Thermodynamics of the Topological Kondo Model

Using the thermodynamic Bethe ansatz, we investigate the topological Kondo model, which describes a set of one-dimensional external wires, pertinently coupled to a central region hosting a set of Majorana bound states. After a short review of the Bethe ansatz solution, we study the system at finite temperature and derive its free energy for arbitrary (even and odd) number of external wires. We then analyse the ground state energy as a function of the number of external wires and of their couplings to the Majorana bound states. Then, we compute, both for small and large temperatures, the entropy of the Majorana degrees of freedom localized within the central region and connected to the external wires. Our exact computation of the impurity entropy provides evidence of the importance of fermion parity symmetry in the realization of the topological Kondo model. Finally, we also obtain the low-temperature behaviour of the specific heat of the Majorana bound states, which provides a signature of the non-Fermi-liquid nature of the strongly coupled fixed point.


I. INTRODUCTION
The possibility of faithfully simulating quantum low dimensional systems gives today the unique opportunity of testing against experiments predictions made by powerful methods -such as bosonization [1] (see also Schulz et al. in [2]), conformal field theory [3], integrable models and Bethe ansatz [4] -developed in theoretical investigations of strongly correlated condensed matter and spin systems. As a result, one may now confidently apply the above methods to newly engineered quantum systems relevant for the fabrication of quantum devices as well as for the realization, in a easily controllable setting, of new phases of matter such as the non-Fermi liquid phases realized in Dirac materials [5] and in overscreened multi-channel Kondo models [6]. Within the new models made accessible to theoretical investigations, great opportunities are offered by the analysis of pertinent junctions of one dimensional systems such as ladders and star junctions [7][8][9][10][11][12].
Networks of one dimensional models attracted much attention over the past few years. In pioneering works [13,14] a star junction of three quantum wires enclosing a magnetic flux was studied: modelling the wires as Tomonaga-Luttinger liquids (TLL), the authors of Ref. [13,14] were able to show the existence of an attractive finite coupling fixed point, characteristic of the geometry of the circuit. Later, a repulsive finite coupling fixed point was found in a T-junction of one dimensional Bose liquids [15]. Crossed TLL were also the subject of many investigations both analytical [16] and numerical [17]: these analyses pointed out that, in crossed TLLs, the junction induces behaviours similar to those arising from quantum Kondo impurities in condensed matter physics [18]. For what concerns the analysis of junctions of spin chains, in ref. [19] it was argued that novel critical behaviours emerge when crossing at a point two spin 1/2 Heisenberg models since, as a result of the crossing, some operators turn from irrelevant to marginal leading to correlation functions exhibiting power law decays with non-universal exponents. Star junctions of Josephson junction arrays were investigated in [20] with the result that a finite coupling fixed point was also emerging in these superconducting systems. With Majorana fermions [21] star junctions of quantum wires become very attractive since these geometries facilitate their braiding [22] allowing, at least in principle, for the engineering of quantum circuits relevant for the implementation of quantum protocols [23].
The very close relation between the phase diagram emerging from the investigation of networks of quantum onedimensional systems (quantum spin chains and quantum wires, essentially) and the one typical of multichannel Kondo models was established only very recently. In the two papers [24,25] it was shown that a star junction of three critical Ising models and a star junction of three XX models may be made equivalent to the two channel and the four channel over-screened Kondo model, respectively. To achieve their exact mapping, these authors used a generalization of the Jordan-Wigner transformation needed to satisfy the anticommutation relations between fermions located on different legs of the junction. For this purpose, one modifies the usual Jordan-Wigner transformation by the addition of an auxiliary space made -for a star junction-by three Klein factors, i.e. three real anti-commuting fields, lying at the inner boundary of each chain. As a result, in these realizations of the multichannel Kondo model, the central spin, with which the Jordan-Wigner fermions interact, is realized as a non-local combination of three Klein factors. The Kondo effect occurring when bulk fermions scatter on a composite "spin" non-locally encoded by any number of Klein factors located at different space points provides a realization of the topological Kondo effect. Since one expects that an extended spin is less sensitive to noise and decoherence, it is generally believed that this realization of Kondo models could "naturally" be much more robust than the one realized by other means (for example with quantum dots [26]).
In an effort to look for new experimental realization of a two-channel Kondo Model the authors of [27,28] showed that, in pertinent circuit regimes, networks of quantum wires supporting edge Majorana modes [29] provide a first experimentally attainable realization of the so-called Topological Kondo effect. Subsequently, the spin dynamics [30], as well as the exact solution for finite number of electrons [31], was investigated. In realistic cases the topological Kondo effect may be realized in networks of quantum wires supporting edge Majorana modes. For instance, it takes place when a mesoscopic superconducting island, capacitively coupled with the ground, hosts a set of Majorana bound states (MBSs), which may be the M Majorana edge modes (MEMs) of a set of spin-orbit coupled nanowires (green in figure 1 in color version) laying on this superconductor. Coulomb interactions among the electrons within the external leads are not essential for the topological Kondo effect and will not be accounted for in the following. However, for the host superconducting region, they play an essential role, since they determine its charging energy E c [27,30]. We focus on the temperature regime T E c , which implies that for all real processes, the number of electrons on the island is conserved. We emphasize that, at least at low temperature, the MEMs must be the only degrees of freedom in the central region which are involved in the dynamics. Under these conditions, the effective Hamiltonian describing the TKM at low energy scales is [27,28,32,33] Here ψ α (x) are the (complex) Fermi fields describing electrons in the wires α = 1, . . . , M and satisfying canonical anticommutation relations while the γ α = γ † α are Majorana fields constrained in a box connected with the wires and satisfying the Poisson algebra The symmetric matrix λ α,β > 0 is the analogue of the coupling with the magnetic impurity in the usual Kondo problem. The couplings h α,β = h β,α represent a direct coupling between only a pair of Majorana fermions. We note, in passing, that these can be made exponentially small by considering the MEMs sufficiently separated in space. This model coincides with the one studied in [31]. A related, yet different model, with real spinless fermions in the bulk, has been analyzed in [34].
The set of M MEMs may be regarded as a "Majorana spin": in contrast to a conventional spin the Majorana spin is non-locally encoded in the spatially separated MEMs and governed by the orthogonal symmetry group SO(M ). It has been found [27,28,32,33] that the topological Kondo model (TKM) provides a natural realization of the quantum critical point described by a Wess-Zumino-Witten-Novikov boundary conformal field theory [3,[35][36][37] and support non-Fermi-liquid ground states.
In this paper, using the thermodynamic Bethe ansatz (TBA), we investigate the thermodynamics of the TKM with an arbitrary number of wires M . For this purpose, we study a system with a thermodynamically large number of electrons in contact with an heat bath at temperature T and at fixed density and perform the thermodynamic limit as in (11). Our analysis allows to derive the free energy of the TKM for arbitrary number of concurring wires, to compute the ground state energy as a function of the number of wires and of their couplings to the Majorana modes, as well as the low temperature behaviour of the specific heat of the central region.
An outline of the structure of the paper and of the presented results is as follows: In section II we review the exact solution of the TKM, as provided in [31].
In section III we derive the exact free energy of the TKM made by M wires ending with M Majorana edge modes in the central region. Here, we write the TBA equations for both even and odd M . In section III C we obtain, in a closed form, the exact ground state energy shift due to the presence of the Majorana modes in the central region as a function of the effective tunnelling strength λ between the wires.
In section IV, we compute the entropy associated with the Majorana degrees of freedom localized in the central region both for T → 0 and T → ∞. This will provide us with the exact expression of the impurity entropy in the TKM. As expected, for T → ∞, the computation of the entropies amounts to determine the dimension of the Hilbert space associated with the Majorana degrees of freedom localized in the central region. We find: which summarizes equations (54) and is in agreement with the conformal perturbation theory predictions of [31] for any M . Finally, we show in section V that the system exhibits non-Fermi liquid behaviour at low temperatures by computing the temperature dependence of the specific heat of the Majorana modes in the central region.We find that the power expansion exhibits a term proportional to T 2 M −2 M . We argue that the power 2(M − 2)/M originates only from the symmetry of the strong-coupling fixed point.

II. THE BETHE ANSATZ SOLUTION
The diagonalization of (1) has been performed in [31] and it is based on the overall SO(M ) symmetry of the problem and on the results of [38] (see also [39]).
The Bethe Ansatz provides the exact quantized momenta {k n } n=1,...,N of the bulk electrons as a function of the solution of a set of nested Bethe ansatz equations. The energy is read from the eigenvalues of the transfer matrix [38] as: where: (where L is the length of each wire). The set of rapidities x (1) j satisfies, for even number of branches M = 2K, the system of coupled algebraic equations (Bethe equations): The set of integers r 1 , . . . , r K , denoting the number of roots in each level of the nested Bethe ansatz, defines a sector corresponding to given eigenvalues of the Cartan operators of the lie algebra D K , generating the group SO(M ) with even M = 2K [38].
For an odd number of branches M = 2K + 1, the energy is given again by (7), but the rapidities x (1) j j=1,...,r1 solve now the system [38]: associated this time with the non simply-laced B K algebra, generating the group SO(M ) with odd M = 2K + 1. We refer to [24] for the Bethe equations of this model. In the following section, we will use the notation and the results shown here to derive the thermodynamics of the TKM.

III. THE THERMODYNAMIC BETHE ANSATZ
In this section, we study the temperature dependence of the free energy of the system by means of the thermodynamic Bethe ansatz, which will allow for considerably simpler exact expressions for the thermodynamic quantities when the system size is large. The thermodynamic limit is defined as: In this limit the roots of the system (9), are known to group into clusters of roots with the same real part and equally spaced imaginary parts, called "string" solutions of the Bethe equations.
A. Even number of branches: M=2K In this case, the roots at each level group as follows: x (j) n;α + i 2 (n − 2l + 1) : x (j) n;α ∈ R, l = 1, . . . , n j = 1, . . . , K (12) in which the real number x (j) n;α is the string "center", i.e., the common real part of the solution, the index j denotes the level, the subscripts n and α denote the string index and the root index within the bound state.
The number of such clusters goes to infinity, while the spacing of the string centers goes to zero in the limit of infinite number of sites: therefore, a more convenient description is given in terms of densities of solutions. In particular, we will denote by ρ (j) n the density of solutions of length n at the j-th level of the nesting and byρ (j) n the density of unoccupied levels (holes) for the same type of configurations.
We first substitute the form (12) for the solutions of the Bethe equations into (9), then we group the terms of the products in the RHS according to their real part (string center). Then, we multiply all the equations corresponding to the different roots in the same string among them and we consider −i times the logarithm of the resulting equation. The kernel associated with the "scattering" of an n-string and an m-string is (hat denotes Fourier transform): On the other hand, the logarithmic derivative of the function e 1 produces the kernel s * A n,m , where the symbol * denotes convolution andŝ As a result, the resulting density equations are: in which there appear the function a n (ω) = e −n |ω| 2 and the K × K matrix: encodes the structure of the D K algebra. It is possible to write the free energy at temperature τ in the usual way as where S is the entropy of the whole system and is computed counting the number of states associated to a given root density as (sum over repeated indexes is implied): The partition function will be dominated by the root configurations corresponding to the densities that make the free energy stationary in the thermodynamic limit. These can be selected by imposing the variation of (18) with respect to ρ (j) n to be vanishing, also taking into account the constraint (16). Defining n (x) (20) such configurations are characterized by the saddle point equations: withÊ (1) . We have chosen to measure all the energies rescaled by 2E F π and defined the dimensionless temperature T = πk B τ 2E F (k B is the Boltzmann constant). The matrix: is the inverse of (13). The saddle point equations can be put in a more explicit form as (assume K ≥ 3): The case K = 2 is instead written as: With the aid of these functions, it is now possible to write the free energy in a more compact way. It differs from the free energy of M uncoupled wires by terms which are O 1 N , which can therefore be associated with the Majorana degrees of freedom in the central region and dubbed "junction" free energy. We write it as: The latter expression contains an infinite sum over the string lengths m and is therefore not very convenient for numerical evaluation. Hence, we multiply the system (23) by the matrixÃ, i.e., the inverse of the matrix M defined in (17) and extract an equality for L (j) +,m (x). In the case K = 2 the matrixÃ reads: For higher K, one has:Â This matrix has already appeared in [40] for the TBA associated to an integrable quantum field theory with G l ×Gm G l+m symmetry, apart from an overall normalization. Substituting into (24), we obtain the final result for the junction free energy in a more compact form: where is the Majorana contribution to the ground state energy (see also [41,42]). The case of K = 3 is the one considered in [31]. The formula (27) is valid for the topological Kondo with any even number of branches.

B. Odd number of branches: M=2K+1
We present now the thermodynamic Bethe ansatz for a system with an odd number of concurring branches. The case K = 1, i.e., three wires connected with three MEMs in the central region, corresponds to the four-channel Kondo model [24,25]. Its Bethe equations and thermodynamics have been studied in [43]. The case K = 2 is somewhat special, as there are only two levels present, and can be read in [31]. The right-hand side of the last of the equations (10) implies that, in the thermodynamic limit, the K-th level roots to group into "close" strings such as: For all the other levels, the strings are of the "wide" type (12). Once again, we substitute the form of the solutions (12) and (29) into (10), group the terms in the RHS according to their real part and multiply among them all the equations which show a root of the same string solution in the RHS. We then take the logarithm and multiply by −i and, when taking the thermodynamic limit, rewrite the resulting equations in terms of string densities as with the matrices (in Fourier transform)Â j,l m,n (ω) =Â m t j,l t j , n t j,l t l ω t j,l andM j,l (ω) = δ j,l −ŝ 1/t j,l δ j,l−1 + δ j,l+1 (32) in which t j = 1 + δ j,K represents the inverse length of the j-th root of the B K algebra and is equal to one for all the long roots and to 2 for the short root associated to the K-th node, while t j,l = max (t j , t l ) andŝ x =ŝ x (ω) =ŝ (xω). The thermodynamics is not very different from the one for even number of wires, overall, and pass through the computation of the variation of the free energy (18) with respect to the string densities ρ (j) m with the constraint (30). Here, however, the root length and the string length are not independents, hence we have a more complicated matrix in the equations for the sting densities, which implies that one will have more than one frequency appearing in the Fourier transform of (30). For this reason, we shall be rather pedantic in the subsequent exposition, even if similar results are already available in literature [24,40,[43][44][45].
From the minimization of the free energy of the system at the effective temperature T (defined in section III A) we obtain a saddle point condition for the function (20): The resulting free energy is (the units and the dimensionless temperature have been defined in section III A) where the second term, subleading in the number of particles, represents the free energy of the MEMs located at the junction. This is a rather inconvenient expression, for we have an infinite sum over all the string lengths. In order to simplify it, we define the matrices:Q and following [40], we multiply the saddle equations above by the matrix Q (j)− m,n and obtain Now consider the functions:ŷ then we multiply the system (33) on the left by the function a m/tj and sum over m, obtaining: The following step is to computeÃ =M −1 , the inverse of the matrix (38), and solve for y (j) . In particular (see also [40]). Now we substitute our findings into equation (34) and isolate the effective temperature dependence of the junction free energy: with pseudoenergies φ (l) m satisfying the system of equations (36). The temperature-independent part of the junction free energy is: having substituted the definition (39).

C. The ground state energy
We have summarized the results for the shift of ground state energy due to the coupling of M ≥ 3 wires with the Majorana bound states in equation (4) which is of the same form of the impurity contribution in the Kondo effect (see e.g. [41]). This expression can be obtained for even and for odd M by making use of a standard integral representation of the logarithm of the Γ-function [46] into the integral expressions (28) and (28).
When the coupling between legs λ goes to zero, the junction energy vanishes linearly in the coupling with the Majorana box with a coefficient which is independent of the number of legs. In figure 2, we show some example of (4) as a function of M and λ. It is interesting to note that the next non vanishing order in the expansion is λ 3 , with coefficient 4+12M −3M 2 48 depending on the number of branches.
In the strong coupling limit λ → ∞, the energy goes to a constant value as 1 λ with ψ being the logarithmic derivative of the Γ function. This quantity is monotonous in the coupling and is plotted in Figure 3. Here, it should be noted that there exists a point at intermediate λ where the scaling with M changes.

IV. THE ENTROPY OF THE MAJORANA EDGE MODES
In this section, we compute the entropy at T → ∞ of the MEMs and determine the dimension of the Hilbert space H J of the additional Majorana degrees of freedom localized at the concurring ends of the wires. We will compute the infinite-temperature limit of the entropy S = −T log (dim H J ). In addition, we compute the entropy of the ground state for T = 0. The residual entropy found at zero temperature shows that the degrees of freedom introduced by the Majorana modes, are in a non-trivial superposition of degenerate states. We will obtain the entropy of the central region at T = 0 S (0) J from the first-order temperature correction to the free energy [42] which in turn is obtained by substituting the zero-temperature solutions to the TBA systems (23) and (36).
Similarly to what observed in [47,48], the systems (23) and (36)  IV D for odd M . Remarkably, the solution of this system can be parametrized in a similar way in the low-and in the high-temperature limits. Therefore, we will be able to study the high-temperature limit in sections IV B and IV E for even and for odd number of wires, respectively. We will also be able to study the low-temperature limit in sections IV C and IV F, again for even and for odd M . The following sections will be devoted to the analysis of the TBA equations in the cases of even and of odd number of branches.
A. The asymptotic TBA: even number of branches Let us analyze the system (23): due to the boundedness of the source term, located in the equation for φ (1) 2 , the infinite temperature limit removes such term from the system. As a consequence, a constant solutionφ (j) m appears. The presence of such solution, which only occurs in the limits of zero and at infinite temperature, greatly simplifies the analysis of the TBA equations. In fact, we plug such constant ansatz into (23) and rewrite the system in terms of the quantityη It is implicit, here, thatη On the other hand, when the temperature goes to zero, the positive-defined driving term diverges to +∞ for all the value of the argument. As a result, it is possible to substitute another rapidity-independent ansatz into the TBA equations and one obtains once again the system (44), this time with the boundary condition: coming from the definition (43). On the contrary, at infinite temperature, since there are no divergences that can force one order of solutions to vanish, we expectη (j) n = 0 for all values of j and m. Therefore, the very same set of algebraic equations (44) holds in the two regimes, high and low temperature, yet these regimes are distinguished by the boundary conditions 45 only. Once we have the asymptotic solutions at infinite or at zero temperature, we insert them into the formula for the free energy (27). Then the constant asymptotic solution factors out of the integral, and one obtains that the temperature dependence of the free energy is given by (46) in which there appear the filling fractions Note that (46) represents both the leading contribution to the free energy as T → ∞ and the first order correction in a low-temperature expansion. Of course, the asymptotic solutions for the f (j) 1 s will be different in the two cases. The solution to (44) is given in [49,50] by the expression: The quantities Q (j) m are objects which only depend on the overall symmetry of the problem and satisfy [51] the system of equations with initial conditions It is understood that Q (0) m = 1 for all m, as well as Q (j) 0 = 1 for all j. The expressions above are written in terms of sums of characters χ ω of the corresponding Lie algebra representations, each built on a given highest weight ω described in [50]. Such characters are specialized to the point yρ, where ρ = K j=1 ω j is the Weyl vector, given by the sum of the fundamental weights {ω j } j=1,...,K The complex number y = 2πi g+l is determined from the dual Coxeter number of the symmetry algebra (g = 2K − 2 for the D K algebra) and the positive integer l; the latter is the order at which the vanishing of the solutionη (1) l is imposed, as described below and in [50], and for the sake of definiteness it will be l = 2 in the low-temperature limit and l → ∞ in the high-temperature limit.
In order to compute the free energy, we solve for the fillings (47) using the system (44): Here below, we specialize the solution to the two limiting cases. These are different only in the value of l, the formulas (51) and (49) holding for both the constant ansatze.

B. The high temperature limit
Analogously to the results for the spin-s Heisenberg spin chain and the multichannel Kondo model, we seek a rational solution to (44), which is non-vanishing for all theη (j) m s. In particular, this is obtained when y → 0 (or l → ∞). Then the characters χ ω (yρ) appearing in (51) yield, by definition, the dimension of the representation of highest weight ω. In particular, we have: By inserting such a solution into (51) and the latter into (52) we obtain the asymptotic fillings, corresponding to the solution at high temperature of the TBA system. These are explicitly listed in appendix A. By substituting these into (46) and evaluating it numerically for several values of M , we conclude that C. The low temperature limit As observed earlier, at T → 0 we need to solve (44) with the constraint (45). This is written in terms of the specialized characters of the irreducible representations of the D K algebra at the point iπ K ρ, which corresponds to the choice l = 2 in (51). These are provided in [51] as: At this particular specialization, these three characters are enough [50] to determine the full solution recursively, thanks to the constraint (45), which implies the property Q (j) 2 = 1, Q (j) 3 = 0, j = 1 < . . . , K, established in [50]. In fact, in (52), we only need the Q (j) 1 , j = 1 < . . . , K. If K > 3, we can specialize the first of (49) for j = 1 and, using Q 1 . Analogously, we obtain recursively the results up to j = K − 2, which is most rapidly done by solving numerically the recurrence.
Finally, evaluating numerically the free energy (46) for several values of M , we conclude that the residual ground state entropy introduced by the interaction of the bulk fermions with the Majorana degrees of freedom is in agreement with the boundary conformal field theory results of [31].
D. The asymptotic TBA: odd number of branches In the same way as for even M , the driving term disappears from the system (36) both in the high-and in the low-temperature limits. Therefore, the TBA system is reduced to: and the two regimes are distinguished by the fact that (45) has to be imposed at T = 0, while we haveη The solution to (57) is again given by (48), but now the Qs are solutions [51] of: with initial conditions given by in which the dual Coxeter number is g = 2K − 1 for the B K algebra. The solution to (58), in turn, is provided in [50]. In order to evaluate the free energy, all we will be needing are the expressions: in which we note that only the K numbers Q (j) 1 appear and they can be read in (59) once specialized to a particular value of l. By using the definition (47 into the free energy (40), one obtains for the junction free energy. This holds both as the leading order as T → ∞ and as the first order in a low-T expansion, provided the corresponding asymptotic fillings f (j) 1 are inserted.

E. The high temperature limit
We select the solution of (58) with y → 0 in the initial values (59). Then, the specialized characters become -by definition -equal to the dimension of the corresponding representation. In particular: Inserting such solutions into (60) we obtain the expressions listed in appendix A. Plugging them into (61) and evaluating numerically for several values of (odd) M , we conclude that for all odd values of M (≥ 3).

F. The low temperature limit
As T → 0, we need to evaluate the solution of (57) with the constraint (45). Hence, we specialize the characters in (59) to the point 2πi 2K+1 , corresponding to l = 2 in the initial values (59). From the expressions: and rewriting the constraint (45) as Q = 1 as in [50], we can solve recursively the system (58). Once again, this is most efficiently done by the aid of a computer. Evaluating the free energy of the MEMs by the use of (60) into (61) for several values of M , we conclude that the ground state entropy produced by the topological Kondo term is which shows that our thermodynamic Bethe ansatz agrees with the results of [31], obtained by boundary conformal field theory.

V. THE SPECIFIC HEAT OF THE MAJORANA EDGE MODES
A signature of the non Fermi liquid nature of the strongly coupled fixed point is given by the next term in the expansion (42). In this section, we shall see that it is indeed of order O T 1+2 M −2

M
. As a consequence, the Majorana contribution to the specific heat behaves at low temperatures as where the (dimensionless) Kondo temperature T K depends on the coupling between legs as A non-integer power is a strong and experimentally detectable signature of the presence of a non Fermi liquid fixed point. In particular, it is related to the operator content of the conformal field theory describing the fixed point at strong coupling, as explained in [37]. The power we find is indeed the same as what derived from the conformal dimension of the leading irrelevant operator in the conformal perturbation theory of [30] around the strongly coupled critical point. Therefore, we confirm such result from our Bethe ansatz computation. Also in this approach, it has its origin in the overall symmetry of the problem only, since what characterizes the Kondo model in the TBA equations, i.e., the functional form of the source term, disappears in the derivation of the Y -system. The latter is uniquely specified by the SO(M ) 2 symmetry of the model, and the periodicity of the associated TBA equations is all we have used to determine the power characterizing the temperature dependence of the specific heat as T → 0.

A. Even number of branches
Following [41,42], we focus on the rapidity regime x → −∞, which introduces the first temperature corrections to the asymptotic values of the TBA functions. In fact, in this regime the source term is very small, being 2e πx , so that the ratio with the temperature in the denominator gives a finite result and the rapidity dependence appears in the solution. For intermediate and large positive values of the rapidity, instead, the solutions of the TBA equations quickly approach their asymptotic zero-temperature value. It is convenient, in this regime, to shift the rapidities as x = ξ + 1 π log T , with T → 0. Similarly to (43), we can define and, starting from (23), we see that the η (j) m s satisfy, for even M and at low temperature, the following TBA system log η (j) n (ξ) = −2e πξ δ j,1 δ n,2 + s * L We now look at the free energy (27) and shift the integration variable as above. In order to express the temperature in terms of the Kondo temperature (67), we need to rescale the integration variable as ξ → (M −2)ξ π , which brings its temperature dependence for T T K into the form To evaluate this expression, being the TBA equations (69) symmetric in the exchange K ↔ K − 1, one only needs the functionsĥ which come directly from the definitions (15) and (26). The case M = 2K = 4 is again written as (71) witĥ Having already isolated the term in the free energy which is linear in the temperature, originating from the asymptotic value of the functions η (j) m s, we now concentrate on the following power, which in turn comes from the low-temperature dependence on the rapidity of these functions.
Being the source term exponential and given the relation s * e bx ∝ e bx , a reasonable ansatz [52] is that, at low enough temperatures, the exponential behaviour of the source term is retained and we can assume the form withη (j) m being the already discussed asymptotic values. Under the ansatz (74), the free energy becomes where the term linear in T has been computed in section IV A. In order to fix the coefficient b, we note that from the TBA systems (69) a universal Y -system for the analytic continuation in the complex plane of the functions η (j) m can be derived, as explained in [53]. Universality, in this context, means that the source term of the integral equations disappear, i.e., that only the SO(M ) 2 structure of the TBA is retained in the Y -system, while the source, characterizing the Kondo models, plays no role. We refer the reader to the original literature [53] for the explicit form of the SO(M ) 2 Y -system and just state here that its solutions have periodicity: which has been obtained in [54,55] for both even and odd M . It follows (see also [56]) that they can be expanded in powers of the variable e For M = 4, 6, this value reproduces the results of [42] for the equivalent Coqblin-Schrieffer models.

B. Odd number of branches
We repeat here the procedure for an odd number of branches. After the change of variable x = ξ + 1 π log T we obtain, again at low temperatures and with the definitions (68) and (70), the TBA system The temperature dependence of the free energy for an odd number of legs and T T K can then be rewritten as which have been obtained from (39) and (15). As explained for even M , substituting the ansatz (74) into (78), the free energy turns into whose linear term has been determined in section IV D and which, using (66), proves that the specific heat vanishes as a power law of T → 0. The power itself is identified through the periodicity of the corresponding Y -system, as found in [54,55]: which proves that the power law is just the one stated in (66) and, therefore, that the system at the strong coupling fixed point cannot be modeled by a Fermi liquid. For M = 3, this agrees with the known results for the four-channel Kondo model. We note in passing that the determination of a the specific heat characteristic power through the periodicity of the associated Y -system is, remarkably, more general than for the model considered. For instance, the formula C J ∝ T T K 2g g+l , which can be proved from the periodicity properties assessed in [54] too, works in general for the classes of impurity models. In fact, for the l-channel Kondo model, the level of the Bethe ansatz is l, while the dual Coxeter number for su(2) is g = 2.
Therefore C J ∝ T T K 4 2+l , which is an elegant way of deriving an established result [37,44,52]. In the l-species su(N ) Coqblin-Schrieffer model, the dual Coxeter number is g = N , hence the specific heat power law at low temperatures reads C J ∝ T T K

VI. CONCLUDING REMARKS
We have used the thermodynamic Bethe ansatz to provide an exhaustive analysis of the thermodynamics of the topological Kondo model. This model describes physical situations where M quantum wires with Majorana edge modes concur in a central region, where the electrons interact with an extended "spin" formed by spatially distant Majorana modes. The model has SO(M ) symmetry. Our study provided the thermodynamics of this model for arbitrary M and a proof that this model represents a natural description of non-Fermi liquid quantum critical points associated with the SO(M ) 2 Wess-Zumino-Witten-Novikov boundary conformal field theory. Indeed, a key result of our work is the exact computation of the low temperature behaviour of the specific heat of the central region. We have found that where T K e − π λ(M −2) is the Kondo temperature and argued that the non-integer power originates only from the symmetry of the strong-coupling fixed point.
We have derived the exact free energy of the topological Kondo model and provided, in a closed form, the exact ground state energy shift due to the presence of the Majorana edge modes in the central region as a function of the effective tunnelling strength between the end of the wired.
We have computed the entropy associated to the Majorana degrees of freedom localized in the central region -the so-called impurity entropy -both for T → 0 and for T → ∞. Our analysis provided the exact asymptotic expressions of the impurity entropy of the topological Kondo model for arbitrary M . For T → ∞, the computation of the entropy determines the dimension of the Hilbert space of the Majorana degrees of freedom localized in the central region.
Throughout this paper, we have assumed that the Majorana bound states in the central region were sufficiently separated in space so as to allow to neglect the direct coupling between pairs of Majorana modes. The inclusion of such terms should be an interesting extension of our work, since these terms are known to induce -for sufficiently low temperature -the M → M − 2 crossover between different non Fermi liquid phases and could provide a signature of the presence of MBSs in the central region.
Networks of quantum wires with Majorana edge modes realize the topological Kondo effect only when the superconducting central region is capacitively coupled with the ground [27,28,32,33]; when the superconducting central region is floating, the system exhibits remarkable even-odd effect in the tunnelling conductance through the central region [29]. Both situations can be tested experimentally with present day technology, leading to the exciting possibility of an experimental test of exact results stemming from Bethe ansatz.
Ising spin chain realizations of the topological Kondo model have been also recently investigated in [34]. It was argued that this kind of system exhibits Fermi liquid behaviour at the strong-coupling fixed point when the number of legs is even, while it produces a non-Fermi liquid when the number of legs is odd. This feature, due to the reality of the fermions in the bulk, is manifest, e. g., in the low temperature behaviour of the specific heat and marks a difference with the model analyzed in this paper, which reaches a non Fermi liquid fixed point at low temperatures for all values of M .
Networks of XX chains could provide a basis for constructing alternative experimental realizations of the topological Kondo model investigated in this paper: their implementation could be realized in ultracold atom platforms and we think that it would be a very interesting line of future research to investigate feasibility and detection schemes in such setups.