A single nodal loop of accidental degeneracies in minimal symmetry: triclinic CaAs$_3$

TL;DR

This paper reports that triclinic CaAs₃, with minimal inversion symmetry, naturally hosts a topologically nontrivial nodal loop of accidental degeneracies, serving as a fundamental example of such phenomena in low-symmetry crystals.

Contribution

It demonstrates the existence of a topological nodal loop in CaAs₃ with minimal symmetry, without the need for tuning, highlighting a new fundamental case in topological materials.

Findings

CaAs₃ has a topological nodal loop centered on a Brillouin zone face.

The loop is very flat in energy and intersects the Fermi level four times.

Spin-orbit coupling lifts the degeneracy, resulting in trivial Kramers pairs.

Abstract

The existence of closed loops of degeneracies in crystals has been intimately connected to associated crystal symmetries, raising the question: what is the minimum symmetry required for topological character, and can one find an example? Triclinic CaAs$_3$, in space group $P{\bar 1}$ with only a center of inversion, has been found to display, without need for tuning, a nodal loop of accidental degeneracies with topological character, centered on one face of the Brillouin zone that is otherwise fully gapped. The small loop is very flat in energy, yet is cut four times by the Fermi energy, a condition that results in an intricate repeated touching of inversion related pairs of Fermi surfaces at Weyl points. Spin-orbit coupling lifts the fourfold degeneracy along the loop, leaving trivial Kramers pairs. With its single nodal loop that emerges without protection from any point group symmetry, CaAs$_3$ represents the primal "hydrogen atom" of nodal loop systems.

Equations

$$\tilde{H}(\vec{k})$$

$$f(k_a,k_b,k_c)$$

$$g(k_a,k_b,k_c)$$

Full text

Nodal loop semimetals (NLSs) represent the most delicate type of topological phase in the sense that they arise from a closed loop of accidental degeneracies in the Brillouin zone. In some ways they complement the topological character of Weyl semimetalswan2011 by displaying surface Fermi arcs or Fermi lines, or both. Several structural classes of NLSs have been identified, always associated with specific space group symmetries that enable, or in common parlance protect, the necessary degeneracies. On the other hand, the early theoretical workHerring1937 ; HerringThesis presumed only the minimum symmetry necessary to allow a nodal loop: time reversal symmetry and a center of inversion. This limiting case of “minimal symmetry” has prompted us to look for an example and understand its behavior.

When the little group at wavevector $\vec{k}$ contains only the identity, the Hamiltonian $H(\vec{k})$ has matrix elements between states with neighboring eigenvalues and anticrossings occur as some parameter of H is varied. von Neumann and Wigner first investigated the conditions under which degeneracies nevertheless occur, so-called accidental degeneracies,Neumann1929 where matrix elements vanish for no physical reason. Herring generalized their arguments to accidental degeneracies in three dimensional (3D) crystals.Herring1937 ; HerringThesis with some extension by Blount.Blount1962 Herring pointed out, for example, that a mirror plane provides a natural platform for a ring of degeneracies. If a band with even mirror symmetry is higher in energy than a band of odd symmetry at $\vec{k}_{1}$ but lower at $\vec{k}_{2}$ (both on the mirror plane), then due to the continuity of eigenvalues and differing symmetry, on any path connecting them there must be a point of degeneracy. The locus of such degeneracies maps out either a loop encircling one of the points, or an extended line from zone to zone separating the two points (which, considering periodicity, also becomes a closed loop topologically).

The topologically singular nature of such nodal loops was established by Berry.Berry1985 Allen demonstratedAllen2007 how, with minimal symmetry available, these loops of degeneracies are destroyed by spin-orbit coupling (SOC). Special symmetries can enable nodal loops in the presence of SOC, for example a screw axis in the example of Fang et al.Fang2015 Burkov et al. made the modern rediscovery of nodal loops and illustrated the type of Weyl-point connected electron and hole Fermi surface that should be expected when the band energies around the loop cross the Fermi energy.Burkov2011 Such nodal loops should be common, and indeed have been found even in high symmetry elemental metals.hirayama2016 Nodal loop semimetals based on crystal symmetries, especially mirror symmetries, have appeared in several modelsBurkov2011 ; Phillips2014 ; Kim2015 ; Fang2015 ; Heikkila2015 ; Mullen2015 and crystal structures.Pardo2010 ; Fang2012 ; Yu2015 ; Huang2015 ; Xu2015 ; Lv2015 ; Shekhar2015 ; Weng2015 ; Ahn2015 ; Sun2015 ; Yang2015 ; Neupane2016 ; Huang2016 ; Zhao2015

Before the discussion of Burkov et al.Burkov2011 , a nodal loop of a pair of coinciding Fermi rings – a nodal ring coinciding with the Fermi energy $E_{F}$ – had been discovered in calculations of a ferromagnetic compensated semimetal SrVO3 quantum confined within insulating SrTiO3.Pardo2010 Mirror symmetry was a central feature in providing compensation and the degenerate nodal loop coinciding with $E_{F}$. What is unlikely but not statistically improbable is: (1) having the nodal loop cut by $E_{F}$ while (2) the remainder of the Brillouin zone is gapped. Such loops will have real impact, and possible applications, when they are the sole bands around $E_{F}$, because they generate topological character with corresponding boundary Fermi arcs or points at zero energy.

Among his several results relating crystal symmetries to accidental degeneracies without consideration of SOC, HerringHerring1937 ; HerringThesis found that inversion symmetry ${\cal P}$ alone is sufficient to allow nodal loops of degeneracies (fourfold: two orbitals times two spins), a result extended recently.Burkov2011 ; Fang2015 ; Kim2015 Simply stated, ${\cal P}$ symmetry leads to a real Bloch Hamiltonian $H(\vec{k})$ if the center of inversion is taken as the origin. The minimal (for each spin) 2$\times$2 Hamiltonian then has the form $H(\vec{k})=f_{k}\tau_{x}+g_{k}\tau_{z}$ (neglecting spin degeneracy for the moment) with real functions $f_{k},g_{k}$; $\vec{\tau}$ represents the Pauli matrices in orbital space. Degeneracy of the eigenvalues $\varepsilon_{k}=\pm(f_{k}^{2}+g_{k}^{2})^{1/2}$ requires $f_{k}=0=g_{k}$, two conditions on the 3D vector $\vec{k}$, giving implicitly (say) $k_{y}={\cal K}(k_{x},k_{z})$ for some function ${\cal K}$. This condition either has no solution, or else corresponds to a loop ${\cal L}$ of degeneracies. Allen has given a constructive prescriptionAllen2007 for following the nodal loop once a degeneracy is located.

Any such loop will not lie at a single energy,Herring1937 ; Allen2007 ; Burkov2011 and as mentioned only acquires impact when the dispersion around the loop crosses $E_{F}$, with a gap elsewhere. This intersection results in a pair (or an even number) of points where, in the absence of spin-orbit coupling, the valence and conduction band Fermi surfaces touch. The dispersion at the Fermi contact points will, barring accidents of zero probability, be massless in all three directionsHerringThesis – Weyl points. Thus at this level (before SOC) the nodal loop semimetal is a special subclass of 3D Weyl semimetal.

Topics that have not been addressed are: how little symmetry is necessary for topological character to be retained, what are the consequences, and can an example with minimum symmetry be found? The line of reasoning above applied to the case of no inversion center (i.e. no crystal symmetry at all) dictates that all of the coefficients of $\sigma_{x},\sigma_{y},\sigma_{z}$ in $H(\vec{k})$ vanish. Accidental point degeneracies are thus possible by tuning, while a line of degeneracies occurs with zero probability.

Discovery and study of topological nodal line semimetals protected by crystal symmetry is developing rapidly.hirayama2016 ; Yu2015 ; Weng2015 ; Ahn2015 ; Yang2015 The class $TPn$ ($T$=Nb, Ta; $Pn$=P, As) lacks an inversion center but contains several crystalline symmetries facilitating nodal loops.Huang2015 ; Xu2015 ; Lv2015 ; Shekhar2015 ; Weng2015 ; Ahn2015 ; Sun2015 ; Yang2015 The cubic antiperovskite Cu3PdN contains nodal loops in a background of metallic bands,Kim2015 ; Yu2015 the BaTaSe4 family has nodal loops in its band structure enabled by symmetry, and as mentioned cubic elemental metals contain loops within their metallic bands.hirayama2016 Here we show that triclinic CaAs3 is an example of a minimal symmetry nodal loop semimetal with a single loop of degeneracies, providing the “hydrogen atom” of the class of nodal semimetals..

CaAs3 and three isovalent tri-arsenides (Ca$\rightarrow$Sr, Ba, Eu) were synthesized more than thirty years ago, with their structure, transport, and optical properties studied by Bauhofer and collaborators.bauhofer1981 ; Oles1981 CaAs3 is the sole triclinic member of this family, with space group $P\bar{1}$ (#2) containing only an inversion center, lying midway between Ca sites.bauhofer1981 Heavily twinned samples of CaAs3 has been reported as insulating in transport measurementsbauhofer1981 but curiously displayOles1981 in far infrared reflectivity a Drude weight corresponding to 1017-1018 carriers per cm3.

The sole symmetry condition in $P\bar{1}$ symmetry on the energy bands is $\varepsilon_{-k}=\varepsilon_{k}$. This simplicity indicates that “symmetry lines” are simply convenient lines with a trivial little group. $P{\bar{1}}$ symmetry does however provide eight inversion symmetry invariant momenta (ISIM) $m\frac{a^{*}}{2}+n\frac{b^{*}}{2}+p\frac{c^{*}}{2},m,n,p=0,1,$ in terms of the primitive reciprocal lattice vectors $a^{*},b^{*},c^{*}$. At these ISIMs, which are the analog of (and equivalent to) the time reversal invariant momenta (TRIMs) important in topological insulator theory,Teo2008 eigenstates have even or odd parity. Isolated nodal loops either (a) must be centered at an ISIM, or (b) they occur in inversion related pairs. Due to the low symmetry, finding unusual characteristics (viz. the occurrence of and center of a nodal loop) necessitates meticulously searching in band inversion regions.

The linearized augmented plane wave method as implemented in WIEN2kwien2k was applied with the generalized gradient approximation (GGA) exchange-correlation potential.GGA $R_{m}K_{max}$=7 is a sufficient cutoff for the basis function expansion in this $sp$ electron material. Studies have shown that GGA may underestimate relative positioning of valence and conduction bands in semiconductors and semimetals, and that the modified Becke-Johnson (mBJ) potential provides a reasonably accurate correction.zhang Thus we rely on the GGA+mBJ combination throughout. The impact of the As SOC is assessed.

The CaAs3 band structure and density of states (DOS) in directions along reciprocal lattice vectors and in the energy range from -2 eV to 2 eV, shown in Fig. 2, suggests small-gap insulating character. Valence and conduction bands are separated in energy except for an evident band inversion at the $Y\equiv\vec{b}^{*}/2$ zone boundary ISIM point. Note that with non-ISIM points having a trivial little group, bands do not cross except at accidental degeneracies, and these will coincide with any given line with zero probability.Allen2007 The combination of ${\cal P}$ symmetry and periodicity is enough to ensure that band energies at $\vec{b}^{*}/2\pm(0,\delta k_{y},0)$ are equal, thus (relative) band extrema occur at the ISIMs, and can be observed at $X,Y,$ $Z$, and $\Gamma$ in Fig. 2.

Searching the band inversion region, a loop ${\cal L}$ of accidental degeneracies centered at $Y$ was mapped out, i.e. there is no gap. Its position in the BZ is shown in Fig. 3 together with two perspective views of the Fermi surfaces (FSs). The loop, resembling a nearly planar lariat, is cut by $E_{F}$ at not two but four points, each point being a touching point for a hole and electron FS (guaranteed by the nodal degeneracies). At this level (no SOC) the spectrum is that of a semimetal with FSs touching at four Weyl points. The loop energy lies in the -20 meV to +20 meV range, making it a very flat nodal loop in the energy domain as well as in momentum space.

The surface Fermi arcs of a few 3D Weyl semimetals are now well studied.wan2011 The analogous states in NLSs were discussed originally by Burkov et al.Burkov2011 Projected onto a surface, ${\cal L}$ will enclose an area (which we call a “patch”) within which topologically-required surface states (“drumhead states”) reside. For CaAs3, ${\cal L}$ projected along $a^{*}$ leaves a roughly circular patch, and along $c^{*}$ roughly elliptical, consistent with what can be surmised from Fig. 3. The $b^{*}$ axis however lies nearly within the plane of the loop, projecting to a very slender patch. A plot along a $\vec{k}$-line crossing the patch will reveal a surface band starting at the edge of this patch and ending when the k-line leaves the patch. Considering the constant energy contours (potential Fermi lines) in the patch, they may be closed lines or isolated arcs that terminate at the boundary of the patch.

Surface band plots along special directions $\bar{Y}-\bar{\Gamma}-\bar{X}$ are shown for the (001) surface in Fig. 4. As mentioned, the Fermi energy cuts the nodal loop, hence it intersects the surface patch band resulting in one or more Fermi lines on each surface. The surface band disperses along these lines shown by 70 meV. We have confirmed other studiesZhao2015 ; Pi2016 that indicate that surface bands obtained from Wannierization followed by truncation to obtain a surface can be sensitive to numerical procedures and the chosen surface termination, so these bands are not a definitive prediction of the physical surface states. Moreover, non-topological surface bands such as from dangling bonds may appear as well.

Effect of spin-orbit coupling. Allen demonstrated in generality the effect of SOC on the nodal loop, using a two band model in the low energy regime.Allen2007 Without any symmetry operation to cause the SOC matrix element to vanish, which is the case in CaAs3, the degeneracy is opened to a $k$-dependent gap $\xi_{k}$ along the entire loop, which retains an inactive Kramers degeneracy. For integration around a circuit surrounding the loop ${\cal L}$, the topological phase $\pm\pi$ is replaced by a non-topological Berry phase that is dependent on the radius of the circuit. A magnetic field coupled to spins splits the Kramers degeneracies everywhere, giving four distinct bands near ${\cal L}$. Allen’s paper should be consulted for specific dependencies on the materials parameters.

The SOC splitting of the atomic As $4p$ level is 270 meV. Since the bands that are inverted at $Y$ are primarily As $4p$ character, the SOC-driven band shifts will be some appreciable fraction of this value. Given the 40 meV span in energy of the nodal loop, large enough SOC can open a gap. The bulk band projection, visible in Fig. 4, is altered little by SOC. Within the accuracy of the Wannier interpolation and surface projection, the result is characteristic of separated valence and conduction bands that however leave little or no gap.

Fig.  4 reveals that the surface spectrum evolves considerably under SOC. Most evidently, the dispersion of the valence (occupied) surface band has decreased from 70 meV to only 10 meV. If SOC coupling is large enough compared to the dispersion around ${\cal L}$, the system will be gapped by SOC, and CaAs3 seems on the borderline of this situation. If a gap opens, it may provide a distinct topological character, signaled by the usual $\nu_{0}(\nu_{1}\nu_{2}\nu_{3})$ indices. We calculate that CaAs3, with SOC taken into account, is a topological phase with indices $\nu_{0}(\nu_{1}\nu_{2}\nu_{3})$=1(010) using the criteria of Fu and Kane.

The spectrum in the right hand panel of Fig. 4 indicates the surface bands that will be topological insulator boundary states if SOC is large enough to give a gap. Otherwise they are topological surface states of a semimetal arising from indirect overlap, that is, a topological semimetal neither Weyl nor nodal loop. Our results in Fig. 4 indicate that CaAs3 is extremely close to the topological semimetal - topological insulator transition.

Topological nodal loop from an effective Hamiltonian. The band structure near EF of CaAs3, with the highest valence band inverted across the lowest conduction band at $Y,$ was fit to a tight-binding model. Away from $Y$ CaAs3 is gapped, making this compound ideal for observing a topological nodal line. For simplicity one can imagine the crystal deformed by an affine transformation to have orthogonal axes with $a$=$b$=$c$=$1$. We consider the following two orbital, non-inversion symmetric Hamiltonian which reproduces the essential features of the inverted band region of CaAs3. It includes nearest neighbor hopping between like orbitals $\{t_{\alpha},\alpha=1-3\}$, and between unlike orbitals $\{t_{\alpha},\alpha=4-6\}$ having differing parity:

[TABLE]

where {$\tau_{j}$},{$\sigma_{j}$} are the 2$\times 2$ matrices in orbital and spin space respectively, and $\xi$ is the SOC parameter. This Hamiltonian describes two particle-hole symmetric bands $\pm|g_{k}|$ with centers separated by $2|m|$, coupled by $f_{k}$, and including intra-orbital SOC, with eigenenergies $\varepsilon_{k,\pm}=\pm\sqrt{g^{2}_{\vec{k}}+f^{2}_{\vec{k}}+\xi^{2}}$. Evidently SOC ($\xi\neq 0$) splits the degeneracy everywhere.Allen2007 ; Fang2015 To mimic CaAs3 we consider the site energy $m$ and hopping parameters (in eV) $m=1.64$, $t_{1}=0.37$, $t_{2}=-0.95$, $t_{3}=0.37$, $t_{4}=-0.18$, $t_{5}=0.12$, $t_{6}=0.38$. Without SOC ($\xi=0$), degeneracy $f_{k}=0=g_{k}$ is realized around a nodal loop flat in energy (at zero energy). The loop, centered at $Y$ but otherwise depending on parameters, and shown in the left panel of Fig. 5, resembles the nodal loop of CaAs3 pictured in Fig. 3.

The evolution of the loop topology can be followed by varying the band separation $2m$. Two types of lines of accidental degeneracies may emerge from the Hamiltonian: a closed nodal loop as in CaAs3, or a line extending from zone to zone, which by zone periodicity become closed lines on the 3D-torus, the difference from the former being that they must occur in pairs. In Fig.  5, the two types of loops are plotted in the first Brillouin zone. On the left, where $m$=1.44, a single loop is centered at $Y$.

Varying $m$ tunes the system through an evolution from an odd number (one) to an even number (four) of nodal loops. The right panel in Fig. 5 ($m$=0) has two pairs of inversion symmetric nodal loops threading through extended Brillouin zones. Because the loop is flat in energy (at zero), adding SOC immediately opens a global gap of $2\xi$. We find the resulting state to have indices 0(000), i.e. a trivial insulator, A different, lower symmetry model would be necessary to produce a topological insulating state such as occurs in 1(010) CaAs3.

In this work we have studied the electronic and topological properties of triclinic CaAs3, which is distinguished by possessing the lowest possible symmetry for a nodal loop semimetal. In the absence of spin-orbit coupling, CaAs3 has a single nodal loop (others have loops occurring in pairs) that is cut by the Fermi level four times. Spin-orbit coupling leads not only to lifting of the nodal loop degeneracies and separation of valence and conduction bands complexes. An effective Hamiltonian demonstrates that a variety of types and numbers of nodal loops will emerge as parameters are varied. This model provides guidance for engineering topological transitions in CaAs3 and related materials by applying external tensile or compressive strains, or by alloying with isovalent atoms on either site.

We have benefited from comments on the manuscript from A. Essin, from discussions of topological aspects with K. Koepernik and J. Kǔnes, and from T. Siegrist and A. P. Ramirez on CaAs3 samples and transport data. The calculations used high performance clusters at the National Supercomputer Center in Guangzhou, and resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.. W.E.P. was supported by the NSF grant DMR-1534719 under the program Designing Materials to Revolutionize and Engineer our Future. Z. P. Y. and Y. Q. were supported by the National Natural Science Foundation of China (Grant No. 11674030) and the National Key Research and Development Program of China (contract No. 2016YFA0302300) .

I CaAs3

The boundary spectra of nodal loop semimetals is has intriguing aspects. Without spin-orbit coupling (SOC) the projection of the nodal loop of accidental degeneracies onto the surface provides a “patch” (an area) in which surface states reside, as opposed to topological insulators where only isolated surface bands appear.

Figure 6 provides three views of the nodal loop (spin0orbit coupling neglected) and surrounding Fermi surface of CaAs3 that indicates what the projected patches for topological surface states are like. Already the loop is small (see main text), and the $b^{*}$ projection extremely narrow, giving a patch with tiny area. Figure XX in the main text indicates how spin-orbit coupling affects the surface spectrum along two special lines, leaving topological surface state and a topological insulator or topological semimetal depending on whether a gap is fully opened, or (indirect band overlap remains.‘

II The two-band model

In Fig. 7 the spectrum projected on the surface perpendicular to lattice vector $b$ is shown for the two-band model in the main text, for two values of the “mass” $m$ (the separation of the two bands). Due to the high (cubic) symmetry of the model, the nodal loop for the surface band along the symmetry lines is flat. and the particle-hole symmetry (evident in the orange projected bulk band structure) puts the Fermi level at the points of degeneracy along these lines.

The insets indicate the surface patches defined by the projection of the nodal loop onto the surface. For $m$=0with the single small loop (see Fig. 5 of the main text) the patch is a small ellipse around $\bar{Y}$. The $m$=1.64 case is instructive, as it contains nodal loops that are only closed by the periodicity of the zone. Unlike the single loop for $m$=0, these nodal lines appear in inversion symmetry dictated pairs. One pair is crossing roughly the middle of the zone, whereas the other pair wiggles along the top.bottom of the zone. The projections gives two pairs of lines, with a chirality that results in the patch of drumhead states choosing to be on one particular side of edge of the patch. Again, the symmetry of the model leads to little if any variation in the intensity of the surface drumhead states across the patches.