A pedagogical derivation of dynamical susceptibilitie

Dynamical two-particle susceptibilites are important for a wide range of different experiments in condensed-matter physics and beyond. Nevertheless, most textbooks avoid describing how to derive such response functions, perhaps because they are viewed as too complex. In the literature, most derivations work with generalized susceptibilities, which are more general, but require an even higher layer of complexity. In this work, we show a more direct derivation in the context of model Hamiltonians which can be mapped directly onto an impurity model. We also present an alternative derivation for the irreducible vertex in the context of the Falicov-Kimball model.


Introduction
One of the authors (J. K. F.), met Andrĳ Shvaika in 2000 at a conference in Lviv. He actually met him virtually before the meeting, serving as a referee for a conference proceeding Andrĳ had submitted earlier in the year. The initial meeting blossomed into a two-decades-long collaboration involving numerous funded grants, over two dozen joint publications, and a lifelong friendship. Time and again during our collaborative work, we would each approach problems from different perspectives. The western approach emphasized equations of motion and functional methods, while the eastern approach emphasized diagrammatic expansions based on Larkin irreducible parts of Green's functions. This worked well, because we usually were able to derive results from two different perspectives -when they agreed, we had good confidence that they were correct.
About fifteen years ago, J. K. F. developed some notes on how to derive dynamical susceptibilities using a direct nonequilibrium perturbation theory approach, but they were never published anywhere. It seemed appropriate to dust them off now and take the opportunity to write an article that does this here and that emphasizes the pedagogical aspect of the work. We then lost the notes during the COVID-19 pandemic and ended up reconstructing them in more detail; the result of these efforts is presented here. We hope it will help others who are looking for simpler ways to understand and work with dynamical susceptibilities.
A dynamical susceptibility satisfies the so-called Bethe-Salpeter equation. This is the two-particle equivalent of the Dyson equation for single-particle Green's functions. The Bethe-Salpeter equation was first derived by Bethe and Salpeter in 1951 [1]. This equation typically has a matrix character to it, but we will be deriving a vector character form of the equation here. Our approach is to begin with a static Hamiltonian and add a time-dependent disturbance to it. We then seek to determine the linear-response under such time-dependent driving. We focus on a single-site lattice impurity; one can then map the impurity to a lattice problem via the dynamical mean-field theory approach, but we do not provide all of those details. We also concentrate on treating spinless fermions, since we are working with the charge susceptibility and Andrĳ Shvaika has worked on many different spinless fermion systems in his career. Of course, our Bethe-Salpeter equation derivation can be easily generalized to more complex cases.
In many-body physics, we need to evaluate expectation values of different types of operator expressions. Within dynamical mean-field theory, we do so in the presence of a dynamical mean-field (denoted by ), which represents how electrons hop onto and off a given lattice site as a function of time. A themodynamic expectation value of a time-independent operatorˆ is then given by where Z = Tr{e − Ĥˆ ( )} is the partition function,ˆ ( ) is a time-dependent evolution operator (describing the dynamical mean-field), and Boltzmann's constant is set equal to one. We will be examining the class of Hamiltonians which can be mapped to the impurity Hamiltonian,Ĥ = − ˆ in the presence of the dynamical mean-field ( , ′ ). This constitutes the set of Hamiltonians that can be treated via dynamical mean-field theory.
Here,ˆ =ˆ †ˆ , and we work in the interaction representation where with T the (imaginary) time-ordering operator and the inverse temperature; all operators evolve in time according to the unperturbed Hamiltonian H . The fermionic operators areˆ andˆ † , which satisfy the canonical anticommutation relation {ˆ , Next, we define the Green's function in imaginary time as Note that we do not assume, nor can one show, that the Green's function is time-translation invariant when the dynamical mean-field is not time-translation invariant (which occurs when we add a time-dependent field to the Hamiltonian). Nevertheless, one can easily show, via the invariance of the trace Trˆ ˆ = Trˆ ˆ , that ( − , ′ ) = − (0 + , ′ ), when 0 < ′ < , and ( , − ) = − ( , 0 + ), when 0 < < . This means that we can still expand the time dependence in terms of a Matsubara-frequency matrix, with a separate Matsubara-frequency expansion for the and ′ dependences, respectively. This yields where i = 2iπ ( + 1/2) is the fermionic Matsubara frequency, for ∈ Z. The Green's function in Matsubara frequency space is expressed as and the dynamical mean-field is given by a similar expression Note that we are working with an imaginary-time formalism here. When one is in equilibrium, with a time-independent Hamiltonian (no external field), the Green's function and the dynamical mean-field become time-translation invariant, and the Matsubara-frequency representation of the Green's function and the dynamical mean-field become diagonal matrices. Finally, we define the frequency-dependent charge susceptibility as where i ℓ is a bosonic Matsubara frequency, i ℓ = 2iπℓ , for ℓ ∈ Z. Next, note that [Ĥ ,ˆ ( )] = 0, so that the second term of the susceptibility in equation (1.7) gives a ℓ,0 when integrating over (a similar result holds for = 0 in the lattice problem, since the total electron number is conserved). One might have thought that the first term also has no imaginary time dependence, but when ≠ 0, the expectation value has oneˆ operator located at = 0 and one located at ; the expectation value has an intrinsic dependence inherited from the dynamical mean-field due to the time ordering. We are interested in the non-zero frequency response, so dropping the zero frequency component, our expression for the susceptibility becomes (1.8) To find the equation which governs the frequency-dependent susceptibility, we introduce a timedependent field ℎ( ) into our impurity Hamiltonian, and take the limit ℎ( ) → 0 at the end of the calculation. This is equivalent to replacing the dynamical mean-field in the evolution operator with so that the evolution operator in equation (1.2) becomes and the Green's function satisfies In this fashion, we find it more convenient to think of the ℎ-field as being added to the dynamical meanfield. It is important to note that the introduction of the ℎ-field breaks the time-translation invariance of the problem. Expressing the auxiliary field ℎ( , ′ ) as a matrix in Fourier space, we have This means that, in Matsubara space, the auxiliary field takes the form of a Toeplitz matrix, ℎ = ℎ − ; a Toeplitz matrix has all the elements on each subdiagonal equal to each other (but they can vary for different subdiagonals). Note that the difference in fermionic Matsubara frequencies is a bosonic Matsubara frequency, so we define the auxiliary field ℎ( ) as a sum over bosonic Matsubara frequencies, and by inversion we have By comparison with the matrix form ℎ = ℎ − , we see that the matrix elements are determined by bosonic Matusbara frequencies, i ℓ = i − i . We rewrite the susceptibility as a functional derivative of the Green's function, so that equation (1.8) becomes Note that after the derivative is taken, and ℎ → 0, the Green's functions in the final expression have their time-translation invariance restored.
The above equation provides an explicit expression for the susceptibility in terms of a functional derivative of the Green's function. In cases where we can explicitly determine how depends on the field ℎ ℓ (for small fields), this equation presents us with an alternative method to determine the dynamic susceptibility without solving the Bethe-Salpeter equation. The equivalence of this and the Bethe-Salpeter equation derived below will be demonstrated in the context of the Falicov-Kimball model, where such an explicit relation can be found.

Determination of the Green's function to linear order in the perturbing field
To begin, we consider the Green's function for a system with an impurity HamiltonianĤ in the presence of a (time-translation invariant) dynamical mean-field ( − ′ ). The full Hamiltonian is given byĤ As stated above, this is equivalent to the system evolving via the unperturbed HamiltonianĤ in the presence of a new dynamical mean-field, Since the time-ordering operator allows us to freely manipulate the exponentials, we find that We expand the exponential of the external field to linear order in the field ℎ( ), then the equation for the Green's function, to linear order in the presence of the ℎ-field, becomes The first term is the equilibrium Green's function. Expanding the perturbing field ℎ( ) in Matsubara frequencies, and assuming a pure tone (only one nonzero Matsubara frequency component to the external field) the above becomes is governed only by the evolution operator ( ) and not the perturbing field ℎ, it is an equilibrium correlation function and must be time-translation invariant. Hence, it only depends on the time differences − ′ , −¯ , ′ −¯ . We now Fourier transform ( , ′ ) to Matsubara space via equation (1.5), with the result Since the correlation function depends only on time differences, shifting → +¯ , ′ → ′ +¯ removes the¯ dependence from the correlation function. We are then able to perform the integral over¯ which results in a delta function (i − i + i ℓ ), forcing i = i + i ℓ . Therefore, the only non-vanishing contribution to which is proportional to ℎ ℓ is along the diagonal , + ℓ. This result is critical to our derivation of a Bethe-Salpeter-like equation below.

Derivation of a Bethe-Salpeter-like Equation
In Fourier space, the matrix inverse of is found from the equation of motion, and is given by This result holds in equilibrium and nonequilibrium (of course, in equilibrium, we have ℎ = 0 and both and Σ are proportional to ). Next, we rewrite the matrix schematically as the matrix multiplication −1 in the expression for the susceptibility. This yields where the derivative of −1 with respect to ℎ ℓ is found explicitly from the field term in equation (2.11) and implicitly from the dependence of the self-energy on the field. After the derivative has been performed, we take the limit ℎ → 0, which restores time-translation invariance to the Green's function, given by → . Taking this limit where appropriate and simplifying gives (2.14) Using the fact that the self-energy is a functional of the Green's function, we employ the chain rule yielding The functional derivative of the self-energy with respect to the Green's function (which is proportional to the irreducible vertex function) does not have all four frequency indices independent of each other. To see this, we write the functional derivative Σ / ′ ′ in terms of integrals over imaginary time, , (2.16)

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and we recall that we must take the limit ℎ → 0 (after taking the functional derivative). When we do, the problem becomes time-translation invariant. Hence, lim ℎ→0 Σ( 1 , 2 ) ( ′ 1 , ′ 2 ) must be independent of absolute time, depending only on relative times 1 − 2 , ′ 1 − ′ 2 and 1 − ′ 1 . Consequently, we can shift the times in equation (2.16), performing the change of variables 1 2 ) becomes independent of ′ 2 and performing the integral over ′ 2 results in a delta function (i Consequently, the susceptibility becomes Note that one cannot still assume time-translation invariance for the ′ ′ in the summation above, until after we have taken the derivative with respect to ℎ . We are only capturing the linear response to the imposed external field. Since the dynamic susceptibility (i ℓ ) depends explicitly on ℎ ℓ ℎ ℓ →0 , only the contributions to proportional to ℎ ℓ survive. This means that, without loss of generality, we again consider the external field to have just one single Fourier component, given by ℎ ℓ with ℓ ≠ 0. As demonstrated in the section above, the only elements of which are linear in ℎ ℓ are those for which = , +ℓ . Hence, revisiting equation (2.17), only elements with i ′ − i ′ = i ℓ are nonzero. Therefore, We make the definitions and then, we have Unlike the conventional Bethe-Salpeter equation, written in terms of matrices, we find an equation that has a vector character to it and is given by with the susceptibility determined from (2.24)

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This vector character can be viewed as the dressed or retarded interaction between a particle-hole pair and the exchanged boson mediating the interaction, as is commonly used in triangular vertices [5]. The generalization to the lattice is more complicated, because of the additional momentum dependence of the external field, which introduces an additional momentum dependence to the susceptibilities. We do not discuss those issues further here.

Application to the Falicov-Kimball Model
For the Falicov-Kimball model, introduced in [4], the impurity Hamiltonian takes the formĤ = H 0 +V, whereĤ with 1 a classical variable equal to 0 or 1 depending on whether a heavy fermion is present or not and the impurity is placed in the presence of the dynamical mean-field ( − ′ ).
In equilibrium, the Green's function can be solved in terms of the expectation value of 1 , denoted 1 , which is the heavy fermion density on the impurity. We find that the equilibrium Falicov-Kimball model Green's function is and the equilibrum self-energy is defined from the Dyson equation ( FK,eq ) −1 = (i + − ) − Σ FK,eq , which becomes This Falicov-Kimball model Green's function can also be expressed as a weighted sum of the Green's function for the impurity and the Green's function for the impurity with → − . This is written as This latter form also holds in nonequilibrium.

The Falicov-Kimball Vertex
We obtain the vertex function for the Falicov-Kimball model without defining an "auxillary Green's function" as in [2] or using diagrammatic techniques as in [3]. We begin with the Green's function for a spinless fermion with HamiltonianĤ 0 = − ˆ in the dynamical mean-field plus the external field, Using the equation of motion and expressing it in Matsubara space, we obtain (3.6) This is recast as a matrix equation

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43703-7 so we can identify the quantity in square brackets as the inverse Green's function of the impurity, −1 , which is given by We want to compute the Green's function as the inverse of −1 to lowest order in the external field ℎ ℓ . We re-arrange the equation as follows, and use the fact that ℎ is small to allow us to invert this matrix inverse via the geometric series and obtain of the impurity as to lowest order in the external field. Note that the first term is the equilibrium result and the second term is the lowest-order correction due to the external field.
The results for the Falicov-Kimball model then follow by a weighted sum of the impurity results for empty and occupied heavy electrons on the impurity site (just as we identified in equilibrium). This yields Thus, to order ℎ ℓ , we have where again, we separate with respect to the equilibrium result and the lowest-order corrections due to the external field. We denote the lowest-order correction via FK and after putting everything over a common denominator, we find (3.14) To calculate the vertex, we also need to determine the lowest-order change to the self-energy, denoted by Σ FK . The total self-energy is defined from the Dyson equation We separate out the equilibrium and lowest-order corrections due to the field via Σ FK = Σ FK,eq + Σ FK , then factor the Falicov-Kimball model Green's function factor out to the left By substituting in FK from equation (3.14) and after some significant algebra, we find (3.20) Finally, by computing the differences FK,eq − FK,eq , Σ FK,eq −Σ FK,eq of the equilibrium Green's function and self-energies, we can replace the complicated expression with a much simpler one given by (3.21) Therefore, we have (3.23) Thus, we see that for the Falicov-Kimball model, the vertex function is diagonal and takes the simple form While the algebra is a bit messy and long, we feel this derivation is simpler than previous derivations.

Conclusion
This paper is primarily pedagogical. It shows a straightforward, simple, and direct methodology for calculating dynamical susceptibilities. We show how the conventional Bethe-Salpeter (matrix) equation can be simplified to a vector-like equation for systems that can be mapped to an impurity Hamiltonian, and hence treated via dynamical mean-field theory (DMFT). The resulting vector equation allows for an immediate solution when the vertex function is diagonal (as it occurs in the Falicov-Kimball model). In addition, in cases where one can determine the functional relationship of the nonequilibrium Green's function with respect to the applied field, the dynamic susceptibility is computed directly avoiding the need for the Bethe-Salpeter-like equation altogether. We also show a new derivation for the vertex function of the Falicov-Kimball model. We hope that Andrĳ enjoys seeing some of his old work re-examined in a new light. We certainly enjoyed preparing this publication in his honor.