Interferometric Thermodynamic Aharonov–Bohm Effect

• The paper demonstrates that magnetic flux controls phase interference, modulating both charge and heat currents via Peltier effects and thermal balance.
• Methodologies employ voltage-biased interferometers with dual coherent paths and quantum dots to achieve temperature modulations of up to 80 mK.
• Implications extend to generating spin-polarized heat currents and enabling thermal switches and nonlocal thermoelectric conversion in mesoscopic systems.

The interferometric thermodynamic Aharonov–Bohm effect denotes a class of phase-coherent phenomena in which an Aharonov–Bohm (AB) phase, imposed by magnetic flux through an interferometer, modulates not only charge transport but also energy transport and thermodynamic observables such as heat current, local temperature, or induced electrostatic response. In the most explicit recent formulation, a voltage-biased AB interferometer acts as a phase-controlled heat modulator: the magnetic flux changes the quantum interference between two transport paths, the interference changes the Peltier heat current, and the heat current shifts the reservoir temperature through a thermal-balance condition (Hwang et al., 2024). More broadly, related work shows that AB phases can control thermal currents, suppress heat flow by destructive interference, generate non-local thermoelectric responses, and produce flux-periodic induced potentials in hybrid interferometers (Balduque et al., 2024, Dolgirev et al., 2018). The common theme is that a gauge-controlled interference phase is transduced into an observable thermodynamic or thermoelectric quantity.

1. Definition and conceptual scope

In its narrowest sense, the term refers to the mechanism analyzed in “Phase-controlled heat modulation with Aharonov-Bohm interferometers” (Hwang et al., 2024). There, the magnetic flux through a loop changes the phase of electron waves propagating along two coherent paths, thereby modulating the electronic transmission. Because both charge current and heat current are functionals of that transmission, the heat flow becomes flux dependent, and a reservoir temperature becomes flux dependent as well through electron–phonon thermal balance. The effect is “thermodynamic” because the interferometric phase controls a thermodynamic state variable rather than only a conductance oscillation (Hwang et al., 2024).

A broader usage is supported by several adjacent developments. In mesoscopic rings, AB flux can be used to manipulate electronic thermal currents, enhance thermoelectric response, and even totally suppress heat transport at destructive-interference points (Balduque et al., 2024). In voltage-driven Andreev interferometers, flux-periodic coherent transport can induce a phase-shifted electrostatic potential in an isolated normal terminal, providing a flux-controlled thermodynamic-like response in a nonequilibrium hybrid structure (Dolgirev et al., 2018). In asymmetric four-terminal AB rings, temperature broadening causes thermal phase averaging, showing that AB interference can encode thermodynamic occupation effects even when the measured observable is resistance oscillation visibility rather than a direct heat current (Buchholz et al., 2010).

These usages should be distinguished from the ordinary AB effect in conductance. Ordinary AB physics concerns the dependence of interference on magnetic flux through the loop. The interferometric thermodynamic variant concerns what happens when that same phase dependence is mapped onto energy flow, temperature, or induced potentials. A plausible implication is that the term designates not a single device architecture but a research direction spanning caloritronics, mesoscopic thermoelectricity, and hybrid quantum transport.

2. Microscopic mechanism in a voltage-biased AB heat modulator

The canonical implementation analyzed in (Hwang et al., 2024) is a two-path AB interferometer with an upper direct tunneling arm of amplitude $W$ and a lower arm through a quantum dot with couplings $t_L,t_R$. The spin-resolved relative phase is

$$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$

with AB phase

$\phi = \frac{e}{h}\Phi_B$

and Rashba phase

$\varphi = \alpha_R l.$

The spin-resolved transmission $\mathcal{T}_\sigma(\varepsilon)$ contains interference terms proportional to $\cos(\varphi_\sigma)$, so constructive or destructive interference can be selected by the magnetic flux and, when present, by Rashba spin-orbit interaction (Hwang et al., 2024).

The central transport relation is

$$\begin{pmatrix} I_{\sigma} \ J_{\sigma} \end{pmatrix} = \frac{1}{h}\int d\varepsilon\, \begin{pmatrix} e \ \varepsilon-\mu \end{pmatrix} \mathcal{T}_{\sigma}(\varepsilon)\,[f_L(\varepsilon)-f_R(\varepsilon)].$$

Thus, the same energy-dependent transmission governs both charge and heat transport. The total heat current and spin heat current are

$J = J_+ + J_- , \qquad J_s = J_+ - J_- .$

The device is voltage biased, with $V_L=V$ and $t_L,t_R$0. Under bias, electrons carry both charge and energy, and the energy dependence of $t_L,t_R$1 produces Peltier heating or cooling of the reservoir (Hwang et al., 2024). The paper explicitly defines the heat current as

$t_L,t_R$2

so the heat current includes both a transport contribution and a Joule correction. This competition yields the key operational behavior: for some $t_L,t_R$3, $t_L,t_R$4 and the left reservoir is cooled; for others, $t_L,t_R$5 and it is heated. Because $t_L,t_R$6 depends on $t_L,t_R$7, the magnetic flux switches both the sign and the magnitude of heat flow (Hwang et al., 2024).

This mechanism differs from direct magnetic heating. The flux does not itself inject thermal energy into the reservoir. Instead, it modifies the phase relation between two coherent paths, thereby changing the transmission profile, which changes the Peltier heat current and only then the local temperature. That distinction is central to the thermodynamic interpretation proposed in (Hwang et al., 2024).

3. Thermal balance, temperature modulation, and observability

To convert the flux-dependent heat current into a measurable temperature shift, (Hwang et al., 2024) uses a simple electron–phonon thermal model. The left reservoir temperature is written as

$t_L,t_R$8

where $t_L,t_R$9 is the background temperature of the right reservoir and the phonon bath. Electron–phonon coupling is modeled by the standard low-temperature hot-electron law

$$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$0

with $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$1 and $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$2 used as realistic mesoscopic values. The stationary state satisfies

$$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$3

The interpretation is straightforward: the flux-dependent electronic heat current $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$4 is compensated by electron–phonon cooling or heating until a steady state is reached, and the calculation is performed self-consistently because changing $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$5 also modifies the Fermi functions entering $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$6 (Hwang et al., 2024).

The reported temperature modulation is sizable on the scale discussed in the paper. The calculations predict temperature modulation up to $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$7 at a base temperature near $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$8, corresponding to a relative variation of about $$\varphi_\sigma = \phi + \mathrm{sgn}(\sigma)\varphi,$$9 (Hwang et al., 2024). The relative modulation is defined as

$\phi = \frac{e}{h}\Phi_B$0

and values around $\phi = \frac{e}{h}\Phi_B$1 are predicted under optimal conditions. The paper states that this is much larger than typical thermometer resolution of $\phi = \frac{e}{h}\Phi_B$2 mK, and that the effect persists over a broad parameter range without requiring fine tuning of the dot level $\phi = \frac{e}{h}\Phi_B$3 (Hwang et al., 2024).

The significance of these numbers is methodological as well as physical. They indicate that the flux dependence is not confined to a tiny perturbation of a transport coefficient; rather, under the model assumptions, it can be amplified into a directly resolvable temperature shift through thermal feedback. This suggests that interferometric phase control can serve as a practical caloritronic transduction mechanism, though that implication remains contingent on maintaining phase coherence and on the validity of the electron–phonon balance model.

4. Spin dependence, broken reciprocity, and multiterminal caloritronics

A notable extension in (Hwang et al., 2024) is the prediction of spin-polarized heat currents without ferromagnetic contacts. The origin is the spin-dependent phase

$\phi = \frac{e}{h}\Phi_B$4

Because the Rashba phase changes sign between spin channels while the AB phase is common to both, the two spins experience different interference conditions. This can generate a nonzero spin heat current $\phi = \frac{e}{h}\Phi_B$5. The paper emphasizes that both a finite magnetic flux $\phi = \frac{e}{h}\Phi_B$6 and a finite Rashba phase $\phi = \frac{e}{h}\Phi_B$7 are required; if either is absent, the spin asymmetry disappears (Hwang et al., 2024). In that sense, flux and Rashba coupling function together as an effective spin-dependent phase shifter for thermal transport.

A complementary line of work considers thermal control by AB phases in two- and three-terminal normal rings (Balduque et al., 2024). There, the flux phase is written as

$\phi = \frac{e}{h}\Phi_B$8

and the ring eigenenergies are

$\phi = \frac{e}{h}\Phi_B$9

The transport formulation is

$\varphi = \alpha_R l.$0

with

$\varphi = \alpha_R l.$1

In the two-terminal configuration, the symmetric transmission contains a factor $\varphi = \alpha_R l.$2, implying

$\varphi = \alpha_R l.$3

independent of energy. This yields complete destructive interference and thus total suppression of both charge and heat currents, which the paper identifies as the basis of a thermal switch (Balduque et al., 2024).

In the three-terminal configuration, the flux breaks reciprocity according to

$\varphi = \alpha_R l.$4

and produces a transmission asymmetry $\varphi = \alpha_R l.$5 in the weak-coupling limit (Balduque et al., 2024). This asymmetry underlies two explicitly thermodynamic consequences. First, heat injected from one hot terminal can be routed asymmetrically between two cold terminals, quantified by the circulation coefficient

$\varphi = \alpha_R l.$6

Second, under open-circuit conditions, a hot third terminal can generate a nonlocal electrical current between the other two terminals; at symmetry points the current reduces to

$\varphi = \alpha_R l.$7

with

$\varphi = \alpha_R l.$8

These results extend the thermodynamic AB idea from local heat modulation to nonlocal thermoelectric conversion and heat circulation (Balduque et al., 2024).

5. Related interferometric thermodynamic responses in hybrid and thermal-interference platforms

The connection between AB phase and thermodynamic-like observables is not confined to purely normal electronic rings. In a five-terminal voltage-driven normal-superconducting nanostructure, (Dolgirev et al., 2018) analyzes the coexistence of a Josephson current $\varphi = \alpha_R l.$9, odd in $\mathcal{T}_\sigma(\varepsilon)$0, and an Aharonov–Bohm current $\mathcal{T}_\sigma(\varepsilon)$1, even in $\mathcal{T}_\sigma(\varepsilon)$2. At sufficiently large voltage $\mathcal{T}_\sigma(\varepsilon)$3, the current-phase relation becomes approximately

$\mathcal{T}_\sigma(\varepsilon)$4

In an asymmetric five-terminal configuration, an isolated normal terminal develops an induced potential $\mathcal{T}_\sigma(\varepsilon)$5 with coherent flux dependence. The even component contains

$\mathcal{T}_\sigma(\varepsilon)$6

This is a direct example of a flux-periodic induced thermodynamic-like response: the applied flux controls not only dissipative and supercurrent components but also a measurable electrostatic potential in a terminal that is isolated from the external circuit (Dolgirev et al., 2018).

Another, conceptually distinct, relation between AB interference and thermodynamics appears in four-terminal GaAs/AlGaAs waveguide rings (Buchholz et al., 2010). There the main observable is the AB resistance oscillation phase and visibility, but the temperature dependence reveals an explicit thermal averaging of phase-coherent transport. The oscillation amplitude decays as

$\mathcal{T}_\sigma(\varepsilon)$7

and is interpreted through

$\mathcal{T}_\sigma(\varepsilon)$8

For device B, the thermal estimate

$\mathcal{T}_\sigma(\varepsilon)$9

is close to the observed non-local slope

$\cos(\varphi_\sigma)$0

The paper concludes that thermal averaging is the dominant dephasing mechanism in the non-local configuration (Buchholz et al., 2010). This does not constitute a heat-current modulation effect, but it demonstrates that AB interference is directly shaped by thermodynamic occupation broadening.

A more distant but relevant analogue is the thermal interferometry of neutral anyons in Kitaev spin liquids (Wei et al., 2021). There the role of ordinary magnetic-flux-induced AB phase is replaced by topological/statistical phases, and the observable is heat current rather than electric current. In the Fabry–Perot geometry, the heat current depends on the trapped topological charge, for example

$\cos(\varphi_\sigma)$1

whereas in the Mach–Zehnder geometry the averaged heat current becomes

$\cos(\varphi_\sigma)$2

This work shows that thermal interferometry can encode phase information even in the absence of charge transport (Wei et al., 2021). A plausible implication is that the “thermodynamic AB” perspective is part of a broader program in which phase-coherent topology controls energy transport across electronic and non-electronic systems.

6. Gauge structure, measurement subtleties, and limitations

A recurrent misconception is that the thermodynamic AB effect means that magnetic flux directly acts on heat as a thermodynamic force. The explicit mechanism in (Hwang et al., 2024) contradicts that reading: flux modifies the phase of coherent electron amplitudes, the phase modifies transmission, the transmission modifies Peltier heat flow, and the heat flow modifies temperature through a balance equation. The causal chain is interferometric, not a direct magnetic coupling to thermal energy.

A second conceptual issue concerns what is actually measurable in AB physics. For charged particles coupled to an abelian gauge field, (Horvat et al., 2020) emphasizes that observable phases are gauge-invariant closed-loop quantities in spacetime, not arbitrary open-path phases. In ordinary recombination experiments the measurable phase is

$\cos(\varphi_\sigma)$3

This matters for thermodynamic AB settings because any flux-controlled heat or voltage response still presupposes gauge-invariant interference encoded by loop geometry rather than a local phase assigned point-by-point to a single path (Horvat et al., 2020).

A third limitation is coherence. In the thermal-junction analysis of (Balduque et al., 2024), dephasing is modeled with Büttiker probes that absorb and re-emit electrons with randomized phase; this suppresses interference and washes out flux-dependent oscillations. The paper states that in the fully incoherent limit, flux control disappears. Likewise, (Buchholz et al., 2010) shows that thermal broadening alone can suppress AB visibility through phase averaging, and that probe-induced potential fluctuations can add further decoherence in local measurement configurations.

Device geometry and symmetry also impose constraints. In (Balduque et al., 2024), complete suppression of heat transport is obtained in the symmetric two-terminal case at $\cos(\varphi_\sigma)$4, while perfect circulation $\cos(\varphi_\sigma)$5 is generally not achieved in the symmetric normal-conductor three-terminal setup. In (Dolgirev et al., 2018), Josephson and AB contributions coexist only in suitable asymmetric or multiterminal topologies; symmetric cross-like interferometers or pure ring geometries isolate one or the other contribution. These results indicate that the thermodynamic manifestation of AB physics is highly topology dependent.

Finally, the term should not be conflated with matter-wave AB sensing in trapped ions (Saito et al., 2024). That work demonstrates a trapped-ion matter-wave interferometer in which the AB phase is read out as a Ramsey fringe shift generated by the magnetic flux enclosed by a multiply-orbiting ion trajectory. It further notes that the interferometric scale factor is common to cyclotron motion and system rotation, yielding a rotation-equivalent sensitivity of approximately $\cos(\varphi_\sigma)$6 (Saito et al., 2024). This is an AB interferometric sensing result rather than a thermal one. Its relevance is conceptual: it reinforces that the AB phase can be transduced into observables other than conductance, here a Ramsey phase and a gyroscopic scale factor.

7. Significance and emerging directions

Taken together, the cited works delineate an emerging view of AB interferometry in which the phase is a control parameter for energy flow, temperature, electrostatic response, and nonlocal thermoelectric conversion, not only for electrical conductance (Hwang et al., 2024, Balduque et al., 2024, Dolgirev et al., 2018). Within that view, the most direct realization of the interferometric thermodynamic Aharonov–Bohm effect is the voltage-biased AB heat modulator of (Hwang et al., 2024), where a flux-controlled interference pattern is converted into a measurable reservoir temperature shift via Peltier transport and electron–phonon thermal balance.

The broader mesoscopic landscape shows several concrete functionalities already formulated at the theory level: phase-controlled heat modulation, spin-polarized heat currents without ferromagnetic contacts, thermal switches based on complete destructive interference, thermal circulators enabled by flux-induced broken reciprocity, and nonlocal thermoelectric engines where a hot terminal drives current elsewhere in the circuit (Hwang et al., 2024, Balduque et al., 2024). In hybrid normal-superconducting structures, flux-periodic induced potentials provide an additional route by which interferometric phase coherence manifests in a thermodynamic-like observable (Dolgirev et al., 2018).

The main constraints identified across the literature are equally clear: coherence must be preserved; dephasing and thermal averaging suppress the effect; topology and symmetry determine whether odd and even phase components coexist; and, in charged systems, measurable phases remain constrained by gauge-invariant loop structure (Buchholz et al., 2010, Balduque et al., 2024, Horvat et al., 2020). Under those conditions, the interferometric thermodynamic Aharonov–Bohm effect can be understood as the extension of AB physics from phase-sensitive charge transport to phase-sensitive thermodynamics.