# Higher-order Fabry-Pérot Interferometer from Topological Hinge States

Chang-An Li<EMAIL_ADDRESS>Institute for Theoretical Physics and
Astrophysics, University of Würzburg, 97074 Würzburg, Germany  Song-Bo Zhang
<EMAIL_ADDRESS>Institute for Theoretical Physics and
Astrophysics, University of Würzburg, 97074 Würzburg, Germany  Jian Li  School
of Science, Westlake University, 18 Shilongshan Road, Hangzhou 310024,
Zhejiang Province, China  Institute of Natural Sciences, Westlake Institute
for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province,
China  Björn Trauzettel  Institute for Theoretical Physics and Astrophysics,
University of Würzburg, 97074 Würzburg, Germany

(August 27, 2024)

###### Abstract

We propose an intrinsic 3D Fabry-Pérot type interferometer, coined “higher-
order interferometer”, that utilizes the chiral hinge states of second-order
topological insulators and cannot be equivalently mapped to 2D space because
of higher-order topology. Quantum interference patterns in the two-terminal
conductance of this interferometer are controllable not only by tuning the
strength but also, particularly, by rotating the direction of the magnetic
field applied perpendicularly to the transport direction. Remarkably, the
conductance exhibits a characteristic beating pattern with multiple
frequencies with respect to field strength or direction. Our novel
interferometer provides feasible and robust magneto-transport signatures to
probe the particular hinge states of higher-order topological insulators.

Introduction.—Higher-order topological insulators (HOTIs) feature gapless
excitations, similar to traditional (first-order) topological insulators, that
are protected by bulk electronic topology but localized at open boundaries at
least two dimensions lower than the insulating bulk (Benalcazar  _et al._ ,
2017a, b; Slager  _et al._ , 2015; Peng  _et al._ , 2017; Langbehn  _et al._ ,
2017; Song  _et al._ , 2017; Schindler  _et al._ , 2018a; Geier  _et al._ ,
2018; Ezawa, 2018; Khalaf, 2018; Park  _et al._ , 2019; Trifunovic and
Brouwer, 2019; You  _et al._ , 2018; Hirosawa  _et al._ , 2020; Franca  _et
al._ , 2018; van Miert and Ortix, 2018). For instance, 3D second-order
topological insulators (SOTIs) host 1D chiral or helical states along specific
hinges of the systems. In recent years, HOTIs have triggered widespread
research interest, owing to their discoveries in a variety of candidate
systems, promotion of our understanding of topological states of matter, and
potential applications (Schindler  _et al._ , 2018b; Imhof  _et al._ , 2018;
Peterson  _et al._ , 2018; Serra-Garcia  _et al._ , 2018; Chen  _et al._ ,
2019; Peng  _et al._ , 2020; Ghosh  _et al._ , 2019; El Hassan  _et al._ ,
2019; Ni  _et al._ , 2019; Xie  _et al._ , 2019; Qi  _et al._ , 2020; Călugăru
_et al._ , 2019; Szabó  _et al._ , 2020; Sheng  _et al._ , 2019; Li and Wu,
2020; Li  _et al._ , 2020a; Li and Sun, 2020; Zhu, 2018; Luo and Zhang, 2019;
Ezawa, 2019; Zhang  _et al._ , 2020a, b; Pahomi  _et al._ , 2020). So far,
most efforts have been put into the potential realization and electronic
characterization of HOTIs. However, the transport properties of HOTIs remain
largely unexplored, despite of a few works associated with superconductivity
(Queiroz and Stern, 2019; Li  _et al._ , 2020b; Choi  _et al._ , 2020).
Indeed, for 3D SOTIs, an intriguing open question is whether the emergent
hinge states can exhibit any particular phenomena in normal-state transport.

Figure 1: (a) Schematic of the higher-order interferometer: a SOTI with four
chiral hinge states (solid red and dashed blue lines) are connected to two
leads (yellow). Adjacent chiral hinge states form interference loops in the
presence of finite reflections at the interfaces. A magnetic field ${\bf B}$
perpendicular to $z$-direction is applied to the SOTI (gray). (b) The hinge
states have linear dispersion and are shifted in $k_{z}$-direction by ${\bf
B}$. (c) Density plot of conductance with respect to the field strength $B$
and direction $\theta$. $B_{0}=\phi_{0}/S_{f}$ with $\phi_{0}$ the flux
quantum and $S_{f}$ the area of the front surface of the SOTI.

One appealing route towards this question involves interferometers built of
SOTIs, which enable us to study quantum-coherent transport of hinge states.
Propagating hinge states that form interference loops enclosing a magnetic
flux applied to the system pick up an Aharonov-Bohm (AB) phase (Aharonov and
Bohm, 1959). In presence of quantum coherence, the AB phase will give rise to
quantum oscillations in transport characteristics such as the charge
conductance. Quantum interference patterns in the two-terminal conductance
have been employed to detect topological phases of matter, for instance,
surface states of topological insulators (Bardarson  _et al._ , 2010; Zhang
and Vishwanath, 2010; Peng  _et al._ , 2010; Bardarson and Moore, 2013),
chiral Majorana modes (Akhmerov  _et al._ , 2009; Fu and Kane, 2009; Li  _et
al._ , 2012, 2019), and topological Dirac semimetals (Wang  _et al._ , 2016).

In this work, we propose a higher-order Fabry-Pérot interferometer to probe
hinge states of SOTIs. Our basic setup, shown in Fig. 1(a), is composed of a
rectangular chiral SOTI in contact with two leads. The chiral hinge states,
existing in 3D space, form interference loops due to finite reflections (not
shown) at the two interfaces, and their energy dispersions split in a non-
uniform manner under magnetic fields, as shown in Fig. 1(b). Particular
quantum interference patterns in the two-terminal conductance, arising from
the AB effect as exemplified by Fig. 1(c), can be observed either by tuning
the field strength $B$ or direction $\theta$. In addition, owing to the
intrinsic 3D nature of the interferometer, there are generally two frequencies
in the magneto-conductance oscillations, leading to a beating pattern. These
features do not depend on the details of the junction, such as the electronic
spectrum of the leads, and are stable against disorder and dephasing. Hence,
they provide robust transport signatures of hinge states in 3D SOTIs.

General analysis based on scattering matrix theory.—Our proposed
interferometer involves a 3D SOTI with four chiral hinge states attached to
two leads in $z$-direction, as sketched in Fig. 1(a). Adjacent chiral hinge
states form interference loops because of finite reflections at the
interfaces, as will be discussed below. A magnetic field ${\bf
B}=B(\cos\theta,\sin\theta,0)$ in $x$-$y$ plane is applied in the SOTI region,
where $B$ measures the field strength and $\theta$ the field direction with
respect to $x$-direction.

Before presenting concrete results based on specific models, it is instructive
to analyze the main transport features of the interferometer using a
phenomenological scattering matrix approach (Büttiker, 1992; Nazarov and
Blanter, 2006; Maciejko  _et al._ , 2010). The transport properties of the
setup are encoded in a scattering matrix that directly connects the conducting
channels in the left and right leads. The scattering processes at the two
interfaces between the leads and the SOTI can be described by two scattering
matrices, respectively. Each matrix consists of four components: transmission
from left to right $t_{L/R}$, transmission from right to left
$t_{L/R}^{\prime}$, reflection from the right $r_{L/R}$ and reflection from
the left $r_{L/R}^{\prime}$, where the subscript ($L$ and $R$) distinguishes
the left and right surfaces. At low energies, the only conducting channels in
the SOTI are the four hinge states which have linear dispersion and are
localized at the four different hinges of the cuboid. In the presence of a
magnetic field, their propagation in the SOTI will pick up AB phases that can
be described by a phase matrix $U\equiv
e^{2i\lambda}e^{i\varphi\sigma_{z}\otimes\sigma_{0}/2}e^{i\phi\sigma_{z}\otimes\sigma_{z}/2}$,
where $\sigma_{z}$ is a Pauli matrix, $\sigma_{0}$ the $2\times 2$ identity
matrix,

$\varphi=BLW_{x}\cos\theta\ \text{and }\phi=BLW_{y}\sin\theta$ (1)

are the magnetic fluxes threading the two surfaces, respectively, with $L$ the
distance between the two leads and $W_{x/y}$ the widths of the sample in
$x/y$-directions. Moreover, $\lambda=k_{F}L$ is the dynamical phase with
$k_{F}$ the Fermi wave number in the absence of magnetic fields. By
eliminating the scattering amplitudes in the SOTI region, we derive
analytically an effective $2\times 2$ scattering matrix that directly connects
the two interfaces (Li2, a)

$\mathcal{S}(B,\theta)=\Phi_{+}(e^{-i\lambda}-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1},$
(2)

where the phase matrices $\Phi_{\pm}\equiv e^{i(\varphi\pm\phi)\sigma_{z}/2}$
account for the AB phase differences between the two right-moving and between
the two left-moving hinge channels, respectively. At zero temperature, the
two-terminal conductance of the setup can be evaluated as

$G(B,\theta)=\frac{e^{2}}{h}\mathrm{Tr}[t_{R}^{\dagger}t_{R}\mathcal{S}(B,\theta)t_{L}t_{L}^{\dagger}\mathcal{S}^{\dagger}(B,\theta)],$
(3)

where $h$ is the Planck constant and $e$ is electron charge. According to Eqs.
(2) and (3), if there is no transmission across any of the two interfaces,
i.e., $t_{L}=0$ or $t_{R}=0$, then $G$ vanishes. In the opposite limit, where
the interfaces are completely transparent for the hinge states,
$r_{L^{\prime}}=0$ and $r_{R}=0$, we find that the matrix $\mathcal{S}$ as
well as $t_{R}^{\dagger}t_{R}$ and $t_{L}t_{L}^{\dagger}$ become diagonal. As
a result, $G$ becomes quantized at $2e^{2}/h$ and is independent of the
magnetic field. These results indicate the necessary condition for a
successful interferometer: non-trivial transmission and reflection at the two
interfaces for the hinge states.

When the interfaces are partially transparent, Eq. (2) indicates the formation
of Fabry-Pérot interference loops. Moreover, the matrix $\mathcal{S}$ contains
explicitly two phases $\varphi\pm\phi$ in general. This indicates the
appearance of beating patterns with two frequencies in the magneto-
conductance. Notably, the two frequencies are intimately connected to the
magnetic fluxes threading the different surfaces of the SOTI. They are solely
determined by the geometry of the sample and insensitive to the details of the
interface barriers. The oscillation pattern of $G$ remains qualitatively the
same even in the presence of a dynamic phase. We verified these results by
properly parametrizing the scattering matrices (Li2, a).

Model simulation and method.—To demonstrate these features of the
interferometer explicitly, we consider an effective model for chiral SOTIs
(Schindler  _et al._ , 2018a)

$\displaystyle H({\bf k})=$ $\displaystyle\Big{(}m+b\sum_{i=x,y,z}\cos
k_{i}\Big{)}\tau_{3}+v\sum_{i=x,y,z}\sin k_{i}\sigma_{i}\tau_{1}$
$\displaystyle+\Delta(\cos k_{x}-\cos k_{y})\tau_{2},$ (4)

where ${\bf k}=(k_{x},k_{y},k_{z})$ is the wave vector.
$\bm{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ and
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices acting on
orbital and spin spaces, respectively; $m$, $b$, $v$ and $\Delta$ are model
parameters. Without loss of generality, we set the lattice constant and the
velocity $v$ to unity hereafter. When $1<|m/b|<3$ and $\Delta=0$, the model
describes 3D topological insulators with gapless surface states (Zhang  _et
al._ , 2009). The surface states are protected by time-reversal symmetry
$\mathcal{T}=i\sigma_{2}\mathcal{K}$, where $\mathcal{K}$ represents complex
conjugation. A finite $\Delta\neq 0$ breaks time-reversal and
$\mathcal{C}_{4}$ rotation (with the rotation axis pointing in $z$-direction)
symmetries individually. It opens gaps in the surface states. However, the
$\Delta$ term preserves the combined symmetry $\mathcal{C}_{4}\mathcal{T}$, as
indicated by
$(\mathcal{C}_{4}\mathcal{T})H(k_{x},k_{y},k_{z})(\mathcal{C}_{4}\mathcal{T})^{-1}=H(k_{y},-k_{x},-k_{z})$.
As a result, the gaps opened by $\Delta$ depend on the surface orientation,
leading to gapless chiral hinge states localized at the hinges connecting
different surfaces.

We take into account the orbital effect of the magnetic field via the Peierls
replacement in the hopping interaction $T_{ij}\rightarrow T_{ij}\exp(2\pi
i\int_{r_{i}}^{r_{j}}d{\bf{\bf r}}\cdot{\bf A}/\phi_{0})$, where $T_{ij}$ is
the hopping amplitude from sites $r_{i}$ to $r_{j}$, $\phi_{0}=h/e$ is flux
quantum. ${\bf A}$ is the vector potential for the magnetic field and it is
chosen as ${\bf A}=B(0,0,y\cos\theta-x\sin\theta)$ for concreteness (Li2, b).

For simplicity, we model the metallic leads with a conventional quadratic
energy dispersion and assume only a few transport channels in both leads such
that considerable reflections for the hinge channels are generated at the
interfaces. Furthermore, we consider the size of the system to be much larger
than the decay length of the hinge states in order to have a well-defined
multiple-loop interferometer based on hinge states. Under these
considerations, we calculate the two-terminal conductance numerically,
employing the standard Landauer-Büttiker approach (Landauer, 1970; Büttiker,
1986; Datta, 1995) in combination with lattice Green functions (see the
Supplemental Material (Li2, a)). We emphasize that our main results
illustrated below remain qualitatively the same if we choose other models for
SOTIs or leads.

Figure 2: (a) Conductance $G$ as a function of field direction $\theta$ at
small field strengths $B=B_{0}$ and $B_{0}/\sqrt{2}$, respectively. (b) $G$ as
a function of $B$ for $\theta=0$ and $\pi/4$, respectively. In these cases,
the oscillations have a single frequency. (c) Particular beating patterns as
varying $B$ at angle $\theta=0.48\pi$. (d) “Irregular” beating patterns as
varying $\theta$ at a large field strength $B=20B_{0}$. (e) The extracted
frequencies (square and circle dots) as a function of $\theta$. The two
frequencies can be described by $\omega_{1}=S|\cos\theta|$ and
$\omega_{2}=S|\sin\theta|$. (f) The low-energy spectrum of the SOTI in the
presence of a magnetic field $B=2B_{0}$ and $\theta=0.15\pi$. Other parameters
are $L_{z}=60a$, $W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$, and the Fermi
energy $E_{F}=0.002$.

Quantum interference pattern.—Now, we analyze the dependence of the
conductance $G$ on the magnetic field, combining general scattering theory and
concrete numerical simulations. Equation (3) implies an oscillation pattern of
$G$ with respect to the field direction $\theta$. As shown in Fig. 2(a),
$G(\theta)$ is periodic in $\theta,$ in accordance with the scattering theory.
Explicitly, we find that for weak magnetic fields $B\leq B_{0}$, $G(\theta)$
is approximately a sinusoidal function of $\theta$ and takes the maximal value
at $\theta=\pi/4+n\pi/2$, $n\in\\{0,1,2,3\\}$, when $W_{x}=W_{y}$. Here,
$B_{0}$ corresponds to the field strength at which the flux enclosed by the
front surface $S_{f}$ is one flux quantum for $\theta=0$. Thus, $G(\theta)$
has a period of $\pi/2$ in $\theta$. Moreover, $G(\theta)$ is minimal at
$\theta=\theta_{c}$ and symmetric in $\theta-\theta_{c}$, where
$\theta_{c}=n\pi/2$. For strong magnetic fields $B>B_{0}$, the number of
conductance peaks increases with increasing $B$, see Fig. 1(c). When
$W_{x}\neq W_{y},$ the period in $\theta$ becomes $\pi$ but $G(\theta)$ is
still symmetric in $\theta-\theta_{c}$.

Equation (3) also indicates an oscillation pattern of $G$ with respect to the
field strength $B,$ which is again fully confirmed by our numerical
simulations. When the magnetic field is applied in $x$\- or $y$-directions, or
at the specific angle $\theta=\pm\text{arctan}(W_{x}/W_{y})$, $G(B)$ exhibits
simple oscillations with a single frequency, see Fig. 2(b). Generally, the
oscillating conductance takes maximal or minimal values when the interference
loop encloses half a flux quantum. In our cases, $G(B)$ takes maximal values
at odd multiples of $B_{0}/2$ for $\theta=0$. The oscillation amplitude is
relatively smaller since only two of the four loops enclose half a flux
quantum at this field direction. For $\theta=\pi/4$, $G(B)$ takes maximal
values at odd multiples of $B_{0}/\sqrt{2}$, where the interference loop also
encloses half a flux quantum, leading to a resonance peak of $G(B)$. These
features signify the interferometer formed by hinge states being of Fabry-
Pérot type, as we further explain below.

Notably, there exist beating patterns, as signified by Eq. (2), where the
matrix $\mathcal{S}$ explicitly contains the two phases $\varphi\pm\phi$. When
the magnetic field deviates away from the special directions at
$\theta=n\pi/2$ (with $n\in\\{0,1,2,3\\}$) and
$\pm\text{arctan}(W_{x}/W_{y})$, beating oscillations of $G(B)$ are clearly
observed, as shown in Fig. 2(c). By performing discrete Fourier transformation
to the beating patterns, we obtain precisely two frequencies $\omega_{1}$ and
$\omega_{2}$. These frequencies depend strongly on the field direction
$\theta$ [dotted lines in Fig. 2(e)]. Explicitly, we find that the two
frequencies can be well described by $\omega_{1}=|S\cos\theta|=|\phi|/B$ and
$\omega_{2}=|S\sin\theta|=|\varphi|/B$ [solid lines in Fig. 2(e)],
respectively, where $S$ is the area of the surfaces of the system (we consider
the case with $W_{x}=W_{y}$ for simplicity). This corresponds exactly to the
two AB phases in Eq. (1), in excellent agreement with the results obtained
from scattering-matrix analysis. When $\theta=n\pi/2$, only one of the two
frequencies survives. When $\theta=\pi/4+n\pi/2$, the two frequencies become
identical. In both cases, the beating behavior in the oscillations disappear.
Similarly, $G(\theta)$ also shows beating-like patterns with respect to the
field direction $\theta$ with irregular peaks and dips for large magnetic
fields $B\gg B_{0}$, as shown in Fig. 2(d). This direction-induced beating
behavior is another manifestation of the two AB phases.

Higher-order Fabry-Pérot interference.—Next, we clarify, in which sense our
quantum interference pattern is a higher-order Fabry-Pérot type interference.
The two frequencies in the beating patterns correspond physically to two areas
of interference loops. As rotating the magnetic field, the two frequencies
match the effective areas of front surface $|S\cos\theta|$ and top surface
$|S\sin\theta|$ quite well, see Fig. 2(e). This fact indicates: (i) the
adjacent hinge states with opposite chirality form effective interference
loops and the interference is typically of Fabry-Pérot type; and (ii) there
are totally four interference loops but any two opposite surfaces of the
sample (namely, the front and back surfaces, or the top and bottom surfaces)
have the same effective area because of the chosen symmetry of the system
(Li2, c). The interference loops are made of chiral hinge modes located in 3D
space, protected by higher-order topology. When rotating the magnetic field,
one of the frequencies increases, whereas the other one decreases. Moreover,
the ratio between the two frequencies depends on $\theta$ as
$S_{f}/S_{t}=W_{y}|\cot\theta|/W_{x}$. Thus, the two frequencies coincide at
the critical field directions $\theta_{c}=\text{arctan}(W_{y}/W_{x})$ and
$\pi-\text{arctan}(W_{y}/W_{x})$, as shown in Fig. 2(e). These features
indicate the 3D nature of the interferometer.

The mechanism of the interferometer can be better understood by analyzing the
splitting of hinge states under magnetic fields. In the absence of magnetic
fields, the four chiral hinge states have a double degenerate linear spectrum
in $k_{z}$-direction, i.e., $\pm vk_{z}$. The magnetic field gives rise to a
spatially varying vector potential. Note that the hinge states are localized
at different hinges of the system. The local vector potential splits the
linear spectrum of the hinge states. Under the chosen gauge, the spectra of
hinge states are split as $+v(k_{z}\pm\delta k_{1}^{z})$ and
$-v(k_{z}\pm\delta k_{2}^{z})$, where the splittings are determined by $\delta
k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)|/2$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)|$/2 (Li2, a). Thus, the hinge states
acquire finite momenta even for vanishing Fermi energy [Fig. 2(f)]. When
propagating across the SOTI region, the hinge channels pick up extra phases,
$\pm\delta k_{1/2}^{z}L_{z}$. Such phases turn out to be exactly the AB phases
$\phi$ and $\varphi$, stemming from the magnetic flux enclosed by each loop.
Explicitly, the flux enclosed by front surface $S_{f}$ and top surface $S_{t}$
of the central SOTI region in Fig. 1(a) are given by $(\delta k_{1}^{z}+\delta
k_{2}^{z})L_{z}$ and $2\delta k_{1}^{z}L_{z}$, respectively. Plugging
$\phi,\varphi=\delta k^{z}L_{z}$ into Eq. (2), this indicates that the
dependence of $G$ on ${\bf B}$ can be attributed to the higher-order Fabry-
Pérot interference of the four hinge states. At special values of $\theta$,
say $\theta=\pi/4$ or $5\pi/4$, one kind of splitting vanishes whereas the
other one remains, $\delta k_{1}^{z}=0$ and $\delta k_{2}^{z}\neq 0$ (similar
results occur for $\theta=+3\pi/4,-\pi/4$). In these cases, we have only one
frequency.

Figure 3: (a) Low-energy spectrum of the SOTI in the presence of a magnetic
field. (b) The three frequencies in the case of $\mathcal{C}_{6}$ symmetric
SOTIs as functions of $\theta$. The lattice model and related parameters can
be found in the Supplemental Material (Li2, a).

Generalization to $\mathcal{C}_{6}$ symmetric SOTIs.—So far, we have focused
on the case of chiral SOTIs with four hinge states and a sample with
(effective) $\mathcal{C}_{4}$ symmetry. However, our scattering theory can be
generalized and applied to SOTIs with more pairs of hinge states. As an
example, we consider a $\mathcal{C}_{6}$ symmetric SOTI with three pairs of
chiral hinge states (Zhang  _et al._ , 2020c) and show the spectrum in Fig.
3(a). The geometry considered here is a hexagonal prism with $\mathcal{C}_{6}$
symmetry in $x$-$y$ plane. The hinge states split generally with different
amounts of momenta under a magnetic field. If we consider an interferometer
similar to the setup in Fig. 1(a), we can also observe characteristic
oscillations and beating patterns in the conductance which depend sensitively
on the field direction $\theta$. In this case, the conductance $G(\theta)$ is
$\pi/3$ periodic in $\theta$. Since there are three pairs of counter-
propagating hinge states, the oscillations can exhibit three frequencies in
general (Li2, a). Figure 3(b) illustrates the three frequencies as a function
of field direction $\theta$. Particularly, the oscillations are described by a
single and two frequencies for $\theta=0$ and $\theta=\pi/6$, respectively.

Discussion and summary.—In realistic samples, disorder and dephasing (Jiang
_et al._ , 2009) due to environmental noises may be detrimental to the
interference pattern of hinge states. However, we show numerically that the
oscillation patterns of the conductance in our setups persist under weak
disorder and dephasing (Li2, a). This indicates the robustness of our
proposal. Our results based on chiral SOTIs can also be applied to helical
SOTIs, which can be regarded as two copies of chiral SOTIs related by time-
reversal symmetry. Recently, SOTIs have been proposed in many candidate
materials. Among these candidates, bismuth (Schindler  _et al._ , 2018b) and
axion insulators including EuIn2As2 and MnBi2Te4 (Xu  _et al._ , 2019; Chen
_et al._ , 2020) provide potential platforms to test our predictions.

In summary, we have proposed a higher-order Fabry-Pérot interferometer and
revealed unique Aharonov-Bohm oscillations arising from topological hinge
states by tuning either strength or direction of an applied magnetic field.
Due to higher-order topology, the interferometer is intrinsically three-
dimensional and features particular beating patterns in the magneto-
conductance. Our results are robust and provide unique transport signatures of
hinge states in higher-order topological insulators.

###### Acknowledgements.

This work was supported by the DFG (SPP1666 and SFB1170 “ToCoTronics”), the
Würzburg-Dresden Cluster of Excellence ct.qmat, EXC2147, project-id 390858490,
and the Elitenetzwerk Bayern Graduate School on “Topological Insulators”. JL
acknowledges support by NSFC under Grants No. 11774317.

## References

  * Benalcazar  _et al._ (2017a) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators”, Science 357, 61 (2017a).
  * Benalcazar  _et al._ (2017b) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators”, Phys. Rev. B 96, 245115 (2017b).
  * Slager  _et al._ (2015) R.-J. Slager, L. Rademaker, J. Zaanen, and L. Balents, “Impurity-bound states and Green’s function zeros as local signatures of topology”, Phys. Rev. B 92, 085126 (2015).
  * Peng  _et al._ (2017) Y. Peng, Y. Bao, and F. von Oppen, “Boundary green functions of topological insulators and superconductors”, Phys. Rev. B 95, 235143 (2017).
  * Langbehn  _et al._ (2017) J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, “Reflection-symmetric second-order topological insulators and superconductors”, Phys. Rev. Lett. 119, 246401 (2017).
  * Song  _et al._ (2017) Z. Song, Z. Fang, and C. Fang, “$(d-2)$-dimensional edge states of rotation symmetry protected topological states”, Phys. Rev. Lett. 119, 246402 (2017).
  * Schindler  _et al._ (2018a) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topological insulators”, Science Advances 4 (2018a).
  * Geier  _et al._ (2018) M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, “Second-order topological insulators and superconductors with an order-two crystalline symmetry”, Phys. Rev. B 97, 205135 (2018).
  * Ezawa (2018) M. Ezawa, “Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices”, Phys. Rev. Lett. 120, 026801 (2018).
  * Khalaf (2018) E. Khalaf, “Higher-order topological insulators and superconductors protected by inversion symmetry”, Phys. Rev. B 97, 205136 (2018).
  * Park  _et al._ (2019) M. J. Park, Y. Kim, G. Y. Cho, and S. Lee, “Higher-order topological insulator in twisted bilayer graphene”, Phys. Rev. Lett. 123, 216803 (2019).
  * Trifunovic and Brouwer (2019) L. Trifunovic and P. W. Brouwer, “Higher-order bulk-boundary correspondence for topological crystalline phases”, Phys. Rev. X 9, 011012 (2019).
  * You  _et al._ (2018) Y. You, T. Devakul, F. J. Burnell, and T. Neupert, “Higher-order symmetry-protected topological states for interacting bosons and fermions”, Phys. Rev. B 98, 235102 (2018).
  * Hirosawa  _et al._ (2020) T. Hirosawa, S. A. Díaz, J. Klinovaja, and D. Loss, “Magnonic quadrupole topological insulator in antiskyrmion crystals”, Phys. Rev. Lett. 125, 207204 (2020).
  * Franca  _et al._ (2018) S. Franca, J. van den Brink, and I. C. Fulga, “An anomalous higher-order topological insulator”, Phys. Rev. B 98, 201114 (2018).
  * van Miert and Ortix (2018) G. van Miert and C. Ortix, “Higher-order topological insulators protected by inversion and rotoinversion symmetries”, Phys. Rev. B 98, 081110 (2018).
  * Schindler  _et al._ (2018b) F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, _et al._ , “Higher-order topology in bismuth”, Nat. Phys. 14, 918 (2018b).
  * Imhof  _et al._ (2018) S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, _et al._ , “Topolectrical-circuit realization of topological corner modes”, Nat. Phys. 14, 925 (2018).
  * Peterson  _et al._ (2018) C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologically protected corner states”, Nature 555, 346 (2018).
  * Serra-Garcia  _et al._ (2018) M. Serra-Garcia, V. Peri, R. Süsstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, “Observation of a phononic quadrupole topological insulator”, Nature 555, 342 (2018).
  * Chen  _et al._ (2019) X.-D. Chen, W.-M. Deng, F.-L. Shi, F.-L. Zhao, M. Chen, and J.-W. Dong, “Direct observation of corner states in second-order topological photonic crystal slabs”, Phys. Rev. Lett. 122, 233902 (2019).
  * Peng  _et al._ (2020) Y. Peng and G. Refael, “Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries”, Phys. Rev. Lett. 123, 016806 (2019).
  * Ghosh  _et al._ (2019) A. K. Ghosh, G. C. Paul, and A. Saha, “Higher order topological insulator via periodic driving”, Phys. Rev. B 101, 235403 (2020).
  * El Hassan  _et al._ (2019) A. El Hassan, F. K. Kunst, A. Moritz, G. Andler, E. J. Bergholtz, and M. Bourennane, “Corner states of light in photonic waveguides”, Nat. Photonics 13, 697 (2019).
  * Ni  _et al._ (2019) X. Ni, M. Weiner, A. Alù, and A. B. Khanikaev, “Observation of higher-order topological acoustic states protected by generalized chiral symmetry”, Nat. Mater. 18, 113 (2019).
  * Xie  _et al._ (2019) B.-Y. Xie, G.-X. Su, H.-F. Wang, H. Su, X.-P. Shen, P. Zhan, M.-H. Lu, Z.-L. Wang, and Y.-F. Chen, “Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals”, Phys. Rev. Lett. 122, 233903 (2019).
  * Qi  _et al._ (2020) Y. Qi, C. Qiu, M. Xiao, H. He, M. Ke, and Z. Liu, “Acoustic realization of quadrupole topological insulators”, Phys. Rev. Lett. 124, 206601 (2020).
  * Călugăru  _et al._ (2019) D. Călugăru, V. Juričić, and B. Roy, “Higher-order topological phases: A general principle of construction”, Phys. Rev. B 99, 041301 (2019).
  * Szabó  _et al._ (2020) A. L. Szabó, R. Moessner, and B. Roy, “Strain-engineered higher-order topological phases for spin-$\frac{3}{2}$ Luttinger fermions”, Phys. Rev. B 101, 121301 (2020).
  * Sheng  _et al._ (2019) X.-L. Sheng, C. Chen, H. Liu, Z. Chen, Z.-M. Yu, Y. X. Zhao, and S. A. Yang, “Two-dimensional second-order topological insulator in graphdiyne”, Phys. Rev. Lett. 123, 256402 (2019).
  * Li and Wu (2020) C.-A. Li and S.-S. Wu, “Topological states in generalized electric quadrupole insulators”, Phys. Rev. B 101, 195309 (2020).
  * Li  _et al._ (2020a) C.-A. Li, B. Fu, Z.-A. Hu, J. Li, and S.-Q. Shen, “Topological phase transitions in disordered electric quadrupole insulators”, Phys. Rev. Lett. 125, 166801 (2020a).
  * Li and Sun (2020) H. Li and K. Sun, “Pfaffian formalism for higher-order topological insulators”, Phys. Rev. Lett. 124, 036401 (2020).
  * Zhu (2018) X. Zhu, “Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields”, Phys. Rev. B 97, 205134 (2018).
  * Luo and Zhang (2019) X.-W. Luo and C. Zhang, “Higher-order topological corner states induced by gain and loss”, Phys. Rev. Lett. 123, 073601 (2019).
  * Ezawa (2019) M. Ezawa, “Braiding of Majorana-like corner states in electric circuits and its non-Hermitian generalization”, Phys. Rev. B 100, 045407 (2019).
  * Zhang  _et al._ (2020a) S.-B. Zhang, A. Calzona, and B. Trauzettel, “All-electrically tunable networks of Majorana bound states”, Phys. Rev. B 102, 100503 (2020a).
  * Zhang  _et al._ (2020b) S.-B. Zhang, W. B. Rui, A. Calzona, S.-J. Choi, A. P. Schnyder, and B. Trauzettel, “Topological and holonomic quantum computation based on second-order topological superconductors”, Phys. Rev. Research 2, 043025 (2020b).
  * Pahomi  _et al._ (2020) T. E. Pahomi, M. Sigrist, and A. A. Soluyanov, “Braiding Majorana corner modes in a second-order topological superconductor”, Phys. Rev. Research 2, 032068 (2020).
  * Queiroz and Stern (2019) R. Queiroz and A. Stern, “Splitting the hinge mode of higher-order topological insulators”, Phys. Rev. Lett. 123, 036802 (2019).
  * Li  _et al._ (2020b) C.-Z. Li, A.-Q. Wang, C. Li, W.-Z. Zheng, A. Brinkman, D.-P. Yu, and Z.-M. Liao, “Reducing electronic transport dimension to topological hinge states by increasing geometry size of dirac semimetal josephson junctions”, Phys. Rev. Lett. 124, 156601 (2020b).
  * Choi  _et al._ (2020) Y.-B. Choi, Y. Xie, C.-Z. Chen, J. Park, S.-B. Song, J. Yoon, _et al._ , “Evidence of higher-order topology in multilayer WTe2 from Josephson coupling through anisotropic hinge states”, Nat. Mater. 19, 974 (2020).
  * Aharonov and Bohm (1959) Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory”, Phys. Rev. 115, 485 (1959).
  * Bardarson  _et al._ (2010) J. H. Bardarson, P. W. Brouwer, and J. E. Moore, “Aharonov-Bohm oscillations in disordered topological insulator nanowires”, Phys. Rev. Lett. 105, 156803 (2010).
  * Zhang and Vishwanath (2010) Y. Zhang and A. Vishwanath, “Anomalous Aharonov-Bohm conductance oscillations from topological insulator surface states”, Phys. Rev. Lett. 105, 206601 (2010).
  * Peng  _et al._ (2010) H. Peng, K. Lai, D. Kong, S. Meister, Y. Chen, X.-L. Qi, S.-C. Zhang, Z.-X. Shen, and Y. Cui, “Aharonov–Bohm interference in topological insulator nanoribbons”, Nat. Mater. 9, 225 (2010).
  * Bardarson and Moore (2013) J. H. Bardarson and J. E. Moore, “Quantum interference and Aharonov–Bohm oscillations in topological insulators”, Rep. Prog. Phys. 76, 056501 (2013).
  * Akhmerov  _et al._ (2009) A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, “Electrically detected interferometry of Majorana fermions in a topological insulator”, Phys. Rev. Lett. 102, 216404 (2009).
  * Fu and Kane (2009) L. Fu and C. L. Kane, “Probing neutral Majorana fermion edge modes with charge transport”, Phys. Rev. Lett. 102, 216403 (2009).
  * Li  _et al._ (2012) J. Li, G. Fleury, and M. Büttiker, “Scattering theory of chiral Majorana fermion interferometry”, Phys. Rev. B 85, 125440 (2012).
  * Li  _et al._ (2019) C.-A. Li, J. Li, and S.-Q. Shen, “Majorana-Josephson interferometer”, Phys. Rev. B 99, 100504 (2019).
  * Wang  _et al._ (2016) L.-X. Wang, C.-Z. Li, D.-P. Yu, and Z.-M. Liao, “Aharonov-Bohm oscillations in Dirac semimetal Cd3As2 nanowires”, Nat. Commun. 7, 10769 (2016).
  * Büttiker (1992) M. Büttiker, “Scattering theory of current and intensity noise correlations in conductors and wave guides”, Phys. Rev. B 46, 12485 (1992).
  * Nazarov and Blanter (2006) Y. V. Nazarov and Y. M. Blanter,  _Quantum Transport: Introduction to Nanoscience_ (Cambridge University Press, 2006).
  * Maciejko  _et al._ (2010) J. Maciejko, E.-A. Kim, and X.-L. Qi, “Spin Aharonov-Bohm effect and topological spin transistor”, Phys. Rev. B 82, 195409 (2010).
  * Li2 (a) See the Supplemental Material including Refs. Landauer (1970); Büttiker (1986); Datta (1995); Zhang  _et al._ (2020c); Jiang  _et al._ (2009); Groth  _et al._ (2014); MacKinnon (1985) for details of (Sec. S1) the general scattering matrix analysis; (Sec. S2) chiral hinge states under magnetic fields; (Sec. S3) numerical simulations; (Sec. S4) trivial cases with perfect transmission; (Sec. S5) parameterizing the scattering matrix; (Sec. S6) the effective model for C6 symmetric SOTIs; (Sec. S7) the influence of disorder and dephasing; and (Sec. S8) multiple frequencies when the cross section is a trapezoid .
  * Zhang  _et al._ (2009) H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, “Topological insulators in $\text{Bi}_{2}\text{Se}_{3}$, $\text{Bi}_{2}\text{Te}_{3}$ and $\text{Sb}_{2}\text{Te}_{3}$ with a single Dirac cone on the surface”, Nat. Phys. 5, 438 (2009).
  * Li2 (b) We ignore the Zeeman interaction in this model because it is expected to have a minor influence on the AB effect .
  * Landauer (1970) R. Landauer, “Electrical resistance of disordered one-dimensional lattices”, Philos. Mag. 21, 863 (1970).
  * Büttiker (1986) M. Büttiker, “Four-Terminal Phase-Coherent Conductance”, Phys. Rev. Lett. 57, 1761 (1986).
  * Datta (1995) S. Datta,  _Electronic Transport in Mesoscopic Systems_ (Cambridge University Press, Cambridge, 1995).
  * Li2 (c) We also consider a geometry with lower symmetry, for instance, the cross section is a trapezoid, see the Supplementary Material Li2 (a) .
  * Zhang  _et al._ (2020c) R.-X. Zhang, F. Wu, and S. Das Sarma, “Möbius insulator and higher-order topology in ${\mathrm{MnBi}}_{2n}{\mathrm{Te}}_{3n+1}$”, Phys. Rev. Lett. 124, 136407 (2020c).
  * Jiang  _et al._ (2009) H. Jiang, S. Cheng, Q.-F. Sun, and X. C. Xie, “Topological insulator: A new quantized spin hall resistance robust to dephasing”, Phys. Rev. Lett. 103, 036803 (2009).
  * Xu  _et al._ (2019) Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, “Higher-order topology of the axion insulator ${\mathrm{EuIn}}_{2}{\mathrm{As}}_{2}$”, Phys. Rev. Lett. 122, 256402 (2019).
  * Chen  _et al._ (2020) R. Chen, S. Li, H.-P. Sun, Y. Zhao, H.-Z. Lu, and X. C. Xie, “Using nonlocal surface transport to identify the axion insulator”, arXiv:2005.14074 .
  * Groth  _et al._ (2014) C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, “Kwant: a software package for quantum transport”, New J. Phys. 16, 063065 (2014).
  * MacKinnon (1985) A. MacKinnon, “The calculation of transport properties and density of states of disordered solids”, Z. Phys. B Condens. Matter 59, 385 (1985).

Supplemental material for “Higher-order Fabry-Pérot Interferometer from
Topological Hinge States”

##  Appendix S1 General scattering matrix analysis

In this section, we present the details for the scattering matrix analysis of
the setup in the main text. Suppose there are $p_{L/R}$ conducting modes at
the Fermi level in the left/right leads. We can define generally
$(p_{L/R}+2)\times(p_{L/R}+2)$ scattering matrices, $S_{L}$ and $S_{R}$, to
describe the scattering at the left and right interfaces, respectively,

$\displaystyle\begin{pmatrix}b_{L}\\\ b_{L^{\prime}}\end{pmatrix}$
$\displaystyle=S_{L}\begin{pmatrix}a_{L}\\\ a_{L^{\prime}}\end{pmatrix},\ \
S_{L}=\begin{pmatrix}r_{L}&t_{L^{\prime}}\\\
t_{L}&r_{L^{\prime}}\end{pmatrix},$ (S1.1)
$\displaystyle\begin{pmatrix}b_{R^{\prime}}\\\ b_{R}\end{pmatrix}$
$\displaystyle=S_{R}\begin{pmatrix}a_{R^{\prime}}\\\ a_{R}\end{pmatrix},\ \
S_{R}=\begin{pmatrix}r_{R}&t_{R^{\prime}}\\\
t_{R}&r_{R^{\prime}}\end{pmatrix}.$ (S1.2)

Here, $a_{L/R}$ and $b_{L/R}$ indicate the incoming and outgoing modes that
propagate in the leads and scatter at the left/right interface, respectively;
$a_{L^{\prime}/R^{\prime}}$ and $b_{L^{\prime}/R^{\prime}}$ indicate the
incoming and outgoing hinge modes that propagate in the SOTI and scatter at
the left/right interface, respectively. The scattering matrix $S_{L/R}$
consists of four components $t_{L/R},$ $t^{\prime}_{L/R},$ $r_{L/R},$ and
$r^{\prime}_{L/R}$, corresponding to the transmission from left to right,
transmission from right to left, reflection from the right, and reflection
from the left, respectively. In the center SOTI region, the conducting chiral
hinge states pick up an AB phases when applying an external magnetic field.
Thus, the incoming and outgoing modes in the SOTI can be connected by a phase
matrix as

$\displaystyle\begin{pmatrix}a_{R^{\prime}1}\\\ a_{R^{\prime}2}\\\
a_{L^{\prime}1}\\\ a_{L^{\prime}2}\end{pmatrix}$
$\displaystyle=e^{i\lambda/2}\begin{pmatrix}e^{i(\varphi+\phi)/2}&0&0&0\\\
0&e^{-i(\varphi+\phi)/2}&0&0\\\ 0&0&e^{i(\varphi-\phi)/2}&0\\\
0&0&0&e^{-i(\varphi-\phi)/2}\end{pmatrix}\begin{pmatrix}b_{L^{\prime}1}\\\
b_{L^{\prime}2}\\\ b_{R^{\prime}1}\\\ b_{R^{\prime}2}\end{pmatrix},$ (S1.3)

where $\lambda=k_{F}L$ is the dynamic phase with $k_{F}$ the Fermi wave number
in $k_{z}$-direction and $L$ the length of the SOTI, and the two phases are
given by

$\phi=BS_{1}\sin\theta,\ \ \varphi=BS_{2}\cos\theta,$ (S1.4)

with $\theta$ the angle between the magnetic field direction and $x$-axis.

Substituting Eq. (S1.3) into Eq. (S1.2), we obtain

$\displaystyle\begin{pmatrix}e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}\\\
b_{R}\end{pmatrix}$
$\displaystyle=S_{R}\begin{pmatrix}e^{i\lambda/2}\Phi_{+}b_{L^{\prime}}\\\
a_{R}\end{pmatrix},$ (S1.5)

where $\Phi_{\pm}\equiv e^{i(\varphi\pm\phi)\sigma_{z}/2}$ and the Pauli
matrix $\sigma_{z}$ acts on (pseudo-)spin space for two left- or right-moving
hinge states. Writing Eqs. (S1.5) explicitly, we have

$\displaystyle e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}$
$\displaystyle=e^{i\lambda/2}r_{R}\Phi_{+}b_{L^{\prime}}+t_{R^{\prime}}a_{R},$
(S1.6) $\displaystyle b_{R}$
$\displaystyle=e^{i\lambda/2}t_{R}\Phi_{+}b_{L^{\prime}}+r_{R^{\prime}}a_{R}.$
(S1.7)

From Eq. (S1.6), we find

$a_{L^{\prime}}=e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}b_{L^{\prime}}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}.$
(S1.8)

Writing Eq. (S1.1) explicitly, we have

$\displaystyle b_{L}$ $\displaystyle=r_{L}a_{L}+t_{L^{\prime}}a_{L^{\prime}},$
(S1.9) $\displaystyle b_{L^{\prime}}$
$\displaystyle=t_{L}a_{L}+r_{L^{\prime}}a_{L^{\prime}},$ (S1.10)

Plugging Eq. (S1.8) into Eq. (S1.10), we obtain

$b_{L^{\prime}}=t_{L}a_{L}+r_{L^{\prime}}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}b_{L^{\prime}}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}),$
(S1.11)

and hence,

$b_{L^{\prime}}=(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}(t_{L}a_{L}+e^{i\lambda/2}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}a_{R}).$
(S1.12)

Plugging this result into Eq. (S1.7), we find $b_{R}$ as

$\displaystyle b_{R}$
$\displaystyle=t_{R}e^{i\lambda/2}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}(t_{L}a_{L}+e^{i\lambda/2}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}a_{R})+r_{R^{\prime}}a_{R}$
$\displaystyle=e^{i\lambda/2}t_{R}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}t_{L}a_{L}+[e^{i\lambda}t_{R}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}}+r_{R^{\prime}}]a_{R}.$
(S1.13)

Plugging Eq. (S1.10) into Eq. (S1.6), we obtain

$e^{-i\lambda/2}\Phi_{-}^{\dagger}a_{L^{\prime}}=e^{i\lambda/2}r_{R}\Phi_{+}(t_{L}a_{L}+r_{L^{\prime}}a_{L^{\prime}})+t_{R^{\prime}}a_{R},$
(S1.14)

and hence,

$a_{L^{\prime}}=(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}t_{L}a_{L}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R}).$
(S1.15)

Plugging Eq. (S1.15) into Eq. (S1.9), we find $b_{L}$ as

$\displaystyle b_{L}$
$\displaystyle=r_{L}a_{L}+t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}(e^{i\lambda}\Phi_{-}r_{R}\Phi_{+}t_{L}a_{L}+e^{i\lambda/2}\Phi_{-}t_{R^{\prime}}a_{R})$
$\displaystyle=[r_{L}+e^{i\lambda}t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}r_{R}\Phi_{+}t_{L}]a_{L}+e^{i\lambda/2}t_{L^{\prime}}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}t_{R^{\prime}}a_{R}.$
(S1.16)

Rewriting Eqs. (S1.13) and (S1.16), we obtain the effective scattering matrix
of the junction

$\displaystyle\begin{pmatrix}b_{L}\\\ b_{R}\end{pmatrix}$
$\displaystyle=S\begin{pmatrix}a_{L}\\\ a_{R}\end{pmatrix},\ \
S=\begin{pmatrix}r&t^{\prime}\\\ t&r^{\prime}\end{pmatrix},$ (S1.17)

where

$\displaystyle t$ $\displaystyle=t_{R}\mathcal{S}t_{L},\ \
r^{\prime}=r_{R^{\prime}}+e^{i\lambda/2}t_{R}\mathcal{S}r_{L^{\prime}}\Phi_{-}t_{R^{\prime}},$
$\displaystyle r$
$\displaystyle=r_{L}+e^{i\lambda/2}t_{L^{\prime}}\mathbb{\mathcal{S}}^{\prime}r_{R}\Phi_{+}t_{L},\
\ t^{\prime}=t_{L^{\prime}}\mathbb{\mathcal{S}}^{\prime}t_{R^{\prime}},$
$\displaystyle\mathbb{\mathcal{S}}$
$\displaystyle=e^{i\lambda/2}\Phi_{+}(1-e^{i\lambda}r_{L^{\prime}}\Phi_{-}r_{R}\Phi_{+})^{-1},$
$\displaystyle\mathbb{\mathcal{S}}^{\prime}$
$\displaystyle=e^{i\lambda/2}(1-e^{i\lambda}\Phi_{-}r_{L^{\prime}})^{-1}\Phi_{-}.$
(S1.18)

With this general scattering matrix, the two-terminal conductance can be
written as

$G(B,\theta)=\frac{e^{2}}{h}\mathrm{tr}(tt^{\dagger})=\frac{e^{2}}{h}\mathrm{tr}(t_{R}^{\dagger}t_{R}\mathcal{S}t_{L}t_{L}^{\dagger}\mathcal{S}^{\dagger}).$
(S1.19)

##  Appendix S2 Chiral hinge states under magnetic fields

In this section, we demonstrate the splitting behavior of the chiral hinge
states when rotating the magnetic field. In the absence of magnetic fields,
the four chiral hinge states have a double degenerate linear spectrum in
$k_{z}$-direction. The left-moving hinge states cross with the right-moving
ones at $k_{z}=0$. The magnetic field gives rise to a spatially varying vector
potential. Remember that the hinge states are localized at different hinges of
the system. The local vector potential splits the linear spectrum of hinge
states. Under the chosen gauge for the vector potential, ${\bf
A}=(0,0,B[y\cos\theta-x\sin\theta])$, the spectrum of hinge states is split as
$+v(k_{z}\pm\delta k_{1}^{z})$ and $-v(k_{z}\pm\delta k_{2}^{z})$, where the
splitting strengths are determined by $\delta
k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)/2|$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)/2|$. Thus, the splitting of hinge state
spectrum strongly depends on the field direction $\theta$.

We focus on the splitting of the chiral hinge states in the lower-energy
spectrum presented in Fig. S1. At $\theta=0\pi,$ two pairs of the hinge states
split by equal value. At $\theta=\pi/4$, only one pair of the hinge states can
be split, whereas the other one remain unaltered, i.e., $\delta k_{1}^{z1}=0$.
The spectrum at $\theta=\pi/2$ looks the same as for $\theta=0$. Another
special field direction is at $\theta=3\pi/4$ at which we have instead $\delta
k_{2}^{z}=0$. At $\theta=\pi$, the spectrum is the same as that at $\theta=0$.
This evolution with rotating the magnetic field is consistent with the
analytical results $\delta k_{1}^{z}=BW_{x}|\sin(\theta-\pi/4)/2|$ and $\delta
k_{2}^{z}=BW_{y}|\cos(\theta+\pi/4)/2|$.

Figure S1: Evolution of the hinge states spectrum when rotating the field
direction from $\theta=0$ to $\pi$. Here, we choose parameters: $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$. The field strength is fixed at
$B=2B_{0}$.

##  Appendix S3 Numerical simulation details

To calculate the conductance, we employ the Landauer-Büttiker formalism
(Landauer, 1970; Büttiker, 1986; Datta, 1995) in combination with lattice
Green functions. The two-terminal conductance is evaluated as

$G=\frac{e^{2}}{h}\mathrm{Tr}[\Gamma_{L}G^{r}\Gamma_{R}G^{a}],$ (S3.1)

where the line width function

$\Gamma_{\beta}=i[\Sigma_{\beta}-\Sigma_{\beta}^{\dagger}]$ (S3.2)

with the $\Sigma_{\beta}$ being the self-energy due to coupling of the lead
$\beta\in\\{L,R\\}$ to the central region of interest. The retarded and
advanced Green function, $G^{r}$ and $G^{a}$, are obtained as

$G^{r}=(G^{a})^{\dagger}=(E_{F}-H_{c}-\Sigma_{L}^{r}-\Sigma_{R}^{r})^{-1}.$
(S3.3)

Here, both the self-energy $\Sigma_{\beta}$ and the Green function $G^{r/a}$
can be calculated by using the recursive method (MacKinnon, 1985).

In the numerical simulations, we choose the parameters $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,$$b=-1$, $v=1$ and $\Delta=1$ for the SOHI, and $m=3,$
$v=0,$ $b=-1$, and $\Delta=0$, and chemical potential $\mu=-0.1$ for the two
leads. Without loss of generality, we set the Fermi energy in the SOHI at
$E_{F}=0.002$.

##  Appendix S4 Trivial cases of perfect transmission

In this section, we demonstrate the behaviors when the interfaces of the
proposed setup are totally transparent. Under this condition, the chiral hinge
states do not talk to each other and thus no interference loop is forming. As
a result, the two-terminal conductance $G$ is quantized at $2e^{2}/h$ and
independent of the magnetic field. Let us consider two scenarios responsible
for such transparent interfaces:

  * •

Case 1: the leads are also made of the same SOTIs. Chiral hinge states exist
in all regions of space and pass from one lead to the other lead directly;

  * •

Case 2: the leads are made of conventional semiconductors or topological
insulators but highly doped. In this case, there are too many channels in the
leads such that chiral hinge states loose quantum coherence once entering the
leads.

The results for these two cases are presented in Fig. S2. The two-terminal
conductance is fixed at $2e^{2}/h$ and has no dependence on neither the field
strength $B$ nor the direction $\theta$.

Figure S2: For the two trivial cases 1 (a) and 2 (b) as discussed in this
section, the conductance is quantized at $2e^{2}/h$ and has no dependence on
either the field strength $B$ or field direction $\theta$.

##  Appendix S5 Parametrizing the scattering matrix

In this section, we parameterize the scattering matrix with the help of the
numerical method. There are two interfaces in our proposed setup. Each
interface is described by a $4\times 4$ scattering matrix, i.e., $S_{L}$ and
$S_{R}$ as listed above, respectively. Due to time-reversal symmetry breaking,
$S_{L}$ and $S_{R}$ are unitary matrices.Parametrizing these scattering
matrices is cumbersome because of the choice of at least 16 free parameters.
Instead, we obtain the scattering matrices directly from numerical simulations
as explained below.

Figure S3: Transport properties obtained using the analytical formula, Eq. (3)
in the main text, after we parametrize the relevant scattering matrix
according to the corresponding parameter settings. (a) Conductance oscillation
pattern as function of field strength $B$ for different field directions
$\theta=0$ and $\theta=0.25\pi$, respectively. (b) Conductance oscillation
pattern as function of field direction $\theta$ for different field strengths
$B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$ and $B=1B_{0}$, respectively. (c)
Beating pattern of conductance as function of field strength $B$. (d) Beating
pattern of conductance as function of field direction $\theta$.

Let us consider a simpler junction with two semi-infinite regions in $z$
direction: one region is made of a trivial insulators in the region $z<0$, and
the other regions made of the SOTI in the region $z>0$. The parameters for
lead and SOTI are taken the same as those in our interference setup. Then, the
interface of this simpler setup mimics the left interface of our interference
setup. The scattering matrix at this interface can be obtained numerically by
calculating the retarded Green functions for the two regions and then
employing the Fisher-Lee relation, or directly using the Kwant algorithm
(Groth  _et al._ , 2014). A similar procedure applies for the right interface.

Known from the conductance, described by Eqs. (2) and (3) in the main text,
the relevant four matrices are $t_{L},t_{R},r_{L^{\prime}}$ and $r_{R}$ (or
another group $t_{L},t_{R},r_{L}$ and $r_{R^{\prime}}$ ). Under the same
parameter setting with the original setup, we obtain the four matrices as

$\displaystyle t_{L}$
$\displaystyle=\left(\begin{array}[]{cc}-0.06198685+0.18465009i,&-0.7498761-0.16515456i\\\
0.60757993+0.469507i,&0.10194116-0.16596995i\end{array}\right),$ (S5.3)
$\displaystyle t_{R}$
$\displaystyle=\left(\begin{array}[]{cc}0.55777311+0.34346096i,&-0.0156968+0.44520283i\\\
-0.26134039-0.36076744i,&-0.27139131+0.59617366i\end{array}\right),$ (S5.6)
$\displaystyle r_{L^{\prime}}$
$\displaystyle=\left(\begin{array}[]{cc}0.17964098-0.37617273i,&0.39605429+0.20453843i\\\
0.25086872-0.36845604i,&-0.33709527-0.2452419i\end{array}\right),$ (S5.9)
$\displaystyle r_{R}$
$\displaystyle=\left(\begin{array}[]{cc}-0.40202362-0.44148246i,&-0.06268933-0.1095996i\\\
0.05582996+0.1132477i,&-0.14131563-0.58013761i\end{array}\right).$ (S5.12)

Figure S3 presents the transport properties obtained using the analytical
formula Eq. (3) in the main text after we parameterize the relevant scattering
matrix according to the corresponding parameter settings. We see that the main
features of the conductance are qualitatively the same as the numerical
results in Fig. 2 of the main text. As shown in Fig. S3 (a), each pattern has
single frequency; the oscillation amplitude is larger at $\theta=\pi/4$; and
the period of red line is about $\sqrt{2}$ times that of the blue line. In
Fig. S3 (b), there are two peaks and the oscillation amplitude is more
pronounced when $B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$. In Fig. S3 (c), the
conductance shows a beating pattern of $B$ at $\theta=0.48\pi$. Finally, in
Fig. S3 (c), the conductance shows an “irregular” beating pattern as a
function of $\theta$ for large $B$.

##  Appendix S6 Model of $\mathcal{C}_{6}$ symmetric SOTIs

The effective model for a $\mathcal{C}_{6}$ symmetric SOTI on a stacked
hexagonal lattice can be written as (Zhang  _et al._ , 2020c)

$H_{\text{hex}}=\begin{pmatrix}h+ms_{z}\sigma_{0}&h_{AB}\\\
h_{AB}^{\dagger}&h+ms_{z}\sigma_{0}\end{pmatrix},$ (S6.1)

where

$\displaystyle h$ $\displaystyle=\tilde{C}-\dfrac{4}{3}C_{2}(\cos k_{1}+\cos
k_{2}+\cos k_{3})+\dfrac{v}{3}(2\sin k_{1}+\sin k_{2}+\sin k_{3})\Gamma_{1}$
$\displaystyle\ \ \ +\dfrac{v}{\sqrt{3}}(\sin k_{2}-\sin
k_{3})\Gamma_{2}+w[-\sin k_{1}+\sin k_{2}+\sin
k_{3}]\Gamma_{4}+\Big{[}M-\dfrac{4}{3}M_{2}(\cos k_{1}+\cos k_{2}+\cos
k_{3})\Big{]}\Gamma_{5}$ $\displaystyle h_{AB}$ $\displaystyle=-2C_{1}\cos
k_{z}+2v_{z}\sin k_{z}\Gamma_{3}-2M_{1}\cos k_{z}\Gamma_{5},$ (S6.2)

and $k_{1}=k_{x},$ $k_{2}=(k_{x}+\sqrt{3}k_{y})/2$ and $k_{3}=k_{1}-k_{2}$.
$h_{AB}$ describes the hopping between neighboring layers. The $\Gamma$
matrices are defined as $\Gamma_{i}=s_{i}\sigma_{1}$ with $i\in\\{1,2,3\\}$,
$\Gamma_{4}=s_{0}\sigma_{2}$ and $\Gamma_{5}=s_{0}\sigma_{3}$ with ${\bf s}$
and ${\bf\sigma}$ the Pauli matrices and $s_{0}$ and $\sigma_{0}$ the
corresponding identity matrices. The model parameters are defined as
$\tilde{C}=C_{0}+2C_{1}+4C_{2}$, and $\tilde{M}=M_{0}+2M_{1}+4M_{2}$. In the
numerical calculations, we set the parameters $C_{0}=0,$ $C_{1}=C_{2}=0.5,$
$M_{0}=-2.5$, $M_{1}=M_{2}=1,$ $v=v_{z}=1$, $m=0.5$ and $w=2$. The perimeter
of the hexagonal prisms is $6\times n_{s}$ with side length $n_{s}=25a$ and
$L_{z}=50a$.

Figure S4: Left panel: influence of disorder on the interference pattern. We
average over 100 disorder configurations. Right panel: influence of dephasing
on the interference pattern. We choose parameters: $L_{z}=60a$,
$W_{x}=W_{y}=12a$, $m=2,b=-1,v=1$, $\Delta=1$. The magnetic field strength is
fixed at $B=B_{0}/\sqrt{2}\thickapprox 0.71B_{0}$. Figure S5: (a) Cross
section of the SOTI in a trapezoid geometry. (b) Four frequencies as functions
of field direction $\theta$. Here, S1 (S2, S3, S4) indicates the bottom
(right-side, top, left-side) surface of the trapezoid. Solid lines are
analytical results, and the dotted lines are obtained by Fourier
transformation from the conductance beating pattern. Discrepancy between them
maybe due to the finite-size effects. We choose parameters for the SOTI as:
$L_{z}=200a$, $m=2,b=-1,v=1$, and $\Delta=1$.

##  Appendix S7 Disorder and dephasing

In this section, we show that the interference pattern of our interferometer
is robust against disorder and dephasing.

To mimic disorder, we consider the onsite type $V_{\mathrm{dis}}=V({\bf
r})I_{4\times 4}$ with random function $V({\bf r})$ distributed uniformly
within the interval $[-U_{0}/2,U_{0}/2]$ and $U_{0}$ being the disorder
strength. It is shown in Fig. S4 that as increasing the disorder strength
$U_{0}$, the oscillation amplitude decreases gradually. However, the
interference pattern of the conductance remains even when the disorder
strength is quite strong (comparable with the bulk gap).

We also consider dephasing in the SOTI region in our setup by attaching each
site in the discretized lattice model with a virtual lead (Jiang  _et al._ ,
2009). These virtual leads are coupled to the system via the self-energy
$-i\Gamma/2$ with $\Gamma$ measuring the dephasing strength ($1/\Gamma$
signifies the quasiparticle life time). It is shown in Fig. S4 that the
interference pattern of the conductance remains under weak dephasing strength.
As increasing dephasing strength $\Gamma$, the electrons loose their phase
memory quickly and thus the oscillation amplitudes decrease accordingly.

The above results shows the oscillation pattern basically remains under weak
disorder and dephasing, which indicates the robustness of our proposal to show
quantum interference of hinge states.

##  Appendix S8 Multiple frequencies when the cross section is a trapezoid

In this section, we present the multiple-frequency case when the cross section
of SOTI is a trapezoid, as shown in Fig. S5(a). In this case, there are
generally four frequencies in the conductance oscillation as function of field
strength $B$. Figure S5(b) shows the four frequencies as varying the field
direction $\theta$. The numerical results are basically consistent with the
analytic ones (obtained by the effective surface areas).

One can see from Fig. S5(b) that the four frequencies do not increase or
decrease simultaneously, which stems from the 3D nature of the interferometer,
as discussed in the main text. Beside, we find that there always exist two
field directions (the field directions along the diagonal of the trapezoid),
at which only two of the four frequencies will survive.


# Stochastic equations and dynamics beyond mean-field theory

Tommaso Rizzo
ISC-CNR, UOS Rome, Università “Sapienza”,
Piazzale A. Moro 2, I-00185, Rome, Italy
Dip. Fisica, Università “Sapienza”,
Piazzale A. Moro 2, I–00185, Rome, Italy

###### Abstract

The dynamical transition occurring in spin-glass models with one step of
Replica-Symmetry-Breaking is a mean-field artifact that disappears in finite
systems and/or in finite dimensions. The critical fluctuations that smooth the
transition are described in the $\beta$ regime by dynamical stochastic
equations. The quantitative parameters of the dynamical stochastic equations
have been computed analytically on the 3-spin Bethe lattice Spin-Glass by
means of the (static) cavity method and the equations have been solved
numerically. The resulting parameter-free dynamical predictions are shown here
to be in excellent agreement with numerical simulation data for the
correlation and its fluctuations.

The idea of a deep connection between structural glasses and spin-glasses (SG)
with one step of Parisi’s Replica symmetry breaking (1RSB) was put forward
more than thirty years ago and has proven to be very influential [7, 2, 23].
Mean-field SG models with 1RSB display a dynamical transition temperature
$T_{d}$ where the Gibbs measure splits into an exponential number of
equilibrium states, i.e.  there is a finite configurational complexity. Often
this is followed by a second (static) transition at $T_{s}$ where the
configurational entropy vanishes. The static transition naturally evokes the
Kauzmann temperature of supercooled liquids, while the dynamical transition
turns out to have the same qualitative features of the Mode-Coupling-Theory
transition [6].

While the existence of the Kauzmann temperature is controversial, the MCT
transition temperature is a very popular concept with both experimentalists
and theorists, indeed MCT captures many qualitative features of the physics of
liquids upon supercooling, notably two-step relaxation and stretched
exponential decay. Furthermore it agrees quantitatively with numerical
simulations [12, 8, 21, 22]. Its main flaw is that in experiments one does not
observe the sharp transition predicted by MCT but rather a crossover from
power-law to exponential increase of the relaxation time. Many authors believe
thus that it should be possible to fix MCT in some way although there is no
agreement on how to do it.

In the context of mean-field 1RSB SG one easily recognizes that the transition
at $T_{d}$ is spurious due to their their mean-field nature and expects that
ergodicity between $T_{d}$ and $T_{s}$ is restored in finite dimensions by
some activated processes: in practice one needs to go beyond mean-field
dynamics. On the other hand it seems that the nature of the problem is
different for temperatures close to $T_{d}$ or deep in the MF glassy phase
between $T_{d}$ and $T_{s}$. In the following we will solely discuss progress
made recently for temperature close to $T_{d}$, and we refer the reader to
[17] for recent work in the MF glassy phase.

In order to go beyond MF and restore ergodicity one has to include
fluctuations neglected at the MF level. In 1RSB SG one sees that the dynamical
transition has the features of a second-order phase transition [5] and thus it
is to be expected that the fluctuations are naturally described by a simple
effective theory. Due to certain non trivial features of the corresponding
theory it turns out that it is equivalent to a set of dynamical stochastic
equations called stochastic-$\beta$-Relaxation (SBR) equations in [15, 16]. In
the following we will demonstrate the validity of SBR in 1RSB SG by comparing
its predictions with numerical simulations for the paradigmatic Ising $p$-spin
model.

We consider a system of $N$ spins each of which interacts with a fixed number
$c=6$ of $p$-spin interactions with $p=3$ and evolve with Metropolis dynamics.
The (random) lattice is such that in the large $N$ limit loops are
increasingly rare and it tends to the corresponding $c=6$ and $p=3$ Bethe
lattice, so that many thermodynamics quantities can be computed analytically
by means of the cavity method 111Note that instead of a random regular graph
we generate the lattice by applying the $M$-layer construction [1, 20] to a
triangular lattice with an interaction for each plaquette. This allows to
obtain a tripartite graph enabling each set of spins to be updated at the same
time.. The $p$-spin interactions are chosen randomly with values $J_{ijk}=\pm
1$ in the annealed ensemble. Instead of the standard white average this
corresponds to weight each disorder instance with a factor proportional to the
partition function of the model. This is convenient for numerical studies
because the averages over the interactions and the configurations can be
exchanged, in particular one can choose a random configuration and then
generate the $J$’s accordingly [5, 9, 10]. The order parameter is the
correlation with the initial condition:

$C(t)={1\over N}\sum_{i}s_{i}(t)s_{i}(0)$ (1)

In the thermodynamic limit the model displays a dynamical transition at a
temperature $T_{d}$, the correlation with the initial equilibrium
configuration $C(t)$ approaches a plateau value $q_{d}$ with a power law

$\langle C(t)\rangle\approx q_{d}+\frac{1}{(t/t_{0})^{a}}$ (2)

where the angle brackets mean average with respect to both the disorder and
different thermal trajectories starting from the same initial configuration
[5]. At finite $N$ one observes instead that even at $T=T_{d}$ the correlation
deviates from the above mean-field expression and crosses the plateau value at
a finite time that increases with the system size. In order to describe this
phenomenon we must compute corrections to mean-field theory. Following the
arguments and computations of [15, 16] one can argue that close to $T_{d}$ the
fluctuation of $g(t)$, defined as

$g(t)\equiv C(t)-q_{d}\ ,$ (3)

are described by SBR, meaning that the generic $K$-point average obeys for
$1\ll N<\infty$:

$\langle g(t_{1})\dots
g(t_{K})\rangle\approx[\hat{g}(t_{1})\dots\hat{g}(t_{K})]\,.$ (4)

where $\hat{g}(x,t)$ in the RHS is the solution of the SBR equations:

$\sigma+s=-\lambda\,\hat{g}^{2}(t)+{d\over
dt}\int_{0}^{t}\hat{g}(t-s)\hat{g}(s)ds\ .$ (5)

The separation parameter $\sigma$ measures the distance from the critical
point and vanishes at $T=T_{d}$. The square brackets mean average with respect
to the field $s(x)$ that is a quenched Gaussian random fluctuation of
$\sigma$:

$[s]=0\,,\ [s^{2}]=\Delta\sigma^{2}\ .$ (6)

the SBR equations have to be solved with the short-time condition

$\lim_{t\rightarrow 0}\hat{g}(t)(t/t_{0})^{a}=1$

where $\lambda$ and $a$ are related by the MCT relationship
$\lambda={\Gamma^{2}(1-a)\over\Gamma(1-2a)}$. In practice for times smaller
than a Ginzburg time $t_{G}\approx N^{1/(4a)}$ the observables on the LHS of
eq. (4) can be accurately approximated with the values they have on the Bethe
lattice while on times of order $t_{G}$ they are described by the RHS [20].
This leads to the initial conditions of the SBR equations: the short-time
behavior on times $O(t_{G})$ matches the long-time behavior for times $1\ll
t\ll t_{G}$, i.e. the mean-field result given by eq. 2.

Figure 1: Ising $p$-spin glass model on fixed connectivity lattice: average
correlation with initial equilibrium condition vs time at $T=T_{d}$. Points
from bottom to top: numerical data for $N=4.5\times 10^{5}$, $N=9\times
10^{5}$, $N=1.8\times 10^{6}$ (Sample numbers are respectively 9554, 8048,
7701, error bars are negligible on the scale of the plot). The data follow the
Bethe lattice $N=\infty$ curve (dashed blue) at initial times and deviate from
it at later times increasing with $N$ eventually crossing the plateau value
$q_{d}=0.78184$. The solid lines are the corresponding SBR predictions
describing the data when they start to deviate from the mean-field curve, see
text.  Figure 2: Ising $p$-spin glass model on fixed connectivity lattice:
$\chi_{4}(t)$ vs. time. Points: numerical data for $N=1.8\times 10^{6}$
(Sample number is 7701). The data follow the mean-field asymptote $t^{2a}$
(dashed blue) at initial times and deviate from it at later times. Solid: SBR
prediction describing the data when they start to deviate from the mean-field
short times asymptote $t^{2a}$ (dashed blue). SBR predicts a large times
asymptotes $t^{2b}$ (dashed, yellow)

Note that on the LHS of eq. ( 4) we have a model with a complex microscopic
dynamics for which no analytic treatment of dynamics is available (not even in
the fully connectec case), on the RHS we have a (numerically) solvable set of
equations that were derived in [15, 16] starting from symmetry considerations
(essentially the detailed balance property of the dynamics) but without
reference to any specific microscopic model. The microscopic details however
determine the actual values of the five SBR parameters $a$, $t_{0}$,
$\Delta\sigma$ and $\sigma$ that are needed to get quantitative predictions.
In order to obtain parameter-free predictions these model-dependent parameters
have been computed analytically using existing [4, 3, 14, 13, 11] and novel
techniques based on the mean-field cavity method on the glassy phase of the
transition (details elsewhere). For the Bethe lattice with $3$-spin
interactions $J=\pm 1$ and connectivity six we have thus obtained (both in the
annealed and quenched ensemble)

$T_{d}=1.087815\ ,\ q_{d}=0.78184\ ,\ a=0.31228\ ,\
\Delta\sigma^{2}={0.02176\over N}\\\ .$ (7)

The microscopic time-scale $t_{0}$ depends on the actual microscopic dynamics
and therefore cannot be estimated by the static cavity method, this is the
only parameter that had to be extracted once and for all by fitting numerical
data in the mean-field limit (i.e.  at very large $N$) with the MF eq. (2),
leading to $t_{0}=0.00866$.

Within SBR, mean-field theory is recovered setting $\Delta\sigma^{2}=0$, in
this case one recovers the critical MCT equation [6], in particular for
$\sigma\geq 0$ ($T<T_{d}$) $C(t)$ never goes below the plateau value. At
finite $N$ there is instead a finite but small $\Delta\sigma$ so that the MCT
transition is avoided and $C(t)$ crosses the plateau at a finite time for all
values of $\sigma$. In figure (1) we compare numerical data at $T=T_{d}$
($\sigma=0$) with the SBR predictions that were obtained solving numerically
(by time discretization) eq. (5) for many instances of the $s$’s. From the
figure we note that the quality of the SBR predictions increases with $N$ and
is excellent for $N=1.8\times 10^{6}$, especially considering that there is no
single fitting parameter as even $t_{0}$ is estimated through an independent
procedure.

SBR provides not only the average correlation but, according to eq. (4), also
its fluctuations of all orders. To demonstrate this in fig. (2) we compare
data and theory for the $\chi_{4}(t)$ function that yields the fluctuations of
the correlation:

$\chi_{4}(t)\equiv N\,(\langle C^{2}(t)\rangle-\langle C(t)\rangle^{2})\ .$
(8)

Within MF theory, at $T=T_{d}$ $\chi_{4}(t)$ should diverge with time as
$t^{2a}$ [20], instead on the Ginzburg time scale $t_{G}$ over which $C(t)$
deviates from MF and reaches the plateau value $q_{d}$, $\chi_{4}(t)$ deviates
from the MF law and in the late $\beta$ regime follows a more pronounced
$t^{2b}$ growth where $b$ is related to $a$ by
$\Gamma^{2}(1+b)/\Gamma(1+2b)=\Gamma^{2}(1-a)/\Gamma(1-2a)$.

Our aim here was to demonstrate quantitatively that the theory is correct for
1RSB SG and we refer instead the reader to [18, 19] for a discussion of the
rich phenomenology displayed qualitatively by SBR when considering the case of
finite dimensions (in which $g(t)$ is promoted to a field $g(x,t)$) and the
case of temperature slightly above and below $T_{d}$. Overall SBR predicts not
only that the transition at $T_{d}$ is avoided as shown by fig. (1) or that
the fluctuations deviate from mean-field theory as in fig. (2) but also that
there is an essential qualitative change of in the structure of the
fluctuations with the appearance of dynamical heterogeneities. The excellent
agreement between numerical data and the parameter-free predictions is
reassuring because SBR is the natural theory for 1RSB SG, however it can be
also obtained solely from the symmetries of the original dynamical problem:
this implies that the same description is potentially valid also for different
models including notably supercooled liquids. In [20] its validity has been
demonstrated in a class of Kinetically-Constrained-Models along the lines
discussed here, by first computing analytically the coefficients of the theory
and then comparing with numerical simulations. Establishing the relevance of
the theory and computing its parameters for actual supercooled liquids is a
promising open problem.

## References

  * [1] Ada Altieri, Maria Chiara Angelini, Carlo Lucibello, Giorgio Parisi, Federico Ricci-Tersenghi, and Tommaso Rizzo.  Loop expansion around the bethe approximation through the m-layer construction.  Journal of Statistical Mechanics: Theory and Experiment, 2017(11):113303, 2017.
  * [2] Giulio Biroli and Juan P Garrahan.  Perspective: The glass transition.  The Journal of chemical physics, 138(12):12A301, 2013.
  * [3] F Caltagirone, U Ferrari, L Leuzzi, G Parisi, F Ricci-Tersenghi, and T Rizzo.  Critical slowing down exponents of mode coupling theory.  Physical review letters, 108(8):085702, 2012.
  * [4] Silvio Franz, Michele Leone, Federico Ricci-Tersenghi, and Riccardo Zecchina.  Exact solutions for diluted spin glasses and optimization problems.  Physical Review Letters, 87(12):127209, 2001.
  * [5] Silvio Franz, Giorgio Parisi, Federico Ricci-Tersenghi, and Tommaso Rizzo.  Field theory of fluctuations in glasses.  The European Physical Journal E, 34(9):102, 2011.
  * [6] Wolfgang Götze.  Complex dynamics of glass-forming liquids: A mode-coupling theory, volume 143.  OUP Oxford, 2008.
  * [7] TR Kirkpatrick and D Thirumalai.  p-spin-interaction spin-glass models: Connections with the structural glass problem.  Physical Review B, 36(10):5388, 1987.
  * [8] Walter Kob.  Computer simulations of supercooled liquids and glasses.  Journal of Physics: Condensed Matter, 11(10):R85, 1999.
  * [9] Florent Krzakala and Lenka Zdeborová.  On melting dynamics and the glass transition. i. glassy aspects of melting dynamics.  The Journal of chemical physics, 134(3):034512, 2011.
  * [10] Florent Krzakala and Lenka Zdeborová.  On melting dynamics and the glass transition. ii. glassy dynamics as a melting process.  The Journal of chemical physics, 134(3):034513, 2011.
  * [11] Carlo Lucibello, Flaviano Morone, and Tommaso Rizzo.  One-dimensional disordered ising models by replica and cavity methods.  Physical Review E, 90(1):012140, 2014.
  * [12] Markus Nauroth and Walter Kob.  Quantitative test of the mode-coupling theory of the ideal glass transition for a binary lennard-jones system.  Physical Review E, 55(1):657, 1997.
  * [13] Giorgio Parisi, Federico Ricci-Tersenghi, and Tommaso Rizzo.  Diluted mean-field spin-glass models at criticality.  Journal of Statistical Mechanics: Theory and Experiment, 2014(4):P04013, 2014.
  * [14] Giorgio Parisi and Tommaso Rizzo.  Critical dynamics in glassy systems.  Physical Review E, 87(1):012101, 2013.
  * [15] Tommaso Rizzo.  Long-wavelength fluctuations lead to a model of the glass crossover.  EPL (Europhysics Letters), 106(5):56003, 2014.
  * [16] Tommaso Rizzo.  Dynamical landau theory of the glass crossover.  Physical Review B, 94(1):014202, 2016.
  * [17] Tommaso Rizzo.  Path integral approach unveils role of complex energy landscape for activated dynamics of glassy systems.  Physical Review B, 104(9):094203, 2021.
  * [18] Tommaso Rizzo and Thomas Voigtmann.  On the nature of the glass crossover.  arXiv preprint arXiv:1504.06263, 2015.
  * [19] Tommaso Rizzo and Thomas Voigtmann.  Qualitative features at the glass crossover.  EPL (Europhysics Letters), 111(5):56008, 2015.
  * [20] Tommaso Rizzo and Thomas Voigtmann.  Solvable models of supercooled liquids in three dimensions.  Physical Review Letters, 124(19):195501, 2020.
  * [21] Francesco Sciortino and Walter Kob.  Debye-waller factor of liquid silica: Theory and simulation.  Physical review letters, 86(4):648, 2001.
  * [22] Fabian Weysser, Antonio Manuel Puertas, Matthias Fuchs, and Th Voigtmann.  Structural relaxation of polydisperse hard spheres: Comparison of the mode-coupling theory to a langevin dynamics simulation.  Physical review E, 82(1):011504, 2010.
  * [23] Peter G Wolynes and Vassiliy Lubchenko.  Structural glasses and supercooled liquids: Theory, experiment, and applications.  John Wiley & Sons, 2012.