${\cal{G}}^{-1}$, i.e. the putative “self energy”, and hence the resulting Schwinger Dyson equation will be ill formed222 It is important to verify that the self energy and vertices are well behaved at high frequencies $\omega$ (i.e. have the form $\sim c_{1}+\ \frac{c_{2}}{\omega}$), and possess proper spectral representations. When these conditions are not satisfied, as with equations formally following from those of ${\cal{G}}$ without the removal of the factor of $\Delta$ as in Eq(31), we denote the resulting equations as ill formed.. The origin of the difficulty is that the coefficient of $\delta[i,f]$ in the RHS of Eq(29) involves $\Delta[i]$ which is time dependent, and essentially made up of ${\cal{G}}$. This in turn is a manifestation of the non canonical nature of the projected electrons. The factor $\Delta[i]$ is physically understandable as arising from the spectral weight contained in the lower Hubbard band, which is less than unity. On the other hand, canonical Greens functions contain a spectral weight of unity. These observations are reflected in the coefficient of $\frac{1}{\omega}$ in the limit of high frequencies of the Greens functions changing from unity in the canonical case to $1-n_{\bar{\sigma}}$ for the  $t$-$J$  model.

We resolve this difficulty by a local space time dependent scale transformation. With this, we eliminate this inconvenient factor of $\Delta[i]$ by a multiplicative process in the time domain. After the removal, we uncover a new Greens function $\hat{G}$ corresponding to effective canonical electrons or quasiparticles (QP) Eq(7) that lie underneath. One could remove this factor $\Delta[i]$ in many equivalent ways, such as symmetrically or from the right, we choose a left sided transformation for maximal ease of computationsymmetrize . Let us write

[TABLE]

so that the EOM Eq(29) becomes after some rearrangement:

[TABLE]

Here $\hat{G}$ is the underlying canonical Greens function (with spectral weight unity), ${V}$ is the transformed source field, and $\Phi$ arises from the time derivative. Appendix B summarizes its properties, and demonstrates that it is negligible on turning off the source terms.

II.3 Calculation of the inverse Greens function

We next turn to the task of finding the inverse Greens function. Using the methodology and results detailed in Appendix A, (especially Eq(34, 154)), and with $\vdots Q\vdots$ symbolizing a right (i.e. normal) ordering of the functional derivative contained in the matrix operator $Q$, we may rewrite Eq(32) as

[TABLE]

Here we denote

[TABLE]

The matrix products are indicated by the center dots, and the terms under the overline symbol indicate the extent of terms over which the derivative acts. Using the results detailed in AppendixA Eq(156) and SectionIV Eq(62), we express various objects in terms of $\hat{G}$ (rather than ${\cal{G}}$):

[TABLE]

With these identifications, we have converted the problem to one only involving $\hat{G}$ and ${V}$ at this point, and jettisoned all reference to the original Greens function ${\cal{G}}$ and the original source ${\cal V}$.

Our next aim is to find an equation for the inverse 333This represents the dynamical inverse and should not to be confused with a matrix inverse. of $\hat{G}$ defined through

[TABLE]

It is useful to define a “susceptibility” type three point object $\chi$

[TABLE]

and a vertex function $\Gamma$

[TABLE]

The relations of these susceptibilities with the physical ones are detailed in Appendix F. Let us right multiply Eq(33) by ${\hat{G}}^{-1}[f,m]$ ( and sum over $f$) so that

[TABLE]

More explicitly, the derivative excludes operating upon the factor ${\hat{G}}^{-1}[f,m]$, since the overline excludes that term. The detailed calculation is presented in Appendix A. We use Eq(167,168) and Eq(A.1) to rewrite this expression. We split the Greens function into the following form with $\hat{G}_{0}$ as the bare quasiparticle Greens function and ${{\cal{S}}}$ as the quasiparticle self energy,

[TABLE]

The self energy is given by

[TABLE]

Regarding nomenclature, we call the expression $\hat{G}_{0}$ as the bare- rather than the unperturbed- Greens function, and the rest as self energy, since there is no perturbation theory involved here. The result Eq(42,43) is the exact Schwinger Dyson equation including the source field, and is central to the calculation of the vertices. Calculating the vertices proceeds by taking functional derivatives of Eq(43), and leads to a standard hierarchy of equations for higher order vertices schwinger_school ; rajagopal . We should also clarify that in contrast to the nomenclature of Fermi liquids, the quasiparticles as defined here are not infinitely sharp in energy, they decay through relaxation processes that we describe via the imaginary part of their (analytically continued) self energy. Thus the quasiparticles of the ECQL have a coherent part as well as a incoherent background part that emerges in complete analogy to the electron Greens function in the FL.

III Schwinger Dyson Equation and the Limit of vanishing sources

Let us gather various objects here when the sources are turned off i.e. $V\to 0$. We recover all the symmetries in this limit, translation invariance allows us to perform Fourier transforms in space and time, and rotation invariance leads to important simplifications of the vertices and susceptibilities. Due to the rotation invariance of the extremely correlated quantum liquid, the three triplet states of the particle hole pair are degenerate. This leads to important identities that parallel the relations in the Fermi liquid. We denote these as Nozières relations and discuss them further in Appendix C, where the abbreviated superscripts and subscripts of the vertices are defined. Therefore as $V\to 0$,

[TABLE]

and for any object $Q=\chi,\;\Upsilon,\;\mbox{or}\;\Gamma$ we have a spin decomposition:

[TABLE]

The object $Q_{s}$ refers to the particle hole singlet channel, corresponding to a charge density variable and $Q_{t}$ to the particle hole triplet channel, i.e. a spin density variable.

Let us note that in the limit of vanishing sources, the physical projected electron Greens function ${\cal{G}}$ and the quasiparticle Greens function $\hat{G}$ are simply related as

[TABLE]

The object $\nu$ is now a function of the difference $a-b$, and due to the multiplying factors $t[a,b],\;J[a,b]$ in the expression for $\hat{G}$, we usually need this at equal times $\tau_{a}=\tau_{b}$. We Fourier transform using the convention in Appendix H as

[TABLE]

Further on turning off the sources, we find from Appendix A.2

[TABLE]

As explained in Appendix C, the superscript $(p)$ stands for Cooper pairing channel and is a specific linear superposition of the singlet and triplet channels $\Gamma^{(p)}=\frac{1}{2}(\Gamma_{s}-3\Gamma_{t})$. Hence we may write the EOM as

[TABLE]

Using the convention for Fourier transforms in Appendix H, and with the Fourier space version of $\hat{G}_{0}$ Eq(42)

[TABLE]

we write the exact Schwinger Dyson equation for the Greens function

[TABLE]

The self energy explicitly contains the electron dispersion $\varepsilon_{q}$ or $t_{ij}$ as a multiplicative factor here. This is quite unlike the standard case of the electron liquid, where the entire dependence on the electronic dispersion is hidden in the form of the bare propagator $G_{0}$, and the self energy is a functional of only the (full) Greens functionluttinger_ward ; agd ; rajagopal ; mahan .

In order to evaluate the first term of ${{\cal{S}}}$ in Eq(LABEL:schwinger_dyson), we need a few definitions at this point. Let us denote

[TABLE]

Similar definitions hold in the triplet channel. In appendix F we show that the local object $\chi_{{\text{\tiny\mbox{loc}}}}$ is identical for both triplet and singlet channels. From Eqs(47, LABEL:schwinger_dyson and 52), and further using the definition of the renormalized energy type variables

[TABLE]

we find

[TABLE]

By construction, the variables $J^{s}_{p},J^{t}_{p},\varepsilon^{s}_{p},\varepsilon^{t}_{p}$ are weighted averages of the bare objects $J_{p},\varepsilon_{p}$ with momentum dependent but static weight factors, and hence it is reasonable to view them as correlation adjusted versions of the bare dispersions. The precise relation between the bare dispersions and $\hat{E}_{k},{\cal J}_{k}$ is particularly simple when the hopping is only nearest neighbour; it involves the the spin spin and density density correlation functions at nearest neighbour distances, and is given explicitly below in Eq(58). Specializing to only nearest neighbour hoppings, we see from symmetry that the form of the band dispersion remains a simple tight binding one. Therefore we write a convenient notation of various objects for the simple cubic lattices:

[TABLE]

The parameters $x_{s}$ and $x_{t}$ can be found in terms of equal time correlations by carrying out the frequency and spatial sums, and using Eqs(LABEL:ups_chi,205), we find

[TABLE]

where $\vec{\eta}$ is a nearest neighbour vector. In terms of these, we may write

[TABLE]

Thus we can express

[TABLE]

where $\vec{\eta}$ is a nearest neighbour vector and the correlations are at equal times. Thus the nearest neighbour charge and spin correlations directly influence the effective band width. The spin correlation has its largest magnitude near half filling, and can be positive or negative depending upon whether ferromagnetic, i.e. Nagaoka Thouless type correlations, or the more usual antiferromagnetic correlations prevail. Near half filling the density term in Eq(58) is suppressed, while the spin correlation term survives and gives the dominant contribution. This correction term to the energy dispersion $\hat{E}$ is of the same form as $\varepsilon$ but has the opposite sign and hence leads to an important band narrowing via the correlation functions indicated in Eq(58) in this theory.

Later we will find equations for the susceptibilities, so that these correlation functions can be found in terms of the vertices and the Greens functions self consistently. Successive approximations for the vertices will be formulated, with each approximation providing a complete calculational scheme. We will find that the susceptibilities $\chi_{s,t}[Q]$ vanish generically as $1-n$ near half filling, in a similar fashion as $\chi_{{\text{\tiny\mbox{loc}}}}$, and hence these renormalized energies tend to a finite non zero limit at half filling.

Eq(54) is the exact Schwinger Dyson equation for the $t$-$J$ model. This breakup of the inverse $\hat{G}$ into $\hat{G}_{0}$ and a self energy type object ${{\cal{S}}}$ is to some extent arbitrary, since there is no a priori notion of an unperturbed Greens function. Our breakup has the natural advantage that the resulting bare vector vertices are frequency independent and in agreement with independent arguments (see next section where the bare current vertex is obtained in Eq(LABEL:kubo-2)). Further they satisfy the Ward identities as noted below Eq(97). The ${{\cal{S}}}$ object, after the standard analytic continuation to the retarded self energy, provides a correction to the quasiparticle energies at the poles of $\hat{G}$ through its real part and more importantly to the decay of the quasiparticles through its imaginary part. The self energy in our formulation is developed in terms of the Greens function $\hat{G}$ and is explicitly a functional of $\hat{G}$. However, and in contrast to the usual weakly interacting Fermi system formulationpothoff luttinger_ward ; agd ; nozieres , it is not a universal functional, i.e. it does depend upon the bare dispersion $t_{ij},J_{ij}$ regardless of how we define the self energy.

Finally it is important to note that this Schwinger Dyson equation is well formed in the sense that the vertices $\Gamma$ are guaranteed to be well behaved (i.e. finite) for $\omega\rightarrow\infty$, since we have avoided the linear $\omega$ dependence of $\Gamma$ that results, if one does not extract the time dependent factor $\Delta[i]$ as in Eq(31). The equations for the vertices are derived and discussed below.

IV Solution of the local Greens function

We next consider the various local Greens functions and their interrelations. Start from

[TABLE]

This equation is easy to solve when we iterate once,

[TABLE]

we see that the inhomogeneous term $\mathds{1}-\hat{G}^{k}$ commutes with $\Delta$ iteratively, so that we may as well rewrite the second term in the RHS as $\Delta[j]\cdot\hat{G}^{k}[j^{-},j]\cdot\hat{G}[j^{-},j]=\det[G]\Delta[j]$, whereby the solution is noted in Eq(37);

[TABLE]

We also require the inverse

[TABLE]

These local objects are the dynamical analogs of the Gutzwiller projection factors Gutz , and we note their physical meaning in terms of the bare (quasiparticle) charges $n[i]$ ($n_{QP}[i]$) and spin densities $\vec{S}[i]$ ($\vec{S}_{QP}[i]$) defined in Eqs(144,151)

[TABLE]

When we turn off the sources, the ECQL state has the following local Greens functions expressed in terms of the number density at each site per spin $n=N/(2N_{sites})$:

[TABLE]

The vanishing near half filling of $\gamma$ and $\gamma 1$, as $(1-n)$ and $(1-n)^{2}$ has important consequences, and leads to the hole density expansion reported below in Section VII.5.

V Calculation of the Vertices

The vertices can be obtained from the general equations for the Greens function Eq(42, 43) as

[TABLE]

Thus the bare vertices are simple, writing them in Fourier space we find

[TABLE]

The vertex corrections arise from the self energy part of Eq(64), and contain several terms. We provide the results of the long calculation in Appendix G. There all terms are retained, with the proviso that the higher order vertex, obtained by differentiating the three point vertex are set to zero, i.e. $\frac{\delta\Gamma}{\delta V}\to 0$.

While this complete solution is available in the Appendix G, it is crucial to understand the relative order of various terms, in order to make sensible approximations. We next provide a set of calculations that give us such an understanding, we evaluate the derivatives of the basic elements $\Delta[i],\ \mu[i,j]$ and $\nu[i,j]$ occurring in the self energy, and show that these have an explicit dependence upon the hole density $\delta=1-n$, measured from half filling. The terms can in fact be grouped in a formal expansion in the inverse hole density $\lambda=\frac{1}{1-n}$. In addition there is an implicit dependence on $n$ in all terms, and certain terms vanish with $\delta$. Thus the final result for the vertex obtained by taking a product of the apparently divergent terms and the coefficients thereof either vanish at half filling $n=1$ or remain finite. This gives us an organizing principle for grouping the contributions, the leading terms near half filling consist of terms that remain finite at $n=1$, and one can throw out the rest, thereby giving us a hole density expansion that has been conjectured earlier early_tJ_1 .

Therefore taking the derivatives of the Schwinger Dyson equation with sources Eq(43), we find that the expansion of the vertices can be arranged as a formal power series in the inverse hole density $\lambda$, in the form $\Gamma=1+\Gamma_{0}+\lambda\Gamma_{1}+\lambda^{2}\Gamma_{2}$. It is implied that the coefficients $\Gamma_{j}$ contain terms that are either vanishing at half filling or are finite, and rules for recognizing this implicit dependence are provided later. In Appendix G we list the vertices $\Gamma_{0},\Gamma_{1},\Gamma_{2}$.

The existence of this expansion is fortunate since the most interesting physical regime for a doped Mott insulator is in the limit $n\to 1$. The factors of $\lambda$ arise from the dynamical projection factors $\Delta[i],\ \mu[i,j]$ and $\nu[i,j]$ in the local Greens functions and the self energy. We next list the leading behaviour of the functional derivatives of the various matrix functions $\mu[i,j],\ \nu[i,j],\ \Delta[i]$ w.r.t. the source terms, so that results for the vertices can be assembled together. For this section, we will use spin diagonal sources ${V}_{m}$ so that the matrices can all be taken as diagonal in spin space. We will need to use the Noziéres relations $\chi^{(1)}\pm\chi^{(2)}=\chi_{s,\ t}$, in order to obtain the complete the set of derivatives in the singlet and triplet channels. After providing the results, we will discuss their relative magnitudes, and also the implicit density dependence of the coefficients.

Let us begin with the less singular terms $\Delta$ and $\mu$, where the functional derivative is explicitly $O(\frac{1}{\delta})$, and then progress to $\nu$ which is $O(\frac{1}{\delta^{2}})$. At some places, we will use the symbol $\sim$ to indicate that the density $n$ has been set at unity in all terms except the singular $\frac{1}{(1-n)^{a}}$ type terms.

V.0.1 $\Delta[i]$ and its derivatives

Let us recall for spin diagonal sources (Appendix E)

[TABLE]

Therefore

[TABLE]

V.0.2 $\mu[i,k]$ and its derivatives

Lets us recall $\mu[i,k]=\Delta^{-1}[i]\cdot\Delta[k]$, and it may be expressed as

[TABLE]

where $g_{2}[i,j]$ and a related object $g_{1}[i,j]$ needed in the next section are given by

[TABLE]

where we used the definitions (recall Eqs(62, 37)).

We first note the derivatives of $g$

[TABLE]

Notice that the derivatives of $g_{1}$ and $g_{2}$ are independent of the spin index $\sigma$. We next compute the derivatives of $\mu[i,k]$ with spin diagonal sources, by using the above equations and the derivative