Quantum Material Defies Theory with ‘giant Bubbles’ of Hidden Energy

Researchers investigating the quantum XY chain have revealed a surprising connection between its energy gaps, dynamics and fundamental excitations. Songtai Lv, Yang Liu (University of California, Riverside), and Erhai Zhao (George Mason University), alongside Haiyuan Zou et al., demonstrate a novel method utilising complex-valued temperatures and partition functions to characterise phases of matter with complex spectra. Their work, employing thermofield dynamics, identifies giant bubbles of Fisher zeros, large-scale closed lines, near the gapless limit of the XY chain, challenging established predictions for Luttinger liquid behaviour. This discovery establishes a clear link between the motion of these bubbles, external fields, and the transfer of spectral weight, offering a promising new pathway to understanding unconventional gap behaviours in strongly correlated systems.

This work introduces an approach leveraging complex-valued inverse temperature and the partition function, utilising thermofield dynamics to quantitatively characterise both quantum and thermal fluctuations.

By exploiting the relationship between low-energy excitations and Fisher zeros, the study offers a new lens through which to examine the behaviour of these systems. The quantum XY chain, subjected to an external field, served as a crucial testbed for this methodology. Investigations revealed that the oscillatory gap behaviour inherent in the XY chain manifests as oscillations within the long-time dynamics of the thermofield dynamics spectral form factor.
Crucially, the research identified the presence of giant bubbles, large-scale, closed lines, of Fisher zeros near the gapless XX limit. These bubbles present a characteristic energy scale that appears to challenge predictions derived from the low-energy theory of a featureless Luttinger liquid. The energy scale associated with these bubbles was identified and linked to the motion of these giant structures with varying external fields, demonstrating a transfer of spectral weight from higher to lower energies.

This work demonstrates that the partition function, traditionally less emphasised in theoretical approaches to strongly correlated quantum systems, can be analytically continued to complex inverse temperatures. This allows for the characterisation of quantum criticality, low-energy excitations, breakdowns of the eigenstate thermalisation hypothesis, and even topological features.

The deep connection established between Fisher zeros, dynamics, and excitations promises to unlock new avenues for understanding unconventional gap behaviours in strongly correlated many-body systems, potentially impacting fields ranging from superconductivity to materials science. The study’s findings offer a powerful new tool for probing the intricacies of quantum matter.

Thermofield dynamics and Fisher zero analysis of the XY chain

Researchers employed a complex-valued inverse temperature and partition function methodology to investigate phases of matter exhibiting nontrivial spectra and dynamics. This work leverages thermofield dynamics to quantitatively characterise thermal fluctuations and establishes a correspondence between low-energy excitations and Fisher zeros.

The XY chain, subjected to an external field, served as a primary testbed for demonstrating how oscillatory gap behaviour manifests as oscillations within the long-time dynamics of the thermofield dynamics spectral form factor. Analysis focused on identifying giant bubbles, defined as large-scale closed lines, of Fisher zeros in proximity to the gapless XX limit.

These bubbles reveal a characteristic energy scale that appears inconsistent with predictions derived from low-energy theory concerning a featureless Luttinger liquid. The study pinpointed this energy scale and correlated the movement of these giant bubbles, influenced by variations in the external field, with the transfer of spectral weight from higher to lower energies.

The research team implemented a precise protocol involving the calculation of the spectral form factor, a measure of the density of states, within the framework of thermofield dynamics. This technique allowed for the observation of subtle oscillations indicative of the gap structure and the identification of Fisher zeros, which are singularities in the partition function.

By meticulously mapping the location and behaviour of these zeros, the study revealed a previously unrecognised connection between Fisher zeros, dynamical properties, and low-energy excitations, opening new avenues for understanding unconventional gap behaviours in strongly correlated systems. This approach provides a novel means of probing quantum criticality and many-body entanglement.

Fisher zero configurations delineate quantum phase transitions in the XY model

Fisher zeros of the XY model exhibit specific configurations that reveal insights into the system’s phases. Exact Fisher zeros were derived off the complex inverse temperature axis in the thermodynamic limit, defined by the transcendental equation Z π 0 log | tanh(βεq)|2dq = 0. Examination of the case with zero transverse field, g = 0, revealed that zeros congregate to form open lines, closed loops, or bubbles, mirroring observations in the transverse field Ising model.

The vertical location, or inverse of the imaginary part of the Fisher zeros, serves as a diagnostic measure for the lowest energy gap of the system. As the parameter γ decreases, open Fisher zero lines shift upwards in the complex β plane, accompanied by a downward movement of closed zeros, consistent with gap closing.

In the limit of γ = 0, the system becomes gapless and the open zero lines vanish completely, demonstrating that the structure change of Fisher zeros can pinpoint quantum criticality. Analysis of the complex Z revealed oscillatory behavior in the long-time dynamics of the thermofield dynamics spectral form factor, particularly within the parameter space γ2 + g2 1−γ2.

Fisher zero structures reveal deviations from Luttinger liquid behaviour

Researchers have demonstrated a novel method employing complex-valued inverse temperature and partition functions to investigate the phases of matter exhibiting nontrivial spectra and dynamics. This approach utilises thermofield dynamics to quantitatively characterise thermal fluctuations and establishes a correspondence between low-energy excitations and Fisher zeros, which are points in the complex plane related to the system’s energy levels.

Applying this technique to the XY chain with an external field, the study reveals that oscillatory gap behaviour manifests as oscillations within the long-time dynamics of the thermofield dynamics spectral form factor. Furthermore, the investigation identifies giant bubbles, large-scale closed lines, of Fisher zeros near the gapless XX limit of the model.

These bubbles exhibit an energy scale that deviates from predictions based on the low-energy theory of a featureless Luttinger liquid, and their movement with varying external fields correlates with the transfer of spectral weight from high to low energies. The findings establish a strong link between Fisher zeros, dynamics, and excitations, potentially offering new insights into unconventional gap behaviours observed in strongly correlated systems.

The authors acknowledge a limitation in the current work, namely that the approach was tested on a one-dimensional model and may require further development for application to higher-dimensional systems. Future research should focus on extending this methodology to two-dimensional strongly interacting models, potentially utilising tensor network algorithms to evaluate complex-valued partition functions with high precision.

Given the connection between the observed phenomena and the pseudogap effect, characterised by a redistribution of spectral weight, this work suggests that Fisher zeros and thermofield dynamics may provide a fresh perspective for understanding the intricate many-body physics underlying unconventional gap behaviours. This complex partition function analysis offers an effective tool for analysing the thermal and dynamical properties of the one-dimensional XY model and accurately captures oscillatory finite-size gap behaviour in the thermodynamic limit.