Loop formulation of supersymmetric Yang-Mills quantum mechanics

We derive the fermion loop formulation of N=4 supersymmetric SU(N) Yang-Mills quantum mechanics on the lattice. The loop formulation naturally separates the contributions to the partition function into its bosonic and fermionic parts with fixed fermion number and provides a way to control potential fermion sign problems arising in numerical simulations of the theory. Furthermore, we present a reduced fermion matrix determinant which allows the projection into the canonical sectors of the theory and hence constitutes an alternative approach to simulate the theory on the lattice.


Introduction
The conjectured holographic duality between supersymmetric Yang-Mills quantum mechanics and the theory of D0 branes of type IIa string theory in the large-N limit in principle allows to probe the physics of certain supergravity black holes by lattice Monte Carlo simulations. In particular, N = 16 supersymmetric Yang-Mills (SYM) quantum mechanics (QM) stemming from the dimensional reduction of N = 1 SYM in d = 10 dimensions is supposed to describe the dynamics of D0 branes which are the degrees of freedom of the underlying M-theory [1]. The connection to so-called black p-branes allows to study the thermodynamics of black holes through the corresponding strongly coupled gauge theory. We refer the reader to the review article [2] for further details. Here we report on our work in this direction on an analogue, but simpler theory, namely N = 4 SYM QM with generic gauge group SU(N ). The model stems from dimensionally reducing N = 1 SYM in d = 4 dimensions, but is expected to share many qualitative features with the 16 supercharge JHEP12(2014)044 model. The aim of this paper is to construct the fermion loop formulation of the strongly coupled gauge theory regularised on the lattice, so as to make it susceptible to numerical simulations.
There have already been a number of nonperturbative investigations of SYM QM using numerical techniques. In [3][4][5][6][7] the Hamiltonian formulation was employed together with the cut Fock space method. This approach also allowed analytic solutions, at least for d = 2 dimensional SYM QM [8][9][10][11][12]. On the other hand, in [13] the Wilson lattice discretization was constructed and the d = 4 SYM QM was simulated in the quenched approximation [13,14]. Further discretizations were proposed and investigated by Monte Carlo simulations in [15][16][17], and it was also shown that the (naive) Wilson discretization does not require any fine tuning to reach the correct continuum limit. A different nonlattice approach has been followed by [18][19][20][21] which used a momentum cutoff regularization while completely fixing the gauge.
Our motivation to study the loop formulation of this model is threefold. Apart from the motivation given by the interesting physics related to the thermodynamics of black holes and the possibility to test the gauge/gravity duality outlined above, the loop formulation provides a new approach to simulate fermions on the lattice [22]. In contrast to standard approaches the fermion loop formulation allows for local fermion algorithms [23], i.e., local updates of the fermionic degrees of freedom. The simulation algorithm applicable to the loop formulation works for massless fermions and appears not to suffer from critical slowing down [23,24]. This is of particular importance in the context of supersymmetric field theories with spontaneously broken supersymmetry, since in such cases one has to deal with a massless fermionic mode, the Goldstino fermion. The third motivation finally stems from the fact that the fermion loop formulation offers the potential to control the fermion sign problem. Again, this is of particular significance in theories with spontaneously broken supersymmery where the partition function for periodic boundary conditions, and hence the fermion determinant (or Pfaffian), averages to zero, since it represents the vanishing Witten index [22,[25][26][27]. The possibility to control the fermion sign then follows from the fact that in the loop formulation the fermionic contribution to the partition function decomposes into contributions from fixed fermion number sectors, each of which has a definite sign depending only on the specific choice of the fermionic boundary conditions. The paper is organised as follows. In section 2 we discuss the d = 4 dimensional SYM QM in the continuum and describe the lattice regularisation using the Wilson derivative. In section 3 we derive a reduction formula for the determinant of the fermion matrix which separates the dependence of the bosonic degrees of freedom from the chemical potential and which then allows the straightforward discussion of the canonical sectors of the theory. In section 4 the fermion loop formulation is introduced and in section 5 we discuss the various fermion sectors emerging from the transfer matrices in the loop formulation. We close the main part of the paper with our conclusions and an outlook in section 6. Finally, in appendix A we review various ways how to determine the canonical determinants from the reduced fermion matrix and prove in appendix B the algebraic equivalence between the reduced fermion matrix approach and the fermion loop formulation.

JHEP12(2014)044 2 Lattice regularisation
We start from N = 1 SYM in d = 4 dimensions with gauge group SU(N ) and dimensionally reduce the theory by compactifying the three spatial dimensions. While the temporal component A(t) of the 4-dimensional gauge field remains unchanged, the three spatial components become bosonic fields X i (t), i = 1, 2, 3. The action of the dimensionally reduced theory then reads where the anticommuting fermion fields ψ(t), ψ(t) are complex 2-component spinors, σ i are the three Pauli matrices and D t = ∂ t − i[A(t), · ] denotes the covariant derivative. All fields in the theory are in the adjoint representation of SU(N ) and the theory possesses a N = 4 supersymmetry. Note that the analogue reduction from N = 1 SYM in d = 10 dimensions yields a very similar action with the only change that there are 9 bosonic fields X i (t), i = 1, . . . , 9 corresponding to the 9 compactified gauge degrees of freedom, the σ i 's are the SO(9) γmatrices and the fermionic Grassmann variables are Majorana, i.e., can be taken to be real. The dimensionally reduced theory then corresponds to N = 16 SYM QM.
Let us now describe the lattice regularised version of the N = 4 SYM QM where the Euclidean time extent is discretised by L t points. The bosonic part of the action is then given by where the gauge field is replaced by the gauge link U (t) living in the gauge group SU(N ) and the covariant lattice derivative is explicitly For the regularisation of the fermionic part we use the Wilson discretisation to get rid of the fermion doublers. Note that in d = 1 dimensions adding a Wilson term with Wilson parameter r = ±1 to the symmetric derivative yields either a forward or backward derivative, Hence, the discretised fermion action reads whereD t is simply the covariant derviative defined above. Note that the Wilson term breaks the time reversal and hence also the charge conjugation symmetry. However, the symmetries are restored in the continuum limit together with the full supersymmetries without any fine tuning since any further symmetry breaking terms are prohibited by the gauge symmetry [15]. For our further discussion of the fermionic part of the theory, it is convenient to work in uniform gauge U (t) = U , although it is not necessary for the derivation of the reduced JHEP12(2014)044 fermion matrix in the next section. In addition, we also include a finite chemical potential term e µ in the forward fermion derivative [28] in order to facilitate our discussion of the canonical fermion sectors in the next section. To be specific, the fermion action then reads where the gauge part of the hopping term connecting the nearest neighbour Grassmann fields ψ a α (t) and ψ b β (t + 1) is given by and is independent of t. Here, T a are the generators of the SU(N ) algebra and are normalised such that det W = 1. The Yukawa interaction between the fermionic and bosonic fields is described by a 2( and the fermion action can be compactly written in terms of the fermion Dirac matrix D p,a , i.e., where the subscripts p,a specify periodic or antiperiodic temporal boundary conditions for the fermions in time, ψ(L t ) = ±ψ(0), respectively. Eventually, the grand canonical partition function reads where the determinant of the fermion Dirac matrix is the result from integrating out the fermionic degrees of freedom ψ and ψ.

Fermion matrix reduction and canonical formulation
In d = 1 dimensions the fermion matrix is particularly simple and takes a cyclic block bidiagonal form, Subsequently, determinant reduction techniques based on Schur complements similar to the ones described in [29] can be applied. As a consequence the grand canonical determinant for the reduced fermion matrix yields

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where T is the simple matrix product For given background fields U and X i (t) the formula allows to calculate the determinant for any value of the chemical potential µ by simply diagonalising T and evaluating the characteristic polynomial of order 2(N 2 − 1) in e µLt . The coefficients of the polynomial are then just the fermion contributions to the grand canonical partition functions [29], which is the conventional fugacity expansion. Note that the computational effort to evaluate eq. (3.2) grows only linearly with the temporal extent of the lattice (through the number of multiplications in the product), for example as one takes the continuum limit L t → ∞.
One can also work in temporal gauge in which all gauge links are transformed to unity except one denoted by W , e.g., the one connecting time slice t = L t − 1 and t = 0. The relation to the uniform gauge is then W = W Lt and the product becomes Lt−1 t=0 Φ(t) · W . Finally we note that for ordinary supersymmetric quantum mechanics the expression for T reduces to the result given in [30].
Next we turn to the explicit evaluation of the canonical determinants. Denoting the eigenvalues of T in eq. (3.3) by τ j , j = 1, . . . , 2(N 2 − 1) we can express the determinants directly in terms of these by comparing the coefficients of the characteristic polynomial which simply reflects the fact that the sector with maximally saturated fermion number is quenched. For the sector with n f = 0 we obtain where we made use of the fact that det W = 1. The formula shows that the fermion contribution in the n f = 0 sector is nontrivial, even though it is independent of the gauge link U .
The sectors with n f = 1 and n f = n max f − 1 fermions are similarly simple, The generic formula for the canonical determinants in terms of the eigenvalues can be expressed by the elementary symmetric functions S k of the n max . The k th elementary symmetric function is defined as where the sum has n max f k summands, and the canonical determinant in the sector with n f fermions eventually reads Of course the coefficients of the characteristic polynomial can be obtained in many other ways. In appendix A we present several alternative methods how to calculate the canonical determinants directly from the matrix T . One method makes use of the traces of powers of T while the other employs the minors of T . The latter turns out to be closely related to the transfer matrices emerging from the fermion loop formulation discussed in the next section.

Fermion loop formulation
In the fermion loop formulation the decomposition into the various fermion sectors are recovered in a completely different and independent way. The formulation is based on the exact hopping expansion of the fermion Boltzmann factor involving the action in eq. (2.5). Since the overall prefactor 1/2g 2 only contributes a trivial factor we suppress it in the following. We apply the expansion not only to the hopping term, but in fact to all terms in the fermion action including the Yukawa term. The expansion is exact because it naturally truncates after the first two terms due to the nilpotency of the Grassmann variables. Such an expansion is most conveniently expressed by i.e., in terms of occupation numbers m. Applying this equation to each term in the fermion action eq. (2.5) characterised by the colour indices a, b, the Dirac algebra indices α, β and the time coordinate t, the expansion of the fermion Boltzmann factor yields JHEP12(2014)044  Here, the terms in the first product follow from the Yukawa interaction while the terms in the second product stem from the hopping terms in which we have put µ = 0 to simplify the discussion. Note that one has a separate expansion for every combination of indices t, a, b, α, β which stops after the first two terms due to the Grassmannian character of the fermionic degrees of freedom. The two terms in each expansion are characterised by the occupation numbers h ab αβ (t) and m ab αβ (t) taking the values 0 or 1. The Grassmann integration over the fermion fields requires that every pair ψ a α (t)ψ a α (t) needs to be saturated by the integration measure in order to give a nonvanishing contribution. This condition yields local constraints on the occupation numbers h ab αβ (t) and m ab αβ (t) separately at each site t, The integration over the fermion fields is then replaced by a summation over all configurations of occupation numbers satisfying the constraints above. The various configurations of occupation numbers and the corresponding constraints can most easily be specified graphically by representing each pair ψ where the sum is over all combinations of occupation numbers satisfying eq. (4.3) and (4.4).

Fermion sectors and transfer matrices
In figure 3 we show three sample configurations consisting of closed oriented fermion loops for four fermionic degrees of freedom (representative for the generic 2(N 2 − 1) ones). One immediately notices that the configurations can be classified according to the number of fermions n f propagating forward in time. For the three examples depicted in figure 3 the fermion numbers are n f = 0, 1 and 4 (i.e. n f = 2(N 2 − 1) for the generic case), respectively. In each sector, the propagation of the n f fermions can be described by transfer matrices where the first transfer matrix describes the various ways how to connect n f fermions entering at time t with n f fermions exiting at t. It depends on the boson field configuration X i (t) through the Yukawa interactions matrix Φ(t) and hence depends on t. The second transfer matrix describes how to connect n f fermions exiting at t and entering at t + 1, and hence depends on the gauge field U through W in eq. (2.6). In uniform gauge, this transfer matrix has no time dependence. Then, for a given gauge and boson field background JHEP12(2014)044 {U, X i (t)} the fermion contribution to the partition function in the sector with n f fermions is simply given by The full contribution is then obtained by adding up all these terms taking into account a factor (∓1)e µLt for each fermion loop winding around the lattice in temporal direction, with the sign depending on whether periodic or antiperiodic boundary conditions are employed.
The expression eventually reads and can directly be compared with eq. (3.4).
Let us now look in more detail at the transfer matrices separately in each sector. First we note that the size of T n f is given by the number of states in sector n f , i.e., The sectors with n f = 0 and n f = 2(N 2 − 1) are therefore particularly simple since in these cases the transfer matrix is just 1 × 1. We will hence first discuss these two sectors, followed by the still rather simple sectors with n f = 1 and n f = 2(N 2 − 1) − 1, before presenting the generic case for arbitrary values of n f .

Sector n f = 0
For n f = 0 we see by inspection of the corresponding configuration in figure 3 that there is no gauge link dependence, and hence T W 0 = 1, while the transfer matrix T Φ 0 (t) must contain the sum of the weights of all fermion loop configurations on a given time slice t. By doing so, we need to take care that each nontrivial fermion loop picks up the usual factor (−1) from the Grassmann integration. It is not difficult to see that a given time slice configuration can be specified by a permutation σ of the indices i = 1, . . . , 2(N 2 − 1) labelling the fermionic degrees of freedom. Each cycle (ijk . . . l) in the permutation then corresponds to a sequence of indices characterising a specific fermion loop and its weight is given by Φ ij Φ jk . . . Φ li . The total sign of the configuration is given by including a factor (−1) for each nontrivial cycle, i.e., counting whether the number of nontrivial cycles in the permutation is even or odd which corresponds to the parity of the permutation. Finally, the sum over all configurations amounts to summing up all permutations including the corresponding weights and the signs given by the parity of the permutation. This prescription is of course nothing else than the definition for the determinant, so the transfer matrix in the n f = 0 sector is simply given by and the total fermion contribution factorises completely,

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Comparing this with eq. (3.7) we obviously find complete agreement. In the fermion loop approach however it is evident from the beginning that the gauge link U does not contribute in the n f = 0 sector. Similarly to the n f = 0 sector we need to take all permutations of the colour indices a, b into account. For each nontrivial permutation of two indices the number of fermion loops winding in temporal direction is reduced by one and we take this into account by including a factor (−1). Summing over all permutations including the sign corresponding to the parity of the permutations again yields the determinant, i.e., This is in accordance with the result from the determinant reduction, cf. eq. (3.6), and it is obvious that the same result would be obtained without referring to a particular gauge. Since the fermions are completely saturated by the temporal hopping terms and contribute only trivially to the canonical determinant, this sector corresponds to the quenched one as noted before.

Sector n f = 1
Next, we look at the sector with n f = 1 fermions. The corresponding transfer matrices T 1 (t) are of size 2(N 2 − 1) × 2(N 2 − 1). Each matrix element (T Φ 1 (t)) ij contains the sum of weights of all configurations at fixed t where the fermion degree of freedom i = (a, α) is entering time slice t and j = (b, β) is leaving. The corresponding degrees of freedom are then already saturated by the corresponding hops in and out of the time slice and hence the weights Φ ki and Φ jk , k = 1, . . . , 2(N 2 − 1) can not appear in any of the configurations. The remaining time slice configurations can be obtained in analogy to the considerations in the n f = 0 sector, that is by constructing all permutations, i.e., cycles of the remaining degrees JHEP12(2014)044 of freedom and taking into account factors of (−1) for each nontrivial cycle. Following the arguments from the n f = 0 sector it turns out that this is again equivalent to taking the determinant of Φ(t), but with row j and column i removed, i.e., which is in fact the (j, i)-cofactor of Φ. This will be discussed in more detail in section 5.5. Similarly, in order to include the colour changing hops due to the gauge link between time slices we multiply with the corresponding gauge link transfer matrix T W which in uniform gauge is constant in time and is in fact the complementary (i, j)-minor det W ij . Eventually, the full fermion contribution in the n f = 1 sector reads and comparing this result to the one in eq. (3.8) from the fugacity expansion, we find a nontrivial relationship between the two expressions. We will comment further on this relation in section 5.5 and establish it in detail in appendix B.

Sector
In the sector where all but one, i.e., n max f − 1 fermions are propagating, the states of the transfer matrices T n max f −1 (t) are most conveniently labelled by the degree of freedom i = (a, α) not occupied by a temporal hopping term. The transfer matrices are hence of size 2(N 2 − 1) × 2(N 2 − 1) = n max f × n max f . The matrix elements (T Φ n max f −1 ) ij are calculated following the arguments outlined above for the n f = 0 and 1 sector, namely to take the determinant of the Yukawa matrix Φ with all columns and rows deleted except i and j, respectively. The reduced Yukawa matrix is then just a single element and hence we have which is just the complementary (i, j)-cofactor of Φ up to an overall sign. The transfer matrix describing all the possible configurations within a time slice needs to be complemented by the one inducing the colour changing hops due to the gauge link between the time slices. If fermion i is not hopping out of t and j not into t + 1 they will not contribute, while the mixing of the remaining degrees of freedom is described as before by taking the determinant of the hop matrix, which is the (i, j)-minor of W . The full fermion contribution in the n f = n max f − 1 sector finally yields (5.14) This can be compared to the one in eq. (3.9) from the fugacity expansion and we find again a nontrivial relationship between the two expressions.
It is easy to check that this generic definition yields the correct expressions for the transfer matrices and canonical determinants for the cases n f = 0, 1, n max Finally, one can show that the canonical determinants obtained in the fermion loop approach are equal to the ones using the fermion matrix reduction, cf. eq. (3.11). Using

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various relations between matrices of minors and cofactors, one can derive that The details of this derivation are given in appendix B. The r.h.s. forms the n × n matrix of cofactors of T of order n f and the trace in eq. (5.17) then yields the sum over the n principal minors of T of order n f denoted by E n f , i.e., Recalling a known relation from linear algebra between the sum of minors of a matrix and its symmetric functions [31] one has which establishes the equivalence between eq. (3.11) and eq. (5.19).

Remarks
We close this section with several remarks. Firstly, we note that in contrast to the full determinant det[U, X i ], which can be proven to be positive [15], the various canonical determinants det D n f [U, X i ] need not necessarily be positive. Obviously, det D n f =2(N 2 −1) [U, X i ] is so and it seems that at least det D n f =0 [U, X i ] is also positive, although we do not have any proof. It would be interesting to study potential fermion sign problems in the canonical sectors in the present model. Despite its simplicity due to the low dimension, it nevertheless contains all the important features of a gauge theory, and hence conclusions can most likely be generalised to more complicated gauge theories in higher dimensions, such as QCD in the canonical formulation [29]. Secondly, we note that the various sectors, in particular the ones with many fermions, can in principle be simulated by open fermion string (fermion worm) algorithms along the lines described in [23,24]. This approach has indeed already been applied successfully in ordinary supersymmetric quantum mechanics [22], in the supersymmetric nonlinear O(N ) sigma model [32] and in the two dimensional N = 1 Wess-Zumino model [25,27] where the transfer matrix techniques discussed here and in [26] are out of reach. Furthermore, for the model discussed in this paper, a discrete bond formulation for the bosonic degrees of freedom is available [33]. Such a formulation promises a huge gain in efficiency for numerical simulations, but it is not clear whether the bosonic bond formulation can be put into practice.
Thirdly, from investigations in the Hamiltonian formulation of the theory [3], where time is treated as a continuous variable, it is known that there is a (spectral) symmetry between the sectors with n f and 2(N 2 − 1) − n f fermions, due to the exchange symmetry between particles and antiparticles. Our results above indicate that the symmetry is not maintained by our choice of the discretisation in the Lagrangian formalism, but the reason for this is clear. As we mentioned earlier the Wilson term needed to control the doubler JHEP12(2014)044 fermions explicitly breaks the time reversal and hence the charge conjugation symmetry which of course is crucial for an exact particle/antiparticle exchange symmetry. However, since the symmetries are restored in the continuum limit without fine tuning, the symmetries between the various canonical sectors will also be mended automatically in the continuum, and the difference between the related sectors will provide a good estimate of the remaining systematic lattice artefacts.

Conclusions and outlook
In this paper we have investigated in detail the structure of the fermionic part of the d = 4 dimensional supersymmetric Yang-Mills quantum mechanics, i.e., N = 4 SYM QM with gauge group SU(N ). On the one hand, we derived a reduced fermion matrix whose size is independent of the temporal extent of the lattice. In addition, the dependence on the chemical potential is factored out and this allows the exact projection of the fermion determinant onto the canonical sectors with fixed fermion number, once the eigenvalues of the reduced matrix are calculated. On the other hand, we have presented the fermion loop formulation of the theory in which the grand canonical fermion determinant naturally decomposes into sectors with fixed fermion numbers. The construction of transfer matrices is rather straightforward in the various fermion sectors and the comparison with the fugacity expansion, accessible via the reduced fermion matrix, yields identical results and interesting relations between the transfer matrices and the eigenvalues of the reduced fermion matrix. In fact, we presented a proof which establishes the equivalence of the canonical determinants from the reduced fermion matrix approach and from the fermion loop formulation on the algebraic level.
Our results open various possibilities for a range of nonperturbative investigations of the theory. This can be done for example by numerical simulations using methods different from the usual Hybrid Monte Carlo approach, either using the transfer matrices in the various canonical sectors with fixed fermion numbers, or using the projection to the sectors with the help of the reduced fermion matrix. Another interesting approach could be the application of mean field methods to the spatial gauge degrees of freedom, again either in the transfer matrix approach or using the reduced fermion matrix. It is even conceivable that the methods presented here and the emerged simple structures lead to new analytic results in some interesting limits. All results obtained either way will provide important insights into the conjectured M-theory and will add to our understanding of the corresponding gauge/gravity duality, besides unveiling interesting physics of the model itself.
Another interesting line of research starting from here concerns the investigation of ordinary, non-supersymmetric gauge field theories in higher dimensions at finite fermion density, such as QCD at finite baryon density. It is notoriously difficult to obtain reliable results in these theories using the known numerical approaches, due to the intrinsic fermion sign problem at finite density, and any insight into how the simulations of these theories could be facilitated would be extremely valuable. The explicit fugacity expansion derived in this paper allows to investigate finite density simulations or canonical simulations in a JHEP12(2014)044 simple setup which nevertheless displays a similar structure, and hence contains all the important features, as the more complicated theories in higher dimensions such as QCD.
Finally, the extension of the loop formulation to N = 16 supersymmetric Yang-Mills quantum mechanics is in principle straightforward but requires special care. This is due to the fact that the corresponding dimensionally reduced model has obviously a different Dirac structure, and it remains to be seen whether the structure is compatible with the requirements for the fermion loop formulation. The fermion matrix reduction on the other hand should be unaffected by the change of the Dirac structure.

A Determinations of canonical determinants
In this appendix we review three alternative methods to calculate the canonical determinants from the matrix T in eq. (3.3). As shown in section 3 the canonical determinants are just the coefficients of the characteristic polynomial of the matrix T . The first method provides recursion relations which yield the coefficients in terms of the eigenvalues τ i of T . The second method evaluates the coefficients in terms of the traces of powers of T and the third makes use of the minors of T . The latter method turns out to be closely related to the transfer matrix approach in the fermion loop formulation and hence deserves special emphasis.
In the following we assume the matrix T to be of size n × n and for simplicity we consider only the case of antiperiodic b.c., hence the relevant characteristic polynomial is where 1 is the n × n unit matrix and the coefficients c k are the canonical determinants det D n f =k in sector k.

A.1 Coefficients from recursion relations
The coefficients can be obtained from the eigenvalues τ i of T using recursive relations [29].
To this end, we first define the partial products which fulfill r+1 (x) = (τ r+1 + x) r (x). Setting c k . After n steps we obtain the coefficients c k ≡ c (n) k of n (x) which are then just the canonical determinants det D n f =k . The generalisation of the recursion to include the minus sign from the periodic b.c. is straightforward.

A.2 Coefficients in terms of traces
Here we review the calculation of the coefficients c k in terms of traces of powers of the matrix T . To do so we introduce the notation Then, Newton's identities (or the Newton-Girard formulae) provide a set of relations between the traces, which can be solved recursively. The solution can conveniently be written down in closed form as and the generalisation to periodic b.c. is again straighforward.

A.3 Coefficients in terms of minors
Instead of computing the traces of the matrices T , T 2 , T 3 , . . . , T n we now present an alternative method for determining the coefficients of the characteristic polynomial which is more interesting from the point of view of the transfer matrix construction discussed in section 5. The method involves the expansion of determinants of order 1 to n [34]. In order to determine the coefficients c k of x k in g(x) it is useful to separate the occurrences of x by introducing f (x 1 , x 2 , . . . , x n ) = det (T + diag(x 1 , x 2 , . . . x n )) . (A.7) One then has g(x) = f (x, x, . . . , x) and c k is the sum of the coefficients of the terms with total degree k in f (x 1 , x 2 , . . . , x n ). Since f (x 1 , x 2 , . . . , x n ) is of degree 1 in each x i , it is straightforward to express the coefficient in terms of derivatives w.r.t. x i 's, where 0 ≤ k ≤ n. As a consequence the coefficients are now expressed explicitly in terms of the matrix elements of T . Denoting them by t ij it turns out that
It is then immediately clear that from which eq. (A.9) follows via eq. (A.10).
On the other hand the rules for the Laplace expansion of a determinant by a row or a column indicate that ∂ det T /∂t ij is the (i, j)-cofactor of T , or in fact the (i, i)-minor when i = j. Therefore, the partial derivatives in eq. (A.9) are simply the subdeterminants of T resulting from crossing out the rows and columns numbered by i 1 , i 2 , . . . , i k , i.e., the principal minors of T of order k.
Denoting the sum of principal minors of order k of T by E k (T ) and keeping in mind that det D k = c k one finds by comparison with eq. (3.11) that for each k = 1, . . . , n, which is a known identity in matrix analysis from linear algebra, see e.g. [31]. Comparing these results with the ones derived in section 5 we immediately notice that the trace over the states of the transfer matrix is represented in eq. (A.9) by the sum i 1 <i 2 <...<i k . The number of summands here is n k and indeed equal to the number of states in the sector with n f = k. Furthermore, the principal subdeterminants (minors) in eq. (A.9) correspond to the diagonal elements of the product of transfer matrices in the given sector.

B Equivalence of canonical determinants
Here we show that the canonical determinants obtained in the fermion loop approach, cf. eq. (5.17), are equal to the ones using the fermion matrix reduction, cf. eq. (3.11).
Following the notation introduced in section 5.5, for two index sets A and B of size n f the transfer matrix T Φ n f in eq. (5.15) is the transposed matrix of cofactors of Φ of order n f and is denoted by i.e., the transfer matrix T W n f can be expressed as a cofactor matrix instead of a complementary minor matrix.
Next, we note that the cofactor matrix C and the corresponding minor matrix M are related by modifying the sign of each element according to C AB = (−1) p(A,B) M AB . The sign change can be achieved by a similarity transformation with the matrix S AB = (−1) i∈A i δ AB , i.e., C = S −1 · M · S. Therefore a product of cofactor matrices becomes a product of minor matrices under a trace, and so we can eventually write det D n f = Tr Note that we have made use of the fact that C(Φ) † = C(Φ) since Φ † = Φ. We can now employ the Cauchy-Binet formula which states in its symmetric form that given the n × n matrices P, Q with R = P Q and two index sets A, B of size 1 ≤ k ≤ n the (AB)-minor of R is where the sum is taken over all index sets D of size k. From the formula it follows that for the matrices of minors (and similarly for the matrices of cofactors) one has Recalling from linear algebra [31] the fact that the sum of all principal minors of order n f of a matrix is equal to the (n max f − n f ) th symmetric function of its eigenvalues, i.e. E n f (T ) = S n max f −n f (T ), eventually proves the equivalence between det D n f from the fermion loop formulation in eq. (5.19) and from the fermion matrix reduction in eq. (3.11).
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