Role of a higher dimensional interaction in stabilizing charge density waves in quasi-1D NbSe$_3$ revealed by angle-resolved photoemission spectroscopy

We revisit charge density wave (CDW) behavior in the archetypal quasi-1D material NbSe$_3$ by high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements utilizing a micro-focused laser with a photon energy of 6.3 eV. We present a detailed view of the electronic structure of this complex multi-band system and unambiguously resolve CDW gaps at the Fermi level ($E_{\mathrm{F}}$). By employing a tight-binding model, we argue that these gaps are the result of inter-band coupling between electronic states that reside predominantly on distinct 1D chains within the material. This inter-band coupling is mediated by a particular electronic state that extends across multiple 1D chains, highlighting the importance of a higher-dimensional interaction in stabilizing the CDW ordering in this material. In addition, the temperature evolution of CDW gaps below $E_{\mathrm{F}}$ deviates from the behavior expected for a Peierls-like mechanism driven by nesting; the upper and lower bands of the re-normalized CDW dispersions maintain a fixed peak-to-peak distance while the gaps are gradually filled by broadening at higher temperatures. This points towards a gradual loss of long-range phase coherence as the dominant effect in reducing the CDW order parameter, which may correspond to the loss of coherence between the coupled chains. Furthermore, one of the gaps is observed above the bulk and surface CDW transition temperatures, implying the persistence of short-range incoherent CDW order. The influence of higher-dimensional interactions likely plays an important role in a range of low-dimensional systems.