Navigating the phase diagram of quantum many-body systems in phase space

TL;DR

This work investigates the phase diagrams of homogeneous spin $\left(\tfrac{1}{2}-\tfrac{1}{2}\right)$ and inhomogeneous spin $\left(\tfrac{1}{2}-1\right)$ Ising-Heisenberg (ATIH) chains through a phase-space lens using the Wigner function $W_{ ho}(\Omega)$ and its negativity $\mathcal{N}_{W}$. It employs Tilma et al.'s discrete Wigner construction and the equal-angle slice approximation, alongside exact solutions via decoration-iteration transformations to map the ATIH chain to an effective Ising model, enabling direct comparisons with lower bound concurrence $\tau_N(\rho)$. Key findings show that the equal-angle slice captures the homogeneous chain's phase diagram but misses several negativity-based features, while full phase-space Monte Carlo integration is essential for the inhomogeneous chain, where Wigner negativity signals entanglement in certain QFO states but not in FRU/QFO III regions. The results highlight the strengths and limitations of phase-space methods for quantum many-body systems and suggest nuanced usage of Wigner negativity as an entanglement indicator, with potential impact on quantum materials design and quantum information processing.

Abstract

We demonstrate the unique capabilities of the Wigner function, particularly in its positive and negative parts, for exploring the phase diagram of the spin$-(\frac{1}{2\!}-\!\frac{1}{2})$ and spin$-(\frac{1}{2}\!-\!1)$ Ising-Heisenberg chains. We highlight the advantages and limitations of the phase space approach in comparison with the entanglement concurrence in detecting phase boundaries. We establish that the equal angle slice approximation in the phase space is an effective method for capturing the essential features of the phase diagram, but falls short in accurately assessing the negativity of the Wigner function for the homogeneous spin$-(\frac{1}{2}\!-\!\frac{1}{2})$ Ising-Heisenberg chain. In contrast, we find for the inhomogeneous spin$-(\frac{1}{2}\!-\!1)$ chain that an integral over the entire phase space is necessary to accurately capture the phase diagram of the system. This distinction underscores the sensitivity of phase space methods to the homogeneity of the quantum system under consideration.

Figures (6)

• Figure 1: The asymmetric tetrahedral Ising-Heisenberg (ATIH) chain as outlined by the Hamiltonian, Eq. \ref{['ham']}. (a) Shows three unit cells of the lattice, where white vertices symbolize the sites connected via Ising interactions, whereas black vertices represent those interacting via Heisenberg interactions, indicated by $J$ for Ising and $J_{\alpha}$ ($\alpha = x, y, z$) for Heisenberg connections, respectively. (b) Single unit cell of the ATIH chain and its effective mapping into an Ising model. Fisher's decoration-iteration transformation (DIT), changes the Heisenberg edge within the tetrahedral structure into "decorative" elements fisher1925theory. This process essentially replaces the Heisenberg interactions with an equivalent Ising-type interaction. The modification preserves the effect of the Heisenberg spins on the system's magnetic behaviors within a simpler analytical framework. The resulting effective Ising model underlies the core physical phenomena of the original configuration into a form that is significantly more tractable for examining phase transitions and critical phenomena. Appendix \ref{['ATIH_thermo']} elaborates on the statistical and thermodynamics of the ATIH chain, achieved through the DIT method.
• Figure 2: Phase diagram for the asymmetric tetrahedral Ising-Heisenberg chain, Eq. \ref{['ham']}, for two different spin configurations. Panel (a) shows the phase diagram for the spin$-(\frac{1}{2}\!-\!\frac{1}{2})$ case, utilizing parameters from Eq. \ref{['phase_diagram_1_parameters']}. The spin alignments are represented by arrows, with up or down arrows indicating the two possible spin states at the Ising and Heisenberg nodes. Panel (b) illustrates the phase diagram for the spin$-(\frac{1}{2}\!-\!1)$ case, using parameters from Eq. \ref{['phase_diagram_2_parameters']}. Here, up and down arrows denote the Ising nodes, while a combination of up, dot, and down illustrates the spin-1 configurations on the Heisenberg edges. Magnetic phases are labeled as FM (ferromagnetic), QFO (quantum ferromagnetic), FRI (ferrimagnetic), and FRU (frustrated ferromagnetism). The FM state is a product state, while the QFO states are entangled and differ due to the distinct orientation probabilities of spins in the Heisenberg edges. The FRI state emerge from the superposition of ferromagnetic and antiferromagnetic state, causing a competition in the spin alignment. The FRU states exhibit non-classical behaviour caused by competing interactions, leading to frustration where classical ordering is absent, suggesting a higher degree of quantum correlation. Appendix \ref{['spect_ATIH']} elaborates on the model's spectrum and the specific states of each phase for both scenarios.
• Figure 3: Figures of merit in the single cell spin$-(\frac{1}{2}-\frac{1}{2})$ ATIH model, Eq. \ref{['ham']}, under the parameters defined in Eq.\ref{['phase_diagram_1_parameters']}. (a) The average Wigner function values, Eq. \ref{['wigner_arbirary_system']}, across the phase space mapping out the complete phase diagram and critical phase boundaries under the equal angle slice approximation. Distinct Wigner function values characterize different phases, varying from positive to negative as influenced by the phase space parameters $(\theta, \varphi)$, demonstrating the utility of the Wigner function in identifying phase transitions within the model. (b) The negativity of the Wigner function, Eq. \ref{['negativity_wigner']}, under the equal angle slice approximation showing its ineffectiveness in this limit in capturing properly the phase diagram \ref{['phase_diagram_1']}. We assess this properly in (c) by adopting a comprehensive mapping in the entire phase space using Monte Carlo (MC) integration, which confirms the predominance of the negative values of the Wigner function across the phase space, especially significant around the phase boundaries, except in the coherence-free FM and FRI phases. (d) Lower bound concurrence, Eq. \ref{['LBCC']}, splitting the phase diagram in three parts: unentangled region which describes the FM and FRI phases. Maximally entangled region describing the QFO III and IV phases, and an intermediate region that identify the FRU III and FRU IV phases.
• Figure 4: Figures of merit in the single cell spin$-(\frac{1}{2}-1)$ ATIH model, Eq. \ref{['ham']}, under the parameters defined in Eq.\ref{['phase_diagram_2_parameters']}. (a) Average of the Wigner function, Eq. \ref{['wigner_arbirary_system']}, under the equal slice approximation showing its limitation in revealing the phase diagram \ref{['phase_diagram_2']} for the system, which is due to the bias introduced by the approximation and the information loss in non-uniform regions of phase space. (b) The average Wigner function, Eq. \ref{['wigner_arbirary_system']}, across the entire phase space, providing a clearer description of phase boundaries, particularly between regions of positive and negative Wigner function values. (c+d) Negativity of the Wigner function, Eq. \ref{['negativity_wigner']}, calculated via Monte Carlo integration and the lower bound concurrence, Eq. \ref{['LBCC']}, respectively. Negative values cluster in phases where Heisenberg nodes align with highly entangled Bell states, such as the QFO I and QFO II states. In contrast, the FRU and QFO III phases, associated with generalized-Bell states, show no such negativity despite also being entangled. This suggests that the absence of negativity does not imply the lack of entanglement.
• Figure 5: Correlation function of the ATIH chain in the spin$-(\frac{1}{2}-\frac{1}{2})$ case