Multi-terminal thermoelectric transport in a magnetic field: bounds on Onsager coefficientsand efficiency View the table of contents for this issue, or go to the journal homepage for mor
Phenomenological constraints
The phenomenological framework of linear irreversible thermodynamics provides two fundamental constraints on the matrix of kinetic coefficients L(B) Firstly, since the entropy production accompanying the transport process described by (2) reads [18]
S˙ =F t J=F t L(B)F, (12) the second law requires L(B) to be positive semi-definite Secondly, Onsager’s reciprocal relations impose the symmetry
Currently, no well-known general relations impose constraints on the elements of L(B) at a fixed magnetic field beyond existing limitations This absence of restrictions has significant implications for the thermodynamic properties of the model, highlighting its unique behavior To analyze these effects, the current vector J is decomposed into reversible and irreversible components, providing deeper insights into the system's thermodynamic dynamics.
Reversible transport components vanish at zero magnetic field due to reciprocal relations, but can become arbitrarily large for non-zero magnetic fields without increasing entropy production In theory, it is possible to achieve complete reversibility (˙S=0) while maintaining finite reversible currents (J_rev ≠ 0), indicating potential for thermoelectric heat engines operating at Carnot efficiency with finite power This raises the question of whether stronger constraints exist among kinetic coefficients beyond the known reciprocal relations In the subsequent analysis, bounds derived from microscopic models demonstrate that such idealized scenarios are prevented by fundamental physical limitations.
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 matrixA∈R m × m we define an asymmetry index by
Some of the basic properties of this asymmetry index are outlined in appendixA We note that a quite similar quantity was introduced by Crouzeix and Gutan [19] in another context.
This article demonstrates that the asymmetry index of the kinetic coefficient matrix L(B) and all its principal submatrices is universally bounded for any finite number of terminals n, ensuring stability across diverse systems Building on this, we derive new bounds on the elements of L(B) that extend beyond traditional second law constraints, offering deeper insights into thermodynamic processes To simplify the notation, the magnetic field dependence is temporarily suppressed throughout the analysis, facilitating clearer mathematical expressions.
+ i L A −L t A z (16) 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.
LA is a principal submatrix of L that maintains the original 2×2 block structure, ensuring structural consistency Comparing equation (16) with the definition in (15) demonstrates that the minimum values for which Q(z, s) is positive semi-definite correspond to the asymmetry index of LA Additionally, by referencing equation (11), we can rewrite the matrix to further analyze its properties in relation to stability and semi-definiteness.
L A in the rather compact form
, (17) where T¯ A (E)∈R m × m is obtained fromT¯(E)by taking the rows and columns indexed by the set A Decomposing the vectorzas z≡z 1 ⊗
0 1 withz 1 ,z 2 ∈C m (18) and inserting (17) and (18) into (16) yields
−∞ dE F(E)y † (E)K(E,s)y(E) (19) Here we introduced the vector y(E)≡z 1 + E−à e z 2 (20) and the Hermitian matrix
∈C m × m (21) which is positive semi-definite for any s⩾S 1− ¯T A (E)
However, since T¯(E) is doubly stochastic for any E, the matrix T¯ A (E) must have the same property and it follows from corollary2proven in appendixB:
Hence, independently of E,K A (E,s)is positive semi-definite for any s⩾cot π m+ 1
Finally, we can infer from (19) thatQ(z,s)is positive semi-definite for anys, which obeys (24).
Consequently, with (16), we have the desired bound on the asymmetry index ofL 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 Lby a two-step procedure, which is schematically summarized in figure2 In the first step, we consider the 4×4 principal submatrix ofLgiven by
Figure 2 presents a schematic illustration of the reduction from \(L\) to \(\tilde{L}_{\alpha\beta}\) The large square represents the matrix \(L\) for the case when \(n=6\), while the smaller squares depict the 2×2 blocks introduced in equation (5) By focusing on the bold-framed squares, the 4×4 matrix \(L_{\{\alpha\beta\}}\) is derived for the specific case \(\alpha=1\) and \(\beta=3\) The filled squares indicate the elements of the 2×2 matrix \(\tilde{L}_{\{\alpha\beta\}}\) introduced in equation (28), with color coding for (i, j)=(1,1) in blue and (i, j)=(2,1) in green This reduction involves selecting only the blocks with row and column indices equal to \(\alpha\) or \(\beta\) From equation (25), we immediately obtain the results for the case when \(m=2\).
Next, from (26), we take a 2×2 principal submatrix
, (28) where L αβ i j with i, j =1,2 denotes the (i, j)-entry of the block matrix L αβ By virtue of proposition3proven in appendixB, the inequality (27) implies
√3 (29) which is equivalent to requiring the Hermitian matrix
A matrix is positive semi-definite when its diagonal entries are non-negative, leading to the condition that the determinant of the matrix, given by \( K_{11}K_{22} - |K_{12}|^2 \), must be greater than or equal to zero This ensures the matrix's non-negative definiteness Additionally, by expressing the matrix elements \( K_{ij} \) in terms of the \( L_{ij} \) components, a new constraint is derived, reinforcing the conditions necessary for the positive semi-definite property.
4L 11 L 22 −(L 12 +L 21 ) 2 ⩾3(L 12 −L 21 ) 2 (31)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 requires L˜ {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 fieldB.
The procedure outlined for 2×2 principal submatrices of L can be easily extended to larger principal submatrices, leading to a hierarchical series of constraints involving increasing numbers of kinetic coefficients Among these, equation (31) represents the strongest bound derived from equation (25), and it can be expressed using only four specific kinetic coefficients This highlights the significance of identifying the most restrictive constraints within complex matrix relations in kinetic theory.
Bounds on efficiencies 9 1 Heat engine
Refrigerator
In the previous section, we examined the multi-terminal model's performance when functioning as a heat engine However, it can be adapted to operate as a refrigerator by consuming electrical power to transfer heat from the cold to the hot reservoir This mode of operation involves reversing the roles of energy input and output, with the affinities Fρ and Fq chosen so that both currents Jρ and Jq are negative, effectively moving heat against the temperature gradient.
The figure illustrates the bounds on the efficiency of a multi-terminal thermoelectric heat engine as a function of the asymmetry parameter x, measured in units of ηC The upper panel displays the maximum efficiency ηmax(x) (see equation (46)), while the lower panel shows the efficiency limit η∗(x) (see equation (49)) In both panels, blue lines represent models with the number of terminals n ranging from 3 to 12, increasing from bottom to top, highlighting how efficiency bounds vary with terminal count The solid black line depicts the fundamental efficiency bound derived from the second law of thermodynamics, as established by Benenti et al [9] Additionally, the dashed line in the lower panel indicates the Curzon–Ahlborn limit ηCA = ηC/2, serving as a benchmark for heat engine performance.
Like heat engines, thermoelectric refrigerators are constrained by bounds on their kinetic coefficients, which ultimately limit their performance Specifically, the bound (39) plays a crucial role in governing the efficiency of these systems To analyze this, we consider the coefficient of performance (COP), defined as ε ≡ − J q, which provides a key metric for assessing the refrigerator's effectiveness This relationship highlights how fundamental thermodynamic limits influence the maximum attainable performance of thermoelectric cooling devices.
Figure 4 illustrates the maximum coefficient of performance, εmax(x), of a thermoelectric refrigerator as a function of the asymmetry parameter x, highlighting the impact of system asymmetry on efficiency The plot features blue lines representing models with varying numbers of terminals (n=3 to 12), demonstrating how increasing terminal count influences performance The black curve indicates the fundamental bound dictated by the second law of thermodynamics, which the εmax(x) approaches asymptotically as n tends to infinity, serving as a key benchmark Additionally, the upper bound defined by the second law, denoted as εC, provides a theoretical maximum efficiency for comparison across different system configurations.
T/1T 2 =1/(TF q ), which is the efficiency of the ideal refrigerator In this sense, εC is the analogue to the Carnot efficiency.
Taking the maximum of ε over F ρ (under the condition J ρ 0), we can derive an upper bound ηmax(x)≡ 1 x
√x 2 −x+ 1 +|x−1| (78) from (76) Again, this bound is independent of the number of probe terminals Figure6shows it as a function of the asymmetry parameter x≡L 0 23 /L 0 32
For completeness, we emphasize that the efficiency (77) used here differs from the coefficient of performance ε≡ J 2 q
L 0 32 F 2 q +L 0 33 F 3 q (79) used as a benchmark parameter in [23,24] Sinceεis unbounded in the linear response regime, maximization with respect toF 2 q orF 3 q would be meaningless.
Our study explores how broken time-reversal symmetry impacts thermoelectric transport within a broad n-terminal model framework, revealing that the asymmetry index of any principal submatrix of the Onsager matrix is bounded, leading to new and stronger constraints on kinetic coefficients beyond the second law These constraints, derived from analytical calculations, are not obtainable through traditional Onsager time-reversal arguments, highlighting novel limitations on thermoelectric performance The methodology used can be extended to larger principal submatrices, generating a hierarchy of relations involving higher-order products of kinetic coefficients, which offers promising directions for future research.
Our analysis of transport processes in multi-terminal systems reveals that the maximum efficiency and efficiency at maximum power in thermoelectric heat engines are constrained by bounds that depend on the number of terminals, n In the minimal case of n=3, these bounds align with previously established limits, confirming their robustness While these bounds weaken as n increases, they demonstrate that reversible transport is impossible with a finite number of terminals, with only the infinite-terminal limit resembling the scenario described by Benenti et al., where the second law of thermodynamics is the sole constraint Notably, for n=3, these bounds can be saturated, as shown by Balachandran et al., but the possibility of saturation for higher values of n remains an open question crucial for future research.
The maximum coefficient of performance for thermoelectric refrigerators becomes less restrictive as the number of terminals increases, similar to the heat engine case However, for isothermal engines and absorption refrigerators—discussed in sections 4.3 and 4.4—the fundamental bounds on their performance remain unchanged regardless of the number of terminals involved These bounds are consistent with the three-terminal case, suggesting a universal limit on device efficiency if inelastic scattering is modeled by numerous probe terminals The findings indicate a fundamental difference between transport processes under broken time-reversal symmetry driven solely by either chemical potential or temperature differences, and those driven by both thermodynamic forces simultaneously.
Our results are fundamentally based on the sum rules (8) for the elements of the transmission matrix, which embody the principle of current conservation—an essential physical law These bounds are universally valid, extending beyond quantum mechanics to any model, quantum or classical, where kinetic coefficients can be expressed in the general form (6) Examples of quantum models satisfying these criteria are discussed in references [17, 25], while a classical model fitting this framework was recently introduced by Horvat et al [26].
We have developed a comprehensive understanding of thermoelectric transport under broken time-reversal symmetry in non-interacting particle systems, with Onsager coefficients described by the Landauer–Büttiker formalism However, simulating fully interacting systems, which necessitate moving beyond the single-particle approximation, remains an open area of research Addressing these complex many-body interactions presents a significant challenge for future studies in thermoelectric phenomena.
We gratefully acknowledge stimulating discussions with K Saito and support of the ESF through the EPSD network.
Appendix A Quantifying the asymmetry of positive semi-definite matrices
We first recall the definition (15)
The asymmetry index of an arbitrary positive semi-definite matrix \(A \in \mathbb{R}^{m \times m}\) is characterized by the inequality \(A - A^T \geq 0\), which highlights its fundamental non-negativity property This measure provides valuable insights into the matrix's asymmetry and can be directly derived from its defining formula Understanding the basic properties of the asymmetry index is essential for analyzing matrix behavior in various mathematical and engineering applications These properties help in assessing the degree of asymmetry in positive semi-definite matrices, making it a useful tool in fields such as matrix analysis, control theory, and data science.
Proposition 1 (Basic properties of the asymmetry index) For any positive semi-definite
S(A)⩾0 (A.3) with equality if and only ifAis symmetric IfAis invertible, it holds additionally
Furthermore, we can easily prove the following two propositions, which are crucial for the derivation of our main results.
Proposition 2 (Convexity of the asymmetry index) Let A,B∈R m × m be positive semi- definite, then
Proof.By definitionA.1the matrices
J(s)≡s(A+A t )+ i(A−A t ) and K(s)≡s(B+B t )+ i(B−B t ) (A.6) withs≡max{S(A) ,S(B)}both are positive semi-definite It follows that
J(s)+K(s)=s(A+B)+s(A+B) t + i(A+B)−i(A+B) t (A.7) is also positive semi-definite and henceS(A+B)⩽s u t
Proposition 3 (Dominance of principal submatrices) Let A∈R m × m be positive semi- definite andA¯ ∈R p × p (p