Free fermions behind the disguise

A notorious challenge for the simulation of quantum many-body systems is the exponential growth of the Hilbert space dimension in the number of constituent degrees of freedom. Systems for which this difficulty can be circumvented via an analytic solution are invaluable for at least two reasons. First, the discovery of a new class of analytic solutions opens up the prospect of tractable simulation to a new family of models, and potentially of new phenomenology. Second, analytic solutions can be taken as starting points for approximations to more realistic models, thus extending the reach of these methods.

For a quantum spin-$\nicefrac{{1}}{{2}}$ (qubit) system, one remarkable type of analytic solution comes in the form of a duality to effective fermions. When the effective fermions are noninteracting, it can be said that we have found a means of restricting the model’s essential behavior to the low-dimensional subspace of a single fermion, and the physics of the model is well-understood. The textbook example of a free-fermion mapping is the Jordan-Wigner transformation jordan1928uber , which was famously employed to solve the one-dimensional XY model by Lieb, Schultz, and Mattis lieb1961two . The key insight is the identification of nonlocal Pauli operators with fermionic ladder operators. In the fermionic picture, the $n$-qubit XY-model Hamiltonian is mapped to a free-fermion Hamiltonian on $2n$ fermionic modes. The model is then completely solved by exactly diagonalizing the model’s $2n\times 2n$ free-fermion Hamiltonian, an exponential simplification from the naive brute-force diagonalization that one might expect to need in the qubit picture. This solution method is generic, meaning it applies regardless of the values taken by the nonzero coupling constants in the Hamiltonian. This is because the Jordan-Wigner transformation maps each term in the Hamiltonian linearly to a fermionic bilinear operator.

One physical signature of models that are solvable in this way is an energy spectrum $\{E_{\boldsymbol{x}}\}$ that is given in terms of a number $\alpha$ of single-particle energies $\varepsilon_{k}$ by

where $\alpha\leq n$ and $\boldsymbol{x}\in\{0,1\}^{\times\alpha}$ is a binary vector describing the occupation of each canonical fermionic mode. We will refer to a spectrum of the form in Eq.˜1 as free. We say that a Hamiltonian is solvable if it has a free spectrum and it can be written in terms of its eigenmodes $\psi_{j}$ as

where the $\psi^{\vphantom{\dagger}}_{k}$ obey the canonical anticommutation relations, $\{\psi^{\vphantom{\dagger}}_{j},\psi^{\vphantom{\dagger}}_{k}\}=0$ and $\{\psi^{\vphantom{\dagger}}_{j},\psi_{k}^{\dagger}\}=\delta_{jk}I$. When $\alpha<n$, a free spectrum will necessarily have degeneracies, since this is equivalent to the case where a subset of the energies $\{\varepsilon_{k}\}_{k=1}^{n}$ are equal to zero.

Recently, two of the authors Chapman2020Characterization have given a simple necessary and sufficient condition for a qubit Hamiltonian $H$ to be solvable via a Jordan-Wigner mapping by looking at properties of the frustration graph of $H$ (see Def.˜2). This gives a complete solution for this best-known class of solvable models. However, there exist models that are free and solvable, but that cannot be solved via any Jordan-Wigner mapping. Such a model has been introduced by Fendley fendley2019free as free fermions “in disguise”. Fendley solves this model by directly defining the single-particle eigenstates for the Hamiltonian through its interaction terms. The free fermions manifest nonlinearly and nonlocally in a basis which is dependent on the specific interaction strengths, but they remain free for all values of the couplings. The solution is therefore generic. This solution method has since been reproduced in a family of generalized spin-chain models Alcaraz2020Free ; Alcaraz2020Integrable , including qudit models, where the system is dual to so-called free parafermions Fendley2014Free . The generic nature of the free spectrum in these models would suggest that the frustration-graph formalism of Ref. Chapman2020Characterization could be applied to understand this solution. However, since these models provably do not admit a Jordan-Wigner mapping, it would seem the solution relies on much subtler commutation structures than previously understood.

In this work, we go behind the disguise and clarify the role that the frustration graph plays in solving these models. We develop an infinite family of free-fermion solutions which generalizes Ref. fendley2019free by finding specific graph-theoretic conditions for when such a solution is possible. Specifically, when the frustration graph of $H$ avoids certain obstructions known as claws and even holes (see Table˜1 and Def.˜3), then $H$ is solvable.

As a corollary to our result, we prove that independent sets in the frustration graph give us a family of conserved charges for the model whenever the frustration graph is only claw-free. That is, that when the frustration graph of $H$ avoids claws, the model is integrable.

The proof proceeds by considering the independence polynomial of the frustration graph. The independence polynomial of a graph is the polynomial generating function that counts the independent sets in the graph. We can incorporate detailed information about the Hamiltonian into the independence polynomial by attaching certain vertex weights given in terms of squares of Hamiltonian coupling strengths. We first prove ˜2 under the less restrictive assumption that the frustration graph is claw-free. We then prove that when the frustration graph is additionally even-hole-free, the independence polynomial factorizes into a quadratic function of a certain transfer operator. The single-particle spectrum can then be derived by looking at the zeros of the vertex-weighted independence polynomial. Given knowledge of the spectrum, the transfer operator then acts like a raising operator when acting on a fiducial mode whose existence is guaranteed by the structure of (even-hole, claw)-free graphs. The modes generated in this way allow us to define the eigenmodes of the free-fermionic Hamiltonian.

As in Fendley’s original model fendley2019free , there are no “physical modes” to speak of from this derivation as there would be from a Jordan-Wigner transformation. The eigenmodes are “disguised”, and then emerge directly as nonlinear, nonlocal combinations of the underlying spin operators. We call the eigenmodes revealed by this procedure incognito modes.

We then describe explicit families of models with frustration graphs that satisfy the (even-hole, claw)-free condition. The first of these examples is a small system, which is chosen explicitly as it has no generalized Jordan-Wigner solution and yet has a free-fermion solution of the form we consider. We show that this particular model is in fact related to one with a Jordan-Wigner solution by a local rotation. We next demonstrate how the family of generalized spin chain models included in Refs. fendley2019free ; Alcaraz2020Free ; Alcaraz2020Integrable fit into our formalism. These models have a particular one-dimensional structure which makes their asymptotic dispersion relations amenable to the techniques of Toeplitz-matrix analysis. Though this is not expected to be true of general (even-hole, claw)-free graphs, a structure theorem for these graphs demonstrates that we should expect their coarse topology to be one-dimensional, or possibly treelike.

The structure of the paper is as follows: in Section˜2, we formally state all definitions and our main results. In Section˜3, we discuss our result in the context of prior work. In Section˜4 we prove the main results as Thm.˜2.1 and Thm.˜2.2. In Section˜5, we demonstrate how specific examples fit into our formalism, and in Section˜6 we close with a discussion of possible future work.

We begin with self-contained statements of our main theorems and necessary supporting definitions. First, let us fix graph-theoretic conventions. A graph $G\coloneqq(V,E)$ consists of a set of vertices $V$ together with a set of 2-element subsets $E\subset V^{\times 2}$ called edges. All graphs we consider are finite and simple: every pair of vertices neighbors by at most one edge, and the graphs contain no self loops. An induced subgraph is a graph $G[S]\coloneqq(S,E\cap S^{\times 2})$ whose vertex set is $S\subseteq V$ and whose edge set consists of all of the edges in $E$ with both endpoints in $S$. We denote the vertex and edge sets of $G[S]$ by $V_{S}$ and $E_{S}$, respectively. We will also use the notation $G-W\coloneqq G[V\backslash W]$ to denote the induced subgraph of the graph $G$ by removing the set of vertices $W$. We will often refer to a subset of vertices interchangeably with the subgraph it induces. The open neighborhood of a vertex, $\Gamma(v)$ is the set of vertices neighboring the vertex, $v$. The closed neighborhood of a vertex, $\Gamma[v]$, is the set of vertices neighboring the vertex, $v$, together with $v$ itself. An independent set, or stable set, of a graph $G=(V,E)$ is a subset of vertices $S\subseteq V$ which induces a subgraph with no edges, $G[S]=(S,\{\})$. Notice that our definition includes the empty set and sets of one vertex as independent sets.

A clique is a graph where every pair of vertices is neighboring. For us, a particularly important type of clique in a graph is a simplicial clique (See Table˜1 (a)):

The claw, $K_{1,3}$, is the complete bipartite graph between one vertex and a set of three non-neighboring vertices (See Table˜1 (b)). A path of length $\ell$ is a connected graph of $\ell+1$ vertices and $\ell$ edges such that every vertex has at most two neighbors. A cycle, $C_{\ell}$, is a connected graph of $\ell$ edges and $\ell$ vertices such that each vertex has exactly two neighbors. Informally, a path of length $\ell$ is a cycle $C_{\ell+1}$ with one edge removed. A hole in a graph $G$ is a subset of $\ell$ vertices $W\subseteq V$ such that $G[W]\cong C_{\ell}$ (i.e. an induced cycle of $G$), where $\ell\geq 4$. A hole is called even if it has an even number of vertices and edges. Importantly, our definition of an even hole includes holes of four vertices (See Table˜1 (c)).

Next we turn to definitions involving a physical many-body qubit model. Consider an $n$-qubit Hamiltonian, $H$, written in a given basis of Pauli operators $\{\sigma^{\boldsymbol{j}}\}$ as

where $V\subseteq\{0,x,y,z\}^{\otimes n}$ is a set of strings labeling the $n$-qubit Pauli operators in the natural way. A frustration graph describes the commutation relations between the Hamiltonian terms as follows:

The frustration graph is the complement of the Pauli graph introduced by Planat Planat2007Pauli . Notice that it is always simple by construction. Where clear from context, we will drop the dependence on the Hamiltonian from $G(H)$.

It can be shown that all (even-hole, claw)-free graphs are simplicial, meaning they contain a simplicial clique CSSS . If a Hamiltonian $H$ is ECF then its frustration graph is necessarily simplicial, so we say that $H$ is simplicial as well.

Our first main result says that the spectrum of an ECF Hamiltonian is free, with single-particle energies given by the roots of a certain polynomial.

An important result that we will show is that even if the frustration graph is only claw-free, then the Hamiltonian is still integrable, as there exists a set of mutually commuting conserved charges. We consider this a result of independent interest to many-body physicists.

As we will prove in Lemma˜1 below, the independent-set charges satisfy

Since $Q^{(1)}=H$, this demonstrates that the charges are conserved. To take advantage of the independent-set charges, we exploit the simplicial property of $H$ and define a fiducial mode, $\chi$, in terms of which we can express the “incognito modes".

Note that we can always construct a simplicial mode for any simplicial Hamiltonian. To do so we introduce an additional (fictitious) spin to the system and augment each Hamiltonian term in the simplicial clique with a Pauli-X applied to the extra spin — notice that this will not affect the frustration graph. The simplicial mode is then defined by a Pauli-Z operator applied to the additional spin; clearly, the simplicial mode will anti-commute with all terms in the simplicial clique, but no other Hamiltonian terms.

Our proofs of Theorem˜2.1 and Theorem˜2.2 closely resemble the solution method introduced by Fendley in Ref. fendley2019free . Importantly, the fact that our Hamiltonians can be written in the form of Eq.˜2 implies that each energy level in the Hamiltonian has the same degeneracy, as similarly shown in Ref. fendley2019free . The operative technical insight is that many of the key properties of that model, and its generalizations in Refs. Alcaraz2020Free ; Alcaraz2020Integrable , are actually special cases of more general recursion relations in the class of models we identify.

Since its discovery, the Jordan-Wigner transformation jordan1928uber and subsequent generalizations Fradkin1989JordanWigner ; Wang1991Ground ; Huerta1993BoseFermi ; Shaofeng1995JordanWigner ; Batista2001Generalized ; Kitaev2006Anyons ; Nussinov2009Bond ; Galitski2010Fermionization ; Cobanera2011Bond ; backens2019jordan have enjoyed great success in probing the fundamental physics of quantum many-body spin models, as well as classical statistical mechanics models through so-called transfer-matrix methods Onsager1944Crystal ; schultz1964twodimensional ; kochmaski1997generalized . An understanding of these mappings has furthermore proven useful for designing fermion-to-qubit mappings with desired properties for simulating fermionic systems on a quantum computer Ortiz2001Quantum ; Bravyi2002Fermionic ; Verstraete2005Mapping ; Nussinov2012Arbitrary ; Bravyi2017Tapering ; Havlek2017Operator ; Steudtner2018Fermion ; Setia2019Superfast ; Jiang2019Majorana ; Jiang2020Optimal . Here, operator locality in the dual spin model is generally enabled through coupling to an auxiliary gauge field Chen2018Exact ; Chen2019Bosonization ; Tantivasadakarn2020JordanWigner , which endows fermionic-pair excitations with the structure of freely deformable strings on the spin lattice Levin2003Fermions . The preponderance of these mappings suggests that a fundamental theory of physics containing fermionic degrees of freedom need not hold fermions as fundamental objects Ball2005Fermions ; Wen2003Quantum .

Free-fermion models have an interesting connection to graph theory. The dynamics of free-fermion models are equivalent to matchgate circuits Knill2001Fermionic ; Terhal2002Classical ; Bravyi2006Universal ; Jozsa2008Matchgates ; Brod2011Extending ; Hebenstreit2019All , which were originally developed in the context of counting perfect matchings in graphs Valiant2002Quantum ; Cai2006Valiants ; Cai2007Theory ; Valiant2008Holographic . Independently, graph-theoretic methods have been utilized in quantum information in the context of variational quantum eigensolvers Crawford2019Efficient ; Izmaylov2019Unitary ; BonetMonroig2019Nearly ; Gokhale2019Minimizing ; Yen2020Measuring ; Verteletskyi2020Measurement ; Zhao2020Measurement , where the frustration graph is commonly known as the anticompatibility graph. Inspired by these methods, two of the authors have shown that a generalized Jordan-Wigner transformation exists for exactly those models for which the frustration graph is a line graph Chapman2020Characterization .

We note that line graphs also play an important role in understanding the spectrum of certain tight-binding models Kollar2019Line ; Kollar2019Hyperbolic , but we will not discuss these models further here.

A generalized Jordan-Wigner transformation maps a spin Hamiltonian of the form in Eq. (3) to one which is quadratic in Majorana fermion modes $\{\gamma_{j}\}$. These are Hermitian operators, which satisfy canonical anticommutation relations

That is, when solving a Hamiltonian of the form in Eq. (3) by Jordan-Wigner, we are asking whether there exists a mapping $\phi:V\mapsto\widetilde{V}^{\times 2}$ acting on the Pauli terms of $H$, and effecting

such that

in a way that preserves the commutation relations between terms, i.e. $G(H)\simeq G(\widetilde{H})$. The coefficient matrix $\mathbf{h}$ — called the single-particle Hamiltonian — is necessarily antisymmetric, since any symmetric part will vanish under the sum in Eq. (13), and we may take $H$ to be traceless without loss of generality. The central theorem of Ref. Chapman2020Characterization gives a necessary and sufficient criterion for a generalized Jordan-Wigner solution to exist for a particular qubit Hamiltonian.

Upon constructing a free-fermion solution for a given Hamiltonian, we find that $\mathbf{h}$ gives an edge-weighted skew-adjacency matrix of the root graph $R$. The graph $R$ may therefore be seen as the Majorana-fermion hopping graph. A full solution for $H$ is found by a linear transformation on the Majorana modes in Eq. (13) to diagonalize the Hermitian matrix $i\mathbf{h}$. Letting the nonzero eigenvalues of $i\mathbf{h}$ be given by $\{\pm\varepsilon_{j}\}_{j=1}^{\alpha}$ with $\varepsilon_{j}>0$ for all $j\in\{1,\dots,\alpha\}$, this brings the Hamiltonian to the form in Eq. (2) with single-particle energies given by the $\{\varepsilon_{j}\}$.

Claw-free graphs were originally investigated as natural generalizations to line graphs chudnovsky2005structure and have since developed into the subject of a rich area of study in graph theory faudree1997clawfree . Line graphs are claw-free, as no three edges in $R$ can be incident to another edge without at least two of them being incident to each other. Remarkably, the free-fermion solution method presented in this work extends the generalized Jordan-Wigner solution in a way that parallels the relationship between claw-free graphs and line graphs. Specifically, the single-particle energies obtained from Eq. (13) satisfy Eq. (5) when $L(R)$ is an ECF graph. Every vertex of $R$ corresponds to a simplicial clique of $L(R)$. Clearly, the edges incident to a given vertex in $R$ are mapped to the vertices of a clique in $L(R)$, and since every edge is incident to two vertices in $R$, the open neighborhood of a vertex in $L(R)$ induces two vertex-disjoint cliques  krausz1943demonstration . When a line graph is even-hole-free, the corresponding root graph is even-cycle-free, as any even cycle in $R$ will be mapped to an even hole in $L(R)$. Suppose that a given Hamiltonian satisfies Eq. (14) with $L(R)$ an ECF graph. The single particle energies are zeros of

Equivalently, we may consider the reciprocal polynomial $f^{*}_{R}$ to $f_{R}$,

where $n$ is the number of vertices in $R$. For an arbitrary simple graph $R$, the characteristic polynomial $f_{R}$ (and thus $f_{R}^{*}$) would only depend on products of elements from $\mathbf{h}$ from matchings and even cycles of $R$ cavers2012skew . Since $R$ is an even-cycle-free graph however, only the matchings are relevant. Let $\mathcal{M}_{k}$ be the set of all $k$-edge matchings $M$ of $R$. We have

The last equality follows because the matchings of a graph correspond to the independent sets of its line graph. Therefore, $\pm\varepsilon_{j}$ are an eigenvalue pair of $i\mathbf{h}$ if and only if Eq. (5) is satisfied for $G(H)\simeq L(R)$. Though even-hole-free line graphs are a rather limited set of frustration graphs, what is incredibly surprising is that Theorem 2.1 holds when $G(H)$ is relaxed to be a general ECF graph, though there is no fermion-hopping graph $R$ for this set of graphs in general. We remark that for any claw-free graph, the vertex-weighted independence polynomial, $P_{G}(x)$, is real-rooted for all values of the Hamiltonian couplings by the results given in Refs. Leake2016Generalizations ; Engstrom07inequalitieson . This generalizes the result for the un-weighted independence polynomial originally proved by Chudnovsky and Seymour Chudnovsky2007Roots . Since $P_{G}(x)$ has non-negative coefficients and real roots, all of its roots must be negative by Descartes’ rule of signs descartesgeometrie , and therefore all of the single-particle energies $\{\varepsilon_{j}\}$ are themselves real. The free-fermion solution method presented here therefore includes systems for which we can prove that no generalized Jordan-Wigner solution is possible, as we shall now see.

Line graphs can be characterized by the set of nine forbidden subgraphs beineke1970characterizations , shown in Table˜2. No model containing a subset of terms inducing any of the frustration graphs in Table˜2 admits a generalized Jordan-Wigner solution. One possible exception is when the Jordan-Wigner mapping is allowed to be non-injective; i.e. there is a mapping satisfying Eq. (12) and preserving the frustration graph which takes multiple Pauli terms to the same fermionic pair. These Pauli terms must then correspond to twin vertices in $G(H)$: vertices with identical open neighborhoods, $\Gamma(v)$. Notice that twin vertices are never neighboring by this definition, as a vertex is not included in its own open neighborhood.¹¹1Here we caution the reader that this definition differs slightly from that used in the graph-theory literature, where vertices with identical closed neighborhoods (which are therefore neighboring) are also referred to as twins. We will return to pairs of vertices with identical closed neighborhoods in Section 5.1. Since operators corresponding to twin vertices anticommute with the same set of operators in the Hamiltonian, the product of any pair of such operators commutes with every term in the Hamiltonian and so constitutes a symmetry. We can thus project onto the eigenspace of this symmetry operator to replace one operator in the set of twins with another, thus removing its vertex from the frustration graph. If twins can be removed in such a way as to change the frustration graph into a line graph, then the Hamiltonian is still solvable via Jordan-Wigner. This will sometimes be possible, as some of the forbidden subgraphs in Table˜2 themselves contain twins. From Table˜2, we see that all forbidden subgraphs for line graphs either contain twins, are simplicial ECF, or both. They therefore surprisingly all have a free spectrum, and it is truly how these graphs are connected to one another that allows us to infer the existence of a free-fermion solution.

The class of ECF graphs is generalized by the set of so-called (even-hole, pan)-free graphs Cameron2017On . A pan is a graph consisting of a hole together with an additional vertex with exactly one neighbor on the hole. A pan contains a claw as an induced subgraph, and so an (even-hole, pan)-free graph is necessarily ECF. The structure of (even-hole, pan)-free graphs has been completely characterized, and this allows the authors of Ref. Cameron2017On to give an $O(mn)$-time algorithm for recognizing them, where $m$ is the number of edges in the graph and $n$ is the number of vertices. Specifically, the authors of Ref. Cameron2017On show that (even-hole, pan)-free graphs either: (i) have a clique cutset, (ii) are unit circular-arc graphs, (iii) are a clique, (iv) are the join of a clique and a unit circular-arc graphs. A unit circular-arc graph is one whose vertices correspond to distinct arcs of unit length on a circle, such that vertices are neighboring if and only if their corresponding arcs intersect. A clique cutset is a subset of vertices inducing a clique whose removal disconnects the graph. The join of two graphs $G_{1}\coloneqq(V_{1},E_{1})$ and $G_{2}\coloneqq(V_{2},E_{2})$ is the graph with vertex-set $V_{1}\cup V_{2}$ and edge-set $E_{1}\cup E_{2}\cup\{(u,v)|u\in V_{1},v\in V_{2}\}$. Though the theorem of Ref. Cameron2017On completely describes the structure of these graphs, we can intuitively expect that the coarse topology of these models is fundamentally one-dimensional or treelike.

As general claw-free graphs can also be recognized efficiently Kloks2000Finding , we have an efficient algorithm for recognizing ECF graphs. In Ref. Chudnovsky2012Growing , a polynomial time algorithm is given for detecting whether a general claw-free graph contains a simplicial clique. It is therefore computationally efficient to recognize a simplicial clique in an ECF graph. Moreover, every (nonempty) ECF graph has at least one simplicial clique CSSS .

In this section we prove the two main results presented in Section˜2. We restate these theorems here for convenience. The first main result tells us that an ECF model will have a free spectrum, of the form Eq.˜1, and provides an explicit form of the single-particle energies.

The second main result gives an explicit realization of the canonical modes of an ECF model in terms of independent sets of Hamiltonian terms, and the simplicial mode, $\chi$.

Recall that the incognito modes are defined in terms of the simplicial mode, $\chi$, as

where $u_{j}\coloneqq 1/\varepsilon_{j}$ for the single-particle energy $\varepsilon_{j}$ satisfying Eq.˜19, $T_{G}(u)$ is a transfer operator

and $N_{j}^{-1}$ is a normalization factor which is computable.

We proceed by making successively more restrictive assumptions on $G(H)$: first that it is claw-free, then (even-hole, claw)-free. We begin by proving the following lemma, regarding claw-free Hamiltonians and the independent-set charges.

Lemma˜1 implies that all claw-free models have a set of conserved quantities whose size generally grows with system size, since $Q^{(1)}\coloneqq H$. Thus, we can conclude that, in the traditional sense, claw-free models are integrable. Consider as an example the frustration graph of the model introduced in Ref. fendley2019free , as shown in Fig.˜1. The graph is always claw-free, although contains even holes when the model has periodic boundary conditions. Two independent sets of vertices (highlighted in red and blue) induce a path in the frustration graph. In a general claw-free graph, the symmetric difference of any pair of independent sets induces a bipartite subgraph of maximum degree two: all of the components of the subgraph are induced paths and even-holes. Lemma˜1 also implies that the transfer operator, $T_{G}(u)$, defined in Eq.˜22, will commute with the Hamiltonian.

Next consider the following lemma regarding the transfer operators, $T_{G}(u)$, of even-hole-free models.

Note that $P_{G}$ has strictly positive coefficients, which do not depend on the signs of the Hamiltonian coefficients $\{b_{\boldsymbol{j}}\}_{\boldsymbol{j}\in V}$. Thus, as discussed in Sect.˜3, $P_{G}(-x)$ will have all positive roots, denoted by $x\coloneqq u_{\ell}^{2}$.

We next consider the commutation of the incognito modes, $\psi_{\ell}$, with the Hamiltonian. Here we further use the fact that an ECF graph, $G(H)$, contains a simplicial clique, $K_{s}$. Recall that the simplicial mode, $\chi$, commutes with all terms in the Hamiltonian outside of $K_{s}$, but anticommutes with all terms in $K_{s}$, $\{\chi,h_{v}\}=0$ for all $v\in K_{s}$. Thus, we can write the commutation of $T_{G}\chi T_{G}^{-}$ with the Hamiltonian for arbitrary $u$ as

For an ECH model, we require that when $u=-u_{\ell}$, the right-hand-side of Eq.˜33 is equal to $2\varepsilon_{\ell}\psi_{\ell}$, where $1/\varepsilon_{\ell}\coloneqq u_{\ell}$ (similarly when $u=u_{\ell}$, it is equal to $-2\varepsilon_{\ell}\psi^{\dagger}_{\ell}$). A crucial step for proving this is the following lemma: