Half-Degeneracy Signals Quantum Phase Transitions

A thorough investigation into the behaviour of momentum-space entanglement entropy during dynamical quantum phase transitions in two-band insulators and superconductors has revealed key insights. Kaiyuan Cao and colleagues at College of Physics Science and Technology demonstrate that a specific geometric condition linking initial and final momenta predicts exact degeneracy within the entanglement spectrum, leading to a maximum entropy value. Using models including the Su-Schrieffer-Heeger and Haldane models, the study shows how critical momenta manifest differently in one and two dimensions and highlights the key influence of the chosen basis for measurement. Momentum-space entanglement entropy is now established as a strong indicator of dynamical quantum phase transitions, offering a new connection between entanglement, topology, and non-equilibrium criticality

Entanglement entropy reveals critical dynamics in two-band systems

Momentum-space entanglement entropy offers a new perspective for examining the critical behaviour of quantum systems undergoing dynamical quantum phase transitions, or DQPTs. This approach focuses on the direction and speed of change within the system, rather than simply its composition. The technique begins by considering a system’s ‘momentum space’, a mathematical representation of particle properties based on motion, not location. Each momentum point is then divided into two interconnected parts, creating a ‘bipartition’ to measure the entanglement between them, similar to assessing the connection between two dance partners.

Researchers investigated dynamical quantum phase transitions, or DQPTs, within two-band insulators and superconductors possessing translational symmetry. These systems were described using a momentum-dependent vector, dk, and a Hamiltonian incorporating the Pauli sigma matrices. The analysis focused on the Su-Schrieffer-Heeger model, the quantum XY chain, and the Haldane model, establishing a geometric DQPT condition where initial and final ‘d-vectors’ are perpendicular. Momentum space proved advantageous because DQPT nonanalyticities localise within specific momentum modes, providing a more direct probe than global measures like the Loschmidt echo. The findings demonstrate how the choice of mathematical framework impacts observed results, revealing that alternative approaches, such as using a sublattice basis, yield time-dependent entropy.

Maximum entanglement entropy of ln 2 pinpoints dynamical quantum phase transitions and reveals

Entanglement measures now reach a maximum value of ln 2, a significant improvement over previous limitations where diagnosing dynamical quantum phase transitions (DQPTs) relied on identifying time-dependent behaviours or local extrema in entanglement entropy. This peak entropy value arises specifically when the geometric condition for DQPTs, orthogonal initial and final momentum vectors, results in exact degeneracy of 1/2 within the entanglement spectrum. This precise value was previously unattainable with other diagnostic methods.

Critical momenta manifest as isolated points in one dimension, but form continuous manifolds in two dimensions, revealing how dimensionality impacts the underlying critical structure. The Su-Schrieffer-Heeger model, the quantum XY chain, and the Haldane model consistently demonstrated this behaviour across different material types. In one-dimensional systems, critical momenta were observed as isolated points, but these points expanded into continuous one-dimensional manifolds in two dimensions, further illustrating how a material’s structure influences its critical behaviour. This demonstrates the method’s robustness across various material types and dimensionalities.

Mathematical viewpoint dictates entropy measurement in dynamical quantum phase transitions

Traditionally, identifying dynamical quantum phase transitions, sudden shifts in a quantum system’s behaviour, has relied on tracking changes over time or pinpointing specific locations where these transitions occur. However, this new work reveals a surprising sensitivity to data interpretation, specifically the mathematical ‘basis’ used for analysis. Acknowledging that different mathematical viewpoints can yield varying interpretations of these quantum shifts is not a weakness, but a key refinement of our understanding.

The choice of ‘basis’, or how data is organised, sharply impacts observed entropy, a measure of disorder. Selecting an appropriate analytical framework is therefore crucial when studying rapidly changing quantum systems; momentum-space entanglement entropy, when correctly applied, remains a reliable indicator of these transitions despite potential ambiguities. This work establishes a strong method for identifying dynamical quantum phase transitions by examining entanglement within momentum space, a representation of particle motion. Achieving a maximal entropy value of ln 2 confirms the reliability of this approach, differing from previous techniques reliant on tracking changes over time or pinpointing specific locations. In particular, this research demonstrates the importance of the mathematical framework used to analyse quantum systems, revealing how the chosen ‘basis’ sharply impacts observed results and underscores the need for careful consideration of analytical methods.

The research demonstrated that dynamical quantum phase transitions can be reliably identified using momentum-space entanglement entropy, achieving a maximum value of ln 2 when analysed in a specific eigenbasis. This is significant because it provides a time-independent method for detecting these transitions, differing from previous approaches that tracked changes over time. The study found that critical behaviour varied with dimensionality, appearing as isolated points in one dimension and continuous manifolds in two dimensions. Researchers suggest this method offers a unified perspective linking entanglement, topology, and non-equilibrium criticality, highlighting the importance of basis selection in quantum analysis.