Mott physics and spin fluctuations: a functional viewpoint
We present a formalism for strongly correlated systems with fermions coupled to bosonic modes. We construct the three-particle irreducible functional by successive Legendre transformations of the free energy of the system. We derive a closed set of equations for the fermionic and bosonic self-energies for a given . We then introduce a local approximation for , which extends the idea of dynamical mean field theory (DMFT) approaches from two- to three-particle irreducibility. This approximation entails the locality of the three-leg electron-boson vertex , which is self-consistently computed using a quantum impurity model with dynamical charge and spin interactions. This local vertex is used to construct frequency- and momentum-dependent electronic self-energies and polarizations. By construction, the method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply it to the Hubbard model on two-dimensional square and triangular lattices. We complement the results of Ref. Ayral and Parcollet, 2015 by (i) showing that, at half-filling, as DMFT, the method describes the Fermi-liquid metallic state and the Mott insulator, separated by a first-order interacting-driven Mott transition at low temperatures, (ii) investigating the influence of frustration and (iii) discussing the influence of the bosonic decoupling channel.
Systems with strong Coulomb correlations such as high-temperature superconductors pose a difficult challenge to condensed-matter theory.
One class of theoretical approaches to this problem emphasizes long-ranged bosonic fluctuations e.g. close to a quantum critical point as the main ingredient to account for the experimental facts. This is the starting point of methods such as spin fluctuation theory Chubukov et al. (2002); Efetov et al. (2013); Wang and Chubukov (2014); Metlitski and Sachdev (2010); Onufrieva and Pfeuty (2009, 2012), two-particle self-consistent theory Vilk et al. (1994); Daré et al. (1996); Vilk and Tremblay (1996, 1997); Tremblay (2011) or the fluctuation-exchange approximation Bickers and Scalapino (1989). These methods typically rely on an approximation of the electronic self-energy as a one-loop diagram with a suitably constructed bosonic propagator, neglecting vertex corrections.
Another class of approaches focuses instead, following Anderson Anderson (1987), on the fact that the parent compounds of high-temperature superconductors are Mott insulators and assumes that Mott physics is essential to describe the doped compounds. In recent years, dynamical mean-field theory (DMFT) Georges et al. (1996) and its cluster extensions like cellular DMFT Lichtenstein and Katsnelson (2000); Kotliar et al. (2001) or the dynamical cluster approximation Hettler et al. (1998, 1999); Maier et al. (2005a) have emerged as powerful tools to capture the physics of doped Mott insulators. Formally based on a local approximation of the two particle-irreducible (2PI, or Luttinger-Ward) functional , they consist in self-consistently mapping the extended lattice problem onto an impurity problem describing the coupling of a small number () of correlated sites with a noninteracting bath. The coarse-grained (short-ranged) self-energy obtained by solving the impurity model is used as an approximation of the lattice self-energy.
Cluster DMFT methods have given valuable insights into the physics of cuprate superconductors, in particular via the study of the Hubbard model: they have allowed to map out the main features of its phase diagram, to characterize -wave superconductivity or investigate its pseudogap phase with realistic values of the interaction strength Kyung et al. (2009); Sordi et al. (2012a); Civelli et al. (2008); Ferrero et al. (2010); Gull et al. (2013); Macridin et al. (2004); Maier et al. (2004, 2005b, 2006); Gull et al. (2010); Yang et al. (2011a); Macridin and Jarrell (2008); Macridin et al. (2006); Jarrell et al. (2001); Bergeron et al. (2011); Kyung et al. (2004, 2006); Okamoto et al. (2010); Sordi et al. (2010, 2012b); Civelli et al. (2005); Ferrero et al. (2008, 2009); Gull et al. (2009). Moreover, they come with a natural control parameter, the size of the impurity cluster, which can a priori be used to assess quantitatively the accuracy of a given prediction as it interpolates between the single-site DMFT solution () and the exact solution of the lattice problem (). Systematic comparisons with other approaches, in certain parameter regimes, have started to appear.Leblanc et al. (2015) Yet, cluster methods suffer from three major flaws, namely (i) they cannot describe the effect of long-range bosonic fluctuations beyond the size of the cluster, which can be experimentally relevant (e.g. in neutron scatteringRossat-Mignod et al. (1991); Keimer et al. (1992); Bourges et al. (1996)) ; (ii) the negative Monte-Carlo sign problem precludes the solution of large impurity clusters, (iii) the cluster self-energy is still quite coarse-grained (typically up to 8 or 16 patches in regimes of interest Gull et al. (2009, 2010); Vidhyadhiraja et al. (2009); Macridin and Jarrell (2008)) or relies on uncontrolled periodization or interpolation schemes (see e.g. Ref. Kotliar et al., 2001).
Recent attempts at incorporating some long-range correlations in the DMFT framework include the GW+EDMFT method Sun and Kotliar (2002); Biermann et al. (2003); Sun and Kotliar (2004); Ayral et al. (2013); Biermann (2014) (which has been so far restricted to the charge channel only), the dynamical vertex approximation (DAToschi et al. (2007); Katanin et al. (2009); Schäfer et al. (2015); Valli et al. (2015)) and the dual fermionRubtsov et al. (2008) and dual bosonRubtsov et al. (2012); van Loon et al. (2014) methods. DA consists in approximating the fully irreducible two-particle vertex by a local, four-leg vertex computed with a DMFT impurity model. This idea has so far been restricted to very simple systemsValli et al. (2015) (“parquet DA”) or further simplified so as to avoid the costly solution of the parquet equations (“ladder DA”Katanin et al. (2009)). This makes either (for parquet DA) difficult to implement for realistic calculations, at least in the near future (the existing “parquet solvers” have so far been restricted to very small systems only Yang et al. (2009); Tam et al. (2013)), or (for the ladder variant) dependent on the choice of a given channel to solve the Bethe-Salphether equation. In either case, (i) rigorous and efficient parametrizations of the vertex functions only start to appearLi et al. (2015), (ii) two-particle observables do not feed back on the impurity model in the current implementationsHeld (2014), and (iii) most importantly, achieving control like in cluster DMFT is very arduous: since both DA and the dual fermion method require the manipulation of functions of three frequencies, their extension to cluster versionsYang et al. (2011b) raises serious practical questions in terms of storage and speed.
The TRILEX (TRiply-Irreducible Local EXpansion) method, introduced in Ref. Ayral and Parcollet, 2015, is a simpler approach. It approximates the three-leg electron-boson vertex by a local impurity vertex and hence interpolates between the spin-fluctuation and the atomic limit. This vertex evolves from a constant in the spin-fluctuation regime to a strongly frequency-dependent function in the Mott regime. The method yields frequency and momentum-dependent self-energies and polarizations which, upon doping, lead to a momentum-differentiated Fermi surface similar to the Fermi arcs seen in cuprates.
In this paper, we provide a complete derivation of the TRILEX method as a local approximation of the three-particle irreducible functional , as well as additional results of its application to the Hubbard model (i) in the frustrated square lattice case and (ii) on the triangular lattice.
In section II, we derive the TRILEX formalism and describe the corresponding algorithm. In section III, we elaborate on the solution of the impurity model. In section IV, we apply the method to the two-dimensional Hubbard model and discuss the results. We give a few conclusions and perspectives in section V.
In this section, we derive the TRILEX formalism. Starting from a generic electron-boson problem, we derive a functional scheme based on a Legendre transformation with respect to not only the fermionic and bosonic propagators, but also the fermion-boson coupling vertex (subsection II.1). In subsection II.2, we show that electron-electron interaction problems can be studied in the three-particle irreducible formalism by introducing an auxiliary boson. Finally, in subsection II.3, we introduce the main approximation of the TRILEX scheme, which allows us to write down the complete set of equations (subsection II.4).
Our starting point is a generic mixed electron-boson action with a Yukawa-type coupling between the bosonic and the fermionic field:
and are Grassmann fields describing fermionic degrees of freedom, while is a real bosonic field describing bosonic degrees of freedom. Latin indices gather space, time, spin and possibly orbital or spinor indices: , where denotes a site of the Bravais lattice, denotes imaginary time and is a spin (or orbital) index ( in a single-orbital context). Barred indices denote outgoing points, while indices without a bar denote ingoing points. Greek indices denote , where indexes the bosonic channels. These are for instance the charge () and the spin () channels. Repeated indices are summed over. Summation is shorthand for . (resp. ) is the non-interacting fermionic (resp. bosonic) propagator.
The action (1) describes a broad spectrum of physical problems ranging from electron-phonon coupling problems to spin-fermion models. As will be elaborated on in subsection II.2, it may also stem from an exact rewriting of a problem with only electron-electron interactions such as the Hubbard model or an extension thereof via a Hubbard-Stratonovich transformation.
ii.1 Three-particle irreducible formalism
In this subsection, we construct the three-particle irreducible (3PI) functional . This construction has first been described in the pioneering works of de Dominicis and Martin.de Dominicis and Martin (1964a, b) It consists in successive Legendre transformations of the free energy of the interacting system.
Let us first define the free energy of the system in the presence of linear (), bilinear (, ) and trilinear sources () coupled to the bosonic and fermionic operators,
We do not need any additional trilinear source term (similar to , and ) since the electron-boson coupling term already plays this role.
is the generating functional of correlation functions, viz.:
The above correlators contain disconnected terms as denoted by the superscript “nc” (non-connected).
ii.1.1 First Legendre transform: with respect to propagators
Let us now perform a first Legendre transform with respect to , and :
with . By construction of the Legendre transformation, the sources are related to the derivatives of through:
where we have defined the connected correlation function:
where we have defined the fermionic and bosonic self-energies and as functional derivatives with respect to :
The functional is the Almbladh functional.Almbladh et al. (1999) It is the extension of the Luttinger-Ward functional ,Luttinger and Ward (1960); Baym (1962) which is defined for fermionic actions, to mixed electron-boson actions. While contains two-particle irreducible graphs with fermionic lines and bare interactions (see e.g. diagram (a) of Fig. 1), contains two-particle irreducible graphs with fermionic () and bosonic () lines, and bare electron-boson interactions vertices (see e.g. diagram (b) of Fig. 1).
Both and can be approximated in various ways, which in turn leads to an approximate form for the self-energies, through Eqs (10a-10b). Any such approximation, if performed self-consistently, will obey global conservation rules.Baym and Kadanoff (1961) A simple example is the approximation,Hedin (1965) which consists in approximating by its most simple diagram (diagram (b) of Fig. 1). The DMFT (resp. extended DMFT, EDMFTSengupta and Georges (1995); Kajueter (1996); Si and Smith (1996)) approximation, on the other hand, consists in approximating (resp. ) by the local diagrams of the exact functional:
The DMFT approximation becomes exact in the limit of infinite dimensions.Georges et al. (1996) Motivated by this link between irreducibility and reduction to locality in high dimensions, we perform an additional Legendre transform to go one step further in terms of irreducibilty.
ii.1.2 Second Legendre transform: with respect to the three-leg vertex
We introduce the Legendre transform of with respect to :
where is the three-point correlator:
We also define the connected three-point function and the three-leg vertex as:
is the amputated, connected correlation function. It is the renormalized electron-boson vertex. These objects are shown graphically in Fig. 2. is a shorthand notation for .
We now define the three-particle irreducible functional as:
Note that in the right-hand site, is determined by , , and (by the Legendre construction). is the generalization of the functional introduced in Ref. de Dominicis and Martin, 1964b to mixed fermionic and bosonic fields. We will come back to its diagrammatic interpretation in the next subsection.
Differentiating with respect to the three-point vertex yields , the generalization of the self-energy at the three-particle irreducible level, defined as:
Before proceeding with the derivation, let us first state the main results: and are related by the following relation:
This is the equivalent of Dyson’s equations at the 3PI level. This relation is remarkably simple: it does not involve any inversion, contrary to the Dyson equations (9a-9b). This relation is illustrated in Figure 3.
The fermionic and bosonic self-energies and are related to by the following exact relations:
The second term in is nothing but the Hartree contribution. These expressions will be derived later. The graphical representation of these equations is shown in Figure 4.
The above equations, Eqs (17-18-19a-19b-9a-9b), form a closed set of equations for , , , , and . The central quantity is the three-particle irreducible functional , obtained from the 2PI functional algebraically by a Legendre transformation with respect to the bare vertex , or diagrammatically by a ’boldification’ of the bare vertex.
has been shown to be made up of all three-particle irreducible (3PI) diagrams by de Dominicis and Martinde Dominicis and Martin (1964b) in the bosonic case. A 3PI diagram is defined as follows: for any set of three lines whose cutting leads to a separation of the diagram in two parts, one and only one of those parts is a simple three-leg vertex . The simplest 3PI diagram is shown in Fig. 5(a). Conversely, neither diagram (b) of Fig. 1 nor diagram (b) of Fig. 5 are 3PI diagrams.
Most importantly, the hierarchy is closed once the functional form of is specified: there is no a priori need for a higher-order vertex. This contrasts with e.g. the functional renormalization group (fRGMetzner et al. (2012)) formalism (which requires the truncation of the flow equations) or the Hedin formalismHedin (1965); Aryasetiawan and Gunnarsson (1998); Aryasetiawan and Biermann (2008) which involves the four-leg vertex via the following Bethe-Salpether-like expression for :
Of course, one must devise approximation strategies for in order to solve this set of equations. In particular, any approximation involving the neglect of vertex corrections, like the FLEX approximationBickers and Scalapino (1989), spin fluctuation theoryMonthoux et al. (1991); Schmalian et al. (1998); Chubukov et al. (2002), the GW approximationHedin (1965) or the Migdal-Eliashberg theory of superconductivityMigdal (1958); Eliashberg (1960) corresponds to the approximation
which yields, in particular, the simple one-loop form for the self-energy:
These approximations only differ in the type of fermionic and bosonic fields in the initial action, Eq. (1): normal/superconducting fermions, bosons in the particle-hole/particle-particle sector, in the spin/charge channel…
The core idea of the DMFT and descendent methods is to make an approximation of (or ) around the atomic limit. TRILEX is a similar approximation for , as will be discussed in section II.3.
ii.1.4 Derivation of the main equations
By construction of the Legendre transform (Eq. (12)),
Let us now derive Eqs (19a-19b). They are well-known from a diagrammatic point of view, but the point of this section is to derive them analytically from the properties of . In order to obtain the self-energy , we use Eq. (10a). We first need to reexpress in terms of using (16): thus
where is a function of , , , . Thus, Eq. (10a) becomes:
This derivative must be performed with care since the electron-boson vertex now appears in its interacting form . This yields:
As a result,
Finally, using (18), we obtain:
Similarly, using (10b), one gets for :
Thus, using (18), we get
i.e. can be written as:
where and are two functions. This is illustrated in Fig. 6 for the simplest diagram of .
From (28a), one gets:
From (28b), in turn, one gets:
where we have used the property that is symmetric twice: first by trivially using , and second to prove that
This latter property can be proven by noticing that when is symmetric, is a homogeneous function of the symmetrized : , with , , and . Then, one has .
We thus obtain the following relations:
Right-multiplying (31a) by and (31b) by and replacing using the definition of (Eq. (17)) shows that the bracketed terms in Eqs (II.1.4-II.1.4) vanish. Thus, these expressions simplify to the final expressions for the self-energy and polarization, Eqs (19a-19b).
Finally, these exact expressions can be derived in alternative fashion using equations of motion, as shown in Appendix D.
ii.2 Transposition to electron-electron problems
In this section, we show how the formalism described above can be used to study electron-electron interaction problems. We shall focus on the two-dimensional Hubbard model, which reads:
denotes a point of the Bravais lattice, , is the tight-binding hopping matrix (its Fourier transform is ), is the local Hubbard repulsion, and are creation and annihilation operators, , with . In the path-integral formalism, the corresponding action reads:
Here, , where denotes fermionic Matsubara frequencies, the chemical potential and the bare dispersion reads in the case of nearest-neighbor hoppings. and are Grassmann fields. We remind that .
The Hubbard interaction in Eq. (33) can be decomposed in various ways. Defining (where and denotes the Pauli matrices), the following expressions hold, up to a density term:
with the respective conditions:
In Eq. (34a), the Hubbard interaction is decomposed on the charge and longitudinal spin channel (“Ising”, or “”-decoupling), while in Eq. (34b) it is decomposed on the charge and full spin channel (“Heisenberg”, or “”-decoupling). The Heisenberg decoupling preserves rotational invariance, contrary to the Ising one. In addition to this freedom of decomposition comes the choice of the ratio of the charge to the spin channel, which is encoded in Eqs. (35a-35b).
where we have used: . Similarly, we can write:
In the Ising decoupling,
We remind that and . The parameter may take the values (Heisenberg decoupling) or a subset thereof (e.g for the Ising decoupling).
In the Hubbard model (Eq.(32)),
We now decouple the interaction (38) with a real111In principle, the interaction kernel should be positive definite for this integral to be convergent. Should it be negative definite, positive definiteness can be restored by redefining and , which leaves the final equations unchanged. After this transformation, the electron-electron action (33) becomes Eq. 1, where we have chosen the minus sign for the Yukawa coupling in Eq. (42). bosonic Hubbard-Stratonovich field :
We have thus cast the electron-electron interaction problem in the form of Eq. (1), namely an electron-boson coupling problem. We can therefore apply the formalism developed in the previous section to the Hubbard model and similar electronic problems. The only caveat resides with the freedom in choosing the electron-boson problem for a given electronic problem: we discuss this at greater length in subsection II.3.4.
For later purposes, let us now specify the equations presented in the previous section for the Hubbard model in the normal, paramagnetic case.