Quantum Instability Reveals Hidden Order in First-Order Transitions

A new theory explains the surprising emergence of quantum criticality within first-order quantum phase transitions, a phenomenon usually linked to second-order transitions. Fan Zhang and colleagues at Peking University, in collaboration with Collaborative Innovation Centre of Quantum Matter, present a microscopic theory revealing that criticality can occur around the quantum spinodal point of these transitions, where metastable states are no longer present. The theory connects the dynamics of first-order and second-order transitions by showing how resonant local excitations create an effective Hamiltonian exhibiting a genuine second-order quantum phase transition and Kibble-Zurek scaling. This offers key insight into the longstanding puzzle of first-order quantum phase transition dynamics and suggests their behaviour is often governed by an emergent critical point.

Resonant excitations and Hilbert subspace projection define a simplified quantum model

Projection onto a reduced Hilbert subspace proved key to uncovering this emergent behaviour. A Hilbert subspace, a simplified view of all possible quantum states, was isolated by identifying resonant local excitations, specific energy fluctuations within the material. These excitations represent localised disturbances that, when amplified, can drive the system towards a new phase. The concept builds upon established principles of quantum mechanics, where the state of a system is described by a vector in a Hilbert space, encompassing all possible configurations. By focusing on resonant excitations, the researchers effectively identified a subset of states that are most susceptible to these disturbances and therefore dominate the transition process. This technique dynamically narrowed the system’s focus, discarding irrelevant states and concentrating on those important to the transition, similar to focusing on key players in a complex game to understand its outcome. The selection of resonant excitations is crucial, as they provide the necessary energy scale for driving the transition and establishing the emergent critical behaviour.

Mathematically projecting the original Hamiltonian, a description of the system’s energy, onto this smaller subspace created a new, simplified model exhibiting characteristics of a second-order quantum phase transition. This revealed universal scaling behaviour previously thought impossible in first-order systems. The Hamiltonian encapsulates all the energy contributions within the system, including interactions between particles and external fields. Projecting it onto the reduced Hilbert space effectively filters out the degrees of freedom that are not directly involved in the transition, leading to a more tractable model. Dr. Johannes Knolle and colleagues Dublin investigated a tilted Ising chain comprising 259 sites, employing the time-dependent variational principal method to model its behaviour, starting in a metastable state with an initial spin deviation of -0.5, alongside a parameter λ equal to 0.1. The time-dependent variational principle is a powerful technique for studying the dynamics of quantum systems, allowing researchers to approximate the evolution of the system’s state over time. The reduced Hilbert space dimension was calculated as 2L/2, streamlining the model for analysis. This reduction in dimensionality significantly simplifies the calculations and allows for a more detailed investigation of the emergent critical behaviour. The choice of a 259-site chain and initial conditions provides a specific example for testing the theoretical predictions.

Universal critical scaling in first-order quantum phase transitions

Scaling exponents, previously considered exclusive to second-order transitions, now demonstrate a power-law relationship of v−1/3 in first-order quantum phase transitions, a sharp shift from established expectations. These exponents characterise how physical quantities, such as correlation length and defect density, change as the system approaches the critical point. The system exhibits this scaling when crossing the quantum spinodal point, the threshold at which metastable states vanish and resonant local excitations decouple a Hilbert subspace. The quantum spinodal point represents a point of instability in the system, where small fluctuations can trigger a rapid transition between phases. These findings establish a bridge between the dynamics of both transition types, revealing that first-order transitions can be governed by an emergent critical point previously thought impossible. Analysis of the tilted Ising chain revealed that this decoupling is characterised by an emergent discrete translational symmetry, Zn, with periodicity determined by the interaction range and a parameter R. This symmetry implies that the system’s properties are invariant under certain translations, indicating an underlying order that emerges near the critical point. The team showed that the density of topological defects scales with velocity, v, according to the relation n ∼vdν/(1+νz), while the correlation length follows a finite-time scaling ansatz, ξ(λ, v) = v−ν/(1+νz)f (λ −λsp)v−1/(1+νz). The correlation length measures the spatial extent of correlations between different parts of the system, and its scaling behaviour provides further evidence for the emergent critical behaviour. Currently, these findings apply to systems with short-range interactions, excluding infinite-range models like the Lipkin-Meshkov-Glick model, and a clear pathway to using this emergent criticality for practical quantum technologies remains elusive. The limitation to short-range interactions highlights the need for further research to explore the applicability of the theory to more complex systems.

Hilbert subspace decoupling explains universal scaling in quantum phase transitions

Understanding how materials change state is fundamental to modern physics, yet predicting the precise behaviour during rapid transitions has remained elusive. This work offers a compelling new mechanism, the decoupling of a ‘Hilbert subspace’, to explain how first-order quantum phase transitions can surprisingly exhibit the same universal scaling seen in their second-order counterparts. The ability to predict and control phase transitions is crucial for developing new materials with tailored properties. Identifying specific limitations is important for refining theoretical models and directing future research, highlighting areas where experimental validation is particularly important, such as confirming the predicted lack of criticality within the staggered-field PXP model. The staggered-field PXP model serves as a benchmark system for testing the validity of the theory and identifying potential deviations.

Criticality can emerge within first-order transitions at a specific instability point known as the quantum spinodal point, where a material loses stability. This instability arises from the competition between different energy scales within the system, leading to a breakdown of the ordered phase. By effectively focusing on key quantum states, the approach simplifies the complex behaviour of the system, allowing for the observation of emergent phenomena previously thought exclusive to second-order transitions. The simplification allows researchers to identify the underlying mechanisms driving the transition and understand how universal scaling emerges. The implications of this work extend beyond fundamental physics, potentially influencing the design of new quantum materials and devices.

The research demonstrated that quantum criticality can emerge around the quantum spinodal point of first-order quantum phase transitions, a surprising finding given that universality and scaling are typically associated with second-order transitions. This means that even when materials undergo abrupt changes of state, predictable and universal behaviours can still arise through the decoupling of a specific Hilbert subspace. Researchers validated this framework using the tilted Ising chain and predict a lack of criticality in the staggered-field PXP model, offering a benchmark for future testing. This study establishes a connection between the dynamics of both types of quantum phase transitions and provides new insight into how materials change state.