Topological invariant for holographic Weyl-Nodal line coexisting semimetal

TL;DR

This work computes a comprehensive set of topological invariants for a strongly coupled Weyl–nodal-line coexisting semimetal using holographic (AdS/CFT) methods, introducing and evaluating mirror-protected invariants $\zeta_0$ and $\widetilde{\zeta}_{2}$ alongside the Weyl charge and nodal-line invariants $\zeta_1$ and $\zeta_2$. By combining weakly coupled field theory with a dual holographic model, the authors show that the nine-phase structure controlled by dimensionless ratios $M_1/b$ and $M_2/c$ persists in the strongly coupled regime, with band-crossing ordering interchange and multi-Fermi-surface features appearing in both realizations. In the Weyl–Nodal coexisting phase, they find Weyl charges $\pm1$ and nodal-line invariants $\zeta_1=1$, $\zeta_2=0$, with mirror-protected invariants $\zeta_0=1$ and $\widetilde{\zeta}_{2}=0$, while critical phases exhibit distinct behavior: Weyl critical points carry zero charge, nodal lines may persist with $\zeta_1=0$, and the Weyl–Critical coexisting phase has Weyl charges $\pm1$ but vanishing $\zeta_1$ and $\zeta_2$. These results validate the holographic framework for characterizing topological phases in strongly correlated semimetals and point to rich, experimentally relevant phenomena such as band-structure reconfigurations and multiple Fermi surfaces at criticality.

Abstract

The presence of a topological phase in a topological many-body system can be distinguished through the analysis of topological invariants. In the present study, the topological invariants for the strongly coupled holographic semimetals have been systematically computed, especially focusing on the holographic Weyl-Nodal line coexisting semimetal. The topological invariants that we calculate include the Weyl charge, the topological charges for a nodal ring $ζ_0$, $ζ_1$, $ζ_2$ and an additional mirror symmetry protected topological invariant, $\widetildeζ_{2}$, that we herein introduce. In addition, the effective band structures and topological invariants in the critical phases of holographic semimetals are investigated, including the case of Weyl, nodal line and Weyl-Nodal line coexisting semimetals. The findings indicate the presence of notable and unique features inherent to strongly coupled topological semimetals, including band-crossing ordering interchange and multi Fermi surfaces, which provide a valuable platform for experimental investigations of strongly coupled semimetals in condensed matter physics.

Figures (12)

• Figure 1: Left: the phase diagram for the weak coupling Weyl-nodal line coexisting model \ref{['WeylNodalCoexistLagrangian']}. Right: the phase diagram for the holographic Weyl-nodal line coexisting model \ref{['CoexistHolographicModel']}. $\hat{M_1}=\frac{M_2}{c}$, $\hat{M_2}=\frac{M_1}{b}$ and $c/b=1$.The red points represent the Critical-Critical phase, where both the Weyl nodes and the nodal line collapse into a single critical point. The cyan dashed line indicates the Weyl-Critical phase, characterized by the vanishing radius of the nodal ring while a pair of Weyl nodes persists. The blue dashed line corresponds to the Critical-Nodal phase, in which the Weyl nodes annihilate into a critical Dirac node, yet the nodal ring remains intact. Finally, the purple dotted lines describe the Critical-Gap (or Gap-Critical) phase, where either the Weyl nodes merge into a critical Dirac point, leaving the nodal line gapped, or the nodal ring collapses to zero radius, while the Weyl nodes become gappedHolographicWeylNodalChu_2024.
• Figure 2: The Berry curvature distributions for the pure Weyl semimetal (left panel) and the nodal-line semimetal (right panel) are shown in the $k_y=0$ plane. In the left panel, a distinct source and sink structure is evident, enabling the evaluation of a non-zero Weyl charge through surface integration of the Berry curvature around the source point. In contrast, the right panel exhibits a trivial Berry curvature distribution, confirming that the Weyl charge must vanish in the nodal-line semimetal.
• Figure 3: Manifolds of different dimensions $(\mathbb{S}^0,\mathbb{S}^0\times\mathbb{S}^1,\mathbb{S}^1,\mathbb{T}^2)$ that enclosed the nodal ring (i) a loop $(\mathbb{S}^1)$ linked to the nodal ring to calculate $\zeta_1$. (ii) a torus $(\mathbb{T}^2)$ surrounding the entire nodal ring to calculate $\zeta_2$. (iii) two points $(\mathbb{S}^0)$ inside and outside the nodal ring pinned to the mirror plane to calculate $\zeta_0$. (iv) two closed ring $(\mathbb{S}^0\times\mathbb{S}^1)$ inside and out side the nodal ring embedding in the mirror plane to calculate $\widetilde{\zeta}_2$.
• Figure 4: (i)When $\zeta_1=0$ the nodal ring is not stable and will be gapped under small perturbations, in this situation the $\zeta_2$ invariant must be $0$. (ii) When $\zeta_1=1, \zeta_2=0$ the nodal ring can be gapped by modifying the parameters to shrink the nodal ring to a critical point. (iii) When $\zeta_1=1, \zeta_2=1$ the nodal ring will not be gapped by shirnking the nodal ring to a critical point but re-expand to a nodal ring.
• Figure 5: (i): The density plot of $\det H$ where $H$ is the Hamiltonian for the weak coupling Weyl-nodal line coexisting semimetal \ref{['WeylNodalCoexistHamiltonian']}: a nodal line in the $k_x-k_y$ plane and a pair of Weyl nodes along the $k_z$ axis. (ii): Integrating the Berry curvature on closed surfaces $\Sigma_1,\Sigma_2$ to get the Weyl charges, which are $\pm 1$ for this weakly coupled field theoretic model. Integrating the Berry connection on the closed curve $C$ to get $\zeta_1$, which is $1\mod 2$, and calculate the winding number for Wannier center on the closed surface $\Sigma_3$ to get $\zeta_2$, which is $0\mod 2$. (iii): Comparing the eigenvalue of mirror operator on the point $p_{in}$ and $p_{out}$ to get $\zeta_0$, which is $1$, and comparing the average phases for the Wilson loop along the loop $C_{in}$ and the loop $C_{out}$ to get $\widetilde{\zeta}_2$, which is $0 \mod 2$.