🔬 mesoscale physics

From Effective Temperature to Non-Boltzmann State Selection in Driven-Dissipative Quantum Criticality

This paper demonstrates that while a voltage-biased driven-dissipative quantum critical system remains organized by an effective temperature, its steady state is selected via a non-Boltzmann rule that shifts first-order transitions away from the equal-depth point of a deterministic potential.

Original authors: Tiago Jorge, Jens Paaske, Pedro Ribeiro

Published 2026-09-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tiago Jorge, Jens Paaske, Pedro Ribeiro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of the quantum world, particles often behave as if they are in a state of perfect balance, settling into patterns dictated by the temperature of their surroundings. This is the realm of equilibrium, where the rules are well-understood and the behavior of matter is predictable. But the universe is rarely so still. When we push a system out of balance—by applying a voltage, shining a light, or forcing a current through it—we enter the chaotic and fascinating territory of nonequilibrium physics. Here, the familiar rules of thermal balance often break down, and scientists have long wondered if a new, simpler set of rules could take their place. Specifically, they have asked whether a system driven far from equilibrium could still be described by a single, effective temperature, as if the chaos of the drive simply heated the system up to a new, uniform level. This question is crucial because if such a description holds, it would allow physicists to use their powerful, established tools to understand complex, driven systems like the electronic components in future quantum computers or the strange states of matter found in high-speed transport.

A team of researchers has now tackled this question using a carefully constructed model of a tiny magnetic dot connected to two metal wires. In this setup, the dot acts as a collection of electrons that can align their spins together, much like a group of tiny magnets, while the wires act as reservoirs that constantly feed electrons in and out. By applying a voltage difference between the wires, the researchers forced the system into a steady state that is constantly being driven and dissipated, never truly resting. They used a sophisticated mathematical approach that treats the system as having a vast number of electrons, allowing them to track how the collective behavior of the spins changes as they vary the temperature and the voltage. Their goal was to see if the messy, driven state could still be tamed by the concept of an effective temperature, or if the drive created something entirely new that defied simple thermal descriptions.

The researchers found that the answer is both yes and no, depending on how hard they push the system. When the voltage is low, the system behaves in a surprisingly orderly way. Even though it is being driven by an electric current, the fluctuations of the electron spins are governed by a single effective temperature that depends on both the actual temperature of the wires and the strength of the voltage. In this regime, the system acts as if it were in thermal equilibrium at this new, combined temperature. The researchers mapped out a region of the system's behavior, known as a quantum-critical fan, where this effective temperature successfully organizes the chaos, predicting how the spins will fluctuate just as a standard thermometer would in a hot room. This confirms that for gentle drives, the complex dance of electrons can indeed be simplified into a thermal picture.

However, the story changes dramatically when the voltage is increased to a strong level. In this regime, the system undergoes a sudden shift, jumping from one state to another in a way that is typical of a first-order transition, like water freezing into ice. Here, the researchers discovered that the effective temperature is no longer a single, global number that applies to the whole system. Instead, the effective temperature varies depending on the specific state the system is in. As the system moves between different possible configurations, the "temperature" it feels changes along the path. This variation means that the system does not settle into the state that would be expected if it were simply following the rules of thermal equilibrium. In a normal thermal system, the most stable state is the one with the deepest energy valley. But in this driven system, the state that wins out is selected by a different rule, one that accounts for how the effective temperature changes across the landscape of possibilities.

This finding reveals a fundamental limit to how far we can simplify nonequilibrium systems. While an effective temperature can describe the fluctuations near a smooth transition, it fails to predict the final state when the system is pushed hard enough to jump between distinct phases. The researchers showed that the steady state is selected by a non-Boltzmann rule, meaning the system chooses its fate not just by looking for the lowest energy, but by navigating a landscape where the rules of thermal balance are constantly shifting. This work provides a clear boundary for when our familiar thermal intuition works and when it breaks down, showing that in the driven quantum world, the path a system takes to reach its final state is just as important as the destination itself. The study does not claim to solve all nonequilibrium mysteries, but it offers a precise, controlled example of how drive and dissipation reshape the very nature of phase transitions, revealing a new layer of complexity in the behavior of quantum matter.

Problem Statement
A central open question in nonequilibrium quantum physics is whether steady states of driven-dissipative quantum-critical systems can be described by an effective temperature ( $T_{\text{eff}}$ ). While fluctuation-dissipation ratios (FDRs) often reduce to an effective equilibrium description for slow degrees of freedom, it remains unclear if this holds universally near quantum critical points (QCPs) in non-Markovian environments, such as electronic transport where reservoirs impose nonthermal occupations. Specifically, the predictive power of an effective-equilibrium description and the extent to which it leaves state selection undetermined in driven open systems are not fully understood.

Methodology
The authors address this by studying an electronic realization of the Lipkin–Meshkov–Glick (LMG) model (eLMG), a minimal driven-dissipative model of a collective spin coupled to voltage-biased metallic leads. The system consists of $N$ orbitals with infinite-range anisotropic exchange interaction and a transverse magnetic field $h$ , tunnel-coupled to two leads at temperature $T$ with chemical potentials $\mu_{L,R} = \pm V/2$ .

The study employs a controlled large- $N$ treatment using a Keldysh path-integral formalism. The exchange interaction is decoupled via a real bosonic Hubbard–Stratonovich field $\phi$ representing the collective $x$ -polarization. After integrating out the fermionic degrees of freedom (both dot and leads), the authors derive an effective bosonic action. They analyze the system through:

Self-consistent saddle-point equations to determine the steady-state order parameter $\phi = \langle \hat{S}_x \rangle / N$ .
Quadratic expansion of the action to analyze collective propagators and fluctuations.
Renormalization-group (RG) analysis at the tree level to establish scaling laws for fluctuations.
Mapping to a stochastic equation (Model A type) to access fluctuations beyond the Gaussian level, specifically in the bistable regime.

Key Contributions and Results

Dissipation as a Relevant Critical Perturbation:
The coupling to leads ( $\Gamma$ ) fundamentally alters the critical behavior compared to the closed LMG model. Even at $T=V=0$ , hybridization shifts the critical field $h_c$ and changes the universality class. The open QCP is overdamped, with fluctuations diverging logarithmically ( $N\langle \delta\phi^2 \rangle \sim -\log|\delta h|$ ) rather than with the power law ( $|\delta h|^{-1/2}$ ) of the closed model. This leads to a distinct scaling regime where the relevant control parameter is $\delta h / \Gamma^4$ , and the finite-size scaling exponent changes from $N^{-2/3}$ (closed) to $N^{-1}$ (open).
Effective Temperature Regime and the Quantum-Critical Fan:
The authors identify a regime where critical fluctuations are governed by a single effective temperature, $T_{\text{eff}}(T, V)$ , defined via the low-frequency FDR. This $T_{\text{eff}}$ organizes the parameter space into a quantum-critical fan covering both ordered and disordered sides.
- $T_{\text{eff}}$ interpolates between the bath temperature $T$ (at low bias) and $V/4$ (at high bias).
- Crucially, this effective temperature description holds even at $T=0$ for the low-frequency Gaussian modes, extending the concept of effective equilibrium beyond the $T \gg V$ limit.
Non-Boltzmann State Selection in the Bistable Regime:
At strong bias, the transition becomes first-order via a tricritical point. In this regime, the steady state is selected by a non-Boltzmann rule.
- While the deterministic potential $F[\phi]$ (derived from the drift term) suggests a transition at the equal-depth point of its minima, the actual steady-state transition is shifted.
- The probability distribution $P_{\text{ss}}(\phi)$ is governed by a large-deviation function $W(\phi)$ , which acts as a Boltzmann weight only if $T_{\text{eff}}$ is constant.
- In the driven regime, $T_{\text{eff}}$ varies across the order-parameter landscape ( $T_{\text{eff}}(\phi)$ ). Consequently, the steady state is selected by the integral of $F'(\phi)/T_{\text{eff}}(\phi)$ along the path connecting competing states, rather than by the depth of the potential minima alone. This decouples stationary-state selection from the deterministic potential.

Significance
The paper demonstrates that driven criticality can remain organized by an effective temperature while simultaneously revealing genuinely nonequilibrium state selection mechanisms. The work provides a controlled framework to distinguish between emergent thermal behavior (where a single $T_{\text{eff}}$ suffices near second-order transitions) and non-thermal selection (where $T_{\text{eff}}$ depends on the order parameter $\phi$ in the bistable regime). This highlights that the order-parameter dependence of the FDR serves as a criterion for identifying when a system's steady state cannot be described by a simple effective equilibrium potential, a finding relevant to the broader understanding of multistability and phase transitions in nonequilibrium quantum matter.

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