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Energy transmutation in nonequilibrium quantum systems

We investigate the particle and heat transport in quantum junctions with the geometry of star graphs. The system is in a nonequilibrium steady state, characterized by the different temperatures and chemical potentials of the heat reservoirs connected to the edges of the graph. We explore the Landauer-Buettiker state and its orbit under parity and time reversal transformations. Both particle number and total energy are conserved in these states. However the heat and chemical potential energy are in general not separately conserved, which gives origin to a basic process of energy transmutation among them. We study both directions of this process in detail, introducing appropriate efficiency coefficients. For scale invariant interactions in the junction our results are exact and explicit. They cover the whole parameter space and take into account all nonlinear effects. The energy transmutation depends on the particle statistics.


Introduction
The study of non-equilibrium quantum systems is among the most rapidly expanding areas of theoretical physics. Triggered by the remarkable experimental progress in manipulating trapped ultra-cold atomic gases, there is recently great interest in the search for universal properties of such systems (see e.g. [1] for a review). Much attention is devoted to the behavior of quantum systems after a quench and in particular, on the nature of the equilibrium state which is approached in this case. Other interesting studies concern the impact of both internal and space-time symmetries (e.g. scale invariance) on the quantum transport in Non-Equilibrium Steady States (NESS).
In this paper we investigate some general features of the energy transport in NESS's, representing non-equilibrium extensions of a Gibbs State (GS). The physical models we focus on are schematically shown in Fig. 1. We are dealing with a multicomponent system represented by the n ≥ 2 semi-infinite leads (edges) L i of a star graph Γ. Each lead L i is attached at infinity to a heat reservoir R i . The interaction between the n components of the system is localized in the vertex of the graph and is defined by a scattering matrix S. Each heat reservoir R i is described by a GS characterized by inverse temperature β i and chemical potential µ i . From Fig. 1 it is evident that the system is away from equilibrium if S admits a nontrivial transmission coefficient between two reservoirs R i and R j with (β i , µ i ) = (β j , µ j ).  Figure 1: A star graph Γ with scattering matrix S at the vertex and leads L i connected at infinity to thermal reservoirs R i . We start our investigation by shortly reviewing the explicit construction of a NESS Ω β,µ , induced by the scattering matrix S from the tensor product of the GS's relative to the heat reservoirs R i . The state Ω β,µ is fully determined by S and (β i , µ i ). We show that Ω β,µ is not invariant under parity and time-reversal transformations and generates therefore a nontrivial orbit O β,µ , consisting of four nonequivalent NESS's. The total energy of the system in these states has two components: heat energy (parametrized by β i ) and chemical potential energy (parametrized by µ i ). Provided that the dynamics of the system is invariant under time translations, the total energy is conserved. For generic values of (β i , µ i ) the two energy components are however not separately conserved and an energy transmutation occurs. The process is controlled by a single parameterQ, describing the heat flow in the junction. For Q < 0 heat energy is transformed in chemical potential energy. The opposite transformation takes place forQ > 0.
The regimeQ < 0 has been investigated mostly by means of linear response theory. For the details we refer to the review paper [2] and the references therein, observing only that in these studies the junction is usually compared to a heat engine, whose efficiency represents the main point of interest. This comparison is very suggestive, keeping always in mind that the junction acts actually as a converter between heat and chemical potential energy. The caseQ > 0 is quite subtle and has been explored only partially in the domain where the junction can be compared to a refrigerator or heat pump. The main goal of the present paper is to study the system in Fig. 1 as an energy converter, thus providing a systematic and unified description of both regimesQ ≶ 0, which are characterized by appropriate efficiency coefficients. Since the energy transport in the junction is strongly influenced by nonlinear phenomena, the well known linear response approximation is not suitable for our investigation because it covers only a little part of the parameter space. For this reason a fundamental objective of our study is to simplify as much as possible the dynamics in order to solve the problem in exact form, preserving at the same time a nontrivial quantum transport. In this respect we demonstrate that assuming free propagation along L i and scale-invariant point-like interactions in the vertex is enough for this purpose. We study both fermionic and bosonic systems, showing that the particle statistics affects the energy transmutation. The detailed comparison between the two cases is very instructive, because the bosonic case is poorly investigated. The impact of parity and time reversal on the quantum transport is discussed in detail as well.
We proceed as follows. In the next section we briefly recall the construction of the NESS Ω β,µ and its orbit O β,µ under parity and time reversal transformations. In section 3 we fix the Schrödinger dynamics on the leads and the interaction at the vertex of Γ. Studying the local conserved currents and the relative densities, we show that the junction converts heat energy in chemical potential energy or vice versa, depending on the values of the parameters characterizing the reservoirs R i . The basic features of this process of energy transmutation are investigated in section 4, where we introduce the efficiency coefficients for both caseṡ Q ≶ 0. We describe here also the scale invariant (critical) interactions in the junction. The efficiency of the quantum transport for a system with two leads is investigated in section 5 both for the LB state Ω β,µ and the orbit O β,µ . We also relate here our approach to some exact results, concerning other systems and NESS's. In section 6 we describe the main features of a junction with 3 leads. Section 7 is devoted to the bosonic case. The conclusions and some future developments form the content of Section 8. The appendix collects some technical results about the 3-lead junction.
2 Non equilibrium states on Γ Following the pioneering work of Landauer [3] and Büttiker [4], non-equilibrium systems of the type in Fig. 1 have been extensively investigated. In this section we briefly recall an explicit quantum field theory construction [5] of the Landauer-Büttiker (LB) state Ω β,µ , which adapts to the star graph case some general ideas [6]- [10] about NESS's. This construction is very convenient for deriving correlation functions and for studying the orbit O β,µ of Ω β,µ under parity and time reversal transformations. As mentioned in the introduction, the elements of this orbit represent new nonequivalent NESS's with interesting physical properties.
The first step in constructing the LB state Ω β,µ is to describe the asymptotic dynamics at t = −∞ (i.e. before the interaction) in terms of the operators which create and annihilate the particle excitations in the reservoir R i . In agreement with the orientation of the leads in Fig.1, the condition k > 0 implies that (2.1) create and annihilate incoming particles. For fermionic systems (2.1) generate therefore an incoming Canonical Anti-commutation Relations (CAR) algebra A in . We denote by Ω β i ,µ i be the GS associated with the heat reservoir R i (see e.g. [11]) and consider the tensor product ⊗ n i=1 Ω β i ,µ i . The next step is to relate A in with the CAR algebra of outgoing excitations A out , generated still by the creation and annihilation operators (2.1), but with k < 0. For this purpose we have to introduce the interaction, connecting different heat reservoirs and driving the system away from equilibrium. We consider the simple case where the incoming particles propagate freely along the leads towards the vertex of the graph, where they are reflected or transmitted with some probability in the rest of the graph. This process is codified in the reflection-transmission equations which relate A in and A out . Here S(k) is the scattering matrix describing the point-like interaction in the vertex of the graph. We assume unitarity and Hermitian analyticity the star * indicating Hermitian conjugation. These conditions imply that S(k)S(−k) = I, which ensures the consistency of the constraints (2.2). The whole algebra A, generated by polynomials involving generators of both A in and Because of (2.3), the right hand side of (2.4) defines the kernel of an integral projection (instead of the usual identity) operator. Hermitian analyticity implies [12] that the * -operation in A is a conjugation. At this point the LB state Ω β,µ is the extension of ⊗ n i=1 Ω β i ,µ i from A in to the whole algebra A, performed by linearity via the reflection-transmission relations (2.2). This construction, which may look at the first sight a bit abstract, is in fact very efficient for deriving the correlation functions defining the LB representation H LB of A. We stress that H LB describes non-equilibrium physics and is not equivalent to the Fock and Gibbs representations H F and H G of A, known [12,14] from the equilibrium case. Denoting by (· , ·) the scalar product in the Hilbert space H LB , one has [5] ( is the Fermi distribution in the heat reservoir R i with dispersion relation ω i (k). Notice that the construction allows for different dispersion relations in the different reservoirs. The explicit form of a i (k)a * j (p) β,µ is obtained from (2.5) by the substitution (2.7) As well known, employing the CAR algebra, one can express a generic correlation n-point correlation function as a polynomial of the two-point correlator (2.5). The non-equilibrium features of the correlation functions are encoded in the mixing of the Fermi distributions, associated with the different heat reservoirs R i , via the scattering matrix S, which is evident already from (2.5). An advantage of the above framework is that it allows to investigate directly the behavior of the LB state Ω β,µ under parity and time reversal. We first recall that these operations are implemented by a unitary operator P and an anti-unitary operator T , acting in the algebra A in the standard way: Apart from the multiplicative phase factors, the action of P and T seems to be the same, but one should remember that P is a linear operator, whereas T is anti-linear. Using the explicit form (2.5) of the two-point function and (2.8), one can explicitly verify that if (β i , µ i ) = (β j , µ j ) for some reservoirs. Therefore, Ω β,µ is not invariant under parity and time reversal and generates the nontrivial orbit O β,µ = {Ω β,µ , Ω X β,µ = XΩ β,µ : X = P, T, P T } . (2.10) The property (2.9) captures the fundamental difference of the LB state Ω β,µ with respect to the Fock vacuum Ω F ∈ H F and the Gibbs state Ω G ∈ H G , which are both P -and Tinvariant. The result (2.9) suggests also that the four states in the orbit O β,µ have different transport properties, which is confirmed by the analysis in section 5.2 below. Since time reversal symmetry is the quantum counterpart of classical reversibility, one can interpret the breakdown of this symmetry in the LB representation H LB as quantum irreversibility. We conclude by observing that above construction can be easily generalized to bosons by replacing the CAR algebra with a Canonical Commutation Algebra (CCA), which implies the substitution of the Fermi distribution in (2.5) with the Bose distribution: As one can expect on general grounds and as shown in section 7, the quantum transport is influenced by the statistics.

The Schrödinger junction
Quantum systems away from equilibrium behave usually in a complicated way. In most of the cases the linear response or other approximations are not enough for fully describing the complexity of this behavior. For this reason the existence of models which incorporate the main non-equilibrium features, while being sufficiently simple to be analyzed exactly, is conceptually very important. An interesting family of such models in s space dimensions is characterized by requiring that the interaction, which drives the system away from equilibrium, is localized on a (s − 1)-dimensional sub-manifold, whereas the propagation in the complementary orthogonal direction is free. In what follows we will consider a special case of this scenario, focussing on a one-dimensional space with the geometry of a star graph Γ. Each point P ∈ Γ is parametrized by the coordinates {(x, i) , : x < 0, i = 1, ..., N}, where |x| is the distance of P from the vertex and i labels the lead. Since s = 1, the interaction is implemented by a point-like defect in the vertex of Γ and the propagation along the leads L i is free. We assume in this paper that it is governed by the Schrödinger equation but other types of time evolution can be considered [5,15] in the same way. The field ψ is complex and the system has a global U(1)-invariance, generating particle number conservation. In the fermionic case ψ satisfies the standard equal-time CAR's. The scattering matrix S in the vertex is fixed by requiring that the bulk Hamiltonian −∂ 2 x admits a self-adjoint extension on the whole graph. These extensions are defined [16]- [18] by where U is a n × n unitary matrix and λ ∈ R is a parameter with dimension of mass. Eq. (3.13) guaranties unitary time evolution of the system on the graph. The matrices U = I and U = −I define the Neumann and Dirichlet boundary conditions respectively. The explicit form of the scattering matrix is [16]- [18] (3.14) The diagonal element S ii (k) represents the reflection amplitude from the vertex on the lead L i , whereas S ij (k) with i = j equals the transmission amplitude from L i to L j . One easily verifies that (3.14) satisfies (2.3) and therefore defines an algebra A of the type introduced in the previous section. Moreover, S(k) is a meromorphic function in the complex k-plane with finite number of simple poles on the imaginary axis. For simplicity we consider in this paper the case without bound states (poles in the upper half plane), referring for the general case to [14], [19]. In this case the solution of equation (3.12) is fixed uniquely by (3.13) and takes the following simple form As expected, the parity and time-reversal transformations (2.8) in the algebra A imply Let us describe now the basic local observables, whose behavior away from equilibrium is the main topic of this paper. The local particle density and relative current are The total energy density is with energy flow The equations of motion lead to the local conservation laws The relation between local conservation laws and the associated charges on a star graph has been investigated in [20]. In the presence of a defect, like the junction in our case, the local conservation (3.20) alone is not enough [20] to ensure the conservation of the relative quantum numbers. One needs in addition the Kirchhoff rules It is worth stressing that the explicit form (3.14) of the S-matrix is fundamental for proving [5] the operator form (3.21) of the Kirchhoff rules, which guaranties the charge conservation in the whole state space H LB of the system. Combining (3.20) and (3.21) one concludes that for these S-matrices the particle number and the total energy in our system are conserved. The heat density q t in the lead L i is obtained (see e.g. [21]) by subtracting from the total energy density the energy density relative to the chemical potential µ i , namely Accordingly, the heat current is is a direct consequence of (3.20), but in general the relative Kirchhoff rule is not at all satisfied. In fact, a key observation is that the heat current obeys the operator Kirchhoff rule if and only if µ i = µ j for all i, j = 1, ..., N. Otherwise, the heat current violates the Kirchhoff rule and the heat energy is therefore not conserved. The chemical potential energy shares the same property, because the total energy is conserved. Summarizing, if µ i = µ j for some i and j, the system converts heat energy in chemical potential energy or vice versa. This is the basic physical process which takes place in the junction. In what follows we will study in detail this phenomenon of energy transmutation.

Currents and efficiency
In order to study the non-equilibrium features of our system, we derive now the expectation values in the state Ω β,µ of the charge densities and currents introduced above. Since these observables are quadratic, the basic input is the non-equilibrium two-point correlation function following from (2.5,3.15). Here t 12 = t 1 − t 2 , x 12 = x 1 − x 2 and x 12 = x 1 + x 2 . The invariance of (4.25) under time translations explicitly confirms that the total energy of the system is conserved. Combining (4.25) with (3.17,3.19) and performing the limits t 1 → t 2 = t and x 1 → x 2 = x one gets the following current expectation values The right hand sides of (4. 26,4.27) are the Landauer-Büttiker expressions [3,4,22] for the particle and energy currents in our case specified by the S matrix (3.14). This is the main reason for referring to Ω β,µ as the LB state. From (3.23) one gets the heat current As already observed, in general the heat and chemical potential currents J Q i and µ i J N i do not satisfy separately the Kirchhoff rule. In fact, the heat flowQ from the junction is given byQ Taking into account the Kirchhoff rule (3.21) for the energy current one getṡ IfQ < 0 the junction transforms heat energy in chemical potential energy. The efficiency of this process can be characterized as follows. Let us denote by K out the subset of heat currents leaving the reservoirs R i . With our choice for the orientation of the leads (see Fig. 1), these currents are positive. With this convention the efficiency of the junction to transform heat energy in chemical potential energy is defined by which is a direct extension of the cases n = 2, 3 [2, 23] to a generic n. Combining (4.30) witḣ Q < 0 one concludes that i∈Kout J Q i > 0, which implies that η is well defined and satisfies 0 < η ≤ 1.
In the regimeQ > 0 of converting chemical potential energy to heat energy the junction is usually compared (see e.g. [2]) to a refrigerator or a heat pump, thus involving in the description the known performance coefficients of these devises. We show below that this approach works in parts of the parameter space, but can not be applied globally to the whole domain withQ > 0. In order to avoid this problem, forQ > 0 we propose and adopt below the efficiency where now the sum in the denumerator runs over the subset L out of outgoing (positive) chemical potential energy currents µ i J N i . The general form of (4.32) is analogous to that of (4.31), but refers to the chemical potential energy. By construction 0 < η ≤ 1 in this case as well.
ForQ = 0 there is no energy transmutation. We conclude this subsection by observing that the heat flow in the system generates the following entropy production [21] Employing (4.26,4.27),Ṡ can be written in the forṁ with Except in section 5.3, we will always assume in this paper that the bulk theory and the heat reservoirs have the same dispersion relation, namely Now, using that (4.37) is a strictly decreasing function of σ j , one can prove [24] that the integrand of (4.34) is nonnegative 2 , implying the second law of thermodynamicsṠ ≥ 0 in the LB state for all (β i , µ i ) and scattering matrices (3.14). As explained in section 5.2 below, this argument does not apply to the other states of the orbit O β,µ and the entropy production there behaves indeed differently. Summarizing, the process of energy transmutation in the junction is controlled by the two parametersQ andṠ and is characterized by the efficiency coefficients η and η. We derive in what follows the explicit form of these physical quantities, assuming that the interaction at the junction is scale invariant.

The scale invariant junction
The great advantage of considering the non-equilibrium Schrödinger junction defined in section 3 is that it provides both exact and explicit results with nontrivial transport. In order to keep this property also after the integration in k, we select in what follows the scaleinvariant elements among the scattering matrices (3.14). They preserve the basic features of the system, while being simple enough to allow the explicit k-integration in (4.26,4.27). The requirement of scale invariance implies that the interaction is k-independent and leads to [25] S The family (4.38) is the orbit of the diagonal matrix S d under the adjoint action of U(n).
We can always enumerate the leads L i in such a way that the first i eigenvalues of S are +1 and the remaining n − i are −1. For i = n and i = 0 one gets S = I and S = −I, which imply vanishing transport because the leads L i are disconnected. One has a nontrivial transport for 0 < i < n.
The scattering matrices of the type (4.38) are in general not symmetric. One can easily prove however that the relative amplitudes are symmetric, namely (4.39) This property of the critical points (4.38) simplifies considerably the study of junctions with n ≥ 3 leads. The k-integration in (4.26,4.27) with the constant scattering matrices (4.38) can be performed explicitly. One finds where Li s is the polylogarithm function. These expressions are the building blocks for the parametersQ andṠ and the efficiency coefficients η and η. They depend on (β i , µ i ) and |S ij | 2 , which (due to the unitarity of S) leads to n(n + 3)/2 independent real parameters. In order to reduce this large number it is instructive to start by considering the system with two leads shown in Fig. 2.
The quantity η * (r) is the counterpart of the concept of efficiency at maximal power, used in the context of heat engines. The plot in Fig. 4 shows that (5.54) satisfies the Curzon-Ahlborn bound for all r ∈ [0, 1], which was known previously from linear response theory only for r ∼ 1. We recall that this bound has been obtained [26,27] for heat engines in the framework of endoreversible thermodynamics. It is known [28,29,30] however that it is not universal. The possibility to enhance η * above η CA in our context is discussed in section 5.3.
We turn now to the case when chemical energy is transformed in heat, namelẏ It is convenient to use in the domain D + the coordinates (β i , µ i ). Keeping β i arbitrary, one can assume without loss of generality that µ 1 > µ 2 and set D + = {β 1 , β 2 , µ 1 > µ 2 }. From (5.56) and the Kirchhoff rule one infers that J N 1 > 0 > J N 2 on D + . Using this information one finds that the efficiency η, defined by (4.32), can be expressed in terms of the parameter u ≡ µ 2 /µ 1 in the simple form fully covering the domain D + . The formula (5.57) describes in exact form the conversion of chemical potential energy in heat. For n = 2 the efficiency η does not depend on the temperatures and the explicit form of the currents µ i J N i , but only on the values of the chemical potentials µ i . The first line of (5.57) resembles the Carnot formula, where the temperature ratio r is substituted by the chemical potential ratio u. The third line instead takes into account that differently from the temperatures, the chemical potentials can take also negative values. Finally, we observe that for µ 1 > 0 > µ 2 both chemical potential currents µ i J N i are flowing towards the junction, transforming the chemical energy in heat completely ( η = 1). Since in this case the domains of µ i are separated by the point µ = 0, this regime of energy transmutation in the junction can not be reached by linear response theory.
We have seen that forQ < 0 the junction can be compared to a heat engine. In analogy, one can attempt to interpret the system forQ > 0 as a refrigerator or a heat pump. A careful analysis shows that this possible only partially because there are subsets in D + where both heat currents have the same sign, i.e. they are both leaving or entering the heat reservoirs. A typical situation is shown in Fig. 5, where J Q 1 and J Q 2 become both negative (i.e. entering the heat reservoirs) for r > 0.75, which is not the case of conventional refrigerators and heat pumps. For this reason the standard coefficients of performance for a refrigerator and a heat pump can not be applied in the whole domain D + . As already mentioned, this is our main motivation to introduce the efficiency η by (4.32), which has the advantage of working everywhere in D + . Let us consider finally the entropy production. Substituting (5.42,5.43) in (4.33) one obtainsṠ confirming the general statement [24] about the entropy production in the LB state.

Transport in the orbit O β,µ
We consider here the quantum transport in the states {Ω X β,µ : X = P, T, P T } ⊂ H LB . Using (3.16), the current expectation values in these states are simply expressed in terms of (4.26,4.27) as follows: Let us observe in passing that the action of the charge conjugation coincides with the P T operation (5.61). In fact, CP T on the above currents is the identity transformation.
The minus sign appearing in some currents affects the quantum transport. Moreover, the entropy productionṠ X in the state Ω X β,µ differs from that (5.58) in the LB state. In fact, combining (4.33) and (5.59) we conclude thatṠ P ≤ 0 for all values of the heat reservoir parameters (β i , µ i ). We see that the state Ω P β,µ , which from the microscopic point of view is a well defined state of the system, violates the second law of thermodynamics and therefore has no admissible macroscopic behavior. The situation with the states Ω T β,µ and Ω P T β,µ is more subtle. The analysis shows that for these two states there exist domains in the whole parameter space (β i , µ i ), whereṠ T > 0 andṠ P T > 0. In these domains the quantum transport is consistent with the laws of thermodynamics. We consider for illustration the family of states {Ω P T β,µ : λ 1 = λ 2 ≡ λ}. The entropy production there iṡ (1 − r) 2 λ ln 1 + e −λ < 0 for λ < 0 , (5.64) and the efficiency of transforming heat into chemical potential energy is given by , This result resembles (5.54), but for a sign in the denumerator. The relative maximum is which coincides also with the efficiency at maximal chemical energy production These results show that the efficiency in the state Ω P T β,µ differs from that in Ω β,µ . We will elaborate more on the value of η * P T few lines below.
Concerning the regimeQ > 0, it is easy to deduce from (5.59-5.61) that all four states in O β,µ have the same efficiency η given by (5.57).
Summarizing, parity and time reversal have an important impact on the quantum transport and efficiency. Indeed, we have shown that there are regions in the parameter space of the states Ω T β,µ and Ω P T β,µ , where the second law of thermodynamics is satisfied and the efficiency η has a physically acceptable value, which differs from that in the LB state Ω β,µ .

Comments about η *
It is instructive to compare now (5.54,5.67) with some exact results about the efficiency at maximal power obtained for other systems. Applying stochastic thermodynamics to a simple model of classical particle transport, the following explicit expression has been derived in [29]. The same expression has been obtained [28] for the Feynman's ratchet model as a heat engine. For a Brownian particle undergoing a Carnot cycle it has been found [31] that which, remarkably enough, coincides with the Schrödinger junction efficiency η * P T given by (5.67). The case of electron transport through a quantum dot has been treated in [32]. It has been observed in [28,29] that away (r < 0.5) from the linear response regime, the efficiency (5.68) exceeds the Curzon-Ahlborn bound. One can wonder if a mechanism exists to enhance the efficiency (5.54) in the LB state Ω β,µ above η CA as well. This question attracted recently some attention [2], the proposal being to couple the system with an appropriate external potential. In [2] the effect of a classical magnetic field has been explored in the linear response approximation. We describe here an alternative, which simplifies our previous construction in [33] both from the technical and physical points of view. The main idea is based on the fact that in our general setting the heat reservoir dispersion relations ω i need not to be equal and/or to coincide with the bulk dispersion relation ω. So, let us perform the shift ω −→ ω − v, which is equivalent to the introduction of a constant potential V = −v in the bulk equation of motion (3.12). If one performs the same shift in ω i the efficiency will not change. One can imagine however to screen the reservoirs R i from the potential V , thus keeping ω i = k 2 /2m invariant. This operation does not affect the particle currents J N i which, according to (4.26), depend only on ω i . From (4.27) one infers however that the shift in ω modifies the energy transport in the following simple way In the domain D − , which is still given by (5.46), one finds the efficiency η(λ 1 , λ 2 ; r, a) = (λ 1 − rλ 2 ) ln 1 + e −λ 1 − r ln 1 + e −λ 2 (λ 1 − a) [ln (1 + e −λ 1 ) − r ln (1 + e −λ 2 )] − [Li 2 (−e −λ 1 ) − r 2 Li 2 (−e −λ 2 )] , (5.71) with a ≡ β 1 v being dimensionless. The maximal efficiency is attained at λ 1 = λ 2 → +∞ and equals η C as before. For η * one finds instead It is easy to deduce from (5.72) that for a ∈ (0, a * ) the efficiency η * (r, a) is enhanced above η * (r, 0) = η * (r) given by (5.54). The opposite effect is observed for a ∈ (−∞, 0). This behavior of the efficiency is explained by the following intuitive physical argument. Since ω i − ω = a/β 1 , positive values of a favor the particle emission from the heat reservoirs R i to the leads L i , thus improving the efficiency. Negative values of a are instead damping this emission, causing the opposite effect. Fig. 6 illustrates the basic properties of η * (r, a). In particular, we see that for a ∼ 1 the Curzon-Ahlborn bound is exceeded.

Junctions with three leads
The new element in the treatment of the case with n > 2 leads is that the transmission amplitudes |S ij | 2 do no longer factorize in front of the currents. For n = 3 the transport properties depend on 9 parameters. The linear response approximation has been studied recently in [23,34]. Here we pursue further the exact analysis of the scale invariant case for Using (4.39), one obtains from (4.40,4.41) the heat currents For the entropy production and heat flow on getṡ (6.79) As expected, the entropy production in the LB state is always positive. For λ > 0 one haṡ Q < 0 and one can study the efficiency (4.31) of transforming heat to chemical potential energy. In order to do that one has to determine first the sign of the currents (6.75-6.77).
Without loss of generality one can assume that r < 1 , s < 1 , r < s , (6.80) which implies the following ordering T 2 < T 3 < T 1 among the temperatures of the heat reservoirs. For simplifying the analysis we assume to end of this section that |S 13 | 2 = 0, devoting the appendix A to the case of generic transmission amplitudes. Then (6.75-6.77) imply J Q 1 > 0, J Q 2 < 0, J Q 3 > 0 for λ > 0. With this information one obtains from (4.31) . which suggests to introduce an effective r-parameter r eff (s) = r |S 12 | 2 (1 − r) + |S 23 | 2 (s − r) |S 12 | 2 (1 − r) + |S 23 | 2 s(s − r) ≤ 1 for r < 1 , r < s (6.83) with the following physical meaning. The hotter reservoirs of our system are R 1 and R 3 because T 1 > T 2 and T 3 > T 2 . Both R 1 and R 3 communicate with the cold reservoir R 2 via S 12 and S 23 . Suppose now we replace R 1 and R 3 with one heat bath R ′ and ask about its temperature T ′ , which gives the same efficiency. It turns out that the answer is T ′ = T 2 /r eff . In fact, one can express the 3-lead efficiency (6.81) in terms of the 2-lead formula (5.51) simply as η(λ; r, s) = η(λ, λ; r eff (s)) . (6.84) We turn now to the regime of heat productionQ > 0 which, according to (6.79), takes place for λ < 0. The corresponding efficiency is defined by (4.32). Let us introduce the parameters Because of (6.74), the conditions (6.80) can be rewritten in the form One can show now that µ 1 J N 1 > 0, µ 2 J N 2 < 0, µ 3 J N 3 > 0. Therefore, using (4.32) one obtains The effective u-parameter now reads and η(u, v) = η(u eff (v)) , (6.89) whose right hand side is given by the first line (since u < 1) of the two-lead expression (5.57). We refer to appendix A for the analysis of the general case, in which all three transmission amplitudes are nontrivial.

Bosonic junctions
We illustrate in this section the influence of the statistics on the transport and efficiency of the Schrödinger junction. Substituting in (4.26,4.27) the Fermi distribution d j (k) with the Bose one b j (k), the corresponding integrands develop a singularity at k 2 = 2mµ i . In order to exclude it from the range of integration, we have to assume µ i < 0. Focussing on the case n = 2, the bosonic counterparts of (5.42,5.43,5.45,5.58) are 93) where one should keep in mind that λ i ≡ −β i µ i > 0. The apex (b) in (7.92,7.93) and below is added in order to distinct the bosonic from the fermionic expressions. The domains D − one finds the bosonic efficiency . Let us derive now the bosonic efficiency at maximal chemical energy production. The maximum of the function −Q (b) (λ 1 , λ 2 , r) is reached at The solution of (7.100) is λ * b = 0.69314 . . . and the efficiency reads The study of (7.101) reveals that at maximal chemical energy production the bosonic junction is slightly less efficient then the fermion one, as shown in Fig. 7. The enhancement mechanism for fermions, described in section 5.3, applies to bosonic junctions as well.
We conclude by observing that in the regimeQ (b) > 0 the bosonic efficiency η (b) of transforming chemical energy in heat precisely coincides with the fermionic one (5.57).

Conclusions
We described in this paper a basic process of transformation of heat in chemical potential energy and vice versa, which takes place in systems away from equilibrium. The phenomenon is universal and stems from the fact that even if the total energy of the system is conserved, the heat and chemical potential energies are in general not separately conserved. Both directions of the process of energy transmutation are characterized by their own efficiency coefficient. The specific features of the phenomenon depend on the particle statistics and on the choice of nonequilibrium state. We illustrated this fact by studying the fermionic and bosonic Schrödinger junctions in the LB state and its orbit under parity and time reversal. The relative quantum transport depends in a complicated nonlinear way on the temperatures and chemical potentials, which parametrize the nonequilibrium states. In order to control these characteristics of the system, we avoided the use of any approximation and in particular, of the linear response theory. Assuming that the interaction which drives the system away from equilibrium is scale invariant, we described the quantum transport exactly and derived the explicit expressions of the efficiency coefficients.
Exploring the orbit of the LB state under parity and time reversal transformations, we have shown that space-time symmetries have an essential impact on the quantum transport and efficiencies. Concerning the internal symmetries, our model has a global U(1)-symmetry associated with the particle number. For systems with a larger internal symmetry group, the process of energy transmutation becomes even more involved, due to the presence of several types of chemical potentials. In this case the Landauer-Büttiker state is induced not by a Gibbs state, but by a generalized Gibbs ensemble [35] generated by a complete set of commuting charges of the extended symmetry group. The study of such nonequilibrium states may provide an important insight in the role of internal symmetries in quantum transport.