#  Quantum oscillations with topological phases in a kagome metal CsTi3Bi5

Yongkang Li  Hengxin Tan  Binghai Yan<EMAIL_ADDRESS>Department
of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001,
Israel

###### Abstract

Quantum oscillations can reveal Fermi surfaces and their topology in solids
and provide a powerful tool for understanding transport and electronic
properties. It is well established that the oscillation frequency maps the
Fermi surface area by Onsager’s relation. However, the topological phase
accumulated along the quantum orbit remains difficult to estimate in
calculations, because it includes multiple contributions from the Berry phase,
orbital and spin moments, and also becomes gauge-sensitive for degenerate
states. In this work, we develop a gauge-independent Wilson loop scheme to
evaluate all topological phase contributions and apply it to CsTi3Bi5, an
emerging kagome metal. We find that the spin-orbit coupling dramatically
alters the topological phase compared to the spinless case. Especially,
oscillation phases of representative quantum orbits demonstrate a strong 3D
signature despite their cylinder-like Fermi surface geometry. Our work reveals
the Fermi surface topology of CsTi3Bi5 and paves the way for the theoretical
investigation of quantum oscillations in realistic materials.

##  I Introduction

Kagome lattice, a 2D corner-sharing triangle lattice, has attracted much
interest due to its geometric frustration and non-trivial band geometry. Among
various materials containing such 2D lattice structure, Kagome material family
$A$V3Sb5 ($A$ = K, Rb, Cs)Ortiz  _et al._ (2019) receives special attention
since it exhibits many exotic quantum phenomena including $\mathbb{Z}_{2}$
topology and flat bandsOrtiz  _et al._ (2020, 2021); Hu  _et al._ (2022a),
possible unconventional superconductivityOrtiz  _et al._ (2020, 2021); Yin
_et al._ (2021); Chen  _et al._ (2021a, b); Tan  _et al._ (2021); Xu  _et al._
(2021) and density wave order Chen  _et al._ (2021b); Ortiz  _et al._ (2020,
2021); Yin  _et al._ (2021); Chen  _et al._ (2021a); Jiang  _et al._ (2021);
Liang  _et al._ (2021); Zhao  _et al._ (2021); Tan  _et al._ (2021); Park  _et
al._ (2021); Hu  _et al._ (2022a); Nie  _et al._ (2022); Li  _et al._ (2023).
However, because of the interplay and competition between different correlated
states, the origin of these physical properties and their relation to the
unique electronic structure remains elusive.

Recently, a new Ti-based Kagome material $A$Ti3Bi5 ($A$ = K, Rb, Cs)
isostructural to $A$V3Sb5 is synthesizedWerhahn  _et al._ (2022) and
investigatedWerhahn  _et al._ (2022); Li  _et al._ (2022); Yang  _et al._
(2022a); Hu  _et al._ (2022b); Yang  _et al._ (2022b); Liu  _et al._ (2022);
Jiang  _et al._ (2022); Zhou  _et al._ (2023); Dong  _et al._ (2023). Unlike
V-based $A$V3Sb5 family, the charge density wave (CDW) order is absent in
$A$Ti3Bi5 family as shown in transport and scanning tunneling microscopy (STM)
experimentsWerhahn  _et al._ (2022); Zhou  _et al._ (2023); Li  _et al._
(2022); Yang  _et al._ (2022c); Hu  _et al._ (2022b); Jiang  _et al._ (2022).
First-principles calculation also shows the absence of lattice structural
instabilityLiu  _et al._ (2022). Hence, $A$Ti3Bi5 could serve as a
complementary system to $A$V3Sb5, in which the origin of these exotic
phenomena and their relation to electronic properties can be investigated
without reference to lattice’s effect. For example, the observed two-fold
rotational symmetry and orbital selectivity in the electronic structure of
$A$Ti3Bi5 Li  _et al._ (2022); Yang  _et al._ (2022c); Hu  _et al._ (2022b);
Jiang  _et al._ (2022) may form a pure electronic nematic phase, similar to
that in Fe-based high-temperature superconductorsChuang  _et al._ (2010).
Understanding the band structure and Fermi surface of $A$Ti3Bi5 is crucial for
further investigating these correlating properties.

Quantum oscillation measurement is one way to measure the Fermi surface
topology as well as its associated properties like cyclotron mass and carrier
mobilityShoenberg (1984). More importantly, the phase of the fundamental
oscillation is related to the band topology. Usually, a $\pi$ phase shift in
the oscillation is regarded as $\pi$ Berry phase which indicates a topological
band structureMikitik and Sharlai (1999); Luk’yanchuk and Kopelevich (2004);
Zhang  _et al._ (2005). The quantum oscillation analysis from this perspective
has been carried out in $A$V3Sb5Shrestha  _et al._ (2022); Fu  _et al._
(2021); Broyles  _et al._ (2022); Chapai  _et al._ (2023) and also recently in
ATi3Bi5Werhahn  _et al._ (2022); Dong  _et al._ (2023), which claims
nontrivial band topology due to this $\pi$ Berry phase.

The topological phase actually has other contributions entangled with the
Berry phaseAlexandradinata  _et al._ (2018); Alexandradinata and Glazman
(2018). Especially in the degenerate case with strong spin-orbit coupling
(SOC), such $\pi$ phase may mainly come from orbital or spin magnetic moment
other than the Berry phase, as revealed recently in CsV3Sb5 Tan  _et al._
(2023). Hence, the analysis of the topological properties based on the phase
shift in quantum oscillation should consider all contributions. Apart from the
experiment, this phase can be independently evaluated from ab-initio band
structures. However, such calculation has to deal with the gauge fixing
problem in the presence of degeneracy which is common for centrosymmetric
nonmagnetic materials. A numerical study for all phase contributions without
gauge ambiguities has not been explored in detail before.

In this work, we develop a Wilson loop method to determine the quantum
oscillation phase and apply it to CsTi3Bi5. We first detail the method which
has explicit gauge independence and can be implemented conveniently in the
case of degenerate bands. Then combining this method with first-principles
calculation we resolve the Fermi surface of CsTi3Bi5 and determine the total
oscillation phase for all quantum orbits. Its relation to the Fermi surface
geometry and band topology is clarified at last. The 3D nature of several
representative quantum orbits present is imprinted in the topological phase,
although related Fermi surfaces show a cylinder-like shape. Our work provides
a useful theoretical tool to investigate the Fermi surfaces and topological
electronic properties in materials.

##  II Overview on the quantum oscillation phase

In the presence of a strong magnetic field, the physical quantities (e.g.,
resistance and magnetization) show oscillation with respect to a magnetic
field ($B$) due to the formation of quantized Landau levels (LLs). Under the
semiclassical limit in which the scale of $k$-space orbit is much larger than
the inverse of magnetic length $l_{B}^{-1}$ ($l_{B}=\sqrt{\hbar/eB}$), the
oscillation is periodic with respect to $1/B$ and can be expanded as a sum of
Fourier series in general:

$\displaystyle\delta
A=\sum_{i}\sum_{r}A_{i,r}\mathrm{cos}\left[r(l_{B}^{2}S_{F,i}+\theta_{i}+\phi_{M,i})+\delta_{i}+\varphi_{A}\right].$
(1)

Here, $A$ is the physical quantity being measured which is usually
magnetization $M$ or longitudinal resistivity $\rho_{xx}$, $\delta A$ is the
oscillation part and $A_{i,r}$ is the oscillation amplitude for the $r$-th
harmonic of the $i$-th extremal orbit. $S_{F,i}$ is the momentum space area of
the $i$-th extremal orbit on Fermi surface and determines the $i$-th
oscillation frequency. Here the total oscillation phase is decomposed into
four parts: $\theta_{i}$ is the first-order correction to the dynamical phase
including the geometry phase and (orbital and spin) magnetic moment phase.
$\phi_{M,i}$ is Maslov correction which equals to $\pi$ for a simple closed
orbit. $\delta_{i}$ is dimension related phase resulting from the integration
over $k_{z}$ if a 3D solid is measured (suppose $B$ is along z direction). The
last term $\varphi_{A}$ is measured quantity ($A$) related phase (see the
following discussion). All phases except $\varphi_{A}$ depend only on the
Fermi surface properties and are universal for any oscillatory quantity. Below
we show that each phase can be determined from first-principles calculations
to understand experiments. We note that a comprehensive theory on quantum
oscillations was established in Refs. Alexandradinata  _et al._ (2018);
Alexandradinata and Glazman (2018). We first overview this theory and then
introduce the Wilson loop method to compute the topological phase.

###  II.1 Phase $\theta$

The first two phases $\theta$ and $\phi_{M}$ (below we focus on a single orbit
and ignore the subscript $i$) are related to LLs. In general, there are no
simple rules to determine the exact LL for arbitrary band structure. However,
in the semiclassical limit, approximate LL can be determined from Bohr-
Sommerfeld-like quantization rules. For a group of $D$-fold degenerate bands,
the $j$-th LLs can be obtained up to leading order in $l_{B}^{-1}$ as,

$l_{B}^{2}S(E_{a,j})+\lambda_{a}+\phi_{M}=2\pi j+O(l_{B}^{-2/3}).$ (2)

$a\in\mathbb{Z}_{D}:=\\{1,\ldots,D\\}$ is the band index among $D$ degenerate
bands and $\lambda_{a}$ is a phase that we are interested. $\lambda_{a}$ is
equivalent to $\theta$ if there is no degeneracy,i.e., $D=1$. $\phi_{M}$ is
Maslov correction and can be determined from the topology of the orbit, which
equals $\pi$ for a simple closed orbit.

Because of degeneracy, $D$ LLs create $D$ oscillation terms with the same
frequency $F=\hbar S_{F}/2\pi e$ by Onsager’s relation but different phase
shift $\lambda_{a}$. It amounts to a single oscillation term with reduced
amplitude $C$ and effective phase shift $\theta$,

$\sum^{D}_{a=1}\mathrm{cos}\left[r(l_{B}^{2}S_{F}+\lambda_{a}+\phi_{M})\right]=C\mathrm{cos}\left[r(l_{B}^{2}S_{F}+\theta+\phi_{M})\right].$
(3)

For example, all bands are doubly degenerate ($D=2$) in the presence of
combined inversion and time reversal ($\mathcal{PT}$) symmetries, which is the
case of kagome metals CsV3Sb5 and CsTi3Bi5. We regulate $\lambda_{1,2}$ in the
range of [$-\pi,\pi$] and then $\mathcal{PT}$ symmetry leads to
$\lambda_{1}=-\lambda_{2}$. Hence, summing two cosine functions in Eq.(3)
leads to

$\displaystyle\theta$ $\displaystyle=\begin{cases}0,&\text{if
}|\lambda_{1}|<\pi/2\\\ \pi,&\text{if }|\lambda_{1}|>\pi/2\\\ \end{cases}$ (4)
$\displaystyle C$ $\displaystyle=|\mathrm{cos}(\lambda_{1})|$ ,

One can find $\theta$ is a quantized topological invariant (0 or $\pi$)
Alexandradinata  _et al._ (2018) insensitive to orbit details.

In general, phase $\lambda_{a}$ can be determined from the spectrum
$\\{e^{i\lambda_{a}}\\}_{a=1}^{D}$ of propagatorAlexandradinata  _et al._
(2018); Alexandradinata and Glazman (2018)

$\mathcal{A}[\mathfrak{o}]=\overline{\exp}\left[i\oint_{\mathfrak{o}}\left\\{(\bm{\mathrm{A}}+\bm{\mathrm{R}})\cdot
d\bm{k}+Z\left(\sigma^{z}/v^{\perp}\right)dk\right\\}\right].$ (5)

Here $\overline{\exp}$ means path-ordered product,
$\bm{\mathrm{A}}(\bm{k})_{mn}=i\left\langle
u_{m\bm{k}}|\nabla_{\bm{k}}u_{n\bm{k}}\right\rangle$ is non-Abelian Berry
connection and

$\displaystyle\bm{\mathrm{R}}_{mn}\cdot d\bm{k}$
$\displaystyle=\sum_{l\notin\mathbb{Z}_{D}}\mathrm{A}_{ml}^{x}\Pi_{ln}^{y}dk_{x}/2v_{y}+(x\leftrightarrow
y)$
$\displaystyle=-i\hbar\sum_{l\notin\mathbb{Z}_{D}}\frac{\Pi_{ml}^{x}\Pi_{ln}^{y}}{\varepsilon_{m\bm{k}}-\varepsilon_{l\bm{k}}}\frac{dk_{x}}{2v_{y}}+(x\leftrightarrow
y)$ (6) $\displaystyle=-(M_{z}/ev^{\perp})dk,$ (7)

is Roth term and represents the orbital correction ($-M_{z}B_{z}$) to the band
energy. $\bm{\Pi}(\bm{k})_{ln}=\left\langle
u_{l\bm{k}}|(1/\hbar)\nabla_{\bm{k}}\hat{H}(\bm{k})|u_{n\bm{k}}\right\rangle$
is velocity matrix element and $\bm{v}=\bm{\Pi}_{nn}$ is group velocity.
$\epsilon_{m\bm{k}}$ is band energy and $v^{\perp}$ is the velocity in $xy$
plane.
$M_{z}=i(e\hbar/2)\sum_{l\notin\mathbb{Z}_{D}}\Pi_{ml}^{x}\Pi_{ln}^{y}/(\varepsilon_{m\bm{k}}-\varepsilon_{l\bm{k}})-(x\leftrightarrow
y)$ is the self rotation part of orbital magnetic momentXiao  _et al._ (2010).
Furthermore, $\sigma_{z,mn}=\left\langle
u_{l\bm{k}}|\hat{\sigma}_{z}|u_{n\bm{k}}\right\rangle$ ($\hat{\sigma}_{z}$ is
spin Pauli matrix) and $Z=g_{0}\hbar/4m$. The last term is the spin Zeeman
term. Once the propagator ($\mathcal{A[\mathfrak{o}]}$) is known, the phase
$\lambda_{a}$ can be easily obtained by diagonalizing it.

Though its formulation is clear in theory, the numerical calculation of this
propagator needs to deal with the derivatives in the Berry connection.
Besides, the multi-band magnetic moment (including orbital and spin) is a
gauge covariant quantity whose matrix elements depend on the gauge. If a
random gauge is chosen, the magnetic moment transforms independently at each
point along the orbit, rendering the (5) meaningless. To deal with these
problems, one can choose a smooth gauge by finding the maximally localized
Wannier functionMarzari and Vanderbilt (1997). Alternatively, the Wilson loop
methodFukui  _et al._ (2005); Yu  _et al._ (2011); Soluyanov and Vanderbilt
(2011) can be applied to avoid the choice of any specific gauge. Below, we
shall use the Wilson loop method for the calculation of $\lambda_{a}$.

In this way, the quantum orbit is discretized into $N$ segments (Fig.1) and
the propagator is written as the product for each segment. If the segment is
small enough, the exponent can be split into Berry connection and magnetic
moment parts.

Figure 1:  The Wilson loop $\mathfrak{o}$ for calculation of propagator
$\mathcal{A}$. Here it’s discretized to N points and the circulation direction
is clockwise.

$\displaystyle\mathcal{A}[\mathfrak{o}]=\prod_{i=1}^{N}\mathrm{exp}\left\\{i\left[(\bm{\mathrm{A}}(\bm{k_{i}})+\bm{\mathrm{R}}(\bm{k}_{i}))\cdot
d\bm{k_{i}}+Z\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]\right\\}$ (8)
$\displaystyle\approx\prod_{i=1}^{N}\mathrm{exp}\left[i\bm{\mathrm{A}}(\bm{k_{i}})\cdot
d\bm{k_{i}}\right]\mathrm{exp}\left[i\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+iZ\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right].$

For numerical calculation, the Berry connection part is usually expressed by
an overlap matrix $M^{i}=\mathrm{exp}\left[i\bm{\mathrm{A}}(\bm{k_{i}})\cdot
d\bm{k_{i}}\right]$. $M_{i}$ is a $D$ by $D$ matrix with
$M^{i}_{mn}=\left\langle u_{m\bm{k_{i+1}}}|u_{n\bm{k_{i}}}\right\rangle$.

The last ingredient for the propagator is an appropriate expression for the
Roth term which shows explicit gauge covariance. It can be written as a
summation of velocity matrix elements over all other states as in (6).
Instead, we propose another method that considers only $D$ degenerate states
on the Fermi surface using the covariant derivativeCeresoli  _et al._ (2006).
The covariant derivative is defined as

$\displaystyle\left|D_{\alpha}u_{n\bm{k}}\right\rangle$
$\displaystyle=Q_{\bm{k}}\left|\partial_{\alpha}u_{n\bm{k}}\right\rangle,\quad
Q_{\bm{k}}:=I-\sum_{a\in\mathbb{Z}_{D}}\left|u_{a\bm{k}}\right\rangle\left\langle
u_{a\bm{k}}\right|.$ (9)

In numerical calculation, it can be evaluated as an appropriate finite
differenceCeresoli  _et al._ (2006)

$\left|D_{\alpha}u_{n\bm{k}}\right\rangle=\frac{1}{2|\bm{q}_{\alpha}|}\left(\left|\overline{u}_{n,\bm{k}+\bm{q}_{\alpha}}\right\rangle-\left|\overline{u}_{n,\bm{k}-\bm{q}_{\alpha}}\right\rangle\right),$
(10)

where the dual state $\left|\overline{u}_{n,\bm{k}+\bm{q}}\right\rangle$ is a
linear combination of $\left|u_{n,\bm{k}+\bm{q}}\right\rangle$ and has the
property
$\left\langle\overline{u}_{m\bm{k}}|\overline{u}_{n\bm{k}+\bm{q}}\right\rangle=\delta_{mn}$.
This ensures the orthogonality between the covariant derivative and states in
degenerate space, i.e. $\left\langle
u_{m\bm{k}}|D_{\alpha}u_{n\bm{k}}\right\rangle=0$. Dual states are constructed
as

$\left|\overline{u}_{n,\bm{k}+\bm{q}}\right\rangle=\sum_{n^{\prime}}\left(S_{\bm{k},\bm{k}+\bm{q}}^{-1}\right)_{n^{\prime}n}\left|u_{n^{\prime},\bm{k}+\bm{q}}\right\rangle$
(11)

and

$\left(S_{\bm{k},\bm{k}+\bm{q}}\right)_{nn^{\prime}}=\left\langle
u_{n\bm{k}}|u_{n^{\prime},\bm{k}+\bm{q}}\right\rangle.$ (12)

Using covariant derivative, Eq. (6) is expressed only by states inside the
degenerate space

$\displaystyle\bm{\mathfrak{\mathrm{R}}}_{mn}\cdot
d\bm{k}=-\frac{i}{\hbar}\sum_{l\notin\mathbb{Z}_{D}}\mathrm{A}_{ml}^{x}(\varepsilon_{n\bm{k}}-\varepsilon_{l\bm{k}})\mathrm{A}_{ln}^{y}dk_{x}/2v_{y}+(x\leftrightarrow
y)$ (13)
$\displaystyle=-\frac{i}{\hbar}\sum_{l\notin\mathbb{Z}_{D}}\left\langle
D_{x}u_{m\bm{k}}|u_{l\bm{k}}\right\rangle(\varepsilon_{n\bm{k}}-\varepsilon_{l\bm{k}})\left\langle
u_{l\bm{k}}|D_{y}u_{n\bm{k}}\right\rangle dk_{x}/2v_{y}$
$\displaystyle\qquad+(x\leftrightarrow y)$
$\displaystyle=-\frac{i}{\hbar}\left\langle
D_{x}u_{m\bm{k}}|\varepsilon_{n\bm{k}}-\hat{H}(\bm{k})|D_{y}u_{n\bm{k}}\right\rangle
dk_{x}/2v_{y}+(x\leftrightarrow y).$

In Appendix we show both Eq. (6) and Eq. (13) are gauge independent, which can
be implemented easily in first-principles calculation. The Eq. (6) is
practical for tight-binding models with a small number of bands but quite
tedious if the total number of bands is large. The Eq. (13) avoids these
problems and focuses only on the degenerate space and it is convenient when
covariant derivatives can be easily calculated.

###  II.2 Phase $\delta$

The above discussion about phase $\theta$ is for a single $k$-plane
perpendicular to the magnetic field. For 3D material, one needs to integrate
over $k_{z}$ to get the contribution from the whole Fermi surface. Extremal
orbits will dominate in the integration and this procedure will introduce
another phase $\delta$ for each of them, which is generally $\pm\pi/4$ ($+$
for minimum cross-section and $-$ for maximum cross-section). $\delta=0$ for
2D material since there is only one k-plane. But for a nearly cylindrical
Fermi surface (e.g., Fig.2(c)), $\delta$ lies between these two limits. Below
we adopt a simple model from Refs. Shoenberg (1984); Shoenberg and Templeton
(1973) to determine $\delta$ for every extremal orbit that lies in a mirror
plane. Here, we assume $\mathcal{PT}$ symmetry for simplicity.

(a)

(b)

(c)

Figure 2:  (a) Spherical Fermi surface, (b) perfect cylindrical Fermi surface,
and (c) nearly cylindrical Fermi surface. In both cases, the red circle shows
the extremal orbit.

The oscillation of 3D Fermi surface is calculated first for a 2D plane with
thickness $dk_{z}$ and then integrate with respect to $k_{z}$, i.e.

$\displaystyle A_{r}$ $\displaystyle=\sum_{a}\int
dk_{z}\,A_{r}(k_{z})\mathrm{cos}\left[r(2\pi\frac{F(k_{z})}{B}+\lambda_{a}(k_{z}))+\phi_{M}\right]$
(14) $\displaystyle\propto\int
dk_{z}\,A_{r}(k_{z})\mathrm{cos}\left[r(2\pi\frac{F(k_{z})}{B}+\theta(k_{z}))+\phi_{M}\right],$

where $A_{r}(k_{z})$ is the oscillation amplitude of 2D plane, which depends
on $k_{z}$ through cyclotron frequency $F(k_{z})$ and cyclotron mass
$m(k_{z})$. The relative change of $F(k_{z})$ and $m(k_{z})$ in the interval
where the integral is appreciable is usually small. Hence in the integration
of Eq. (14), $A_{r}(k_{z})$ can be treated approximately as a constant while
$F(k_{z})$ in the cosine function can’t be treated as fixed because
$F(k_{z})\gg B$. Maslov phase $\phi_{M}$ remains constant as long as the orbit
on the Fermi surface doesn’t change its topology. Moreover, $\mathcal{PT}$
symmetry cause the phase $\theta(k_{z})$ quantized to 0 or $\pi$ as in Eq.
(4). So the $k_{z}$ dependence of $\theta$ can also be ignored and only the
$k_{z}$-variation of $F(k_{z})$ needs to be considered.

We expand $F(k_{z})$ near its extremal value to the fourth order and all odd
orders are zero due to mirror symmetry.

$\displaystyle
F(k_{z})=F_{0}+\frac{1}{2}F_{2}k_{z}^{2}+\frac{1}{24}F_{4}k_{z}^{4}.$ (15)

Introducing dimensionless variable $x=(2r|F_{2}|/B)^{1/2}k_{z}$ and
$\alpha=\mathrm{sgn}(F_{2})\frac{F_{4}B}{24r|F_{2}|^{2}}$ then the integration
can be calculated as

$\displaystyle\begin{aligned}
A_{r}&\propto\mathrm{Re}\int\mathrm{exp}\left[i(2\pi
r\frac{F(k_{z})}{B}+r\theta+\phi_{M})\right]dk_{z}\\\ &\propto\mathrm{Re}\
\mathrm{exp}\left[i(2\pi
r\frac{F_{0}}{B}+r\theta+\phi_{M})\right]\int\mathrm{exp}\left[\mathrm{sgn}(F_{2})i\frac{\pi}{2}x^{2}(1+\alpha
x^{2})\right]dx\\\
&\propto\mathrm{cos}\left[r(2\pi\frac{F_{0}}{B}+\theta)+\phi_{M}+\delta\right].\end{aligned}$
(16)

where phase $\delta$ is the argument of the last integral

$\displaystyle\delta$
$\displaystyle=\mathrm{arg}\left\\{\int^{x_{m}}_{x_{m}}\mathrm{exp}\left[\mathrm{sgn}(F_{2})i\frac{\pi}{2}x^{2}(1+\alpha
x^{2})\right]dx\right\\}.$ (17)

$\delta$ was numerically determined by carrying out the integral with given
value $\alpha$ Shoenberg (1984); Shoenberg and Templeton (1973), for which
$F_{2}$, $F_{4}$ can be found from the polynomial fitting of $F(k_{z})$ around
the extremal orbit. The integral limit $x_{m}$ can be taken as $\infty$ when
$\alpha>0$ because the main contribution comes from $x\approx 0$. However,
this argument does not apply when $\alpha<0$ due to the two extra artificial
extrema. Since the real cross-section varies monotonically on either side of
$x=0$, $x_{m}$ should be taken less than the turning point
$1/\sqrt{2|\alpha|}$ to avoid these artificial extrema. In calculation, the
argument of the integral goes to a steady value before the turning point,
which should be assigned as $\delta$. It’s obvious that $\delta=0$ from Eq.
(14) when $F(k_{z})=F_{0}$. For a general 3D material, if $\alpha\rightarrow
0$ (i.e., $F_{4}\rightarrow 0$ and $F_{2}k_{z}^{2}$ is the leading
dispersion), one can get $\delta=\pm\pi/4$. Otherwise, $\delta$ may take a
value between 0 and $\pm\pi/4$.

###  II.3 Phase $\varphi_{A}$

The last phase $\varphi_{A}$ depends on the type of physical quantity $A$.
When $A$ is the density of states (DOS), this phase vanishes
$\varphi_{DOS}=0$. For other quantities, $\varphi_{A}$ represents the
connection between the oscillation of $A$ and the oscillation of DOS. For
example, $\varphi_{M}=\pi/2$ if $A$ is sample magnetization, and
$\varphi_{\chi}=\pi$ if $A$ is magnetic susceptibility. In four terminal
devices, the longitudinal conductivity $\sigma_{xx}$ oscillates in phase with
DOS hence $\varphi_{\sigma}=0$. But since
$\sigma_{xx}=\rho_{xx}/(\rho_{xx}^{2}+\rho_{xy}^{2})$, the resistivity
$\rho_{xx}$ can be in phase (if $\rho_{xx}\ll\rho_{xy}$) or out of phase (if
$\rho_{xx}\gg\rho_{xy}$) with $\sigma_{xx}$, so $\varphi_{\rho}=0$ if
$\rho_{xx}\ll\rho_{xy}$ or $\varphi_{\rho}=\pi$ if
$\rho_{xx}\gg\rho_{xy}$Xiang  _et al._ (2015); Fu  _et al._ (2021).

To summarize, all the phases in the oscillation term Eq. (1) have the
following intuitive explanations. First, the magnetic-field-dependent term
$l_{B}^{2}S_{F}$ is given by the combination of the de Broglie phase
(determined by the number of wavelengths in an orbit) and the Aharonov–Bohm
phase. Then there is a phase $\lambda_{a}$ associated with each orbit and each
band coming from geometric effects and magnetic moment energy. $\lambda_{a}$
of degenerate bands for the same orbit will combine to give the phase
$\theta$. The reflection of the wave packet at turning points in the orbit
causes phase $\phi_{M}$. These phases are the total phase for a single orbit
lying in the $kx-ky$ plane. For 3D materials, $k_{z}$ integration needs to be
carried out to incorporate the whole Fermi surface’s contribution, which gives
phase $\delta$. At last, depending on what quantity $A$ is measured, there
will be another phase $\phi_{A}$ if the oscillation of $A$ is not synchronized
with the oscillation of DOS.

##  III Results and Discussions

The crystal structure of CsTi3Bi5 is fully relaxed within the Density
Functional Theory (DFT) as implemented in the Vennia $ab$-$inito$ Simulation
Package Kresse and Furthmüller (1996a, b). The cutoff energy for the plane-
wave basis set is 300 eV. The force convergence criteria is 5 meV/Å. The
electronic structure is calculated with the full-potential local-orbital
minimum-basis code (FPLO) Koepernik and Eschrig (1999). The default atomic
basis sets are employed for the wave function expansion. The generalized
gradient approximation parameterized by Perdew, Burke, and Ernzerhof (PBE)
Perdew  _et al._ (1996) is employed to mimic the exchange-correlation
interaction between electrons throughout. The Brillouin zone is sampled by a
$k$-mesh of 12$\times$12$\times$6\. The tight-binding Hamiltonian of CsTi3Bi5
is extracted via the maximally localized Wannier functions Marzari and
Vanderbilt (1997) as implemented in FPLO, which enforces all crystal
symmetries. The Wannier basis set is composed of the Ti $d$ and Bi $p$
orbitals. The Fermi surface is calculated with the tight-binding Hamiltonian
on a $k$-mesh of $300\times 300\times 100$.

We mention that the above Wilson loop method for the total oscillation phase
shift has been successfully applied to the $\mathcal{PT}$ symmetric kagome
metal CsV3Sb5 Tan  _et al._ (2023), which predicted consistent results with
experiments. In the following, we will apply the Wilson loop method to the
recently discovered kagome superconductor CsTi3Bi5 Werhahn  _et al._ (2022) to
further demonstrate the reliability of this method. We note here that the
characterization of the dimensionality of the quantum orbit by the phase
$\delta$ has not been discussed in our previous work on CsV3Sb5.

The band structure of CsTi3Bi5 with spin-orbit coupling is plotted in
Fig.3(a), which contains rich topological properties. Due to the
$\mathcal{PT}$ symmetry in CsTi3Bi5, each band is doubly degenerate.
Characteristic features of the kagome lattice, such as Dirac points at K/H
points away from the Fermi level which are gapped by SOC, van Hove
singularities at M/L, and flat bands along M-K/L-H lines Hu  _et al._ (2022b);
Yang  _et al._ (2022b); Jiang  _et al._ (2022), are shown. There are also type
II Dirac crossings on the $\Gamma$-M and A-L lines, which form a Dirac nodal
line Hu  _et al._ (2022b); Yang  _et al._ (2022b); Jiang  _et al._ (2022) in
the $\Gamma$-M-A plane. Besides, both the experiment and theory have shown
that CsTi3Bi5 has topological Dirac surface states at the $\overline{\Gamma}$
point on the (001) surface Yang  _et al._ (2022a, b); Jiang  _et al._ (2022);
Hu  _et al._ (2022b); Zhou  _et al._ (2023).

Figure 3:  (a) Band structure of CsTi3Bi5 with SOC. (b) 3D Fermi surface of
CsTi3Bi5, where the color representing the Fermi velocity is used to
distinguish between different Fermi surfaces. (c) Fermi surfaces in $k_{z}=0$
and (d) $k_{z}=0.5$ (in units of $2\pi/c$, $c$ is the lattice constant) mirror
plane at the Fermi energy. The grey hexagon is the first Brillouin zone. Fermi
surfaces with the same color come from the same band. The cyclotron
frequencies (in units of T) are given.

The band structure on the $k_{z}=0$ plane looks similar to the band structure
on the $k_{z}=0.5$ plane (in units of 2$\pi$/$c$, $c$ is the lattice
constant), which indicates the quasi-two-dimensional feature of the electronic
structure of CsTi3Bi5. Indeed, the 3D Fermi surface shown in Fig.3(b) shows a
good cylindrical shape for all pieces. There are totally four bands crossing
the Fermi level creating five pieces of the Fermi surface. By sweeping
$k_{z}$, all extremal quantum orbits perpendicular to the $z$-direction are
found to locate at the two mirror planes $k_{z}=0$ and $k_{z}=0.5$, shown in
Fig.3(c) and (d). The initial experiment reported an oscillation frequency of
200 T Werhahn  _et al._ (2022). A more recent transport experiment Dong  _et
al._ (2023) reported a series of oscillation frequencies, ranging from 217 to
1013 T. Our calculations show agreement with the experiments in the low-
frequency region. For example, the calculated frequencies of 213, 336, and 542
T might correspond to the observed frequencies of 200/217, 281, 498 or 594 T,
respectively. We notice that our calculated frequencies are slightly different
from the calculations in Ref. Dong  _et al._ , 2023, which might be induced by
the mismatch of Fermi energy and/or different calculation parameters employed.

The cyclotron masses $m^{*}$ of all calculated quantum orbits are summarized
in Table 1. Except for the two small pockets (336 and 213 T) around M/L
points, all other orbits are electron pockets, whose cyclotron masses are
defined as positive. The two largest hexagonal orbits centered around the
$\Gamma$ point (7488 and 8111 T) have the largest cyclotron masses
(1.6$\sim$1.7) while others have relatively small cyclotron masses.

The different quantum oscillation phases, as mentioned above, of all orbits
are calculated and listed in Table.1. Here every cyclotron orbit is a simple
closed curve; thus the Maslov correction $\phi_{M}=\pi$ is omitted in the
table. The phase $\lambda_{a}$ is calculated by Eq. (13) with random gauge
choices to test the gauge invariance, which presents the same results. We also
confirm the relation $\lambda_{1}=-\lambda_{2}$ for any two degenerate quantum
orbits imposed by the $\mathcal{PT}$ symmetry. Thus only the positive one
$\lambda_{1}$ is listed. The Berry phases without ($\phi_{B0}$) or with SOC
($\phi_{B}$) are also listed for comparison. According to our previous
discussion of Eq. (4), the final phase shift of the quantum orbit $\theta$
must be quantized to either 0 or $\pi$, depending on the magnitude of
$\lambda_{1}$, as listed in Table.1. From these phases, it’s clear that phase
$\lambda_{1}$ is in general different from the Berry phase $\phi_{B}$ due to
the orbital and spin magnetic moment contribution. Also for the
$\mathcal{PT}$-symmetric system, the topology of the quantum orbit is not
equivalent to the band topology of the individual Fermi surface. For example,
the quantum orbits of 336 T (around M) and 8111 T (around $\Gamma$) have Berry
phases $\phi_{B}$ close to 0 but the oscillation phase shifts are $\pi$. On
the contrary, the quantum orbit of 4907 T (around $A$) has a Berry phase close
to $\pi$ but a zero oscillation phase shift. We note here that the strong SOC
is important because these orbits have only a trivial Berry phase in the
spinless case. Therefore, the incorporation of the magnetic moment
contribution in the oscillation phase by SOC is crucial and the quantum phase
shift extracted from the Landau fan diagram should be interpreted more
carefully, rather than just interpreting it as the Berry phase. The recent
experiment Dong  _et al._ (2023) finds that the quantum orbit of 281 T is non-
trivial with a $\pi$ phase shift ($\theta=\pi$), which is consistent with our
calculated non-trivial quantum orbit of 336 T.

Because the 3D Fermi surface is nearly cylindrical, the dimension-related
phase $\delta$ should be determined by considering higher order terms in the
expansion of $F(k_{z})$ in Eq. 15. From numerical calculations, the frequency
$F$ and cyclotron mass $m^{*}$ of all extremal orbits have a small relative
change on the Fermi surface (less than $5\%$ in the interval $|\Delta
k_{z}|\leq 0.1$). Since CsTi3Bi5 has $\mathcal{PT}$ symmetry and all extremal
orbits locate in mirror planes, Eq. (17) applies, which is used to calculate
phase $\delta$. The phase $\delta$ is calculated with the magnetic field $B$
varying from 5 T to 40 T, covering the range of $B$ in general oscillation
experiments Fu  _et al._ (2021); Shrestha  _et al._ (2022); Broyles  _et al._
(2022). The variation of $\delta$ is very small in the considered $B$ range.
Thus the $\delta$ can be approximately treated as a constant, whose average
value is listed in Table 1. It shows that all quantum orbits except for the
213 and 802 T ones have a phase $\delta$ quite close to $\pm\pi/4$. Therefore,
most orbits should be classified as 3D cases in quantum oscillation, even
though the Fermi surfaces in Fig. 3(b) show a strong quasi-2D feature. On the
other hand, the Fermi surface around A is almost dispersionless along $k_{z}$,
so the $\delta$ for the quantum orbit of 802 T is closer to zero than others.
As a result, this quantum orbit is 2D. However, the quantum orbit of 713 T
which comes from the same Fermi surface as the 802 T orbit but on the
$k_{z}=0$ plane, has a $\delta=\pi/4$. Consequently, the character (2D or 3D)
of a quantum orbit should not be simply determined from the appearance of the
related Fermi surface in the 3D $k$ space.

Table 1:  Extremal orbits of Fermi surfaces of CsTi3Bi5 at Fermi energy. Frequency (Freq.) is in units of T. $k_{z}$ refers to the $k_{z}$ plane (in units of $2\pi/c$, $c$ is the lattice constant) where the corresponding extremal orbit is located in. The underlined frequency indicates a minimal Fermi surface cross-section and the others correspond to a maximal cross-section. The cyclotron mass $m^{*}$ is in units of bare electron mass $m_{0}$, where positive and negative values are for electron and hole pockets respectively. All orbits have Maslov correction $\phi_{M}=\pi$. $\phi_{B0}$ ($\phi_{B}$) is Berry phase without (with) SOC and $\lambda_{1}$ is the phase of one of the band between two degenerate bands, defined in Eq. (5). $\delta$ is the phase related to the Fermi surface dimensionality. All phases are in units of $\pi$.  Freq. | $k_{z}$ | $m^{*}$ | $\phi_{B0}$ | $\phi_{B}$ | $\lambda_{1}$ | $\theta$ | $\delta$
---|---|---|---|---|---|---|---
(T) | ($2\pi/c$) | ($m_{0}$) | ($\pi$) | ($\pi$) | ($\pi$) | ($\pi$) | ($\pi$)
213 | 0.5 |  $-$0.24 | 0 | 0.08 | 0.40 | 0 | 0.22
336 | 0 |  $-$0.22 | 0 | 0.16 | 0.59 | 1 | -0.25
542 | 0 | 0.22 | 0 | 0.50 | 0.23 | 0 | 0.25
713 | 0 | 0.26 | 0 | 0.30 | 0.33 | 0 | 0.25
802 | 0.5 | 0.24 | 0 | 0.33 | 0.12 | 0 | -0.14
889 | 0.5 | 0.32 | 1 | 0.26 | 0.10 | 0 | -0.25
4569 | 0 | 0.72 | 0 | 0.74 | 0.42 | 0 | 0.25
4907 | 0.5 | 0.78 | 0 | 0.92 | 0.22 | 0 | -0.25
7488 | 0.5 | 1.62 | 0 | 0.49 | 0.81 | 1 | 0.25
8111 | 0 | 1.68 | 0 | 0.38 | 0.82 | 1 | -0.25

##  IV Conclusion

We theoretically studied the quantum oscillations by revealing their
frequencies and topological phases through a Wilson loop method in CsTi3Bi5.
We revealed three quantum orbits with $\theta=\pi$ phase shift. Despite most
Fermi surfaces are quasi-2D, the dimensional-related phase $\delta$, beyond
the angle-dependent frequency, clearly indicates their 3D nature. Our method
can be applied to other quantum materials and provides a general way to study
quantum oscillations assisted by first-principles calculations.

## Acknowledgement

B.Y. acknowledges the financial support by the European Research Council (ERC
Consolidator Grant “NonlinearTopo”, No. 815869) and the ISF - Personal
Research Grant (No. 2932/21).

## Appendix

The most general gauge transformation is a $U(D)$ basis transformation among
the degenerate bands

$\displaystyle\left|u_{n\bm{k}}\right\rangle$
$\displaystyle\rightarrow\sum_{m=1}^{D}U(\bm{k})_{mn}\left|u_{m\bm{k}}\right\rangle,\quad
U^{-1}=U^{\dagger},$ (18)

It has already been shown that the propagator $\mathcal{A}[\mathfrak{o}]$ is
gauge covariant under such transformationAlexandradinata and Glazman (2018)
provided that the same wave function is used at the initial point and the
final point, i.e. $|u(\bm{k}_{N+1})\rangle=|u(\bm{k}_{1})\rangle$. Here we use
the same way to show our numerical formula inherits this property so it’s
appropriate for calculation.

First, covariant derivatives transform as states under the $U(D)$ gauge
transformation

$\displaystyle\left|\overline{u}_{n,\bm{k}+\bm{q}}\right\rangle$
$\displaystyle\rightarrow\sum_{n^{\prime}}\left(U(\bm{k})^{\dagger}S_{\bm{k},\bm{k}+\bm{q}}U(\bm{k}+\bm{q})\right)^{-1}_{n^{\prime}n}\left|u_{n^{\prime},\bm{k}+\bm{q}}\right\rangle$
(19)
$\displaystyle=\sum_{n^{\prime},m,l,m^{\prime}}U(\bm{k}+\bm{q})^{-1}_{n^{\prime}m}(S_{\bm{k},\bm{k}+\bm{q}}^{-1})_{ml}U(\bm{k})_{ln}$
$\displaystyle\qquad\qquad
U(\bm{k}+\bm{q})_{m^{\prime}n^{\prime}}\left|u_{m^{\prime},\bm{k}+\bm{q}}\right\rangle$
$\displaystyle=\sum_{m,l}(S_{\bm{k},\bm{k}+\bm{q}}^{-1})_{ml}U(\bm{k})_{ln}\left|u_{m,\bm{k}+\bm{q}}\right\rangle$
$\displaystyle=\sum_{l}U(\bm{k})_{ln}\left|\overline{u}_{l,\bm{k}+\bm{q}}\right\rangle$

which makes the covariant derivative expression of Roth term (13) transform
covariantly. This is also true for the matrix elements expression of Roth term
(6) and spin matrix $\sigma_{z}$, meaning that

$\displaystyle\bm{\mathrm{R}}(\bm{k}_{i})_{mn}\cdot d\bm{k}_{i}$
$\displaystyle\rightarrow
U(\bm{k}_{i})^{-1}\bm{\mathrm{R}}(\bm{k}_{i})_{mn}\cdot
d\bm{k}_{i}U(\bm{k}_{i})$ (20) $\displaystyle\sigma_{z}(\bm{k}_{i})_{mn}\cdot
d\bm{k}_{i}$ $\displaystyle\rightarrow
U(\bm{k}_{i})^{-1}\sigma_{z}(\bm{k}_{i})_{mn}\cdot d\bm{k}_{i}U(\bm{k}_{i})$

Therefore, the second term in (8) is also gauge covariant

$\displaystyle\mathrm{exp}\left[i\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+iZ\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]$ (21)
$\displaystyle\rightarrow\mathrm{exp}\left\\{iU(\bm{k}_{i})^{-1}\left[\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+Z\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]U(\bm{k}_{i})\right\\}$
$\displaystyle=U(\bm{k}_{i})^{-1}\mathrm{exp}\left[i\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+iZ\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]U(\bm{k}_{i})$

Besides, the overlap matrix $M^{i}_{mn}=\left\langle
u_{m\bm{k_{i+1}}}|u_{n\bm{k_{i}}}\right\rangle$ transforms like

$\displaystyle M^{i}\rightarrow U(\bm{k}_{i+1})^{-1}M^{i}U(\bm{k}_{i})$ (22)

Hence, the covariance of discretized propagator (8) follows from the
transformation properties of the two separate terms as

$\displaystyle\begin{aligned}
\mathcal{A}[\mathfrak{o}]&\rightarrow\prod_{i=1}^{N}U(\bm{k}_{i+1})^{-1}M^{i}U(\bm{k}_{i})\cdot
U(\bm{k}_{i})^{-1}\\\
&\qquad\mathrm{exp}\left[i\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+iZ\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]U(\bm{k}_{i})\\\
&=U(\bm{k}_{N+1})^{-1}\\\
&\left\\{\prod_{i=1}^{N}M^{i}\cdot\mathrm{exp}\left[i\bm{\mathrm{R}}(\bm{k}_{i})\cdot
d\bm{k_{i}}+iZ\frac{\sigma^{z}}{v^{\perp}}|d\bm{k}_{i}|\right]\right\\}U(\bm{k}_{1})\\\
&=U(\bm{k}_{1})^{-1}\mathcal{A}[\mathfrak{o}]U(\bm{k}_{1})\end{aligned}$ (23)

Since propagator $\mathcal{A}[\mathfrak{o}]$ transforms covariantly, its
spectrum $\\{e^{i\lambda_{a}}\\}_{a=1}^{D}$ is gauge invariant. In other
words, the phase $\lambda_{a}$ obtained through these numerical formulas is
uniquely determined (module $2\pi$) independent of gauge choice in the
calculation.

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