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Sparsity Pattern of the Self-energy for Classical and Quantum Impurity Problems

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Abstract

We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero- and finite-temperature, non-equilibrium systems, and superconducting systems.

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Notes

  1. The fact that \(\left( E_0^{(N)},\big \vert \Psi _0^{(N)} \big \rangle \right) \) is the eigenpair corresponding to the ground state plays no role in the proof; it can be replaced by any other eigenpair of \({\hat{H}}\), and the statement remains valid. It is only natural due to physical reasons to consider the ground state.

  2. Usually \(G^{\pm }\) carry the extra information that their poles are viewed as being located infinitesimally below/above the real axis. The choices that yield the ‘time-ordered’ Green’s function are described in ‘Appendix B.1.’ However, this extra information is irrelevant for the purpose of our results.

  3. The reader will find that from the point of view of this paper, these objects can be thought of as more natural than the non-equilibrium self-energy itself, and indeed, all of our sparsity results are proved by considering their analogs.

  4. In Sects. 3.3 and 3.4, the notion of the Hamiltonian will be somewhat modified.

  5. See ‘Appendix A’ for a details.

  6. The same comments as given in Sect. 3.1 apply here as well, though instead see ‘Appendix C.1’ for details relevant to this setting.

  7. For the zero vector, we forgo the bra–ket notation here to avoid confusion.

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Acknowledgements

This work was partially supported by the Department of Energy under Grant Nos. DE-SC0017867, DE-AC02-05CH11231, by the Air Force Office of Scientific Research under Award No. FA9550-18-1-0095 (L. L.), by the National Science Foundation Graduate Research Fellowship Program under Grant DGE-1106400 (M. L.), and by the National Science Foundation under Award No. 1903031. We thank Garnet Chan, Jianfeng Lu, Nicolai Reshetikhin, Reinhold Schneider, and Lexing Ying for helpful discussions.

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Appendices

A Second Quantization

Here we introduce the formalism of second quantization, with the aim of providing enough background and results to make the results of this paper rigorous. We limit our discussion to fermionic and bosonic Fock spaces with finitely many states, i.e., finitely many creation and annihilation operators. This setting can directly describe lattice models such as the Hubbard model in addition to tight-binding approximations of continuum systems. In this sense, we can view the set \(\{1,\ldots ,d\}\) as indexing sites in a lattice model. More generally, one can reduce a continuum problem to this setting via the choice of a finite orbital basis [19].

1.1 A.1 The Occupation Number Construction

Let \({\mathcal {N}}_{-1} = \{0,1\}\) and \({\mathcal {N}}_{+1} = \{0,1,2,\ldots \}\). These are the sets of allowable occupation numbers of a state in the fermionic and bosonic cases, respectively. (Recall that the cases \(\zeta = -1\) and \(\zeta = +1\) indicate, respectively, the cases of fermions and bosons.)

Let d be a positive integer, the number of states, and consider the collection:

$$\begin{aligned} {\mathcal {B}}_{\zeta ,d} = \left\{ \vert n_1, n_2, \ldots , n_d \rangle {:}\, n_i \in {\mathcal {N}}_\zeta \right\} . \end{aligned}$$

This set will be the occupation number basis for our Fock space \({\mathcal {F}}_{\zeta ,d}\). For short, we may indicate the basis elements by \(\vert {\mathbf {n}} \rangle \) for \({\mathbf {n}} \in {\mathcal {N}}_{\zeta }^{d}\).

To define the Fock space, consider the set \({\mathcal {V}}_{\zeta ,d}\) of finite formal linear combinations of elements of \({\mathcal {B}}_{\zeta ,d}\). Then, \({\mathcal {V}}_{\zeta ,d}\) is a vector space, and it can be endowed with an inner product by stipulating that the elements of \({\mathcal {B}}_{\zeta ,d}\) are orthonormal. (Hence, \({\mathcal {B}}_{\zeta ,d}\) is an orthonormal basis of \({\mathcal {V}}_{\zeta ,d}\).) For fermions, \({\mathcal {V}}_{\zeta ,d}\) is finite-dimensional and therefore a Hilbert space, but this is not the case for bosons. Therefore, we define \({\mathcal {F}}_{\zeta ,d}\) to be the completion of \({\mathcal {V}}_{\zeta ,d}\) with respect to the metric induced by its inner product, so \({\mathcal {F}}_{\zeta ,d}\) is a Hilbert space, and \({\mathcal {B}}_{\zeta ,d}\) is a complete orthonormal set (in fact, a basis if \(\zeta = -1\)).

In accordance with Dirac’s bra–ket notation, we will denote elements of \({\mathcal {F}}_{\zeta ,d}\) with the notation \(\vert \phi \rangle \) (where \(\phi \) can be thought of as a symbolic label), and we denote the adjoint of an element \(\vert \phi \rangle \in {\mathcal {F}}_{\zeta ,d}\) by \(\langle \phi \vert \). Inner products may then be denoted \(\langle \psi \vert \phi \rangle \), and we denote the induced norm on \({\mathcal {F}}_{\zeta ,d}\) by \(\Vert \phi \Vert = \sqrt{ \langle \phi \vert \phi \rangle }\).

The reader should be careful to distinguish between the vacuum state\(\vert 0,\ldots ,0\rangle \), denoted \(\vert -\rangle \) for short, and the zero vector of \({\mathcal {F}}_{\zeta ,d}\), denoted simply as 0,Footnote 7 which is the linear combination of elements of \({\mathcal {B}}_{\zeta ,d}\) in which all coefficients are zero. In particular, the vacuum state has norm 1 and the zero vector has norm 0.

1.2 A.2 Creation and Annihilation Operators

The annihilation operators \(a_i\) are linear operators \({\mathcal {V}}_{\zeta ,d} \rightarrow {\mathcal {V}}_{\zeta ,d}\), defined by their action on the basis \({\mathcal {B}}_{\zeta ,d}\):

$$\begin{aligned} a_i \vert {\mathbf {n}} \rangle = {\left\{ \begin{array}{ll} 0, &{}\quad n_i =0 \\ \zeta ^{\sum _{j<i} n_j} \sqrt{n_i}\ \vert n_1, \ldots , n_{i-1}, n_i - 1, n_{i+1} \ldots n_d \rangle , &{}\quad n_i \ne 0. \end{array}\right. } \end{aligned}$$

Meanwhile, the creation operators \(a_i^\dagger \) are linear operators \({\mathcal {V}}_{\zeta ,d} \rightarrow {\mathcal {V}}_{\zeta ,d}\), defined by their action on the basis \({\mathcal {B}}_{\zeta ,d}\):

$$\begin{aligned} a_i^\dagger \vert {\mathbf {n}} \rangle = {\left\{ \begin{array}{ll} 0, &{}\quad \zeta = -1, n_i = 1 \\ \zeta ^{\sum _{j<i} n_j} \sqrt{n_i + 1}\ \vert n_1, \ldots , n_{i-1}, n_i + 1, n_{i+1} \ldots n_d \rangle , &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

In the case of fermions, \({\mathcal {V}}_{\zeta ,d} = {\mathcal {F}}_{\zeta ,d}\) is finite-dimensional, so the creation and annihilation operators are defined on \({\mathcal {F}}_{\zeta ,d}\), and they are in fact Hermitian adjoints of one another in the usual sense as the notation suggests. Moreover, \(a_i\) and \(a_i^\dagger \) have operator norm 1, which in particular remains bounded in the limit of infinitely many states. In fact, as this observation suggests, the appropriate creation and annihilation operators on fermionic Fock spaces generated by infinitely many states (which we do not define or consider these here) are bounded operators with operator norm 1.

By contrast, in the case of bosons, \(a_i\) and \(a_i^\dagger \) are unbounded operators, even for finite d. Thus, \(a_i\) and \(a_i^\dagger \) are only densely defined (unbounded) operators on \({\mathcal {F}}_{\zeta ,d}\). In fact, adjoint operators can be defined even for operators that are only densely defined on a Hilbert space [11, 21], and in this sense, \(a_i\) and \(a_i^\dagger \) are Hermitian adjoints of one another. For the reader familiar only with adjoints of bounded operators, one can merely consider the ‘\(\dagger \)’ as a notation.

The most important feature of the creation and annihilation operators is the (anti-)commutation relations. Denoting commutator of operators AB by \([A,B]_{+1} := AB - BA\) and the anticommutator by \([A,B]_{-1} = AB + BA\), we have

$$\begin{aligned}&{}[a_i, a_j]_{\zeta } = [a_i^\dagger , a_j^\dagger ]_{\zeta } = 0,\quad [a_i,a_j^\dagger ]_{\zeta } = \delta _{ij} \,\mathrm {Id}, \end{aligned}$$
(A.1)

on \({\mathcal {V}}_{\zeta ,d}\). These relations can be readily verified from the definitions of \(a_i\) and \(a_i^\dagger \).

We say that a composition of creation and annihilation operators such as \(a_i^\dagger a_j\) is normally ordered if all of the creation operators appear to the left of all of the annihilation operators. Any composition of creation and annihilation operators can be converted to a linear combination of normally ordered operators via the (anti)commutation relations.

For a unitary transformation \(T{:}\,{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\), one can define new operators \({\tilde{a}}_i^\dagger = \sum _{j=1}^d T_{ij} a_j^\dagger \) and \({\tilde{a}}_i = \sum _{j=1}^d {\overline{T}}_{ij} a_j\). These can be viewed as creation and annihilation operators, respectively, in that they satisfy the same commutation relations as in (A.1). One can in turn view these as generators for our Fock space inducing a different occupation number basis.

1.3 A.3 Number Operators and Eigenspaces

For each state, we define a number operator

$$\begin{aligned} {\hat{n}}_i := a_i^\dagger a_i, \end{aligned}$$

which is a linear operator \({\mathcal {V}}_{\zeta ,d} \rightarrow {\mathcal {V}}_{\zeta ,d}\). In the case of bosons, \({\hat{n}}_i\) can be viewed as an unbounded, self-adjoint, densely defined operator on \({\mathcal {F}}_{\zeta ,d}\). Note that the number operators all commute, i.e., \([n_i, n_j]_{+} = 0\) for all ij.

We also define the total number operator by

$$\begin{aligned} {\hat{N}} := \sum _{i=1}^d {\hat{n}}_i. \end{aligned}$$

The set of eigenvectors of \({\hat{n}}\) (as a linear transformation \({\mathcal {V}}_{\zeta ,d} \rightarrow {\mathcal {V}}_{\zeta ,d}\)) is precisely \({\mathcal {B}}_{\zeta ,d}\), and each eigenvector \(\vert {\mathbf {n}} \rangle \) has eigenvalue \(\sum _{i=1}^d n_i\). Thus, the set of eigenvalues is given by \(\{0,1,\ldots ,d\}\) in the case of fermions and \(\{0,1,\ldots \}\) in the case of bosons.

Then, we define the N-particle subspace to be the N-eigenspace of \({\hat{N}}\), which is finite-dimensional (even for bosons), and we denote it by \({\mathcal {F}}_{\zeta ,d}^{(N)}\). Then, we can write

$$\begin{aligned} {\mathcal {V}}_{\zeta ,d} = \bigoplus _{N=0}^{\infty } {\mathcal {F}}_{\zeta ,d}^{(N)}. \end{aligned}$$

The N-eigenspace is understood to be \(\{ 0\}\) for any integer \(N \notin \{0,1,\ldots ,d\}\) in the case of fermions and for any \(N \notin \{0,1,\ldots \}\) in the case of bosons. Notice that \(a_i\) maps \({\mathcal {F}}_{\zeta ,d}^{(N)} \rightarrow {\mathcal {F}}_{\zeta ,d}^{(N-1)}\) and \(a_i^\dagger \) maps \({\mathcal {F}}_{\zeta ,d}^{(N)} \rightarrow {\mathcal {F}}_{\zeta ,d}^{(N+1)}\).

We say that an operator A on \({\mathcal {F}}_{\zeta ,d}\)conserves particle number if A maps \({\mathcal {F}}_{\zeta ,d}^{(N)} \rightarrow {\mathcal {F}}_{\zeta ,d}^{(N)}\) for all integers N. Evidently any operator such as \(a_i^\dagger a_j\) in which an equal number of creation and annihilation operators appear, as well as any sum of such operators, must conserve particle number.

1.4 A.4 Hamiltonians

For convenience we shall let \(a = (a_1, \ldots , a_d)\) denote the vector of annihilation operators, and accordingly \(a^\dagger = (a_1^\dagger , \ldots , a_d^\dagger )^T\). Then, consider a Hamiltonian\({\hat{H}} = H(a^\dagger , a)\) that is a normally ordered polynomial of creation and annihilation operators. As an operator on \({\mathcal {V}}_{\zeta ,d}\), we stipulate that \({\hat{H}}\) commutes with the total number operator \({\hat{N}}\). Hence \({\hat{H}}\) conserves particle number and is an operator \({\mathcal {F}}_{\zeta ,d}^{(N)} \rightarrow {\mathcal {F}}_{\zeta ,d}^{(N)}\) for all N, and we demand that \({\hat{H}}\) is self-adjoint as such.

In general, we can write

$$\begin{aligned} {\hat{H}} = {\hat{H}}_0 + {\hat{U}}, \end{aligned}$$

where

$$\begin{aligned} {\hat{H}}_0 := a^\dagger h a = \sum _{i,j=1}^d h_{ij} a_i^\dagger a_j \end{aligned}$$

is the single-particle (or non-interacting) part of the Hamiltonian, specified by a Hermitian \(d\times d\) matrix h, and \({\hat{U}} = U(a^\dagger , a)\) is the interacting part. Here \({\hat{U}}\) is normally ordered and commutes with \({\hat{N}}\) (since \({\hat{H}}_0\) does), and moreover, \({\hat{U}}\) is self-adjoint on \({\mathcal {F}}_{\zeta ,d}^{(N)}\) for all N (since \({\hat{H}}_0\) is).

Via a unitary transformation of the creation and annihilation operators, one can without loss of generality assume that h is diagonal. However, the utility of this manipulation is limited outside of the non-interacting setting because such a transformation may complicate the representation of the interaction term \({\hat{U}}\).

Though we need not define \({\hat{U}}\) more explicitly for the purposes of this paper, for concreteness one might keep in mind the two-body interaction

$$\begin{aligned} {\hat{U}} = \frac{1}{2} \sum _{ijkl} (ij \vert U \vert kl) a_i^\dagger a_j^\dagger a_l a_k. \end{aligned}$$
(A.2)

We comment more concretely on how such a second-quantized two-body operator may arise from the finite-dimensional approximation of a two-body potential in real space. To construct a finite-state Fock space, one first replaces the single-particle Hilbert space \(\mathcal {H} := H^1({\mathbb {R}}^3,\pm \frac{1}{2}; {\mathbb {C}}) \subset L^2({\mathbb {R}}^3, \pm \frac{1}{2}; {\mathbb {C}})\) with a finite-dimensional subspace \(\mathcal {H}_{\mathcal {D}}\) spanned by an orthonormal single-particle basis \(\mathcal {D} := \{ \varphi _1 ,\ldots , \varphi _d\}\). One then defines the N-particle space as \(\mathcal {F}^{(N)}_{\zeta ,\mathcal {D}} := \mathbf {\Lambda }^N (\mathcal {H}_{\mathcal {D}})\) if \(\zeta = -1\) and as \(\mathcal {F}^{(N)}_{\zeta ,\mathcal {D}} := {\mathbf {S}}^N (\mathcal {H}_{\mathcal {D}})\) if \(\zeta = +1\), where \(\mathbf {\Lambda }^N\) and \({\mathbf {S}}^N\) denote the Nth exterior and symmetric powers, respectively. Then \(\mathcal {F}_{\zeta ,\mathcal {D}}^{(N)}\) so constructed is isomorphic to \(\mathcal {F}_{\zeta ,d}^{(N)}\) as above via, in the case \(\zeta = -1\), the isomorphism \(\vert {\mathbf {n}} \rangle \mapsto \bigwedge _{i=1}^d \varphi _{i}^{\wedge n_i} \), where the wedge in the exponent indicates a wedge power. The analogous isomorphism holds in the case \(\zeta = +1\), with wedges replaced by symmetric products. A change of the basis \(\mathcal {D}\) to some \(\tilde{\mathcal {D}} = \{ {\tilde{\varphi }}_1 ,\ldots , {\tilde{\varphi }}_d \}\) that is induced by a unitary transformation \(T: {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) corresponds to a transformation of the annihilation operators by \({\tilde{a}}_i = \sum _{i=1}^d {\overline{T}}_{ij} a_j\).

Under this correspondence, a translation-invariant two-body potential \(v(x - y)\) in real space yields the tensor elements \((ij \vert U \vert kl)\) can be computed via the following two-body integrals [19]

$$\begin{aligned} (ij \vert U \vert kl) = \int _{{\mathbb {R}}^3} \int _{{\mathbb {R}}^3} v(x-y) \varphi _i^*(x) \varphi _j^* (y) \varphi _k (x) \varphi _l (y)\,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$
(A.3)

The elements of h are obtained by suitable one-body integrals; see, e.g., [19].

In the case that \({\hat{U}} = U(a^\dagger , a)\) depends only on \(a_i^\dagger , a_i\) for \(i=1,\ldots , p\), we say that \({\hat{H}}\) is an impurity Hamiltonian, with a fragment specified by the indices \(1,\ldots ,p\). The rest of the indices correspond to the environment.

B The Zero-Temperature Ensemble

At zero temperature, typically one first fixes a particle number N, and attention is restricted to the N-particle subspace. Let \(\big \vert \Psi _0^{(N)} \big \rangle \in \mathcal {F}_{\zeta ,d}^{(N)}\) be the N-particle ground state of \({\hat{H}}\), i.e., the minimal normalized eigenvector of \({\hat{H}}\) considered as an operator on the N-particle subspace. The role of the density operator is assumed by the orthogonal projector \(\big \vert \Psi _0^{(N)} \big \rangle \big \langle \Psi _0^{(N)} \big \vert \) onto the ground state \(\big \vert \Psi _0^{(N)} \big \rangle \), i.e., the statistical average of a linear operator \({\hat{A}}\) (with respect to the N-particle canonical ensemble) is given by

$$\begin{aligned} \langle {\hat{A}} \rangle _{N} = \big \langle \Psi _0^{(N)} \big \vert {\hat{A}} \big \vert \Psi _0^{(N)} \big \rangle . \end{aligned}$$

1.1 B.1 Green’s Functions and the Self-energy at Zero Temperature

For \(t \in {\mathbb {R}}\), we denote the annihilation and creation operators in the Heisenberg representation by

$$\begin{aligned} a_i(t) := e^{i {\hat{H}} t} a_i e^{-i {\hat{H}} t}, \quad a_i^\dagger (t) := e^{i {\hat{H}} t} a_i^\dagger e^{-i {\hat{H}} t}. \end{aligned}$$
(B.1)

Then, for a zero-temperature ensemble with N particles, the time-ordered, single-body, real-time Green’s function (which we call the Green’s function for short) is a function \(G{:}\,{\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} G_{ij} (t,t') = -i\, \big \langle \Psi _0^{(N)} \big \vert \,{\mathcal {T}}\big \{ a_i(t) a_j^\dagger (t') \big \} \,\big \vert \Psi _0^{(N)} \big \rangle , \end{aligned}$$
(B.2)

where \(\mathcal {T}\) is the time-ordering operator, formally defined by

$$\begin{aligned} {\mathcal {T}}\big \{ a_i(t) a_j^\dagger (t') \big \} = {\left\{ \begin{array}{ll} a_{i}(t)a_{j}^{\dagger }(t'), &{}\quad t' < t\\ \zeta a_{j}^{\dagger }(t') a_{i}(t), &{}\quad t' \ge t. \end{array}\right. } \end{aligned}$$

Note that \({\mathcal {T}}\) is not really an operator and it is interpreted merely via the symbolic content of its argument.

We can write

$$\begin{aligned} G(t,t') = G^{+} (t,t') + G^{-} (t,t'), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} i G^{+}_{ij} (t,t')&:= \big \langle \Psi _0^{(N)} \big \vert a_{i}(t)a_{j}^{\dagger }(t') \big \vert \Psi _0^{(N)} \big \rangle \theta (t-t'), \\ i G^{-}_{ij} (t,t')&:= \zeta \big \langle \Psi _0^{(N)} \big \vert a_{j}^{\dagger }(t') a_{i}(t) \big \vert \Psi _0^{(N)} \big \rangle (1-\theta (t-t')), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \theta (s) := {\left\{ \begin{array}{ll} 1, &{}\quad s > 0\\ 0, &{}\quad s \le 0. \end{array}\right. } \end{aligned}$$
(B.3)

It is easy to show that \(G(t,t')\), \(G^{+}(t,t')\), and \(G^{-}(t,t')\) depend only on \(t-t'\), so we can define \(G(t) := G(t,0)\), \(G^{+}(t): =G^{+}(t,0)\), and \(G^{-}(t): =G^{-}(t,0)\) and consider these objects without any loss of information. It is then equivalent to consider the Fourier transforms

$$\begin{aligned} G(\omega ) := \int _{\mathbb {R}}G(t) e^{i\omega t - \eta \vert t \vert } \,\mathrm{d}t \end{aligned}$$

and likewise \(G^{+}(\omega )\) and \(G^{-}(\omega )\) defined similarly, so

$$\begin{aligned} G(\omega ) = G^{+}(\omega ) + G^{-}(\omega ). \end{aligned}$$

Here \(\eta \) is interpreted as a positive, infinitesimally small quantity needed to ensure the convergence of the relevant integrals, and \(G(\omega )\), \(G^{+}(\omega )\), and \(G^{-}(\omega )\) are not really functions, but rather distributions on \({\mathbb {R}}\) defined via the limit \(\eta \rightarrow 0^+\).

One can show that

$$\begin{aligned} G_{ij}^{+} (\omega ) = \big \langle \Psi _0^{(N)} \big \vert a_i \frac{1}{\omega - ({\hat{H}} - E_0^{(N)}) + i \eta } a_j^{\dagger } \big \vert \Psi _0^{(N)} \big \rangle \end{aligned}$$

and

$$\begin{aligned} G_{ij}^{-} (\omega ) = -\zeta \big \langle \Psi _0^{(N)} \big \vert a_j^{\dagger } \frac{1}{\omega + ({\hat{H}} - E_0^{(N)}) - i \eta } a_i \big \vert \Psi _0^{(N)} \big \rangle , \end{aligned}$$

where \(E_0^{(N)}\) is the energy of the N-particle ground state, i.e., \({\hat{H}} \big \vert \Psi _0^{(N)} \big \rangle = E_0 \big \vert \Psi _0^{(N)} \big \rangle \).

Now we can think of \(G^{\pm }\) as the restriction to the real axis of the rational function \(G^{\pm }{:}\,{\mathbb {C}} \rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} \begin{aligned} G_{ij}^{+} (z)&:= \big \langle \Psi _0^{(N)} \big \vert a_i \frac{1}{z - ({\hat{H}} - E_0^{(N)}) } a_j^{\dagger } \big \vert \Psi _0^{(N)} \big \rangle \\ G_{ij}^{-} (z)&:= \ -\zeta \big \langle \Psi _0^{(N)} \big \vert a_j^{\dagger } \frac{1}{z + ({\hat{H}} - E_0^{(N)}) } a_i \big \vert \Psi _0^{(N)} \big \rangle , \end{aligned} \end{aligned}$$

and we can define \(G(z) := G^{+}(z) + G^{-}(z)\) accordingly to be rational on \({\mathbb {C}}\).

Note that here we have left out the \(\pm i \eta \) in the denominators, which specified whether poles should be viewed as being infinitesimally above or below the real axis. This erases the distinction between the time-ordered Green’s function and the advanced and retarded Green’s functions, which we do not define here, though see [19] for details. In fact, the distinction does not matter for our sparsity results, which applies equally well in all of these cases.

The self-energy is the rational function \(\Sigma {:}\,{\mathbb {C}} \rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} \Sigma (z) := z - h - G(z)^{-1}. \end{aligned}$$

C The Finite-Temperature Ensemble

At inverse temperature \(\beta \in (0,\infty )\), the partition function is defined by

$$\begin{aligned} Z := \mathrm {Tr}[e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}]. \end{aligned}$$
(C.1)

where ‘\(\mathrm {Tr}\)’ indicates the Fock space trace. Here \(\mu \in {\mathbb {R}}\) is the chemical potential, but before commenting on its role, some further elaboration on the trace is owed in the bosonic case, in which the Fock space is infinite-dimensional.

By assumption, \({\hat{H}}\) conserves particle number, i.e., it maps \(\mathcal {F}_{\zeta ,d}^{(N)}\) to itself for all N. Thus, \(e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}\) does as well and can be viewed as a positive-definite operator on each \(\mathcal {F}_{\zeta ,d}^{(N)}\). The trace can then be constructed as

$$\begin{aligned} \mathrm {Tr}[e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}] = \sum _{N=0}^\infty \mathrm {Tr}_{N} [e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}] = \sum _{N=0}^\infty e^{\beta \mu N} \, \mathrm {Tr}_{N} [e^{-\beta {{\hat{H}}}}], \end{aligned}$$

where ‘\(\mathrm {Tr}_N\)’ indicates the trace on the N-particle subspace. Since each of the summands is positive, \(\mathrm {Tr}[e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}] \in (0,+\infty ]\) is well defined.

More generally, the trace is defined for all operators in the trace class of \(\mathcal {F}_{\zeta ,d}\), i.e., the set of bounded linear operators \({\hat{O}}{:}\,\mathcal {F}_{\zeta ,d} \rightarrow \mathcal {F}_{\zeta ,d}\) for which

$$\begin{aligned} \sum _{ {\mathbf {n}} \in \mathcal {N}_{\zeta }^d} \langle {\mathbf {n}} \vert \,({\hat{O}}^\dagger {\hat{O}})^{1/2} \, \vert {\mathbf {n}} \rangle < +\infty , \end{aligned}$$

in which case

$$\begin{aligned} \mathrm {Tr}[{\hat{O}}] = \sum _{ {\mathbf {n}} \in \mathcal {N}_{\zeta }^d} \langle {\mathbf {n}} \vert {\hat{O}} \vert {\mathbf {n}} \rangle . \end{aligned}$$

See, e.g., [20] for further details on trace class operators.

Now since the partition function can be viewed as a normalization factor, the scenario \(Z = +\infty \) is to be avoided. It is now that we turn to the chemical potential. We can view Z as defined above as a function of \(\mu \). Evidently \(\mu \mapsto Z(\mu )\) is non-decreasing.

First we want to rule out the case that \(Z \equiv +\infty \). Unfortunately, this case cannot be ruled out without further assumptions! To see why, suppose that \(d = 1\) (so write \(a = a_1\)), and let \({\hat{H}} = -a^\dagger a - a^\dagger a^\dagger a a = - a^\dagger a a^\dagger a = - \hat{N}^2\). Then,

$$\begin{aligned} \mathrm {Tr}[e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}]= & {} \sum _{N=0}^\infty e^{\beta (N^2 + \mu N)}\, \mathrm {Tr}_{N} \left[ \mathrm {Id}_{\mathcal {F}_{\zeta ,d}^{(N)}}\right] \\= & {} \sum _{N=0}^\infty e^{\beta (N^2 + \mu N)}\,\left( {\begin{array}{c}N + d-1\\ d-1\end{array}}\right) = +\infty , \end{aligned}$$

for all\(\mu \in {\mathbb {R}}\).

We conclude that such a choice of \({\hat{H}}\) is unphysical, and to rule out such pathologies, we adopt the following:

Assumption C.1

We assume, in the case of bosons, that there exist some positive integer \(N_0\) and some \(\mu \in {\mathbb {R}}\) such that \({\hat{H}} - \mu {\hat{N}} \succeq 0\) as an operator on all N-particle subspaces for \(N\ge N_0\). (It is equivalent to require that there exist \(N_0, \mu \) such that \({\hat{U}} - \mu {\hat{N}} \succeq 0\) on all N-particle subspaces for \(N\ge N_0\).)

This condition is satisfied in particular if \({\hat{U}}\) is a two-body interaction as in (A.2) such that \(U_{ik,jl} := (kj\vert U\vert il)\), interpreted as a \(d^2 \times d^2\) matrix, is positive semidefinite. Indeed, in this case, \({\hat{U}}\) is equal to (up to a single-body term)

$$\begin{aligned} \frac{1}{2} \sum _{ijkl} U_{ik,jl} \left[ a_i^\dagger a_k\right] ^\dagger \left[ a_j^\dagger a_l\right] \succeq 0. \end{aligned}$$

If the \((ij\vert U\vert kl)\) are derived from a real-space two-body potential v that is a positive semidefinite function (i.e., has nonnegative Fourier transform), then it follows from (A.3) that the matrix \((U_{ik,jl})\) is positive definite as desired. Note that it is possible for v to be positive definite but take negative values at long ranges, i.e., v can act attractively at long range.

Now that we have argued that Assumption C.1 is natural, let us see how it guarantees that the set \(\mathrm {dom}\,Z := \{ \mu {:}\, Z(\mu ) < +\infty \}\) is non-empty. Indeed, choose \(\mu '\) negative enough such that \({\hat{H}}- \mu ' {\hat{N}} \succeq 0\) as an operator on all N-particle subspaces, and let \(\mu = \mu ' - \delta \), where \(\delta >0\). Then,

$$\begin{aligned} \mathrm {Tr}[e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}] \le \sum _{N=0}^\infty \mathrm {Tr}_{N} [e^{-\beta \delta {\hat{N}}}] = \sum _{N=0}^\infty e^{-\beta \delta N} \left( {\begin{array}{c}N + d-1\\ d-1\end{array}}\right) . \end{aligned}$$

Now the binomial coefficient in the last expression is \(O(N^{d-1})\) as \(N\rightarrow +\infty \), so the sum converges.

We will always assume in the finite-temperature setting that \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\). It can be shown that if \({\hat{U}} = 0\), then \(\mathrm {dom}\,Z = \{ \mu {:}\, h \succ \mu \,I_d \}\). Moreover, if there exist \(N_0, \delta > 0\) such that \({\hat{U}} \succeq \delta {\hat{N}}^2\) on all N-particle subspaces for \(N \ge N_0\) (which holds in particular if \({\hat{U}}\) is a two-body interaction as in (A.2) where the \(d^2 \times d^2\) matrix \(U_{ki,jl} := (ij\vert U\vert kl)\) is positive definite), then \(\mathrm {dom}\,Z = {\mathbb {R}}\).

Notice that if \(\mathrm {dom}\,Z\) is open, then since Z is increasing we can write \(\mathrm {dom}\,Z = (-\infty , \mu _{\mathrm {c}})\) for some \(\mu _{\mathrm {c}}\) possibly infinite. If \(\mu _{\mathrm {c}} < +\infty \), then by Fatou’s lemma we have that \(\liminf _{\mu \rightarrow \mu _{\mathrm {c}}^-} Z(\mu ) \ge Z(\mu _{\mathrm {c}}) = +\infty \), so \(Z(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}^-\). (And in any case it follows from the definition of Z that \(Z(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow +\infty \), so we can write more compactly that \(Z(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}\), no matter whether \(\mu _{\mathrm {c}}\) is finite or infinite.)

The grand canonical ensemble is defined by the density operator

$$\begin{aligned} \rho := Z^{-1} e^{-\beta ({{\hat{H}}} - \mu {\hat{N}})}, \end{aligned}$$

and the statistical average of an operator \({\hat{A}}\) with respect to the grand canonical ensemble is denoted

$$\begin{aligned} \langle {\hat{A}} \rangle _{\beta ,\mu } = \mathrm {Tr}[ {\hat{A}} \rho ] \end{aligned}$$

whenever \({\hat{A}} \rho \) is in the trace class. Note that if \({\hat{A}}\) conserves particle number, then

$$\begin{aligned} \mathrm {Tr}[{\hat{A}} \rho ] = \sum _{N=0}^\infty \mathrm {Tr}_{N} [{\hat{A}} \rho ] = Z^{-1} \sum _{N=0}^\infty e^{\beta \mu N} \, \mathrm {Tr}_{N} [{\hat{A}} e^{-\beta {\hat{H}}}] \end{aligned}$$

is defined under the condition that the sum is absolutely convergent, which holds in particular if the norm of \({\hat{A}}\) as an operator on the N-particle subspace grows only polynomially with N, via the assumption that \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\). When the context is clear we simply write \(\langle \, \cdot \, \rangle \).

Of particular interest is the expected particle number

$$\begin{aligned} \langle {\hat{N}} \rangle = \frac{\sum _{N=0}^\infty N e^{\beta \mu N} \, \mathrm {Tr}_{N} [e^{-\beta {\hat{H}}}] }{\sum _{N=0}^\infty e^{\beta \mu N} \, \mathrm {Tr}_{N} [e^{-\beta {\hat{H}}}] }. \end{aligned}$$

Observe that \(\langle {\hat{N}} \rangle _{\beta ,\mu } \rightarrow 0\) as \(\mu \rightarrow -\infty \). Also note that if \(\mathrm {dom}\,Z = {\mathbb {R}}\), then \(\langle {\hat{N}} \rangle _{\beta ,\mu } \rightarrow +\infty \). Defining the free energy\(\Omega (\mu ) := \beta ^{-1} \log Z(\mu )\), we see that \(\langle {\hat{N}} \rangle _{\beta ,\mu } = \Omega '(\mu )\).

It is not hard to check that \(\Omega \) is (strictly) convex, i.e., \(\langle {\hat{N}} \rangle _{\beta ,\mu }\) is increasing in \(\mu \) for \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\). Recall that if \(\mathrm {dom}\,Z = (0,\mu _{\mathrm {c}})\), then \(Z(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}\), hence \(\Omega (\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}\). If \(\mu _{\mathrm {c}} < +\infty \), it follows that \(\Omega '(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}^-\). (Otherwise, since \(\Omega '\) is increasing, it approaches a finite limit \(\mu \rightarrow \mu _{\mathrm {c}}^-\). But in this case it would follow from the fundamental theorem of calculus that \(\Omega \) approaches a finite limit as well: contradiction.) In summary, we have established that if \(\mathrm {dom}\,Z\) is open, then \(Z(\mu ) \rightarrow +\infty \) as \(\mu \rightarrow \mu _{\mathrm {c}}\), no matter whether \(\mu _{\mathrm {c}}\) is finite or infinite. It follows that in this case \(\mu \mapsto \langle {\hat{N}}_{\beta ,\mu } \rangle \) is a bijection from \(\mathrm {dom}\,Z = (-\infty , \mu _{\mathrm {c}})\) to \((0,+\infty )\). Thus, one can select the chemical potential \(\mu \) to yield a predetermined expected particle number.

1.1 C.1 Green’s Functions and the Self-energy at Finite Temperature

Recall our definition (B.1) of the annihilation and creation operators \(a_i(t)\) and \(a_i^\dagger (t)\) in the Heisenberg representation. Then, at finite inverse temperature \(\beta \in (0,\infty )\) and chemical potential \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\), the time-ordered, single-body, real-time Green’s function (which we call the Green’s function for short when the context is clear) is a function \(G{:}\,{\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} G_{ij} (t,t') = -i\, \big \langle \, {\mathcal {T}}\big \{ a_i(t) a_j^\dagger (t') \big \} \,\big \rangle _{\beta ,\mu }. \end{aligned}$$

We can write

$$\begin{aligned} G(t,t') = G^{+} (t,t') + G^{-} (t,t'), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} i G^{+}_{ij} (t,t')&= \frac{1}{Z} \mathrm {Tr}\left[ a_{i}(t)a_{j}^{\dagger }(t') e^{-\beta ({\hat{H}} - \mu {\hat{N}})}\right] \theta (t-t'), \\ i G^{-}_{ij} (t,t')&= \frac{\zeta }{Z} \mathrm {Tr}\left[ a_{j}^{\dagger }(t') a_{i}(t) e^{-\beta ({\hat{H}} - \mu {\hat{N}})}\right] (1-\theta (t-t')), \end{aligned} \end{aligned}$$

where \(\theta \) is defined as above in (B.3).

Once again it is easy to show that \(G(t,t')\), \(G^{+}(t,t')\), and \(G^{-}(t,t')\) depend only on \(t-t'\), so we can define \(G(t) := G(t,0)\), \(G^{+}(t): =G^{+}(t,0)\), and \(G^{-}(t): =G^{-}(t,0)\) and consider these objects without any loss of information. It is then equivalent to consider the Fourier transforms

$$\begin{aligned} G(\omega ) := \int _{\mathbb {R}}G(t) e^{i\omega t - \eta \vert t\vert } \,\mathrm{d}t \end{aligned}$$

and likewise \(G^{+}(\omega )\) and \(G^{-}(\omega )\) defined similarly, so

$$\begin{aligned} G(z) = G^{+}(\omega ) + G^{-}(\omega ). \end{aligned}$$

Now since \({\hat{H}}\) preserves particle number, we can safely diagonalize \({\hat{H}}\) as an operator on each of the N-particle subspaces separately. Then, the spectrum of \({\hat{H}}\) consists of the union of its spectra on the N-particle subspaces. It follows from Assumption C.1 that \({\hat{H}} - \mu {\hat{N}}\) has a ground state, i.e., that its spectrum is bounded from below, for \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\). Let \(m=0,1,\ldots , \) (terminating at \(m=2^d\) in the case of fermions) index the spectrum of \({\hat{H}}\), and let \(\vert \Psi _m \rangle \) denote the mth eigenstate. Let \(N_m\) be the particle number of \(\vert \Psi _m \rangle \) (which is an eigenstate of \({\hat{N}}\)), and let \(E_m\) be defined by \( {\hat{H}} \vert \Psi _m \rangle = E_m \vert \Psi _m \rangle \).

One can show that

$$\begin{aligned} G_{ij}^{+} (\omega ) = \frac{1}{Z} \sum _m e^{-\beta (E_m - \mu N_m)} \big \langle \Psi _m \big \vert a_i \frac{1}{\omega - ({\hat{H}}- E_m) + i \eta } a_j^{\dagger } \big \vert \Psi _m \big \rangle \end{aligned}$$

and

$$\begin{aligned} G_{ij}^{-} (\omega ) = \frac{-\zeta }{Z} \sum _m e^{-\beta (E_m - \mu N_m)} \big \langle \Psi _m \big \vert a_j^{\dagger } \frac{1}{\omega + ({\hat{H}}- E_m) - i \eta } a_i \big \vert \Psi _m \big \rangle . \end{aligned}$$

Recall from (C.1) that

$$\begin{aligned} Z = \sum _m e^{-\beta (E_m - \mu N_m)}. \end{aligned}$$

Now we can think of \(G^{\pm }\) as the restriction to the real axis of the rational function \(G^{\pm }{:}\,{\mathbb {C}} \rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} \begin{aligned} G_{ij}^{+} (z)&:= \frac{1}{Z} \sum _m e^{-\beta (E_m - \mu N_m)} \big \langle \Psi _m \big \vert a_i \frac{1}{z - ({\hat{H}}- E_m) } a_j^{\dagger } \big \vert \Psi _m \big \rangle \\ G_{ij}^{-} (z)&:= \frac{-\zeta }{Z} \sum _m e^{-\beta (E_m - \mu N_m)} \big \langle \Psi _m \big \vert a_j^{\dagger } \frac{1}{z + ({\hat{H}}- E_m) } a_i \big \vert \Psi _m \big \rangle , \end{aligned} \end{aligned}$$

and we can define \(G(z) := G^{+}(z) + G^{-}(z)\) accordingly to be rational on \({\mathbb {C}}\). Once again we have ignored the infinitesimal \(\eta \) in this definition; the same comments made in ‘Appendix B’ apply here.

The self-energy is the rational function \(\Sigma {:}\,{\mathbb {C}} \rightarrow {\mathbb {C}}^{d\times d}\) defined by

$$\begin{aligned} \Sigma (z) := z - h - G(z)^{-1}. \end{aligned}$$

D Non-equilibrium Setting and the Kadanoff–Baym Contour

Here we briefly discuss one main non-equilibrium setting of interest, called the Kadanoff–Baym formalism. One considers an initial time \(t_0\) and a final time \(t_1\), with \(t_1 > t_0\), and for \(t \in [t_0,t_1]\), \({\hat{H}}(t)\) denotes the Hamiltonian at time t. This Hamiltonian determines the evolution, starting at time \(t_0\), of a prepared grand canonical ensemble defined by a density operator \(\rho \), i.e., a positive semidefinite operator on the Fock space of unit trace. Assuming, for simplicity, strict positive definiteness, we can write

$$\begin{aligned} \rho = \frac{1}{\mathrm {Tr}[e^{-\beta {\overline{H}}}]} e^{- \beta {\overline{H}}} \end{aligned}$$

for some Hamiltonian \({\overline{H}}\) and inverse temperature \(\beta \). Of course, this form leaves freedom in choosing \(\beta \), but it is good to think of \(\beta \) as a free parameter. Often \({\overline{H}}\) may be thought of as \({\hat{H}}(t_0) - \mu {\hat{N}}\), but this need not be the case. To ensure that Assumption 3.5 holds, it will suffice to assume that \(\mathrm {Tr}[e^{-\beta {\overline{H}} + \varepsilon {\hat{N}} }] < +\infty \) for some \(\varepsilon > 0\) sufficiently small. This condition is analogous to the condition \(\mu \in \mathrm {int}\,\mathrm {dom}\,Z\) discussed in Appendix C for the equilibrium finite-temperature ensemble. Assuming the condition, let \({\hat{O}}_N\) denote the restriction of \(e^{-\beta {\overline{H}}}\) to the N-particle subspace. Then, it follows that \(\mathrm {Tr}[{\hat{O}}_N]\) decays exponentially in N; hence, \(\Vert {\hat{O}}_N \Vert _2\) does as well.

Here the contour is the Kadanoff–Baym contour \({\mathcal {C}}^{\mathrm {KB}}\), specified by the path \(\gamma ^{\mathrm {KB}}\), which can be written as a concatenation

$$\begin{aligned} \gamma ^{\mathrm {KB}} = \gamma ^{-} + \gamma ^{+} + \gamma ^{\mathrm {M}}. \end{aligned}$$

Here \(\gamma ^{-}:(0,t_1-t_0) \rightarrow {\mathbb {C}}\) is defined by \(s \mapsto s + t_0\), \(\gamma ^{+}:(0,t_1-t_0) \rightarrow {\mathbb {C}}\) is defined by \(s \mapsto t_1 - s\), and \(\gamma ^{\mathrm {M}}{:}\,(0,\beta ) \rightarrow {\mathbb {C}}\) is defined by \(s \mapsto t_0 - is\). Accordingly, we define sub-contours, \({\mathcal {C}}_{\pm }\) and \({\mathcal {C}}_{\mathrm {KB}}\). The concatenation \(\gamma ^{\mathrm {KB}}\) is viewed as a function \((s_0, s_1) \rightarrow {\mathbb {C}}\), where \(s_0 = 0\) and \(s_1 = 2(t_1 - t_0) + \beta \).

We have already defined the contour Hamiltonian \({\hat{H}}(z)\) for \(z \in {\mathcal {C}}_{\pm }\). To complete the specification of our ensemble we stipulate that \({\hat{H}}(z) = {\overline{H}}\) for \(z \in {\mathcal {C}}_{\mathrm {M}}\) . For contour times \(s,s' < t_1 - t_0\), the contour-ordered Green’s function \(G(s,s')\) recovers the appropriate notion of the real-time-ordered non-equilibrium Green’s function; similarly, appropriate notions of advanced and retarded Green’s functions can be recovered from the contour-ordered Green’s function. However, only the contour-ordered Green’s function admits a favorable perturbation theory, and this remarkable fact is one motivation for considering it. See [22] for further details. In this work, we additionally see that the contour-ordered setting is also the natural setting in which to recover a sparsity result for the self-energy of impurity problems in the non-equilibrium setting.

Now one can readily check that the partition function is given by \(Z = \mathrm {Tr}[e^{-\beta {\overline{H}}}] > 0\) (so Assumption 3.6 is satisfied). Now we verify Assumption 3.5. For \(s' \le s \le s_1 - \beta \), note that \(U(s,s')\) is unitary, hence bounded. Moreover, for \(s_1 - \beta \le s' \le s\), we have \(U(s,s') = e^{- (s-s') {\overline{H}}}\), which is trace class (by our assumption), hence bounded. It follows that for any \(s_0 \le s' \le s \le s_1\), the operator \(U(s,s')\) is bounded. In fact, \(U( s_1, s_1 - \beta ) = e^{-\beta {\overline{H}}}\), and as mentioned above, the operator norm of this operator restricted to the N-particle subspace decays exponentially in N. Thus, Assumption 3.5 is satisfied.

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Lin, L., Lindsey, M. Sparsity Pattern of the Self-energy for Classical and Quantum Impurity Problems. Ann. Henri Poincaré 21, 2219–2257 (2020). https://doi.org/10.1007/s00023-020-00917-1

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