Quantum Ramps Achieve Zero-Temperature Universality Class Probing at Finite Temperature

Understanding how quantum systems respond to rapid changes is a fundamental challenge in physics, and recent research explores this behaviour in systems driven across a critical point. Johannes N. Kriel of Stellenbosch University, Emma C. King from the Universität des Saarlandes, and Michael Kastner, along with their colleagues, investigate nonlinear quantum ramps , specifically how simultaneously varying temperature and a Hamiltonian parameter impacts a system’s evolution. Their work utilises an open-system Kitaev wire model to demonstrate that these protocols enable the probing of universality classes, normally associated with zero-temperature equilibrium, in realistic, out-of-equilibrium and finite-temperature scenarios. This research is significant because it identifies specific ramp conditions that allow scientists to dynamically measure critical exponents, offering a pathway to understanding complex quantum phenomena in experimentally achievable conditions.

Simultaneously, systems often evolve in a nonlinear fashion towards a quantum critical point. Researchers utilised an open-system version of a Kitaev quantum wire to demonstrate that, unlike finite-temperature protocols at fixed temperature, these protocols enable the probing of an out-of-equilibrium situation at finite temperature. This allows investigation of the universality class, characterised by the critical exponents ν and z, of an equilibrium quantum phase transition at zero temperature. Crucially, this is achieved through the identification of ramps where both coherent and incoherent aspects of the open-system dynamics significantly affect excitation density, and specific ramps were identified for subleading corrections.

Quantum Kibble-Zurek Mechanism and Scaling Behaviour

Scaling and universality are well-established phenomena near continuous equilibrium phase transitions, with initial experimental observations dating back to the 1940s. Subsequent theoretical development, including the renormalization group approach, provided a framework for understanding these concepts from microscopic theory. While scaling and universality have also been observed in out-of-equilibrium systems, a comprehensive theoretical framework remains elusive, although several promising ideas have been proposed. A common strategy for inducing scaling behaviour in non-equilibrium scenarios involves driving a system out of equilibrium in a manner that reflects an underlying equilibrium phase transition.

The quantum Kibble-Zurek mechanism exemplifies this approach; it begins with a system in equilibrium at zero temperature, with a parameter value μ initially far from its critical value μc. A gradual change in μ results in adiabatic dynamics, maintaining equilibrium until μ approaches μc. As μ nears μc, critical slowing down, the power-law divergence of the relaxation time near a continuous phase transition, prevents continued adiabatic evolution. This causes an approximate “freeze-out”, where the system’s evolution becomes insignificant compared to the timescale of external driving, forcing it out of equilibrium.

Consequently, appropriately selected non-equilibrium quantities exhibit universality and scaling laws dictated by the critical exponents of the underlying quantum phase transition. The goal of protocols like the Kibble-Zurek mechanism is to identify observables that not only capture equilibrium critical exponents but also present them through simple, clean scaling laws. The research focuses on systems where temperature and a Hamiltonian control parameter are simultaneously ramped towards a critical point, unlike previous finite-temperature protocols. Experiments utilising an open-system Kitaev wire demonstrate the ability to probe the universality class, characterised by critical exponents, even in out-of-equilibrium and finite-temperature scenarios. This advancement hinges on identifying ramps where both coherent and incoherent dynamics significantly influence excitation density.

The team measured the total excitation density, E, in an open Kitaev chain subjected to temperature ramps defined as T(t) = Ti − vt, where Ti is the initial temperature, v is the ramp velocity, and t ranges from 0 to tf = Ti/v. Results demonstrate a homogeneity relation, E lzT, lzTi, l−z(s+1)γ/v = lE(T, Ti, γ/v), holding true for arbitrary scaling factor ‘l’, with γ representing the coupling strength to the bath. This relation establishes a foundation for deriving power-law scaling, dependent on the exponent ‘s’ of the bath spectral density and the dynamical critical exponent ‘z’ of the underlying quantum phase transition. Further investigation revealed that while previous work focused on ramps yielding scaling governed by exponent ‘z’, a complete characterisation of the universality class requires incorporating the quantum critical exponent ‘ν’, which quantifies the divergence of the correlation length.

The breakthrough delivers a Kibble-Zurek-type protocol that generates clean scaling laws governed by both ‘z’ and ‘ν’ while operating at nonzero temperatures. This was accomplished by implementing nonlinear, two-parameter ramps of both temperature and the Hamiltonian control parameter, approaching the quantum critical point. Measurements confirm that by carefully selecting power-law ramps, the excitation density, E, exhibits clean power-law scaling, incorporating both quantum critical exponents. Tests prove this approach successfully extends the quantum Kibble-Zurek protocol to finite temperatures, enabling a comprehensive characterisation of the universality class of the underlying equilibrium quantum phase transition based on nonequilibrium data. The study provides a complete solution for the asymptotic properties of these two-parameter ramps in the slow-ramp limit, paving the way for experimentally probing critical exponents in realistic finite-temperature conditions.

Ramping Reveals Open System Criticality

This work details a new method for investigating quantum critical properties in open systems at non-zero temperatures. By analysing systems undergoing simultaneous and nonlinear temperature and Hamiltonian parameter ramps towards a critical point, researchers demonstrated the ability to probe the universality class of an equilibrium phase transition, something not possible with fixed-temperature protocols. The central achievement lies in the asymptotic characterization of excitation density in free fermionic chains coupled to thermalizing baths, revealing a power-law scaling dependent on both equilibrium critical exponents and the bath’s spectral properties. The study identifies specific ramping paths, termed ‘class A’ and ‘class C’, which exhibit distinct scaling behaviours.

Class C ramps, in particular, offer a potential route to extract both the dynamical critical exponent, z, and the correlation length critical exponent, ν, thereby characterizing the universality class of the quantum phase transition from out-of-equilibrium data. The authors acknowledge limitations in fully addressing convergence rates and extending the analysis to super-ohmic bath conditions, suggesting these as avenues for future investigation. Furthermore, they note that experimental constraints may necessitate alternative protocols, and a deeper understanding of scaling law corrections is needed to optimise experimental design.