Non-hermitian Spin Chains Demonstrate Criticality Signs Via Non-Stabilizerness, Revealing Exceptional Points

The behaviour of complex quantum systems at critical points, where properties change dramatically, remains a central challenge in physics, and researchers now investigate how to identify these points in systems that defy conventional rules. Cătălin Paşcu Moca from University of Oradea and Budapest University of Technology and Economics, Doru Sticlet from National Institute for R and D of Isotopic and Molecular Technologies, and Balázs Dóra from Budapest University of Technology and Economics, explore a property called ‘non-stabilizerness’ to pinpoint both critical behaviour and unusual points called exceptional points in non-Hermitian spin chains. Their work demonstrates that non-stabilizerness acts as a sensitive diagnostic, exhibiting distinct signatures depending on the specific quantum model, and revealing the presence of these critical points even as systems become increasingly complex. This research establishes non-stabilizerness as a valuable tool for understanding the behaviour of non-Hermitian matter and examining phenomena like symmetry breaking and quantum complexity.

Non-Hermitian Systems and Quantum Complexity Measures

This research investigates non-stabilizerness, a measure of quantum complexity, to understand quantum phases in non-Hermitian systems, which deviate from traditional quantum rules. Scientists explore whether this measure can reliably pinpoint quantum critical points, the points where materials undergo dramatic changes, even in these unusual systems. The study focuses on characterizing complex quantum states, those more difficult for classical computers to simulate, and assessing their potential for advanced computation. Quantum states are categorized by their complexity, with stabilizer states being relatively simple and non-stabilizer states requiring significantly more computational power.

Non-stabilizerness quantifies how far a quantum state is from being a stabilizer state, with higher values indicating greater complexity and potential for quantum advantage. Non-Hermitian Hamiltonians allow for unique phenomena like exceptional points and complex energies, offering new avenues for exploring quantum behavior. Exceptional points represent singularities where system properties change dramatically, leading to enhanced sensitivity and unusual dynamics. The research combines theoretical calculations with numerical simulations to explore these concepts. Scientists derive mathematical expressions for non-stabilizerness in various models, including the transverse-field Ising model and its non-Hermitian counterparts.

They then employ computational techniques, such as Density Matrix Renormalization Group and Time-Evolving Block Decimation, to simulate the behavior of quantum systems, using the ITensor library to analyze excited states and examine system properties in momentum space. The results demonstrate that non-stabilizerness effectively identifies quantum critical points in both standard and non-Hermitian systems, showing clear changes in behavior at these points. The scaling behavior of non-stabilizerness near critical points appears universal, independent of specific model details. The research reveals the phase diagrams of non-Hermitian models, mapping out different phases and transitions between them, and characterizes exceptional points and their influence on system properties.

Analyzing non-stabilizerness in momentum space provides a more complete picture of system correlations and critical behavior, establishing a connection to correlation functions and offering a physical interpretation of the measure. Importantly, non-stabilizerness appears robust, less susceptible to noise and imperfections than some other indicators of criticality. This research contributes to a better understanding of quantum complexity and the characterization of non-stabilizer states. It provides a new tool for identifying and characterizing quantum critical points, even in complex systems, and sheds light on the unique properties of non-Hermitian systems and their potential applications. Understanding non-stabilizer states is crucial for harnessing the full power of quantum computation, potentially leading to advancements in quantum algorithms and computer design, and has implications for materials science, potentially aiding the study of novel materials with exotic quantum properties.

Non-Hermitian Spin Chains via Matrix Product States

Scientists investigated non-stabilizerness, also known as “magic”, to understand criticality and exceptional points within non-Hermitian many-body systems, focusing specifically on parity-time (PT) symmetric spin chains, namely the non-Hermitian transverse-field Ising and XX models. They pioneered a methodology employing non-Hermitian matrix product state methods to calculate stabilizer Rényi entropies in the ground states of these systems. Researchers utilized a Density Matrix Renormalization Group (DMRG) algorithm, adapted for non-Hermitian systems and implemented within the ITensor library, to compute the ground state wavefunctions. The team represented right eigenstates as matrix product states with open boundary conditions, calculating the reduced density matrix for a subregion A of length l and its complement using the right ground state.

This approach allows quantification of non-stabilizerness, revealing how much a quantum state deviates from the set of classically simulatable stabilizer states. To achieve precise results, scientists employed open boundary conditions and adapted the DMRG algorithm to handle the unique properties of non-Hermitian Hamiltonians. Furthermore, the study performed finite-size scaling to demonstrate that the observed effects become more pronounced with larger systems, confirming the sensitivity of non-stabilizerness as a marker for both quantum criticality and non-Hermitian spectral degeneracies. Researchers also analytically investigated magic in momentum space for the XX model, discovering that it reaches a minimum around exceptional points, further solidifying the connection between non-stabilizerness and the characteristics of non-Hermitian systems. This detailed analysis positions non-stabilizerness as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter.

Magic Peaks Reveal Criticality in Spin Chains

Scientists investigated non-stabilizerness, also known as “magic”, to understand criticality and exceptional points within non-Hermitian many-body systems, focusing specifically on parity-time (PT) symmetric spin chains, including the transverse-field Ising and XX models. They calculated stabilizer Rényi entropies in the ground states of these systems using non-Hermitian matrix product state methods, revealing model-specific signatures of magic. Results demonstrate that magic peaks along the regular Hermitian-like critical line in the Ising chain, but completely disappears at exceptional points, indicating a strong connection between non-stabilizerness and phase transitions. In contrast, experiments with the XX chain showed that magic reaches its maximum precisely at the exceptional line where parity-time symmetry is broken, highlighting a distinct behavior compared to the Ising model.

Finite-size scaling analysis confirmed that these effects become more pronounced as system size increases, establishing non-stabilizerness as a sensitive marker for both quantum criticality and non-Hermitian spectral degeneracies. Researchers also analytically investigated magic in momentum space for the XX model, discovering that it reaches a minimum around exceptional points, further solidifying the link between magic and non-Hermitian behavior. Measurements confirm that magic takes extremal values at exceptional points, both in real and momentum space, positioning it as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter. The study demonstrates that the ground-state density plot of stabilizer Rényi entropy exhibits a peak along the Ising transition line and vanishes at the exceptional points, providing a clear visualization of the relationship between magic and phase transitions. These findings establish non-stabilizerness as a powerful diagnostic for identifying both quantum criticality and phase transitions.