Explicit diagrammatic solution of normalised, nondegenerate Rayleigh-Schr\"odinger perturbation theory

We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh-Schr\"odinger perturbation theory under the constraint that the approximate eigenvector be normalised to $1$ in every order. The series can be expressed in terms of diagrams that were first introduced by C. Bloch [Nucl. Phys. 6, 329 (1958)] for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams' topology to the coefficients.

Huby [6] restated Bloch's results, in a form suggested by Brueckner [9], where the same terms are constructed in a different way. He can give explicit formulas for eigenvectors and not merely projectors because he considered the nondegenerate case. These expressions for the eigenvectors were not normalised. Salzman [7] similarly focused on the nondegenerate, unnormalised case, and developed a new diagrammatic formalism, set up to collect equivalent terms. This in principle allows for a further reduction in the number of terms. The rules he gave for constructing diagrams do not provide this reduction automatically however.
Equivalent terms still had to be collected together manually. From the more mathematical direction, the equivalence of Bloch's diagram counting with that of the leaves of ordered trees can be found in the work of Stanley [11]. More recent surveys have related term counting in Rayleigh-Schrödinger perturbation theory to other combinatorial objects [12].
Silverstone and Holloway [8] derived alternative formulas for the nondegenerate eigenvalues and their unnormalised eigenvectors that formally lead to the least number of terms, however, at the price of evaluating a large number of derivatives. Quantifying the number of terms in the resulting Silverstone-Holloway expression, beyond 'more than the number of partitions of n into positive integers,' is a non-trivial task. More recently, Magesan and Gambetta [13] developed a formalism that preserves norms exactly by perturbatively series expanding the generator of a unitary operator. For a given order, the canonical transformation of the Hamiltonian by that unitary is then in turn series expanded. This method does not directly give an explicit series for eigenvectors and -values. Bloch's original work still finds application in the context of effective Hamiltonians in Jordan and Farhi's arbitrary order perturbative gadgets [14].
In the present work, we consider anew the perturbation of a nondegenerate eigenvalue in standard RSPT. The phase and normalisation freedom of the eigenvector significantly influences the expansion. To the best of our knowledge, here we give the first explicit solution which preserves the norm of the eigenvector at 1 in every order.
In Sec. II, we briefly review nondegenerate RSPT and the main results of Bloch [5] that we build upon. Our new results are derived in Sec. III. We comment on their efficiency and how they can be improved in Sec. IV. Finally, in Sec. V, we focus more on the diagrammatic aspect and show how our results work in practice, going up to fourth order, and conclude in Sec. VI.

A. Recursive definition
Given a Hamiltonian H = H 0 + ǫV parametrised by ǫ ∈ [0, 1], we assume we can expand any of its eigenvalues λ, and the corresponding eigenvector |λ , in a power series in ǫ, i.e. they satisfy ǫ n+m |λ m λ n .
Sorting Eq. (2) by powers of ǫ and equating the coefficients gives in zeroth order H 0 |λ 0 = λ 0 |λ 0 . Usually H 0 is chosen to be analytically diagonalisable; we call λ 0 and |λ 0 the unperturbed eigenenergies and -vectors, respectively. Here we further assume that they are discrete and nondegenerate, and we define the complementary projectors For nonzero powers n ∈ N of ǫ, Eq. (2) gives H 0 |λ n + V |λ n−1 = n m=0 |λ n−m λ m .
We can then consider the P λ -and Q λ -components of Eq. (4) separately to derive equations for λ n and |λ n respectively. For the energies we get Note, here and throughout the paper, the convention M n=N A n = 0 for any sequence A n if M < N applies. The Q λ -component gives where on the right-hand side we see appearing the reduced resolvent which, since by assumption λ 0 is nondegenerate, is well-defined. For the sake of a compact notation, there is no index λ on S, but it should be remembered as implicit. For later reference we also define powers of S, where it will be convenient to define S 0 separately [3] Clearly λ 0 |λ n is not constrained by Eq. (7) or Eq. (5), and by extension Eq. (2). The simplest choice, and one employed by many authors [6][7][8], is λ 0 |λ n = 0. But it can be more convenient to use this degree of freedom to normalise the eigenvector to 1 in every order, i.e. N n,m=0 This way the calculated eigenvectors immediately form an orthonormal basis (up to higher order terms) and can be used straightforwardly to calculate expectation values without having to manually renormalise. Equation (10) requires the unperturbed eigenvector to be normalised, λ 0 |λ 0 = 1, and fixes the real part of λ 0 |λ n . Choosing to set the imaginary part to 0, we arrive at This is not a unique phase choice (see Sec. IV), though it is the conventional one [2,Ch. 11].
In Theorem 1, we collect Eqs. (5), (7) and (11). It is not a new result but rarely stated explicitly for arbitrary orders.
Theorem 1 (Cohen-Tannoudji et al.). The sequences λ n and |λ n , n ∈ N that satisfy the coupled recurrence relations where the starting values λ 0 , |λ 0 are an eigenvalue and corresponding unit eigenvector of H 0 , respectively, solve Eq. (2) while preserving the normalisation of N n=0 ǫ n |λ n for every On the left are all diagrams for n = 1 and n = 2, including the non-convex (0, 2). On the right is some order n diagram. Following Bloch [5], we sometimes use curved, dashed lines to represent an arbitrary diagram.

B. Bloch sequences and diagrams
Bloch's seminal paper on degenerate RSPT [5] was the main inspiration for this paper.
In this subsection, we summarise the relevant definitions and results that we adopt from there. These also apply to the nondegenerate case straightforwardly, see e.g. [6], and are adapted to our notation accordingly. The result for the eigenvector is where the sum is over Bloch sequences of length n defined by and the prime indicates it is restricted to those sequences that satisfy λ n is a solution to Eq. (2); it is not normalised, but satisfies the condition λ 0 λ n = 0, so that λ 0 λ = 1. We distinguish it from the normalised one defined in Theorem 1 with an overline.
Using Eq. (14), Eq. (5) becomes Bloch sequences can be represented graphically as staircase diagrams where step i has height k i and width 1, as illustrated in Fig. 1

III. STEPWISE DIAGRAMMATIC SOLUTION
We find that when we require a normalised state vector, given the phase choice Eq. (11), the basic structure of the solution is retained: The coupled recurrence relations in Theorem 1 are solved by λ n , |λ n of the where c, e are rational-valued functions.
Note that the absence of the primes on the sums in Eqs. (18) and (19) means that we must also allow non-convex diagrams (recall Eq. (16)).
Note further that we have introduced again a series for the eigenvalue Eq. with e(∅) = c(1) = 1, proving the base case.
with the primed sum restricted to k m+1 ≥ m − K m ≥ 0 so that all the arguments of c are non-negative, and K m = m i=1 k i . For m = 1, in some of the argument lists, the initial index is smaller than the final index; as in Eq. (6) such an argument list should be interpreted as an empty set. The starting values of c and e (n = 1) are Note that functions c and e are independent of the Hamiltonian. We will refer to the three cases in Eq. (24) as the k 1 = 0, k 1 = 1, and k 1 > 1 rules. For a diagrammatic explanation of the recurrence relations, refer to Sec. V A.
In Eqs. (14) and (17), all diagrams are summed up with a coefficient of 1, or 0 if the sum is extended to non-convex diagrams. The same cannot be true for c and e because of the factor 1/2 in λ 0 |λ n . A nonzero λ 0 |λ n means that some diagrams start below the diagonal, so are definitely not convex. And the factor 1/2 means their coefficient is generally unequal to 1.
From calculating c for all diagrams up to fourth order, cf. Sec. V, and selected higherorder diagrams, we anticipate that it will have the following property, which will be useful in the subsequent analysis: Definition 1 (Crossing Property). For any Bloch sequence (k 1 , . . . , k n ) let x(k 1 , . . . , k n ) be the number of times its associated diagram intersects the main diagonal. We say a function f has the crossing property if there is another function g such that for all Bloch sequences (k 1 , . . . , k n ), i.e. f depends only on the number of times a diagram crosses from below to above the main diagonal. Here ⌈·⌉ is the ceiling function.
If the function c has the crossing property, the problem of evaluating it only needs to be performed on a set of representative diagrams. These diagrams can be taken to be the ones with the Bloch sequences (cf. Eq. (15)) meaning 0,2 repeated n times. It is helpful below to have a separate symbol for these specific instances of the c function: t(n) can be computed: Proof. The generalised binomial coefficient is defined in the usual way We use the k 1 = 0 and k 1 = 1 rules of Eq. (24) to derive a recurrence relation for t(n), t(n) Here ⌊·⌋, ⌈·⌉ are floor and ceiling function, respectively, rounding to the nearest integer lesser/greater than the argument. The summands are symmetric under reversing the order of summation, so for all but one term the factor 1/2 cancels. But we find it convenient to instead keep all terms and group them by even and odd indices, as done in the last line of Eq. (30). Then by bringing the latter sum to the left hand side, which we can also write as t(n) = 1 2 t(n)t(0) + 1 2 t(0)t(n), and multiplying by 2, we can rewrite Eq. (30) as i.e. we find that the sum is independent of n, so we can set e.g. n = 1 to evaluate it.
To complete the proof constructively, 1 consider that Eq. (31) has the form of a discrete convolution, so we can restate it in terms of the (ordinary) generating function of t, as with a geometric series, so we can express the generating function as a binomial series to determine t, which gives Eq. (28).
The solution for e is slightly more complicated as, in contrast to c, it does not depend solely on the diagram's topology w.r.t. the main diagonal, but also w.r.t. the upper diagonal, which is defined as the diagonal line one unit higher than the main diagonal, as illustrated in Fig. 2.
Definition 3 (Crossing Numbers). We say a Bloch sequence (k 1 , . . . , k n ) has crossing numbers N 1 , n 1 , N 2 , n 2 , . . . , N m , n m if its associated diagram crosses, in order, above the upper diagonal N 1 times, below the main diagonal n 1 times, then above the upper diagonal A canonical diagram that has crossing numbers N 1 , . . . , n m is We will now proceed to the main result of the paper, Theorem 3, in which explicit formulas for c and e are obtained. We first briefly review the heuristics that led us to the formulation of this theorem. We noted that if we assumed that c had the crossing property, Lemma 1 would be sufficient to calculate c for all diagrams. By calculating a number of examples, we made observations about the structure of the solution, noting the dependence on the crossing numbers only, and used these to allow further simplification of the recurrence relations. We came to an ansatz for solving the coupled recurrence relations, guided by the observation that our solution for e has to be consistent with c having the crossing property.
In the end, the ansatz is proved in the following theorem by induction, accompanied by a straightforward algebraic analysis: with t(x) = 2x x 2 −2x as given in Eq. (28), i.e. c has the crossing property and e is a function of the crossing numbers only.
Proof. We verify that Eqs.
where in Eq. (39b) we introduce a i = i j=1 N j and b i = m j=i n j to simplify notation, and shift the summation index k by a i−1 . Then in Eq. (39c) we switch the sums over i and l.
Note that a 0 = 0, a 1 = N 1 ≥ 0, and a i+1 > a i for i > 0. So in Eq. (39d) we can combine the double sum over i, k into one over k. And finally in Eq. (39e) we add and subtract the k = 0 terms, then use Eq. (31) and get Eq. (38).
Similarly, we calculate c(k 1 , . . . , k n ) using Eq. (24) c(k 1 , . . . , k n ) In Eq. . Since we are in the k 1 > 1 term, we know that a 1 = N 1 ≥ 1, and a i+1 > a i for i ≥ 1 still holds, so the only term Eq. (6) applies to is the second j sum for k = 1 since a 0 = 0.

IV. ON NON-UNIQUENESS OF DIAGRAMMATIC REPRESENTATIONS
By stating a recurrence relation and initial conditions we uniquely define a quantity.
For example, combined with the initial conditions, Eqs. (12) and (13)  That does not mean, however, that c and e are necessarily the unique solutions of Eqs. (12) and (13), or that Eqs. (12) and (13) are the unique solutions of Eq. (2).
Since we are considering the nondegenerate case, eigenvectors are determined up to a factor. We are fixing the normalisation with Eq. (10), but that still leaves a phase freedom.
The zeroth order phase is set by our choice of |λ 0 . We can modify this phase in higher orders of ǫ by adding an imaginary part to Eq. (11). An arbitrary imaginary part would generally change the structure of Eq. (19), but we could preserve it, e.g. by setting This reduces the number of diagrams but comes at the cost of a more complicated rule requiring an even/odd distinction.
Note that if the Hamiltonian is real-symmetric, Eqs. This brings us to the main point of this section: Once norm and phase are fixed, the eigenvector is uniquely defined, but the representation in terms of diagrams is not. The eigenenergy is of course independent of the factor in front of the eigenvector but has a similar freedom with regard to the decomposition into diagrams.
We also define the operators T , L, and D ("total", "length", and "difference") applied to string z: As an example for the map Z, we can write Here n = 5, q = 3, and we see the appearance of integer strings z i of varying length, including the null string.
Equation (45) implies that (z 1 , . . . , z M , z M +1 , . . . , z q ) has the same operator content as (z M +1 , . . . , z q , z 1 , . . . , z M ), i.e. the operator content is invariant under cyclical permutation of strings. Now, let k j be the Jth zero, w.l.o.g. assume j > m For the eigenvector, the calculation works out analogously with the only difference that here the operator content does not start with a λ 0 | V , so we can never permute z 1 . The rest of the diagram (z 2 , . . . , z q ) has the same structure 2 as a term in the energy expansion, so the same permutation rules apply.
Part of the redundancy identified by Theorem 4 already appears when we define the recurrence relations for c and e. For example, in the last term of Eq. (12), we can switch the order to λ m λ 0 |λ n−m , which would change Eq. (20) and would lead to a different recurrence relation for e and thus different values for c and e.
We could also compare to the Bloch style result for the energy, Eq. (17), which is a sum over all convex diagrams, and note that in our language, a convex diagram is described as N 1 , 0 with e = t(N 1 ) > 0, but there are also many non-convex diagrams for which e = 0. So our result for the energy is less efficient. But even when restricting to convex diagrams, Theorem 4 still leads to a lot of redundancy. Salzman [7] addresses this for the The canonical representative of a permutation group of strings has z 1 ≥ z 2 ≥ · · · ≥ z q .
Giving the sum over all c or e for an arbitrary representative diagram is generally not an easy task. Of course, given a string representation z m 1 1 . . . z m k k (k distinct strings z i with multiplicity m i ), we can write down all the ( k i=1 m i )!/ k i=1 m i ! permutations, calculate their c and e and sum them up to get a c eff and e eff . The difficulty lies in automating this, i.e. listing only the canonical diagrams and finding an explicit function on them that gives c eff and e eff directly, preferably without having to invoke Eq. (35) for the whole permutation group. This is less of a concern for the energy where we can alternatively start from Eq. (17).
Then the problem becomes counting all the convex permutations, a nested sum for which can be written down but perhaps cannot be evaluated explicitly without specifying a Bloch sequence first.

A. Number of terms
We take a look at how many diagrams are generated by our method and other previous methods, and how many of them may correspond to distinct operator expressions. This subsection is summarised in Table I. At order n, there are 2n−1 n distinct Bloch sequences [5]. This can easily be seen by considering that to construct all diagrams we have to list all distinct arrangements of n unit vertical steps and n − 1 unit horizontal steps (the n-th horizontal step is always fixed at the end). This is the number of terms in our perturbation expansion for |λ n and λ n+1 (though e can be 0). If we apply Theorem 4, it becomes an upper bound for the number of canonically ordered diagrams, i.e. the minimum number of terms required to cover all distinct operators. Asymptotically it scales as 4 n /2 √ πn.
A lower bound for the minimum number of diagrams is the number of partitions of n into positive integers, cf. [8]. There is no known explicit expression for this partition function, but it has a generating function, recurrence relations, and an asymptotic expression exp(π 2n/3)/4 √ 3n = 4 π ln 4 √ 2n/3 /4 √ 3n [15, p.41].
As stated above, Salzman [7] grouped diagrams by z 1 (the string of positive integers before the first 0 in the Bloch sequence) and counted 2 n − n distinct groups within convex diagrams of length n. We can view this as a lower bound on the number of terms in the unnormalised eigenvector correction λ n , since z 1 cannot be permuted with the other strings without changing the operator content. By adding the number of z 1 's leading to a non-convex diagram, we can generalise this to a lower bound for the number of terms in Yet, any of the 2 n − n z 1 's that can start a convex diagram can be the greatest string of a canonically ordered diagram, so this lower bound also applies to λ n+1 after all.
Clearly none of the bounds are tight for sufficiently large n, though the latter set of lower bounds show that the minimal number of diagrams scales as exp(cn) rather than exp(c √ n).
a Up to fourth order, λ n has the same number of terms, but in higher orders it has more than λ n+1 .  is the number of terms in |λ n and λ n+1 , and in column 5, the number of convex diagrams gives the number of terms in λ n and λ n+1 following Bloch [5]. If V is purely off-diagonal (columns 8 and 9), none of the lower bounds apply. For e, we take c of the same diagram, then for every horizontal intersection with the upper diagonal we subtract a decomposition where we take c of a diagram beginning with a 0-step followed by everything before the intersection, multiplied by e of the part of the original diagram following the 0-step after the intersection, see Fig. 3. We split up the c recurrence relations, Eq. (24), into three parts again. For k 1 = 0, see Fig. 4, we sum over all intersections with the main diagonal. There is at least one such intersection, since we start below the diagonal and end above it. The part of the diagram before the intersection is read backwards, or equivalently is rotated by π. The part after the intersection is left as is. We multiply c of both diagram parts and divide by 2. Horizontal intersections get a minus sign. (The example in Fig. 4 shows a vertical intersection.) The k 1 = 1 rule remains the simplest. If a diagram starts with a 1-step, remove it, see  The decomposition of the diagram has similarities with the one in the e-recurrence (Fig. 3).
The second part of the diagram is treated the same but in the first part, instead of adding a 0-step the first step is lowered by 1. Another difference is that here we are guaranteed to have at least one summand since the diagram starts above the upper diagonal. Fourth order perturbation theory is not exactly an outlandish endeavour, yet has sufficient complexity that even though the associated Talk page has since 2010 noted that there are mistakes in the expressions listed on Wikipedia, to date no one has corrected them [16].
There are 35 Bloch sequences for n = 4, of which 14 are convex, 13 need to appear in the energy series (4 if V completely off-diagonal), and 26 need to appear in the normalised eigenvector series (12 if V is completely off-diagonal).

VI. CONCLUSION
We have shown how to explicitly solve the conventionally normalised Rayleigh-Schrödinger perturbation series to arbitrary order. The structure of earlier results for unnormalised vectors is readily adapted to this problem. The normalisation necessarily increases the number of terms in the expansion. We surveyed how the number of terms varies between differ-n = 4 : (4, 0, 0, 0) c, e = 1, 1 ent methods, and how to identify equivalent diagrams. An efficient summation of these equivalent diagrams remains an open problem, and there is likely no simple solution.
No matter how efficiently terms are summarised, their number grows exponentially with the order of perturbation.
Counting and analysing Bloch diagrams and associated quantities offers a rich trove of combinatorics problems, many of which may have already been studied in the context of paths, random walks, and bridges.