Ring frustration and factorizable correlation functions of critical spin rings

Abstract: Basing on the exactly solvable prototypical model, the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by defining the nonlocal factors embedded in its factorizable correlation functions. In the context of nonlocality, the valuable traditional finite-size scaling analysis is reappraised. The factorizable correlation functions of the isotropic $XY$ and the spin-1/2 Heisenberg models are also demonstrated with the emphasis on the effect of ring frustration.

• The paper provides an exact framework to extract nonlocal factors from correlation functions in critical spin rings affected by geometric frustration.
• It employs exact diagonalization and the Jordan-Wigner transformation on models like the transverse Ising, isotropic XY, and spin-1/2 Heisenberg rings.
• The analysis demonstrates that finite-size scaling can approximate nonlocal contributions, offering insights for experimental designs in quantum systems.

Ring Frustration and Factorizable Correlation Functions of Critical Spin Rings

Introduction

The study presented in "Ring frustration and factorizable correlation functions of critical spin rings" (1810.07463) focuses on the examination of quantum spin systems within a ring configuration, highlighting the concept of nonlocality arising from ring frustration. Geometric frustration in spin systems is a well-explored phenomenon, often leading to exotic ground states. This paper contributes to the existing body of knowledge by proposing a framework for identifying and quantifying the nonlocal contributions embedded in the correlation functions of critical spin rings. The authors utilize exactly solvable models, such as the transverse Ising, isotropic $XY$, and spin-1/2 Heisenberg rings to elucidate the impact of ring frustration on these systems.

Nonlocality and Correlation Functions

The traditional approach to analyzing spin chains often involves the assumption of the thermodynamic limit, where the size of the system $N \rightarrow \infty$ is taken at the early stages of calculation. This common practice simplifies the mathematical treatment but can obscure the inherently nonlocal effects present in these systems. In contrast, the authors suggest deferring the application of the thermodynamic limit to the final stages of analysis. By doing so, it becomes possible to isolate and investigate nonlocality within the correlation functions.

The study defines three regimes based on the relation between spin separation $r$ and system size $N$: local ($r \approx 1$), near local ($r \gg 1$ and $\alpha = 0$), and nonlocal ($\alpha \neq 0$ where $\alpha = \lim_{N \rightarrow \infty} \frac{r}{N}$). Nonlocal factors are defined for systems with odd and even numbers of spins, capturing the effect of ring frustration in altering the correlation function in spin rings significantly.

Methodology and Prototype Models

The paper utilizes the transverse Ising ring at its phase transition point as a prototype model to establish a framework for extracting nonlocal factors, employing exact diagonalization techniques. The authors work through the Jordan-Wigner transformation to express the two-point correlation functions as Toeplitz determinants, isolating the effects of nonlocality when the thermodynamic limit is applied at different stages. This approach is comprehensive and highlights the differences in correlation functions due to ring topology in infinite spin systems.

Further analysis is extended to the isotropic $XY$ and spin-1/2 Heisenberg rings, where numerical results support the hypothesis that finite-size scaling (FSS) can approximate nonlocal factors, thus offering an alternative viewpoint of FSS analysis as a method for probing nonlocal properties in many-body systems.

Implications and Future Research

The implications of this research are multifold. From a theoretical perspective, the introduction of nonlocal factors redefines our understanding of frustration-induced quantum states, presenting a new lens through which many-body interactions and criticality can be examined. Practically, these findings could inform experimental designs in quantum information systems where maintaining coherence over large distances in a constrained geometry is crucial.

Looking forward, speculation on future developments includes the potential application of this framework to more complex spin systems and higher-dimensional models, where frustration and nonlocality might play a more pronounced role. The methodologies set forth in this work could be instrumental in uncovering new quantum phases and transitions sensitive to the topology and symmetry in quantum simulators.

Conclusion

The paper presents a novel framework for understanding nonlocality in critical spin rings affected by ring frustration. Through an exact and detailed investigation of prototype models, the work underscores the significance of geometric constraints in spin systems, thus enriching the theoretical landscape of quantum many-body physics and offering direction for future studies in the interplay between frustration, topology, and nonlocality.