Topological Josephson Heat Engine

The promise of fault-tolerant quantum computing has made topological superconductors the focus of intense research during the past decade. In this context, topological Josephson junctions based on nanowires or on topological insulators provide an alternative route for probing topological superconductivity. As a hallmark of their topological nature, such junctions exhibit a ground-state fermion parity that is $4\pi$-periodic in the superconducting phase difference $\phi$. Finding unambiguous experimental evidence for this $4\pi$-periodicity still proves a difficult task, however. Here we propose a topological Josephson heat engine implemented by a Josephson-Stirling cycle as an alternative thermodynamic approach to test the ground-state parity. Using a Josephson junction based on a quantum spin Hall (QSH) insulator, we show how the thermodynamic cycle can be used to test the $4\pi$-periodicity of the topological ground state and to distinguish between parity-conserving and non-parity-conserving engines. Interestingly, we find that parity conservation generally boosts both the efficiency and power of the topological heat engine with respect to its non-topological counterpart. Our results, applicable not only to QSH-based junctions but also to any topological Josephson junction, demonstrate the potential of the intriguing and fruitful marriage between topology and coherent thermodynamics.

The promise of fault-tolerant quantum computing has made topological superconductors the focus of intense research during the past decade [1,2]. In this context, topological Josephson junctions based on nanowires [3,4] or on topological insulators [5][6][7][8][9][10][11] provide an alternative route for probing topological superconductivity. As a hallmark of their topological nature, such junctions exhibit a ground-state fermion parity that is 4π-periodic in the superconducting phase difference φ. Finding unambiguous experimental evidence for this 4π-periodicity still proves a difficult task, however [12][13][14][15][16]. Here we propose a topological Josephson heat engine implemented by a Josephson-Stirling cycle as an alternative thermodynamic [17][18][19][20][21][22][23] approach to test the groundstate parity. Using a Josephson junction based on a quantum spin Hall (QSH) insulator, we show how the thermodynamic cycle can be used to test the 4π-periodicity of the topological ground state and to distinguish between parity-conserving and non-parity-conserving engines. Interestingly, we find that parity conservation generally boosts both the efficiency and power of the topological heat engine with respect to its non-topological counterpart. Our results, applicable not only to QSH-based junctions but also to any topological Josephson junction, demonstrate the potential of the intriguing and fruitful marriage between topology and coherent thermodynamics.
In our proposed setup [ Fig. 1(a)], an external magnetic flux controls the superconducting phase bias φ across the junction. The temperature T of the QSH system is assumed to be externally modulated compared to the bath temperature T b . For example, this could be done with radiative heating of the system [24][25][26] or by having the superconductors acting as reservoirs whose temperature is controlled via resistors or superconductor/insulator/superconductor tunnel junctions [17].
A Josephson-Stirling cycle [23] is composed by a sequence of i) an isothermal phase change of φ = 0 → φ f at an externally set temperature T = T e , followed by ii) an isophasic temperature change T = T e → T b at constant φ = φ f , iii) an isothermal phase change of φ = φ f → 0 at T = T b , and iv) an isophasic temperature change T = T b → T e at φ = 0 to complete the cycle [ Fig. 1(b)]. If the reference phase φ f is chosen as (an integer multiple (d) no parity plane for φ f = 2π with and without parity conservation, respectively. Here, S is the total entropy of the system. The area enclosed by the cycle in the (T, S) plane corresponds to the total heat Q absorbed, which is equivalent to the total work W done during the cycle. In this setup, W = 0 only if parity is conserved due to the trivial 2π-periodicity of the non-topological engine.
of) φ f = 2π, the work released by the engine crucially differs between a setup without fermion-parity constraints and a setup with constant fermion parity. In the former case, the free energy and other thermodynamic quantities are 2π-periodic. This requires that no work or heat is generated or absorbed during each of the isothermal phase changes φ = 0 → 2π and φ = 2π → 0. If we assume, on the other hand, that the fermion parity can be kept constant throughout all processes, the thermodynamic quantities are 4π-periodic. Work and heat are then exchanged with the reservoirs during the isothermal phase changes φ = 0 → 2π and φ = 2π → 0. Hence, for φ f = 2π a topological heat engine releases work only when parity can be conserved [Figs. 1(c,d)].
While the concepts outlined above are expected for any topological Josephson junction, we will discuss them ex-arXiv:2002.05492v1 [cond-mat.supr-con] 13 Feb 2020 plicitly for the example of a short, topological Josephson junction based on a QSH insulator. Here, the pairing in the superconducting (S) regions is induced from nearby s-wave superconductors [see Fig. 1(a), also for the coordinate system]. Assuming two independent edges of the QSH system, the corresponding Bogoliubov-de Gennes (BdG) Hamiltonian for the QSH edge states then readŝ where s =↑ / ↓≡ ±1 describes the natural (out-of-plane) spin projection, σ = t/b ≡ ±1 the top and bottom edges, and τ j (with j = x, y, z) denote Pauli matrices of the particle-hole degrees of freedom.
We study a short junction with a normal (N) QSH region of width L N , approximated by a δ-like profile. The proximity-induced pairing amplitude is ∆ and we use the phase convention Φ(x) = Θ(x)φ to describe the superconducting phase difference φ between the two S regions. Furthermore,p x denotes the momentum operator, and V 0 is the potential difference between the N and proximitized S regions. We employ a scattering approach to determine the ABS and the continuum spectrum of Eq. (1), from which we obtain the free energy-up to some additive φ-independent contributions-as with the Boltzmann constant k B and the temperature T of the QSH states [27]. Here, Eq. (2) arises solely from the ABS energies and the prefactor 2 takes into account contributions from the top and bottom edges [28]. Equation (2) describes a situation where the states of the system have equilibrium occupations without any external constraints. If fermion-parity conservation is enforced, the free energy acquires an additional term and becomes [28] where we use the convention that p = ±1 corresponds to even and odd ground-state parity, respectively. In Eq.
(3), we again omit additive φ-independent contributions to F p , which are also parity independent. The contribution originates from the superconducting electrodes, where the energy scale E S = v F /L S is related to the total  3)]. If parity constraints are enforced, we choose the branch p = 1. In contrast to the current I, the exact values of F and S (as well as C) at a given φ also require knowledge of the φ-independent contributions omitted in Eqs. (2) and (3). To overcome this difficulty, we measure F and S with respect to their values at φ = 0, thereby canceling the offset due to the φ-independent contributions.
length L S of the superconducting QSH edge [7]. Following Refs. [7,29], we have assumed rigid boundary conditions in writing down Eqs. (2) and (3) and do therefore not take into account the inverse proximity effect since L S L N [21][30]. The total free energy F of the QSH junction, given by Eqs. (2) or (3), allows us to calculate the total Josephson current via [29] where e is the elementary charge, and the entropy via From S, one can subsequently obtain the heat capacity of the junction [31] Importantly, Eq. (2) is 2π-periodic in φ, while Eq. (3) is 4π-periodic. Consequently, the quantities derived from Eqs. (2) or (3) inherit the respective periodicities. This is illustrated by Fig. 2, which shows F and S for junctions without and with parity constraints. For the Josephson-Stirling cycle, we need to describe different thermodynamic processes. We study quasistatic processes, during which the system passes through quasi-equilibrium states. Then, the work done and heat released during a process i → f are W i→f = − /(2e) dφ I(φ, T ) and Q i→f = dS T , respectively. The sign convention is such that W i→f is positive when the system releases work to the environment, while Q i→f is positive when the system absorbs heat from the environment.
For an isothermal process where φ is changed from can be directly obtained from Eqs. (2) and (3) and their temperature derivatives. During an isophasic process, T is changed from T i → T f at constant φ. In this case, W i→f = 0, while can be calculated from the total heat capacity. The φdependent contribution δC(φ, T ) = C(φ, T ) − C 0 (T ) can be directly calculated from Eqs. (2) or (3) and its derivatives and is measured with respect to φ = 0. In principle, we also need to account for the φ-independent contribution C 0 (T ) arising from the terms omitted in Eqs. (2) and (3). For additional details, we refer to Ref. [28], where C 0 (T ) is calculated using the BCS DOS.
We are now in a position to explicitly compute the total work and heat produced during each of the processes of the Josephson-Stirling cycle introduced above [ Fig. 1(b)]. As mentioned above, W 1→2 and W 3→4 correspond to integrals over the current-phase relation, but can also be computed directly from F . The total work W = W 1→2 + W 3→4 of each cycle thus coincides with the difference between the integrated areas over the currentphase relation [ Figs. 3(a,b)]. The heat exchanged with the hot (T = T e ) and cold reservoirs (T = T b ) is Q e = Q 1→2 + Q 4→1 and Q b = Q 2→3 + Q 3→4 , respectively. Conservation of energy dictates W = Q, where Q = Q e + Q b is the total heat exchange during the cycle. Note that in our setup, there are no separate hot and cold reservoirs, but the environment acts successively as hot and cold reservoir.
In Fig. 3(c), we show W as a function of the reference phase φ f and compare the case without and with parity constraints. Without parity conservation, W is maximal for φ f = π, whereas W = 0 for φ f = 2π. The latter is a consequence of the 2π-periodicity of F 0 (φ, If fermion parity is kept constant, on the other hand, F p (φ, T ) = F p (φ + 4π, T ) and W is maximal for φ f = 2π. A topological heat engine with φ f = 2π thus releases work only if parity is conserved and can thus serve as a test for the 4π-periodicity of the ground-state fermion parity.
Until now, we have focused only on an engine. Depending on the relative values of T e and T b , the Josephson- Stirling cycle can, however, exhibit also other operating modes. This is illustrated by Figs. 4(a,c), which show W for different combinations of T e and T b . Here, φ f is chosen to yield the maximal work, that is, φ f = π without parity constraints [ Fig. 4(a)] and φ f = 2π if parity is conserved [ Fig. 4(c)].
For T e > T b , the Josephson-Stirling cycle/machine acts as an engine: The machine absorbs Q e > 0 from the hot reservoir and releases |Q b | < Q e to the cold reservoir. Hence, W > 0 is done on the environment and the engine efficiency is given by η = W/Q e . A comparison of the engine efficiency and maximal power shows that a parity-conserving engine is on average more efficient and more powerful than its non-parity-conserving implementation [ Figs. 4(b,d)]. We interpret the stronger power as due to an increased phase space available: To obtain a finite work, the work integral can be integrated over a 0 − 2π range if parity is preserved, whereas one needs to remain within the 0 − π range without parity conservation. Secondly, the lower efficiency of the nonparity-conserving engine can be understood as due to the competition between mutually exclusive processes with opposite parities [32]. Indeed, Eq. (3) shows that the additional parity-related terms contribute with different Te and releases heat Q b < 0 with |Q b | > Qe to the heat sink with temperature T b . If Te < T b , Qe < 0, and Q b < 0, the machine is a Joule pump that completely converts work into heat released to the reservoirs. On the other hand, if Te < T b , Qe < 0, and W < 0, while Q b > 0, the machine acts as a cold pump transferring heat from the hot (T = T b ) to the cold reservoir (T = Te).
signs to F p , implying opposite contributions to the heat exchange. Consequently, the non-parity-preserving engine can be interpreted as a thermal machine composed of two mutually exclusive engines working in an opposite manner, thereby reducing the total efficiency.
If T e < T b , the systems with and without parity conservation act as refrigerators with a Coefficient of Performance COP = Q e /|W | [23] or as Joule or cold pumps. Controlling T e vs T b thus enables multiple operating modes of the Josephson-Stirling cycle. If the cycle is set up as in Fig. 1(b), refrigerators as well as Joule and cold pumps require that T e < T b . While it is possible by superconductor/insulator/superconductor cooling to bring T e below T b [17], a more promising way to realize refrigerators, Joule or cold pumps is to shift the cycle by interchanging the initial and final phases, φ = 0 and φ = φ f . Such a setup implies the same phase diagrams as in Fig. 4 but with T e and T b interchanged [28]. Hence, the 'shifted' Josephson-Stirling cycle can be used to realize operating modes other than engines, making it a highly versatile thermodynamic machine.
Importantly, a topological Josephson heat engine implemented as a Josephson-Stirling cycle can be used to test the hallmark 4π-periodicity of the phase-dependent ground-state fermion parity. In this implementation, a major challenge is to fastly modulate the temperature of the proximitized QSH junction while preserving its fermion parity. While this condition precludes electronic channels of heat transfer to the QSH system, others such as phononic [33], photonic [34,35] or radiative [24] channels could be used. We have discussed topological Josephson heat engines for the example of a short QSH-based Josephson junction. Since the concept is only based on the 4π-periodicity of the ground-state parity, it is also applicable to long as well as nanowire-based topological Josephson junctions.