Multi-terminal Thermoelectric Transport in a Magnetic Field: Bounds on Onsager Coefficients and Efficiency

Thermoelectric transport involving an arbitrary number of terminals is discussed in the presence of a magnetic field breaking time-reversal symmetry within the linear response regime using the Landauer-B\"uttiker formalism. We derive a universal bound on the Onsager coefficients that depends only on the number of terminals. This bound implies bounds on the efficiency and on efficiency at maximum power for heat engines and refrigerators. For isothermal engines pumping particles and for absorption refrigerators these bounds become independent even of the number of terminals. On a technical level, these results follow from an original algebraic analysis of the asymmetry index of doubly substochastic matrices and their Schur complements.


Introduction
Thermoelectric devices use a coupling between heat and particle currents driven by local gradients in temperature and chemical potential to generate electrical power or for cooling [1,2,3,4]. Since they work without any moving parts, such machines have a lot of advantages compared to their cyclic counterparts, which rely on the periodic compression and expansion of a certain working fluid [5]. However, so far their notoriously modest efficiency prevents a wide-ranging applicability. Although it has been shown that proper energy filtering leads to highly efficient thermoelectric heat engines [6], which, in principle, may even reach Carnot efficiency [7,8], so far no competitive devices coming even close to this limit are available. Consequently, the challenge of finding better thermoelectric materials has attracted a great amount of scientific interest during the last decades.
Recently, Benenti et al discovered a new option to enhance the performance of thermoelectric engines [9]. Their rather general analysis within the phenomenological framework of linear irreversible thermodynamics reveals that a magnetic field, which breaks time reversal symmetry, could enhance thermoelectric efficiency significantly. In principle, it even seems to be possible to get completely reversible transport, i.e., devices that work at Carnot efficiency while delivering finite power output. This spectacular observation prompts the question, whether this option can be realized in specific microscopic models.
An elementary and well established framework for the description of thermoelectric transport on a microscopic level is provided by the scattering approach originally pioneered by Landauer [10]. The basic idea behind this method is to connect two electronic reservoirs (terminals) of different temperature and chemical potential via perfect, infinitely long leads to a central scattering region. By assuming non interacting electrons, which are transferred coherently between the terminals, it is possible to express the linear transport coefficients in terms of the scattering matrix that describes the motion of a single electron of energy E through the central region. Thus, the macroscopic transport process can be traced back to the microscopic dynamics of the electrons. This formalism can easily be extended to an arbitrary number of terminals [11,12].
Within a purely coherent two-terminal set-up, current conservation requires a symmetric scattering matrix and hence a symmetric matrix of kinetic coefficients, even in the presence of a magnetic field [13]. Therefore, without inelastic scattering events the broken time reversal symmetry is not visible on the macroscopic scale. An elegant way to simulate inelastic scattering within an inherently conservative system goes back to Büttiker [14]. He proposed to attach additional, so-called probe terminals to the scattering region, whose temperature and chemical potential are adjusted in such a way that they do not exchange any net quantities with the remaining terminals but only induce phase-breaking.
The arguably most simple case is to include only one probe terminal, which leads to a three-terminal model. Saito et al [15] pointed out that such a minimal set-up is sufficient to obtain a non-symmetric matrix of kinetic coefficients. However, we have shown in a preceding work on the three-terminal system [16] that current conservation puts a much stronger bound on the Onsager coefficients than the bare second law. It turned out that this new bound constrains the maximum efficiency of the model as a thermoelectric heat engine to be significantly smaller than the Carnot value as soon as the Onsager matrix becomes non-symmetric. Moreover, Balachandran et al [17] demonstrated by extensive numerical efforts that our bound is tight.
The strong bounds on Onsager coefficients and efficiency obtained within the threeterminal set-up raise the question whether they persist if more terminals are included. This problem will be addressed in this paper. We will derive a universal bound on kinetic coefficients that depends only on the number of terminals and gets weaker as this number increases. Only in the limit of infinitely many terminals, this bound approaches the well-known one following from the positivity of entropy production. By specializing these results to thermoelectric transport between two real terminals with the other n − 2 acting as probe terminals, we obtain bounds on the efficiency and the efficiency at maximum power for different variants of thermoelectric devices like heat engines and cooling devices.
Our results follow from analyzing the matrix of kinetic coefficients in the n-terminal set-up and its subsequent specializations to two real and n − 2 probe terminals. On a technical level, we introduce an asymmetry index for a positive semi-definite matrix and compute it for the class of matrices characteristic for the scattering approach. These calculations involve a fair amount of original matrix algebra for doubly substochastic matrices and their Schur complements, which we develop in an extended and self-contained mathematical appendix.
The main part of the paper is organized as follows. In section 2, we introduce the multi-terminal model and recall the expressions for its kinetic coefficients. In section 3, we derive the new bounds on these coefficients. In section 4, we show how these bounds imply bounds on the efficiency and the efficiency at maximum power for heat engines, for refrigerators, for iso-thermal engines and for absorption refrigerators. In contrast to the former two classes, the latter two involve only one type of affinities, namely chemical potential or temperature differences, respectively, which implies even stronger bounds. We conclude in section 5.  We consider the set-up schematically shown in figure 1. A central scattering region equipped with a constant magnetic field B is connected to n independent electronic reservoirs (terminals) of respective temperature T 1 , . . . , T n and chemical potential µ 1 , . . . µ n . We assume non interacting electrons, which are transferred coherently between the terminals without any inelastic scattering. In order to describe the resulting transport process within the framework of linear irreversible thermodynamics, we fix the reference temperature T ≡ T 1 and chemical potential µ ≡ µ 1 , and define the affinities (α = 2, . . . n) .
By J ρ α and J q α we denote the charge and the heat current flowing out of the reservoir α, respectively. Within the linear response regime, which is valid as long as the temperature and chemical potential differences ∆T α and ∆µ α are small compared to the respective reference values, the currents and affinities are connected via the phenomenological equations [18] Here, we introduced the current vector with the respective subunits Analogously, we divide the matrix of kinetic coefficients into the 2 × 2 blocks L αβ ∈ R 2×2 (α, β = 2, . . . , n), which can be calculated explicitly. By making use of the multi-terminal Landauer formula [11,12], we get the expression where h denotes Planck's constant, e the electronic unit charge, the negative derivative of the Fermi function and k B Boltzmann's constant.
The expression (6) shows that the transport properties of the model are completely determined by the transition probabilities T αβ (E, B), which obey two important relations. First, current conservation requires the sum rule i.e., the transition matrix is doubly stochastic for any E ∈ R and B ∈ R 3 . Second, due to time reversal symmetry, the T αβ (E, B) have to posses the symmetry Notably, for a fixed magnetic field B, the transition matrix T(E, B) does not necessarily have to be symmetric. This observation will be crucial for the subsequent considerations.
For later purpose, we note that, by combining (5) and (6), L(B) can be expressed as an integral over tensor products given by Here, ½ denotes the identity matrix andT(E, B) arises from T(E, B) by deleting the first row and column. Consequently, the matrixT(E, B) must be doubly substochastic, which means that all entries ofT(E, B) are non-negative and any row and column sums up to a value not greater than 1.

Phenomenological Constraints
The phenomenological framework of linear irreversible thermodynamics provides two fundamental constraints on the matrix of kinetic coefficients L(B). First, since the entropy production accompanying the transport process descibed by (2) reads [18] the second law requires L(B) to be positive semi-definite. Second, Onsager's reciprocal relations impose the symmetry Apart from these constraints, no further general relations restricting the elements of L(B) at fixed magnetic field B are known. We will now demonstrate that such a lack of constraints leads to profound consequences for the thermodynamical properties of this model. To this end, we split the current vector J into an irreversible and a reversible part given by respectively. The reversible part vanishes for B = 0 by virtue of the reciprocal relations (13). However, in situations with B = 0 it can become arbitrarily large without contributing to the entropy production (12). In principle, it would be even possible to haveṠ = 0 and J rev = 0 simultaneously, i.e., completely reversible transport, suggesting inter alia the opportunity for a thermoelectric heat engine operating at Carnot efficiency with finite power output [9]. This observation raises the question, whether there might be stronger relations between the kinetic coefficients going beyond the well known reciprocal relations (13). In the next section, starting from the microscopic representation (6), we derive bounds on the kinetic coefficients, which prevent this option of Carnot efficiency with finite power.

Bounds following from Current Conservation
These bounds can be derived by first quantifying the asymmetry of the Onsager matrix L(B). For an arbitrary positive semi-definite matrix A ∈ R m×m we define an asymmetry index by Some of the basic properties of this asymmetry index are outlined in Appendix A. We note that a quite similar quantity was introduced by Crouzeix and Gutan [19] in another context.
We will now proceed in two steps. First, we show that the asymmetry index of the matrix of kinetic coefficients L(B) and all its principal submatrices is bounded from above for any finite number of terminals n. Second, we will derive therefrom a set of new bounds on the elements of L(B), which go beyond the second law. We note that from now on we notationally suppress the dependence of any quantity on the magnetic field in order to keep the notation slim.
For the first step, we define the quadratic form for any z ∈ C 2m and any s ∈ R. Here, A ⊂ {2, . . . , n} denotes a set of m ≤ n−1 integers. The matrix L A arises from L by taking all blocks L αβ with column and row index in A, i.e., L A is a principal submatrix of L, which preserves the 2 × 2 block structure shown in (5). Comparing (16) with the definition (15) reveals that the minimum s for which Q(z, s) is positive semi-definite equals the asymmetry index of L A . Next, by recalling (11) we rewrite the matrix L A in the rather compact form whereT A (E) ∈ R m×m is obtained fromT(E) by taking the rows and columns indexed by the set A. Decomposing the vector z as and inserting (17) and (18) into (16) yields Here we introduced the vector and the Hermitian matrix which is positive semi-definite for any However, sinceT(E) is doubly stochastic for any E, the matrixT A (E) must have the same property and it follows from Corollary 2 proven in Appendix B Hence, independently of E, K A (E, s) is positive semi-definite for any Finally, we can infer from (19) that Q(z, s) is positive semi-definite for any s, which obeys (24). Consequently, with (16), we have the desired bound on the asymmetry index of L A as This bound, which ultimately follows from current conservation, constitutes our first main result.
We will now demonstrate that (25) puts indeed strong bounds on the kinetic coefficients. To this end, we extract a 2 × 2 principal submatrix from L by a twostep procedure, which is schematically summarized in figure 2. In the first step, we consider the 4 × 4 principal submatrix of L given by which arises from L by taking only the blocks with row and column index equal to α or β. From (25) we immediately get with m = 2 Next, from (26), we take a 2 × 2 principal submatrix where (L αβ ) ij with i, j = 1, 2 denotes the (i, j)-entry of the block matrix L αβ . By virtue of Proposition 3 proven in Appendix B, the inequality (27) implies which is equivalent to requiring the Hermitian matrix to be positive semi-definite. Since the diagonal entries ofK {α,β} are obviously nonnegative, this condition reduces to DetK {α,β} = K 11 K 22 − |K 12 | 2 ≥ 0. Finally, expressing the K ij again in terms of the L ij yields the new constraint This bound that holds for the elements of any 2×2 principal submatrix of the full matrix of kinetic coefficients L, irrespective of the number n of terminals is our second main result. Compared to relation (31), the second law only requiresL {α,β} to be positive semi-definite, which is equivalent to L 11 , L 22 ≥ 0 and the weaker constraint Note that the reciprocal relations (13) do not lead to any further relations between the kinetic coefficients contained inL {α,β} for a fixed magnetic field B.
At this point, we emphasize that the procedure shown here for 2 × 2 principal submatrices of L could be easily extended to larger principal submatrices. The result would be a whole hierachy of constraints involving more and more kinetic coefficients. However, (31) is the strongest bound following from (25), which can expressed in terms of only four of these coefficients.

Bounds on Efficiencies
In this section, we explore the consequences of the bound (25) on the performance of various thermoelectric devices.

Heat engine
A thermoelectric heat engine uses heat from a hot reservoir as input and generates power output by driving a particle current against an external field or a gradient of chemical potential [5]. Such an engine can be realized within the multi-terminal model by considering the terminals 3, . . . , n as pure probe terminals, which mimic inelastic scattering events while not contributing to the actual transport process. This constraint reads  By assuming the matrix to be invertible, we can solve the self-consistency relations (33) for F 3 , . . . , F n obtaining After inserting this solution into (2) and identifying the heat current J q ≡ J q 2 leaving the hot reservoir and the particle current J ρ ≡ J ρ 2 , we end up with the reduced system of phenomenological equations. Here, the effective matrix of kinetic coefficients is given by and the affinities F ρ ≡ F ρ 2 = ∆µ 2 /T < 0 and F q ≡ F q 2 = ∆T 2 /T 2 > 0 have to be chosen such that J ρ , J q ≥ 0 for the model to work as a proper heat engine.
L HE is not a principal submatrix of the full Onsager matrix L and therefore the bound (25) does not apply directly. However, L HE can be written as the Schur complement L/L {3,...,n} (see Appendix C for the definition), the asymmetry index of which is dominated by the asymmetry index of L as proven in Proposition 4 of Appendix C. Consequently, we have or, equivalently, This constraint shows that whenever L ρq = L qρ , the entropy production (12) must be strictly larger than zero, thus ruling out the option of dissipationless transport generated solely by reversible currents for any model with a finite number n of terminals. For any n > 3 this constraint is weaker than (31). The reason is that the Onsager coefficients in (39) are not elements of the full matrix (5) but rather involve the inversion of L {3,...,n} defined in (34). Still, this constraint is stronger than the bare second law, which requires only Figure 3. Bounds on the efficiency of the multi-terminal model as a thermoelectric heat engine as functions of the asymmetry parameter x and in units of η C . The upper panel shows η max (x) (see (46)), the lower one η * (x) (see (49)). In both panels, the blue lines, from bottom to top, belong to models with n = 3, . . . , 12 terminals and the solid, black line corresponds to the bound following from the bare second law as obtained by Benenti et al [9]. The dashed line in the lower panel marks the Curzon-Ahlborn limit irrespective of whether or not L HE is symmetric.
The constraint (39) implies a constraint on the efficiency η of such a particleexchange heat engine [5], which is defined as Like for any heat engine, this efficiency is subject to the Carnot-bound η C ≡ 1 − T /T 2 , which, in the linear response regime, is given by , we now introduce the dimensionless parameters which allow us to write the maximum efficiency of the engine η max (under the condition J q > 0) in the instructive form [9] η max (x, Restating the new bound (39) in terms of x and y yields Consequently, maximizing (43) with respect to y yields the optimal y * (x) = h n (x) and the maximum efficiency This bound is plotted in figure 3 as a function of x for an increasing number n of terminals. For n = 3, we recover the result obtained in our preceding work on the three terminal model [16]. In the limit n → ∞, η max (x) converges to the bound derived by Benenti et al [9] within a general analysis relying only on the second law. However, for any finite n, η max (x) is constrained to be strictly smaller than η C , as soon as x deviates from 1. Thus, from the perspective of maximum efficiency, breaking the time reversal symmetry is not beneficial.
As a second important benchmark for the performance of a heat engine, we consider its efficiency at maximum power η * [20,21,22] obtained by maximizing the power output with respect to F ρ for fixed F q . In terms of the dimensionless parameters (42), it reads [9] η * (x, y) = η C xy 4 + 2y and attains its maximum at y * (x) = h n (x). In the lower panel of figure 3, η * (x) is plotted as a function of the asymmetry parameter x. For x = 1, this bound acquires the Curzon-Ahlborn value η CA ≡ η C /2. For x = 1, however it can become significantly higher even for a small number n of terminals. Specifically, we observe that η * (x) exceeds η CA for any n ≥ 3  (53)) of a thermoelectric refrigerator as a function of the asymmetry parameter x. The blue lines from bottom to top represent models with n = 3, . . . , 12 terminals. The black curve shows the bound required by the bare second law, which is asymptotically reached in the limit n → ∞.

Refrigerator
In the preceding section, we discussed the performance of the multi-terminal model if it is operated as a heat engine. Quite naturally, we can change the mode of operation of this engine such that it functions as a refrigerator. The resulting device consumes electrical power from which it generates a heat current from the cold to the hot reservoir.
Thus, compared to the heat engine, input and output are interchanged and the affinities F ρ < 0 and F q > 0 have to be chosen such that both currents J ρ and J q are negative.
Analogously to the case of the heat engine, we will now show that the bound (39) on the kinetic coefficients constrains the performance of the thermoelectric refrigerator described above. To this end, we will use the coefficient of performance [18] as a benchmark parameter. Its upper bound following from the second law is given by , which is the efficiency of the ideal refrigerator. In this sense, ε C is the analogue to the Carnot efficiency.
Taking the maximum of ε over F ρ (under the condition J ρ < 0) while keeping F q fixed, yields the maximum coefficient of performance [9] Here, we used again the dimensionless parameters defined in (42). Since y is subject to the constraint (44), ε max (x, y) attains its maximum with respect to y at y * (x) = h n (x), where h n (x) was introduced in (45). Figure 4 shows η max (x) for models with an increasing number of probe terminals n. For any finite n, ε C can only be reached for the symmetric value x = 1. The black line follows solely from the second law (40) and would in principle allow to reach ε C with finite current for x between −1 and 1. However, like for the heat engine, our analysis reveals that such a high performance refrigerator would need to be equipped with an infinite number of terminals.

Isothermal Engine
By an isothermal, thermoelectric engine, we understand in this context a device in which one particle current driven by a (negative) gradient in chemical potential drives another one uphill a chemical potential gradient at constant temperature T . In order to implement such a machine within the multi-terminal framework, we put F q 2 = · · · = F q n = 0. The remaining affinities F ρ 2 , . . . , F ρ n are connected to the particle currents via a reduced set of phenomenological equations given by  where (L αβ ) 11 denotes the (11)-entry of the block matrix L αβ defined in (6). We note that the heat currents J q 2 , . . . , J q n do not necessarily have to vanish. However, since they do not contribute to the entropy production (12), they are irrelevant in the present analysis. Similar to the treatment of the heat engine, we put J ρ 4 = · · · = J ρ n = 0, thus considering the terminals 4, . . . , n as pure probe terminals simulating inelastic scattering events. Consequently, (54) can be reduced further to the generic form Here, we have introduced the matrix again using the Schur complement defined in Appendix C. The affinities F ρ 2 , F ρ 3 > 0 have to be chosen such that J ρ 2 is negative and J ρ 3 is positive to ensure that the device pumps particles into the reservoir 2 against the gradient in chemical potential ∆µ 2 .
We will now derive a bound on the elements of L IE . By employing expression (11), we can write    and SinceT(E) is doubly substochastic for any E, the matrix T is also doubly substochastic. Therefore, by applying Corollary 3 of Appendix C, we find where ½ − T {3,...,n−1} denotes the principal submatrix of ½ − T consisting of all but the first two rows and columns. Expressing (60) in terms of the elements of L IE gives the bound We emphasize that, in contrast to the bound (39) we derived for the heat engine, the bound (61) is independent of the number of probe terminals involved in the device.
In the next step we explore the implications of (61) for the performance of the isothermal engine. To this end, we identify the output power of the device as and correspondingly the input power as Consequently, the efficiency of the isothermal engine reads  Figure 5. Bounds on benchmark parameters for the performance of the isothermal, thermoelectric engine as functions of the asymmetry parameter x. The right panel shows the maximum efficiency η max (x) (see (71)), the left one efficiency at maximum power η * (x) (see (72)). The black lines follow from the bare second law, the blue lines from the stronger constraint (61). Both, η max (x) and η * (x) asymptotically reach the value 1/4. The dashed line in the right plot marks the value 1/2 of η * (x) at the symmetric value x = 1.
We note that, in the situation considered here, the entropy production (12) reduces tȯ and thus the second lawṠ ≥ 0 requires η ≤ 1 for isothermal engines [22].
Optimizing η and P out (under the condition J ρ 3 > 0) with respect to F ρ 2 while keeping F ρ 3 fixed yields the maximum efficiency and the efficiency at maximum power where we have introduced the dimensionless parameters analogous to (42). Using these definitions, the bound (61) translates to and η max (x, y) as well as η * (x, y) attain their respective maxima with respect to y at y * = h(x). The resulting bounds are plotted in figure (5). We observe that the η max (x) reaches 1 only for x = 1 and decreases rapidly as the asymmetry parameter x deviates from 1, while η * (x) exceeds the Curzon-Ahlborn value 1/2 for x between 1 and 2 with a global maximum η * * = 4/7 at x = 4/3. In contrast to the non-isothermal engines analyzed in the preceding sections, all these bounds do not depend on the number of probe terminals.

Absorption Refrigerator
By an absorption refrigerator, one commonly understands a device that generates a heat current cooling a hot reservoir, while itself being supplied by a heat source [23,24].
The multi-terminal model allows to implement such a device by following a very similar strategy like the one used for the isothermal engine, i.e., we put F ρ 2 = . . . = F ρ n = 0 and end up with the reduced system of phenomenological equations connecting the heat currents with the temperature gradients. Assuming the terminals 4, . . . , n to be pure probe terminals then leads to where F q 2 < 0, F q 3 > 0 have to be adjusted such that J 2 q > 0 and J 3 q > 0. The matrix L AR is given by and by following the reasoning of the last section, we can derive the bound The efficiency of the absorption refrigerator can be consistently defined as Just like for the isothermal engine, after maximizing this efficiency over F q 2 (under the condition J q 2 > 0), we can derive an upper bound from (76). Again, this bound is independent of the number of probe terminals. Figure  6 shows it as a function of the asymmetry parameter x ≡ L ′ 23 /L ′ 32 .
For completeness, we emphasize that the efficiency (77) used here differs from the coefficient of performance used as a benchmark parameter in [23] and [24]. Since ε is unbounded in the linear response regime, maximization with respect to F q 2 or F q 3 would be meaningless.

Conclusion and Outlook
We have studied the influence of broken time reversal symmetry on thermoelectric transport within the quite general framework of an n-terminal model. Our analytical calculations prove that the asymmetry index of any principal submatrix of the full Onsager matrix defined in (5) is bounded according to (25). This somewhat abstract bound can be translated into the set (31) of new constraints on the kinetic coefficients. Any of these constraints is obviously stronger than the bare second law and can not be deduced from Onsagers time reversal argument. Furthermore, we note that it is straight forward to repeat the procedure carried out in section 3.2 for larger principal submatrices, thus obtaining relations analogous to (31), which involve successively higher order products of kinetic coefficients. Investigating this hierarchy of constraints will be left to future work.
After the general analysis of the transport processes in the full multi-terminal setup, we investigated the consequences of our new bounds on the performance of the model if operated as a thermoelectric heat engine. We found that both the maximum efficiency as well as the efficiency at maximum power are subject to bounds, which strongly depend on the number n of terminals. In the minimal case n = 3, we recover the strong bounds already discussed in [16]. Although our new bounds become successively weaker as n is increased, they prove that reversible transport is impossible in any situation with a finite number of terminals. Only in the limit n → ∞ we are back at the situation discussed by Benenti et al [9], in which the second law effectively is the only constraint. We recall that for n = 3 our bounds can indeed be saturated as Balachandran et al [17] have shown within a specific model. Whether or not it is possible to saturate the bounds for higher n remains open at this stage and constitutes an important question for future investigations.
Like in the case of the heat engine, the bound on the maximum coefficient of performance we derived for the thermoelectric refrigerator becomes weaker as n increases. Interestingly, the situation is quite different for the isothermal engine and the absorption refrigerator considered in the sections 4.3 and 4.4. The bounds on the respective benchmark parameters equal those of the three-terminal case irrespective of the actual number of terminals involved. If one assumed that any kind of inelastic scattering could be simulated by a sufficiently large number of probe terminals, one had to conclude that the results shown in figures 5 and 6 were a universal bound on the efficiency of any such device. At least, the results of sections 4.3 and 4.4 suggest a fundamental difference between transport processes under broken time-reversal symmetry that are driven by only one type of affinities, i.e., either chemical potential differences or temperature differences, and those, which are induced by both types of thermodynamic forces.
We emphasize that technically all our results ultimately rely on the sum rules (8) for the elements of the transmission matrix. These constraints reflect the fundamental law of current conservation, which should be seen as the basic physical principle behind our bounds. Therefore the validity of these bounds is not limited to the quantum realm. It rather extends to any model, quantum or classical, for which the kinetic coefficients can be expressed in the generic form (6). Some specific examples for quantum mechanical models which fulfil this requirement are discussed in [17] and [25]. A classical model belonging to this class was recently introduced by Horvat et al [26].
In summary, we have achieved a fairly complete picture of thermoelectric transport under broken time reversal symmetry in systems with non-interacting particles for which the Onsager coefficients can be expressed in the Landauer-Büttiker form (6). However, fully interacting systems, which require to go beyond the single particle picture, are not covered by our analysis yet. Exploring these systems remains one of the major challenges for future research.

Appendix A. Quantifying the asymmetry of positive semi-definite matrices
We first recall the definition (15) of the asymmetry index of an arbitrary positive semi-definite matrix A ∈ R m×m . Below, we list some of the basic properties of this quantity, which can be inferred directly from its definition. Appendix B. Bound on the asymmetry index for special classes of matrices Theorem 1. Let P ∈ {0, 1} m×m be a permutation matrix and ½ the identity matrix, then the matrix ½ − P is positive semi-definite on R m and its asymmetry index fulfils Proof. We first show that ½ − P is positive semi-definite. To this end, we note that the matrix elements of P are given by (P) ij = δ iπ(j) , where π ∈ S m is the unique permutation associated with P and S m the symmetric group on the set {1, . . . , m}.
We now turn to the second part of Theorem 1. For any z ≡ (z 1 , . . . , z m ) ∈ C m and s ≥ 0, we define the quadratic form To this end, we make use of the cycle decomposition π = i 1 , π(i 1 ), . . . , π n 1 −1 (i 1 ) . . . i k , π(i k ), . . . , π n k −1 (i k ) (B.6) of π, where i 1 , . . . , i k ∈ {1, . . . , m}, π l (i) is defined recursively by π l (i) ≡ π π l−1 (i) and π 0 (i) = i, (B.7) k denotes the number of independent cycles of and n r the length the r th cycle. By virtue of this decomposition, (B.4) can be rewritten as where, for convenience, we introduced the notation z[x] ≡ z x . Next, we define the vectors z r ∈ C nr with elements (z r ) j ≡ z [π j−1 (i r )] and the Hermitian matrices H nr (s) ∈ C nr×nr with matrix elements where periodic boundary conditions n r + 1 = 1 for the indices i, j = 1, . . . , n r are understood. These definitions allow us to cast (B.10) in the rather compact form Obviously, any value of s for which all the H nr (s) are positive semi-definite serves as a lower bound for S (½ − P). Moreover, we can calculate the eigenvalues of H nr (s) explicitly. Inserting the Ansatz v ≡ (v 1 , . . . , v nr ) t ∈ C nr into the eigenvalue equation where again periodic boundary conditions v nr+1 = v 1 are understood. This recurrence equation can be solved by standard techniques. We put v j ≡ exp (2πiκj/n r ) with (κ = 1, . . . , n r ) and obtain the eigenvalues Proof. The Birkhoff-theorem (see p. 549 in [27]) states that for any doubly stochastic matrix T ∈ R m×m there is a finite number of permutation matrices P 1 , . . . P N ∈ {0, 1} m×m and positive scalars λ 1 , . . . , λ N ∈ R such that N k=1 λ k = 1 and Hence, we have and consequently ½ − T must be positive semi-definite by virtue of Theorem 1.
Furthermore, using Proposition 2 and again Theorem 1 gives the bound (B.21).
Theorem 2. LetP ∈ {0, 1} m×m be a partial permutation matrix, i.e., any row and column ofP contains at most one non-zero entry and all of these non-zero entries are 1. Then, the matrix ½ −P is positive semi-definite and its asymmetry index fulfils Proof. Let q be the number of non-vanishing entries ofP. If q = 0,P equals the zero matrix and there is nothing to prove. If q = m,P itself must be a permutation matrix and Lemma 1 provides that ½ −P is positive semi-definite as well as the bound This definition naturally leads to the cycle decomposition π = i 1 ,π(i 1 ), . . . ,π n 1 −1 (i 1 ) · · · i k ,π(i k ), . . . ,π n k −1 (i k ) j 1 ,π(j 1 ), . . . ,πn 1 −1 (j 1 ) · · · jk,π(jk), . . . ,πnk −1 (jk) . Here, we introduced two types of cycles. The ones in round brackets, which we will term complete, are just ordinary permutation cycles, which close by virtue of the condition π nr (i r ) = i r and therefore must be contained completely in the set The cycles in rectangular brackets, which will be termed incomplete, do not close, but begin with a certain jr taken from the set Appendix C. Bound on the asymmetry index of the Schur complements For A ∈ C m×m partitioned as Regarding the asymmetry index, we have the following proposition. For the special class of matrices considered in Corollary 2, the assertion of Proposition 4 can be even strengthened. Before being able to state this stronger result, we need to prove the following Lemma. for p = 1. We now continue by induction. To this end, we assume that Lemma 1 is true for p = q. For p = q + 1 the matrix S 22 ∈ R (q+1)×(q+1) can be partitioned as  [29] for details). Furthermore, by the induction hypothesis, there is a doubly substochastic matrixT m−q ∈ R (m−q)×(m−q) , such that S/W 22 = ½ −T m−q . (C.21) Thus, (C.20) reduces to the case p = 1, for which we have already proven Lemma 1.
From Lemma 1 and Corollary 2, we immediately deduce