Quantum Coherence of Topologically Frustrated Spin Chains

TL;DR

The paper investigates whether quantum coherence (QC) in topologically frustrated (TF) one-dimensional spin chains shares the same universal two-term structure observed for entanglement and magic, namely a local, Hamiltonian-dependent contribution plus a topological, parameter-independent contribution. Using a DMRG-driven matrix product state framework combined with Tensor Cross Interpolation, the authors compute the Relative Entropy of Coherence (REC) in the computational basis for TF Ising and ANNNI chains, comparing TF and non-TF cases with matched correlation lengths where appropriate. They show that, in the thermodynamic limit, REC decomposes into a local term and a topological term; the topological part is constant across Hamiltonian parameters and scales logarithmically with system size (e.g., $C(\rho_{\textrm{fr}}^{h\rightarrow0^+})=\log_2[L(L+2)]-\log_2(e)$ for ANNNI near the classical point), while the local term carries the extensive, $h$-dependent contribution. The ratio $R$ of the topological difference to its classical-point counterpart approaches unity in the TF phase and vanishes in the paramagnetic phase, reinforcing the universality of TF-induced structure across quantum resources. The work also confirms robustness under basis rotations and demonstrates a viable numerical route to probe large systems, suggesting avenues for formal proofs and extensions to other models and resources.

Abstract

The study of entanglement and magic properties in topologically frustrated systems suggests that, in the thermodynamic limit, these quantities decompose into two distinct contributions. One is determined by the specific nature of the model and its Hamiltonian, and another arises from topological frustration itself, resulting in being independent of the Hamiltonian's parameters. In this work, we test the generality of this picture by investigating an additional quantum resource, namely quantum coherence, in two different models where topological frustration is induced through an appropriate choice of boundary conditions. Our findings reveal a perfect analogy between the behavior of quantum coherence and that of other quantum resources, particularly magic, providing further evidence in support of the universality of this picture and the topological nature of this source of frustration.

Figures (5)

• Figure 1: (a) Quantum Coherence of the ground state of the TF Ising chain (red full markers) and of the corresponding unfrustrated FM model (blue empty markers) as a function of the magnetic field and for different values of the system size. (b) The same quantities are plotted as a function of the system length for different values of the magnetic field. (c) The ratio $R$ in eq. \ref{['eq:R_def']} for the Ising model as a function of the magnetic field and for different system sizes.
• Figure 2: (a) Quantum Coherence of the ground state of the TF ANNNI chain (red full markers) and of the corresponding unfrustrated OBCs model (blue empty markers) as a function of the magnetic field and for different values of the system size. (b) The same quantities are plotted as a function of the system length for different values of the magnetic field. (c) The ratio $R$ in eq. \ref{['eq:R_def']} for the ANNNI model as a function of the magnetic field and for different system sizes.
• Figure 3: Comparison of the different results for the REC close to the classical point of the ANNNI model. The black line stands for the analytic expression in Eq. \ref{['analytic_ANNNI']}. The blue circular points indicate the results obtained from the exact diagonalization of the ANNNI in the first-order perturbation theory ($h \rightarrow 0^+$). The red squares indicate the results of a numerical analysis that exploit the procedure outlined in Sec. \ref{['DMRG_TCI_Section']} with Hamiltonian parameters equal to $h=10^{-3}, \kappa=1.0$ and tolerance set equal to $\epsilon = 10^{-12}$
• Figure 4: (a) Angular plot of the topological contribution to REC from Eq. \ref{['DeltaCtheta']}, as a function of the rotation angle $\theta$. Shaded areas represent the range of validity for the ratio in Eq. \ref{['eq:R_def']}, that is with a finite denominator: at the classical point $h\rightarrow 0^+$(i.e. $h=0.001$) the denominator vanishes for angles between $\pi/6 \lesssim \theta \lesssim\pi/3$. (b) QC as function of $L$ for the Ising chain with $h=0.5$, $J=\pm1$ (red and blue respectively), and $\theta = \pi/8, 7\pi/16$. Inset is the corresponding ratio $R$ from Eq. \ref{['eq:R_def']} for the same angles $\theta = \pi/8,7\pi/16$. (c) QC as function of $L$ for the Ising chain with $h=0.5$ and $J=\pm1$ (purple and green respectively) for angles $\theta=\pi/5,2\pi/7$. Inset is the corresponding value of $|C_{h=0.5} - C_{h=0.001}|$ for $J=\pm1$ (purple and green curves respectively) for the same angles.
• Figure 5: Maximum bond dimension $\xi$ of the TCI MPS required to reach precision $\epsilon=10^{-6}$ for coherence $C(\rho)$ at the classical point $h\rightarrow 0^+$ for $J=\pm1$ as a function of rotation angle $\theta$ for system sizes $L$ running from 5 to 15.