Extended Landauer-Büttiker Formula for Current through Open Quantum Systems with Gain or Loss

This paper extends the Landauer-Büttiker formula to open quantum systems with gain or loss using the Lindblad-Keldysh formalism, revealing novel transport phenomena such as current generation from symmetry breaking or disorder, and the restoration of the Wiedemann-Franz law beyond the energy band.

The Gist

Imagine a busy highway connecting two cities (the "leads"). In the classic version of this story, cars (particles) drive from one city to the other based on how crowded the cities are. If City A is packed and City B is empty, traffic flows from A to B. This is the famous Landauer-Büttiker formula, a rulebook physicists have used for decades to predict how electricity or heat moves through tiny wires.

However, this old rulebook assumes the highway is a perfect, sealed tube. It doesn't account for what happens if cars are randomly dropped off or picked up along the way by mysterious roadside stations.

This paper introduces a new, expanded rulebook for "open" highways where cars can be added or removed at any point. Here is the breakdown of their findings using simple analogies:

1. The New Rulebook: The "Gain and Loss" Highway

The authors created a new formula to describe what happens when a quantum system (like a tiny wire) is connected not just to two cities, but also to a bunch of "Markovian reservoirs."

• The Analogy: Imagine the highway has side roads where trucks can dump extra cars onto the road (Gain) or where cars can vanish into a black hole (Loss).
• The Result: The old formula only counted cars driving straight through. The new formula accounts for the traffic chaos caused by these side roads. It shows that these "gain and loss" stations fundamentally change how traffic flows, sometimes creating currents even when the two main cities are identical.

2. Breaking the Mirror: Creating Current from Nothing

Usually, if a road is perfectly symmetrical (a mirror image on the left and right), traffic flows equally in both directions, resulting in zero net movement.

• The Discovery: The paper finds that if you break this symmetry—either by having the "dump trucks" (gain) and "black holes" (loss) arranged unevenly, or by having the road itself be uneven—you can generate a current even if the two main cities have the same population and temperature.
• The Analogy: Imagine a perfectly flat, symmetrical playground slide. If you stand in the middle, you don't slide. But if you put a small bump on one side (breaking the symmetry) or have someone push you from one side (asymmetric gain/loss), you start sliding. The paper shows that "pushing" or "pulling" particles unevenly creates a flow where there was none before.

3. Disorder as a Driver: Chaos Creates Order

In normal physics, "disorder" (like potholes or debris on a road) usually stops traffic. It makes cars get stuck.

• The Discovery: In this specific open system with gain and loss, disorder can actually create current.
• The Analogy: Imagine a chaotic construction zone. Normally, this would stop traffic. But in this new setup, the combination of random potholes (disorder) and the side-road trucks (gain/loss) creates a weird, self-sustaining flow of cars that wouldn't exist on a smooth road. The chaos itself becomes the engine.

4. The "Skin Effect": The Crowd Huddles at the Edge

The paper also looks at a phenomenon called the "Non-Hermitian Skin Effect."

• The Analogy: Imagine a crowd of people in a long hallway. In a normal hallway, they spread out evenly. But in this special "gain/loss" hallway, the crowd gets pushed so hard by the physics of the walls that they all huddle up against one specific end of the hall, leaving the middle empty.
• The Result: The paper shows that this "huddling" creates a current that fights against the usual flow caused by voltage differences. Depending on how long the hallway is, the "huddling" current might win, or the "voltage" current might win. It's a tug-of-war between the size of the system and the push of the batteries.

5. Heat and Electricity: The Golden Ratio

Physicists have a famous rule (the Wiedemann-Franz law) that says electrical conductivity and thermal conductivity (how well heat moves) are usually locked in a specific ratio.

• The Discovery: In normal systems, this rule breaks down at the edges of energy bands (like the very top or bottom of a speed limit). However, the paper shows that adding gain and loss "smooths out" the road.
• The Analogy: Think of the road as having a "speed limit" zone. Without gain/loss, the rule about speed and heat breaks down at the very edges of the zone. With gain/loss, the road becomes so smooth that the rule holds true everywhere, even at the very edges and outside the zone.

Summary

The paper essentially says: We have updated the master formula for quantum traffic. By adding the ability for particles to be created or destroyed along the way, we find that:

1. Symmetry breaking creates flow.
2. Disorder can drive flow.
3. Edge-huddling (Skin Effect) can fight against voltage.
4. Heat and electricity stay perfectly linked even in strange conditions.

This provides a new toolkit for understanding how particles move in open, messy, real-world quantum systems where things are constantly being added or lost.

Technical Summary

Technical Summary: Extended Landauer-Büttiker Formula for Current through Open Quantum Systems with Gain or Loss

Problem Statement
The Landauer-Büttiker formula is a foundational framework for describing quantum transport in coherent systems connected to leads, expressing current via transmission coefficients and lead distributions. However, this original formulation assumes the central system is isolated from dissipation other than coupling to the leads. In reality, open quantum systems frequently experience particle gain and loss (e.g., via engineered coupling in cold atoms, photonic lattices, or uncontrolled leakage). While dissipation is known to induce novel phenomena and phase transitions, a general transport formula that uniformly captures the effects of arbitrary gain and loss on particle and energy currents has been lacking. This gap limits the understanding of transport in non-Hermitian and open systems and hinders the prediction of nonequilibrium phenomena.

Methodology
The authors derive a general transport formula by extending the Landauer-Büttiker formalism using the Lindblad-Keldysh framework.

• Model: They consider a one-dimensional central system coupled to left and right leads (with temperatures $T_{L,R}$ and chemical potentials $\mu_{L,R}$) and exchanging particles with Markovian reservoirs.
• Dynamics: The system's evolution is governed by a Lindblad master equation where the reservoir effects are captured by gain ($L_2$) and loss ($L_1$) operators. These operators can be on-site or non-local.
• Derivation:
  1. The authors define particle ($J^0$) and energy ($J^1$) currents flowing out of the leads.
  2. They employ the Keldysh formalism, utilizing a rotation to the Keldysh basis ($\psi_1, \psi_2$) to rewrite the action.
  3. By inverting the matrix form of the action, they obtain the system's retarded, advanced, and Keldysh Green's functions ($G^R_S, G^A_S, G^K_S$).
  4. Using the Langreth theorem, they relate the system-lead coupling Green's functions to the system Green's functions, incorporating self-energies from the leads and the gain/loss terms (represented by matrices $P$ for loss and $Q$ for gain).
  5. This leads to a generalized expression for the current that separates contributions from direct lead-to-lead transmission and indirect transmission via the reservoirs.

Key Contributions and Results
The derived formula decomposes the total current $J^\lambda$ into three components:
$$J^\lambda = J^\lambda_{LR} + J^\lambda_{LS} - J^\lambda_{RS}$$
where $J^\lambda_{LR}$ is the standard Landauer-Büttiker term, and the remaining terms describe current exchange between the leads and the reservoirs through the central system.

The paper identifies several non-trivial transport phenomena based on this formula:

1. Current Generation via Symmetry Breaking:

   • Even in the absence of chemical potential or temperature gradients between leads ($f_L = f_R$), gain and loss can generate a net current.
   • This current vanishes only if the particle-to-hole ratio in the leads matches the gain-to-loss ratio in the system, or if the system and the gain/loss terms possess inversion symmetry (IS).
   • Breaking IS in the gain/loss terms (e.g., asymmetric monitoring) or in the system Hamiltonian (e.g., introducing disorder) induces a current. Specifically, disorder can induce current in systems where it would otherwise suppress transport in closed systems.

2. Impact on Conductance and the Wiedemann-Franz Law:

   • The formula reveals that gain and loss modify electrical ($G$) and thermal ($K$) conductances.
   • In the presence of gain/loss, the transmission function becomes smooth near band edges. Consequently, the Wiedemann-Franz law ($K/G \approx LT$) holds both inside and outside the energy band, whereas in coherent systems, it typically fails outside the band or at edges.
   • The conductance in the thermodynamic limit is dominated by indirect transmission via reservoirs ($\tau_{12}$), which remains size-independent, unlike the direct transmission ($\tau_{11}$) which decays exponentially due to particle scattering into reservoirs.

3. Non-Hermitian Skin Effect (NHSE) and Current Competition:

   • The authors apply the formula to a system with bond loss inducing the NHSE, where eigenstates localize at one boundary.
   • They find a competition between the equilibrium current ($J_0^0$) driven by the skin effect and the response current ($\delta J_1^0$) driven by chemical potential bias.
   • Depending on system size and bias, the total current direction can be reversed. Notably, the skin-effect-induced current vanishes as system size tends to infinity for certain parameter regimes, while the bias-induced response current remains size-independent.

4. Bosonic vs. Fermionic Transport:

   • The framework is extended to bosonic systems. A key distinction is found in the gain term: for fermions, gain is suppressed by Pauli blocking (proportional to hole density $1-f$), whereas for bosons, gain is enhanced by occupation (proportional to $\tilde{f}+1$), leading to opposite trends in response currents under increasing gain rates.

Significance
The paper claims to establish a versatile and foundational framework for studying transport in open quantum systems with arbitrary forms of gain and loss. By providing a unified formula, it deepens the understanding of how dissipation shapes transport properties beyond simple decoherence. The work highlights that gain and loss are not merely detrimental but can actively generate currents, induce transport in disordered systems, and alter fundamental transport laws like the Wiedemann-Franz law. The authors position this work as a necessary extension to the Landauer-Büttiker paradigm, enabling the prediction of novel nonequilibrium phenomena in non-Hermitian and open physics.