# Thermodynamic contacts and breathing mode physics of 1D p-wave Fermi gases
in the high temperature limit

Jeff Maki  Department of Physics and HKU-UCAS Joint Institute for Theoretical
and Computational Physics,
Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
The University of Hong Kong, Hong Kong, China

(August 28, 2024)

###### Abstract

An important tool for understanding the effects of interactions in
harmonically trapped atomic gases is the examination of their collective
modes. One such mode is the breathing or monopole mode, which is special as it
is constrained to occur at twice the harmonic trapping frequency when the
interactions are scale invariant. When the interactions are not scale
invariant, the frequency of the breathing mode will deviate from twice the
trap frequency. The deviation itself depends on the thermodynamic contacts,
which describe how the energy changes with the interactions. In this work I
examine how the thermodynamic contacts and the breathing mode frequency of a
spin-polarized one-dimensional (1D) p-wave Fermi gas depend on the 1D
scattering volume, $\ell$, and the effective range, $r$, in the high
temperature limit. Such dynamics can be studied in experiments and provide a
tool for understanding how the dynamics depend on interactions with a finite
effective range.

##  I Introduction

One fundamental type of dynamics in a many-body system is the collective mode.
Most often collective modes are described using linearized hydrodynamic
equations, while their finite lifetime arises from higher-order effects [1,
2]. Although they are not exact excitations as they possess a finite lifetime,
collective modes provide insights on how the dynamics of the system depend on
the interactions. An excellent platform for studying this physics is an atomic
gas system. The experimental control over atomic gases [3] allows one to study
thoroughly the collective mode physics for different particle statistics,
dimensions, interactions, and even in different phases [4, 5, 6, 7, 8, 9, 11,
12, 10, 13, 14].

A particular atomic gas system of recent interest is a spin-polarized Fermi
gas confined to one spatial dimension (1D) [15, 16, 19, 20, 21, 23, 24, 22,
17, 18]. Since the gas is spin-polarized, the s- or even-wave interactions are
suppressed, and the leading interactions are p- or odd-wave in character. From
the effective-range expansion, one expects that this gas is universal and can
be described by a zero-range theory [15]. In other words, the physics can be
described by a single low-energy scattering volume, $\ell$, while the
contribution to the energetics and dynamics from the effective range, $r$,
should be small and perturbative. This is in contradistinction to a 3D p-wave
Fermi gas where the effective range although small, is pivotal for
understanding both the energetics and dynamics [27, 28, 25, 26, 29].

For a harmonically trapped 1D spin-polarized p-wave Fermi gas, the most basic
collective mode is the breathing mode. The breathing mode is unique in that it
can be shown to occur at exactly twice the harmonic trap frequency and to be
undamped when the interactions are scale-invariant [30, 31]. For a zero-range
1D spin-polarized p-wave Fermi gas, scale invariance arises when the gas is at
resonance, that is when the 1D scattering volume is divergent, or when the gas
is non-interacting, $\ell=0$.

When one explicitly breaks the scale invariance either by a finite $\ell$ or
by the presence of an effective range, $r$, the frequency of the breathing
mode will deviate from its scale invariant value. As discussed for s-wave
interactions [32, 8, 14] and for p-wave interactions in 2D [11, 33], the shift
in the breathing mode frequency in fact depends on the thermodynamic contacts,
which characterize the change in the energy of the system with respect to the
low-energy scattering parameters, $\ell$ and $r$.

In this article I evaluate both the thermodynamic contacts, as well as the
shift in the breathing mode frequency for a p-wave Fermi gas confined in 1D.
In the high-temperature limit, one can obtain analytical results to leading
order in the virial expansion, which is to leading order in $n\lambda_{th}$,
where $n$ is the density and $\lambda_{th}=\sqrt{2\pi\hbar^{2}/(mk_{B}T)}$ is
the thermal de Broglie wavelength for a gas of atoms with mass $m$ and
temperature $T$. The shift in the breathing mode frequency depends on both the
scattering length and effective range contacts, $C_{\ell}$ and $C_{r}$,
respectively. I then apply these results to experimentally applicable
situations, in the hopes of future experimental studies.

The remainder of this article is organized as follows. Sec. II provides the
basics of the 1D p-wave Fermi gas using a two-channel model. Sec. III
evaluates the thermodynamic contacts using the virial expansion. Sec. IV then
proceeds with a calculation of the frequency shift of the breathing mode
within the virial expansion. The experimental applicability of these results
is then discussed in Sec. V. Finally, I conclude in Sec. VI.

##  II The two-channel model

In this work I use a two-channel model for the 1D p-wave Fermi gas. The
Hamiltonian is given by:

$\displaystyle H$ $\displaystyle=\sum_{k}\xi_{k}\psi^{\dagger}(k)\psi(k)$
$\displaystyle+\sum_{Q}\left(\frac{1}{2}Q^{2}+\nu_{0}-2\mu\right)\phi^{\dagger}(Q)\phi(Q)$
$\displaystyle+\sum_{Q,k}\frac{gk}{2\sqrt{L}}\left[\phi(Q)\psi^{\dagger}\left(\frac{Q}{2}+k\right)\psi^{\dagger}\left(\frac{Q}{2}-k\right)+h.c\right]$
(1)

In Eq. (1), $\psi^{(\dagger)}(k)$ is the fermionic annihilation (creation)
operator, $\phi^{(\dagger)}$ is the annihilation (creation) operator for the
closed channel molecules, $\xi_{k}=k^{2}/2-\mu$ is the single particle
dispersion, $\mu$ is the chemical potential, $\nu_{0}$ is the bare detuning of
the molecular channel, and $g$ is the atom-molecule coupling. I also define
$L$ as the length of the system, and both $\hbar$ and the bare atomic mass $m$
have been set to unity for the time being. The factor of $1/2$ in the
interaction is to account for the fact spin polarized fermions are
indistinguishable.

Figure 1: Feynman diagrams for a) the dressed molecular propagator and for b)
the T-matrix. The single lines are the free fermion propagators. The double
dashed (full) lines correspond to the bare (dressed) molecular propagator. a)
and b) apply both in the presence of the vacuum and the many-body background.

Technically this theory has a ultraviolet (UV) divergence. As is custom, one
can remove this divergence by examining the two-body scattering properties in
the vacuum. The two-body scattering can be calculated according to Fig. (1).
Consider two spin polarized fermions with momenta, $Q/2\pm k$, and total
energy $Q_{0}=Q^{2}/4+k^{2}$. As shown in Appendix A, the $T$-matrix depends
on the dressed molecular propagator, $D(Q,Q_{0})$, via:

$\displaystyle\langle\frac{Q}{2}\pm k|T|\frac{Q}{2}\pm l\rangle$
$\displaystyle=klg^{2}D(Q_{0}-Q^{2}/4)$
$\displaystyle\left(g^{2}D(Q_{0}-Q^{2}/4)\right)^{-1}$
$\displaystyle=\left[\frac{-\nu_{0}}{g^{2}}+\frac{1}{2L}\sum_{k}+\frac{Q_{0}-Q^{2}/4}{g^{2}}\right.$
$\displaystyle\left.-\frac{\sqrt{-Q_{0}+Q^{2}/4-i\delta}}{4}\right]$ (2)

Given the $T$-matrix one can show that the scattered relative wavefunction has
the form:

$\displaystyle\psi(x)$ $\displaystyle=e^{ikx}+f_{k}\frac{x}{|x|}e^{ik|x|}$
$\displaystyle f_{k}$ $\displaystyle=-ik\frac{g^{2}D(k^{2})}{4}$ (3)

where $Q_{0}-Q^{2}/4=k^{2}$, and $f_{k}$ is the scattering amplitude. The
scattering amplitude can then be expanded in terms of $k^{2}$, according to
the effective range expansion:

$f_{k}=ik\left[-\frac{1}{\ell}-rk^{2}-ik+O(k^{4})\right]^{-1}$ (4)

This expansion is valid for low-energies, and in particular for energies
$k^{2}\ll r^{-2}$.

From Eq. (2) one can then identify the scattering volume, $\ell$, and the
effective range, $r$:

$\displaystyle\frac{1}{4\ell}$
$\displaystyle=-\frac{\nu_{0}}{g^{2}}+\frac{1}{2L}\sum_{k}$
$\displaystyle\frac{r}{4}$ $\displaystyle=\frac{1}{g^{2}}$ (5)

Here it is important to note that $r$ is strictly positive in this two-channel
model, and both $\ell$ and $r$ have units of length.

##  III Thermodynamic Contacts in the High Temperature Limit

As discussed previously [32, 8, 14, 11, 33], the shift in the breathing mode
frequency depends on the thermodynamic contacts. The thermodynamic contacts
are related to the change in the energy with respect to the low-energy
scattering parameters, $\ell$ and $r$:

$\displaystyle\langle C_{\ell}\rangle$
$\displaystyle=\left\langle-\frac{\partial H}{\partial\ell^{-1}}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx\frac{g^{2}}{4}\langle\phi^{\dagger}(x)\phi(x)\rangle$
$\displaystyle\langle C_{r}\rangle$ $\displaystyle=\left\langle-\frac{\partial
H}{\partial r}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx\frac{g^{2}}{4}\langle\phi^{\dagger}(x,t)\left(i\partial_{t}+\frac{\partial_{x}^{2}}{4}+2\mu\right)\phi(x,t)\rangle$
(6)

The full derivation of the microscopic expressions for the contacts is
presented in Appendix B. Eq. (6) is also consistent with previous definitions
of the contacts [15], and give consistent results for the momentum
distribution and radio-frequency spectroscopy [15, 17].

Eq. (6) can be evaluated using finite temperature field theory [34]. Since the
contact operators are bilinear in the molecular degrees of freedom, they will
be functions of the dressed molecular propagator, calculated in the presence
of the many-body background:

$\displaystyle\frac{\langle C_{\ell}\rangle}{L}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dQ}{2\pi}\int_{-\infty}^{\infty}\frac{dz}{\pi}n_{B}(z)\Im\left[\frac{g^{2}D(Q,z-i\delta)}{4}\right]$
$\displaystyle\frac{\langle C_{r}\rangle}{L}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dQ}{2\pi}\int_{-\infty}^{\infty}\frac{dz}{\pi}n_{B}(z)\left(z-\frac{Q^{2}}{4}+2\mu\right)$
$\displaystyle\Im\left[\frac{g^{2}D(Q,z-i\delta)}{4}\right].$ (7)

In Eq. (7), $n_{B(F)}(z)$ is the Bose-Einstein (Fermi-Dirac) distribution at
$\beta=1/(k_{B}T)$, and the molecular propagator has been evaluated in the
presence of the many-body background:

$\displaystyle\left(\frac{g^{2}D(Q,z-i\delta)}{4}\right)^{-1}=\frac{1}{\ell}+\left(z-\frac{Q^{2}}{4}+2\mu\right)r$
$\displaystyle+\int_{-\infty}^{\infty}\frac{dk}{\pi}\
k^{2}\left[\frac{1-n_{F}\left(\xi_{\frac{Q}{2}-k}\right)-n_{F}\left(\xi_{\frac{Q}{2}+k}\right)}{k^{2}-\left(z-\frac{Q^{2}}{4}+2\mu\right)+i\delta}-\frac{1}{k^{2}}\right]$
(8)

which follows from Fig. (1).

In the high temperature limit the chemical potential is large and negative,
and thus the fugacity, $e^{\beta\mu}$, is very small: $e^{\beta\mu}\ll 1$. The
expansion of Eqs. (7-8) in terms of the fugacity is known as the virial
expansion. As the contacts are related to the change in energy with respect to
the scattering parameters, the contacts are proportional to $e^{2\beta\mu}$ at
leading order in the virial expansion. Performing the virial expansion to this
order, one obtains the following expressions for the contacts:

$\displaystyle\frac{\langle C_{\ell}\rangle}{L}$
$\displaystyle=\frac{e^{2\beta\mu}}{\sqrt{\pi\beta}}\int_{0}^{\infty}\frac{dz}{\pi}e^{-\beta
z}\frac{\sqrt{z}}{\left(\frac{1}{\ell}+zr\right)^{2}+z}$
$\displaystyle\frac{\langle C_{r}\rangle}{L}$
$\displaystyle=\frac{e^{2\beta\mu}}{\sqrt{\pi\beta}}\int_{0}^{\infty}\frac{dz}{\pi}e^{-\beta
z}z\frac{\sqrt{z}}{\left(\frac{1}{\ell}+zr\right)^{2}+z}$ (9)

Eq. (9) only accounts for the scattering states contribution to the contacts
and ignores the bound state contribution which only exists for $\ell>0$. As
discussed in Appendix C, the bound-state contribution is only relevant near
resonance, where the bound state energy is small but finite. Near resonance,
the wavefunctions of the closed channel molecules have substantial overlap
with that of the scattering states. Therefore one expects a finite number of
closed channel molecules, and a finite contribution to the contacts from the
bound-state. In the opposite limit of weak interactions, the two-body bound
state is quite deep. If one is concerned with the upper-branch physics, the
relaxation rate for pairs of individual atoms to form closed-channel molecules
is quite small [3]. The number of closed-channel molecules is then small and
the bound-state contribution to the contacts is negligible. For the simplicity
of the presentation, I have relegated the molecular contribution to the
contacts to Appendix C.

Eq. (9) also suggests a relationship between the two contacts:

$\left[e^{-2\beta\mu}\sqrt{\pi\beta}\frac{\langle
C_{r}\rangle}{L}\right]=-\frac{\partial}{\partial\beta}\left[e^{-2\beta\mu}\sqrt{\pi\beta}\frac{\langle
C_{\ell}\rangle}{L}\right]$ (10)

This relationship is only approximate as it neglects the many-body effects
from higher orders in the virial expansion. That being said, Eq. (10) may
provide a method for evaluating the effective range contact for a thermal gas,
or to quantify the significance of many-body effects by comparing the results
of Eq. (10) to the experimentally measured contacts.

For arbitrary $\ell$ and $r$, Eq. (9) does not have a closed solution, and has
to be determined numerically. However, there are a number of limits which
allow for analytical results. First consider the gas near resonance, and with
an effective range that is small compared to the thermal de Broglie
wavelength: $r/\lambda_{th}\ll 1$. After restoring $\hbar$ and the atomic mass
$m$, one obtains:

$\displaystyle\left.\frac{\langle C_{\ell}\rangle}{L}\right|_{\text{n}ear\
res.}$ $\displaystyle\approx
2\frac{\hbar^{2}n^{2}}{m}\left[1-\frac{1}{\sqrt{2}}\frac{\lambda_{th}}{|\ell|}-2\frac{r}{\ell}+O\left(\frac{\lambda_{th}^{2}}{a^{2}}\right)\right]$
$\displaystyle\left.\frac{\langle C_{r}\rangle}{L}\right|_{\text{n}ear\ res.}$
$\displaystyle\approx\frac{n^{2}}{\beta}\left[1-2\frac{r}{\ell}+O\left(\frac{\lambda_{th}^{2}}{a^{2}}\right)\right]$
(11)

where again $\lambda_{th}=\sqrt{2\pi\hbar^{2}/(mk_{B}T)}$ is the thermal de
Broglie wavelength. I have also replaced the fugacity with the density, $n$,
via: $n\lambda_{th}=e^{\beta\mu}$, which is valid to leading order in the
virial expansion.

Similarly in the weakly interacting limit the contacts are given by:

$\displaystyle\left.\frac{\langle C_{\ell}\rangle}{L}\right|_{\text{w}eak\
int.}$ $\displaystyle\approx
2\pi\frac{\hbar^{2}n^{2}}{m}\frac{\ell^{2}}{\lambda^{2}_{th}}\left[1-6\pi\frac{\ell
r}{\lambda_{th}^{2}}+O\left(\frac{\ell^{2}}{\lambda^{2}_{th}}\right)\right]$
$\displaystyle\left.\frac{\langle C_{r}\rangle}{L}\right|_{\text{w}eak\ int.}$
$\displaystyle\approx
3\pi\frac{n^{2}}{\beta}\frac{\ell^{2}}{\lambda_{th}^{2}}\left[1-10\pi\frac{\ell
r}{\lambda_{th}^{2}}+O\left(\frac{\ell^{2}}{\lambda_{th}^{2}}\right)\right]$
(12)

A final important limit is to consider exactly at resonance, when
$\ell^{-1}=0$. In this case the scale invariance is solely broken by the
effective range. In this limit the contacts are given by:

$\displaystyle\left.\frac{\langle C_{\ell}\rangle}{L}\right|_{\text{r}es.}$
$\displaystyle\approx
2\frac{\hbar^{2}n^{2}}{m}\left[1-\pi\frac{r^{2}}{\lambda_{th}^{2}}+O\left(\frac{r^{4}}{\lambda_{th}^{4}}\right)\right]$
$\displaystyle\left.\frac{\langle C_{r}\rangle}{L}\right|_{\text{r}es.}$
$\displaystyle\approx\frac{n^{2}}{\beta}\left[1-3\pi\frac{r^{2}}{\lambda_{th}^{2}}+O\left(\frac{r^{4}}{\lambda_{th}^{4}}\right)\right]$
(13)

Eqs. (11-13) constitute one of the main results of this paper, and
characterize the dependence of the thermodynamic contacts on the scattering
parameters, $r$ and $\ell$.

##  IV Shift in the frequency of the breathing mode

With the contacts in hand, it is now possible to examine the shift in the
breathing mode frequency. Therefore, consider the spin-polarized 1D Fermi gas
in the presence of a harmonic potential with frequency $\omega$. For this
discussion I integrate out the molecular degrees of freedom, and focus only on
the fermions. The total Hamiltonian is then equivalent to a one-channel model
of interacting spin-polarized fermions inside a harmonic trap:

$\displaystyle H_{trap}$ $\displaystyle=H+\omega^{2}\int_{-\infty}^{\infty}dx\
\frac{x^{2}}{2}\psi^{\dagger}(x)\psi(x)$ (14)

In writing Eq. (14), it is important to note that the scattering properties of
the one-channel Hamiltonian, $H$, are still described by the T-matrix shown in
Eq. (8).

The breathing mode for this system is an oscillatory motion associated with
the moment of inertia:

$\langle x^{2}\rangle(t)=\int dx\ x^{2}n(x,t)$ (15)

In order to evaluate the dynamics of the moment of inertia, consider the
following commutators:

$\displaystyle\left[H,C\right]$ $\displaystyle=-iD$
$\displaystyle\left[C,D\right]$ $\displaystyle=2iC$
$\displaystyle\left[H,D\right]=-i\Pi$ (16)

where $C$, $D$, and $\Pi$ are the generator of conformal transformations, the
generator of scale transformations, and the trace of the stress energy tensor
respectively [35]. They are defined as:

$\displaystyle C$ $\displaystyle=\int_{-\infty}^{\infty}dx\
\frac{x^{2}}{2}\psi^{\dagger}(x)\psi(x)$ $\displaystyle D$
$\displaystyle=-i\int_{-\infty}^{\infty}dx\
\psi^{\dagger}(x)\left(x\partial_{x}+\frac{1}{2}\right)\psi(x)$
$\displaystyle\Pi$ $\displaystyle=2H+\frac{1}{\ell}C_{\ell}-rC_{r}$ (17)

From Eq. (16) and the Heisenberg equation of motion, one can obtain the
following dynamic equation for the moment of inertia:

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}\langle
x^{2}\rangle(t)+4\omega^{2}\langle x^{2}\rangle(t)$ $\displaystyle=4\langle
H+\omega^{2}C\rangle(0)$ $\displaystyle+\frac{2}{\ell}\langle
C_{\ell}\rangle(t)-2r\langle C_{r}\rangle(t)$ (18)

If one neglects the contacts, Eq. (18) has the analytical solution:

$\displaystyle\langle x^{2}\rangle(t)$ $\displaystyle=\langle
x^{2}\rangle(0)\left[\frac{\langle H+\omega^{2}C\rangle(0)}{\omega^{2}\langle
x^{2}\rangle(0)}\right.$ $\displaystyle\left.+\left(1-\frac{\langle
H+\omega^{2}C\rangle(0)}{\omega^{2}\langle x^{2}\rangle(0)}\right)\cos(2\omega
t)\right]$ (19)

Thus in the absence of the contact terms, the breathing mode oscillates
indefinitely at a frequency: $\omega_{B}=2\omega$. This was first pointed out
in Ref. [30], and is a consequence of the non-relativistic conformal symmetry
[30, 31, 32, 36, 37]. Physically, this situation occurs when the Hamiltonian
is scale invariant, i.e. when $\ell^{-1}=0$ and $r=0$, or when $\ell=0$.

The difficulty in evaluating Eq. (18) in general is in determining the time-
dependence of the thermodynamic contacts. To this end it is important to note
that the breathing mode motion is dilatory in nature and can be described with
a time-dependent density of the form:

$n(x,t)=\frac{1}{\lambda(t)}n\left(\frac{x}{\lambda(t)}\right)$ (20)

Such a scaling ansatz presupposes the density profile does not depend on the
scattering parameters $\ell$ and $r$. In other words, Eq. (20) assumes that
the gas follows a Boltzmann distribution for a non-interacting gas, which is
valid to lowest order in the virial expansion where:
$n\lambda_{th}=e^{\beta\mu}$. Since the shift in the breathing mode frequency
is solely due to the contacts, which are already second order in the virial
expansion, it is sufficient to assume that the dynamics of the system are also
dilatory in nature.

From Eq. (20), the dynamics of the moment of inertia are also equivalent to a
time-dependent rescaling:

$\langle x^{2}\rangle(t)=\lambda^{2}(t)\langle x^{2}\rangle(0)$ (21)

Furthermore, due to number conservation, one can show that the scaling
dynamics of the density, Eq. (20), is equivalent to a time-dependent
temperature and chemical potential: $\beta(t)=\beta\lambda^{2}(t)$ and
$\mu(t)=\mu/\lambda^{2}(t)$. This allows one to explicitly evaluate the time-
dependence of the contacts by simply replacing $\beta$ and $\mu$ in Eq. (9)
with their time-dependent counterparts. The result can be written in the form:

$\displaystyle\langle C_{\ell}\rangle(t)$
$\displaystyle=\frac{1}{\lambda(t)}\langle
C_{\ell}\rangle\left(\frac{\lambda(t)}{\ell},\frac{r}{\lambda(t)}\right)$
$\displaystyle\ \langle C_{r}\rangle(t)$
$\displaystyle=\frac{1}{\lambda^{3}(t)}\langle
C_{r}\rangle\left(\frac{\lambda(t)}{\ell},\frac{r}{\lambda(t)}\right)$ (22)

This conclusion ought to be contrasted with the arguments presented in Ref.
[37]. In Ref. [37], the dilatory motion of the contacts, Eq. (22), follows
from an expansion of the contacts around a scale invariant point, and is only
valid to leading order in the breaking of scale invariance. In this work the
dilatory motion of the contacts follows from the trivial dynamics of the
density, which is valid to leading order in the virial expansion. Thus, in
this work the scaling ansatz is valid for arbitary interaction strengths
provided $n\lambda_{th}\ll 1$.

Substituting Eqs. (21) and (22) into Eq. (18), one obtains a second order
differential equation for $\lambda(t)$:

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}\lambda^{2}(t)$
$\displaystyle+4\omega^{2}\lambda^{2}(t)=4\omega^{2}$
$\displaystyle+\frac{2}{\ell}\frac{1}{\langle
x^{2}\rangle(0)}\frac{1}{\lambda(t)}\langle
C_{\ell}\rangle\left(\frac{\lambda(t)}{\ell},\frac{r}{\lambda(t)}\right)$
$\displaystyle-2r\frac{1}{\langle
x^{2}\rangle(0)}\frac{1}{\lambda^{3}(t)}\langle
C_{r}\rangle\left(\frac{\lambda(t)}{\ell},\frac{r}{\lambda(t)}\right)$

where I have used $\langle H+\omega^{2}C\rangle(0)=\omega^{2}\langle
x^{2}\rangle(0)$. The breathing mode frequency can be easily obtained by
considering a small change in $\lambda^{2}(t)$:

$\lambda^{2}(t)=1+\delta\lambda^{2}(t)$ (24)

Substituting Eq. (24) into Eq. (LABEL:eq:lambda_eom) and expanding to lowest
order in $\delta\lambda^{2}(t)$ gives:

$\displaystyle 0$ $\displaystyle=\frac{\partial^{2}}{\partial
t^{2}}\delta\lambda^{2}(t)+\omega_{B}^{2}\delta\lambda^{2}(t)$ (25)

which states $\delta\lambda^{2}(t)$ oscillates at a frequency $\omega_{B}$.
The breathing mode frequency $\omega_{B}$ satisfies the following relation:

$\displaystyle\frac{\omega_{B}^{2}-4\omega^{2}}{\omega^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell}$
$\displaystyle\left[1-\frac{1}{\ell}\frac{\partial}{\partial\ell^{-1}}+r\frac{\partial}{\partial
r}\right]\frac{\langle C_{\ell}\rangle(0)}{\omega^{2}\langle x^{2}\rangle(0)}$
$\displaystyle-r$
$\displaystyle\left[3-\frac{1}{\ell}\frac{\partial}{\partial\ell^{-1}}+r\frac{\partial}{\partial
r}\right]\frac{\langle C_{r}\rangle(0)}{\omega^{2}\langle x^{2}\rangle(0)}$
(26)

Eq. (26) is the second main result of this work. It relates the shift of the
breathing mode frequency to the thermodynamic contacts. It is straightforward
to see that in the case of resonance and zero effective range, i.e. when
$\ell^{-1}=0$ and $r=0$, and in the non-interacting limit, i.e. when $\ell=0$,
$\omega_{B}=2\omega$, which is consistent with non-relativistic conformal
symmetry.

In order to evaluate Eq. (26), I employ a local density approximation (LDA) in
the high-temperature limit. The details of the LDA are shown in Appendix D.
Combining the LDA with the high-temperature expansion of the contacts, Eq.
(9), one obtains the following formula for the shift of the breathing mode
frequency:

$\displaystyle\frac{\omega_{B}^{2}-4\omega^{2}}{\omega^{2}}=N\frac{\hbar\omega}{k_{B}T}$
$\displaystyle\left[\frac{1}{\tilde{\ell}}\left(1-\frac{1}{\tilde{\ell}}\frac{\partial}{\partial\tilde{\ell}^{-1}}+\tilde{r}\frac{\partial}{\partial\tilde{r}}\right)\int_{0}^{\infty}\frac{dz}{\pi}e^{-z}\frac{\sqrt{z}}{\left(\frac{1}{\tilde{\ell}}+z\tilde{r}\right)^{2}+z}\right.$
$\displaystyle\left.-\tilde{r}\left(3-\frac{1}{\tilde{\ell}}\frac{\partial}{\partial\tilde{\ell}^{-1}}+\tilde{r}\frac{\partial}{\partial\tilde{r}}\right)\int_{0}^{\infty}\frac{dz}{\pi}e^{-z}\frac{z^{3/2}}{\left(\frac{1}{\tilde{\ell}}+z\tilde{r}\right)^{2}+z}\right]$
(27)

where $\tilde{\ell}=\ell/(\sqrt{2\pi}\lambda_{th})$,
$\tilde{r}=\sqrt{2\pi}r/\lambda_{th}$, and $N\hbar\omega/k_{B}T\ll 1$.

Figure 2: Shift in the breathing mode frequency, Eq. (27), for a spin
polarized 1D p-wave Fermi gas as a function of the dimensionless scattering
volume, $\tilde{\ell}=\ell/(\sqrt{2\pi}\lambda_{th})$ where $\lambda_{th}$ is
the thermal de Broglie wavelength, and for various values of the dimensionless
effective range, $\tilde{r}=\sqrt{2\pi}r/\lambda_{th}$. This figure disregards
the molecular contribution. The inset shows the shift in the breathing mode
frequency for $\tilde{r}=0.3$ (green solid line), and the asymptotic
expansions from Eqs. (28-29) (black dashed lines).

Eq. (27), has to be evaluated numerically, but one can obtain analytical
expressions for the shift of the breathing mode frequency near the resonantly
interacting limit:

$\left.\frac{\omega_{B}^{2}-4\omega^{2}}{\omega^{2}}\right|_{\text{n}ear\
res.}\approx\frac{\hbar\omega
N}{k_{B}T}\left[\frac{1}{\tilde{\ell}\sqrt{\pi}}\left(1-\frac{2\tilde{r}}{\tilde{\ell}}\right)-\frac{3\tilde{r}}{2\sqrt{\pi}}\right]$
(28)

and for weak interactions:

$\left.\frac{\omega_{B}^{2}-4\omega^{2}}{\omega^{2}}\right|_{\text{w}eak\
int.}\approx\frac{\hbar\omega
N}{k_{B}T}\frac{3\tilde{\ell}}{2\sqrt{\pi}}\left(1-\frac{15}{2}\tilde{r}\tilde{\ell}\right)$
(29)

For a more complete picture, consider Fig. (2). In this figure the shift in
the breathing mode frequency is presented as a function of $\tilde{\ell}^{-1}$
for various values of the effective range, $\tilde{r}$. One prominent feature
of Fig. (2) is that the shift in the breathing mode becomes more negative as
the effective range becomes larger. Conversely, the contribution due to the
scattering volume, $\tilde{\ell}$, depends on the sign of $\tilde{\ell}$. This
can be qualitatively understood by examining Eq. (18) and the pressure, $P$.
As shown in Appendix B, the pressure is given by:

$PL=\langle\Pi\rangle=2\langle H\rangle+\frac{1}{\ell}\langle
C_{\ell}\rangle-r\langle C_{r}\rangle$ (30)

Since the contacts in Eq. (9) are positive semi-definite, the shift in the
pressure from its scale invariant value due to the scattering volume is
positive (negative) for positive (negative) $\ell$, which is a signature of
repulsive (attractive) interactions. On the other hand, the contribution due
to a finite effective range, $r$, is always negative and hence attractive in
nature. This shift in the pressure is directly related to the shift in the
breathing mode frequency via Eq. (26), and explains the general features of
Fig. (2).

One can also see that the shift in the breathing mode frequency is finite at
resonance due to the presence of the effective range. Away from resonance, the
the shift in the breathing mode frequency is more substantial for
$\tilde{\ell}<0$, as some cancellation occurs for $\tilde{\ell}>0$. In the
weakly interacting limit, the contribution from the effective range becomes
minuscule in comparison to the leading contribution governed by
$\tilde{\ell}$, which reinforces the idea that the effective range merely
gives a perturbative correction.

##  V Discussion

The results obtained thus far are for strictly one-dimensional systems.
Realistically, however, the one-dimensional physics described above is a low-
energy approximation for a three-dimensional spin-polarized Fermi gas placed
in a cylindrical trapping potential with tight radial confinement. The low-
energy scattering volume, $\ell$, and the effective range, $r$, in the ground
state of the radial confinement are related to their 3D counterparts, $v$ and
$R$ respectively, via [16, 39, 24]:

$\displaystyle\frac{1}{\ell}$
$\displaystyle=\frac{a_{\perp}^{2}}{6}\left(\frac{1}{v}+\frac{2}{Ra_{\perp}^{2}}\right)-\frac{2}{a_{\perp}}\zeta(-1/2)$
$\displaystyle r$
$\displaystyle=\frac{a_{\perp}^{2}}{6R}+\frac{a_{\perp}}{4}\zeta(1/2)$ (31)

where $a_{\perp}=\sqrt{2\hbar/m\omega_{\perp}}$ is the harmonic oscillator
length for the radial confinement of frequency $\omega_{\perp}$, and
$\zeta(s)$ is the zeta-function.

Since the 3D scattering volume can be tuned via a p-wave Feshbach resonance
[28], the confinement strength can tune both $\ell$ and $r$; a phenomenon
called the confinement induced resonance [40, 41]. Across the confinement
induced resonance $\ell$ can be tuned continuously from positive to negative
infinity via the confinement induced resonance. The 1D effective range is more
or less constant, but its value can be enhanced when $a_{\perp}$ is larger
than the 3D effective range, $R$. Hence the energy dependence of the 1D
scattering can be enhanced by the confinement. For example the effective range
for spin-polarized ${}^{40}K$ is [28]: $R\approx 3.8$ nm. For a radial
confinement of $\omega_{\perp}\approx 2\pi*400$ kHz, or equivalently
$a_{\perp}\approx 25$ nm, the 1D effective range is $r\approx 18$ nm, which is
larger than the 3D effective range by almost a factor of $5$. In fact this
effect becomes stronger for weaker radial confinement. Although the 1D
effective range is an irrelevant quantity to the energetics and dynamics in
the renormalization group sense [38], the effective range may be important to
the dynamics due to its enhancement from the radial confinement. The
importance of the effective range then depends not only on the typical energy
scale in the problem, according to the effective range expansion Eq. (4), but
also this enhancement from the radial confinement.

To further examine the experimental applicability of these results, the shift
in the breathing mode frequency for a 1D spin polarized Fermi gas confined in
an axial harmonic potential with frequency $\omega\approx 2\pi*200$ Hz is
presented in Fig. (3). For a radial confinement with frequency
$\omega_{\perp}\approx 2\pi*400$ kHz, the quasi 1D regime is defined via:
$\hbar\omega_{\perp}\gg k_{B}T$, while the high temperature limit is given by:
$T\gg N\hbar\omega$, see Appendix D. For the parameters under consideration
this leads to $40\ \mu K>T>50\ nK$, where I have used $N\approx 10$.

The results in Fig. (3) are for temperatures $5\ \mu K>T>0.4\ \mu K$, which
sits within the quasi 1D-regime and high-temperature limits. At resonance, the
shift in the contact is small and only a few percent. However slightly away
from resonance and for $\ell<0$, the shift can be about $3-5\%$ depending on
the temperature. The asymmetry in Fig. (3) also implies that the effective
range is important in describing the shift in the breathing mode frequency.

Figure 3: Shift in the breathing mode frequency at various temperatures across
the confinement induced resonance, Eq. (31), for ${}^{40}K$. Here
$\lambda_{th}$ is the thermal de Broglie wavelength, the axial trapping
frequency is $\omega=2\pi*200$ Hz, the transverse harmonic length is
$a_{\perp}\approx 25$ nm, $N=10$, and I have used the parametrization of the
p-wave Feshbach resonance at $198.3$ G [28, 42].

##  VI Conclusions

The main results of this work are the evaluation of the thermodynamic contacts
and the shift of the breathing mode frequencies in the high temperature limit,
see Eqs. (11-13, 27-29). These results are valid to lowest non-trivial order
in the virial expansion, i.e. to second order in
$\left(n\lambda_{th}\right)^{2}$.

Although the above results extend to arbitrary interactions strengths, it is
important to note that this collective mode does possess a finite lifetime
when the scattering volume is finite. The finite lifetime of the breathing
mode can be shown to be related to the finite bulk viscosity, $\zeta$, and
scales as $\zeta^{-1}$ [37]. In the high-temperature limit, the bulk viscosity
will also occur at order $O\left(n\lambda_{th}\right)^{2}$ [29, 43, 44, 45].
Moreover, since the bulk viscosity must be positive, as it is related to
entropy production, the bulk viscosity ought to be proportional to $\ell^{2}$
and $r^{2}$ for a weakly interacting and resonantly interacting gas,
respectively. Practically one expects the breathing mode picture breaks down
around $\tilde{\ell}\approx 1$. A more in-depth study of the bulk viscosity is
beyond the scope of this work and will be presented in the future.

One interesting part of this study is that the effective range for a 1D spin-
polarized Fermi gas can be enhanced by the presence of radial confinement, see
Eq. (31). This is due to the competition between the validity of the low-
energy three-dimensional scattering, set by $R$, and the validity of the
quasi-1D regime, set by $a_{\perp}$. This increase of the effective range can
lead to deviations from the zero-range limit at smaller energy scales. This is
evident in Fig. (3). There the contribution to the shift in the breathing mode
frequency due to the effective range is important for $\ell<0$. A similar
phenomenon has also been emphasized for the case of quasi two-dimensional
Fermi superfluids in Ref. [46].

This situation ought to be contrasted to the spin 1/2 s-wave Fermi gas in 3D.
This system has the same renormalization group structure as the spin-polarized
1D Fermi gas with p-wave interactions [38]. In both cases, the effective range
is an irrelevant quantity, and leads to perturbative effects; a conclusion
consistent with effective field theory [25, 26]. As the effective range is
irrelevant, the 3D s-wave Fermi gas has a well defined zero-range limit, where
the physics are universally described by the s-wave scattering length. In this
case, the effective range can often be ignored as it is generally quite small.
For the spin-polarized 1D Fermi gas, although a well-defined zero-range limit
exists, the effective range can be important due to its enhancement from the
transverse confinement, as evidenced by Figs. (2-3).

I would like to thank Shizhong Zhang and Joseph H. Thywissen for useful
discussions. This work was supported through the Research Grants Council of
the Hong Kong Special Administrative Region, China (General research fund, HKU
17304719, 17304820 and collaborative research fund C6026-16W and C6005-17G),
and the Croucher Foundation under the Croucher Innovation Award.

##  Appendix A Derivation of the Two-Body T-Matrix

Here I evaluate the two-body T-matrix in the presence of the vacuum. The two-
body T-matrix for identical fermions depends on the dressed propagator for the
closed channel molecular state. The equation for the dressed molecular
propagator is:

$\displaystyle D^{-1}(Q,Q_{0})$
$\displaystyle=D^{-1}_{0}(Q,Q_{0})-\Sigma(Q,Q_{0})$ $\displaystyle
D^{-1}_{0}(Q,Q_{0})$ $\displaystyle=Q_{0}-\frac{1}{4}Q^{2}-\nu_{0}+i\delta$
(A.1)

Pictorially, this equation is shown in Fig. (1). In Eq. (A.1)
$\Sigma(Q,Q_{0})$ is the fermion-fermion bubble which is given by:

$\displaystyle\Sigma(Q,Q_{0})$
$\displaystyle=-\frac{g^{2}}{2}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\int_{-\infty}^{\infty}\frac{dk_{0}}{2\pi
i}\ k^{2}$
$\displaystyle\left[\frac{1}{\frac{Q_{0}}{2}+k_{0}-\frac{1}{2}\left(\frac{Q}{2}+k\right)^{2}+i\delta}\right.$
$\displaystyle\left.\frac{1}{\frac{Q_{0}}{2}-k_{0}-\frac{1}{2}\left(\frac{Q}{2}-k\right)^{2}+i\delta}\right]$
$\displaystyle=\frac{g^{2}}{2}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\frac{1}{Q_{0}-\frac{1}{4}Q^{2}-k^{2}+i\delta}$
(A.2)

In defining Eqs. (A.1-A.2), I have ignored the many-body background by setting
the chemical potential to zero.

One can see from Eq. (A.2) that the integrand is divergent at large momenta.
To this end, add a ultraviolet cut-off scale, $\Lambda$ to regularize the
integrals. A direct evaluation of Eq. (A.2) then gives:

$\displaystyle\Sigma(Q,Q_{0})$
$\displaystyle=-\frac{g^{2}}{2\pi}\Lambda+\frac{g^{2}}{4}\sqrt{-Q_{0}+\frac{Q^{2}}{4}-i\delta}$
$\displaystyle=-\frac{g^{2}}{2L}\sum_{k}+\frac{g^{2}}{4}\sqrt{-Q_{0}+\frac{Q^{2}}{4}-i\delta}$
(A.3)

The resulting dressed molecular propagator is then given by:

$\displaystyle D^{-1}(Q_{0},Q)$
$\displaystyle=-\nu_{0}+\frac{g^{2}}{2L}\sum_{k}+Q_{0}-\frac{1}{4}Q^{2}$
$\displaystyle-\frac{g^{2}}{4}\sqrt{-Q_{0}+\frac{Q^{2}}{4}-i\delta}$ (A.4)

where I have written $\Lambda/\pi=L^{-1}\sum_{k}$. Eq. (A.4) is equivalent to
Eq. (2).

##  Appendix B Derivation of the Contacts

In this appendix I show how to obtain the microscopic definitions of the
contact operators, and how they relate to the pressure. First note that the
pressure can be written as:

$P=\frac{1}{\beta^{3/2}}G\left(\beta\mu,\beta^{1/2}\frac{1}{\ell},\beta^{-1/2}r\right)$
(B.1)

where $\beta=1/T$ is the temperature, $\mu$ the chemical potential, $\ell$ the
scattering volume, $r$ the effective range, and finally $G$ is a dimensionless
function. In writing Eq. (B.1) I have used the fact both $\ell$ and $r$ have
dimensions of length. More formally, the pressure is defined in terms of the
partition function, $Z$, via:

$P=\frac{1}{\beta L}\ln\left(Z\right)=\frac{1}{\beta
L}\ln\left(Tr\left[e^{-\beta(H-\mu N)}\right]\right)$ (B.2)

From Eqs. (B.1-B.2), and by considering the derivative of $P$ with respect to
$\beta$, one can show that the pressure must satisfy:

$PL=2\langle H\rangle+\frac{1}{\ell}\langle C_{\ell}\rangle-r\langle
C_{r}\rangle$ (B.3)

where $\langle\cdot\rangle$ denotes a thermal average. In Eq. (B.3), the
thermal average of the scattering volume contact, $C_{\ell}$, and the
effective range contact, $C_{r}$, are defined via:

$\displaystyle\langle C_{\ell}\rangle$
$\displaystyle=\left\langle-\frac{\partial H}{\partial\ell^{-1}}\right\rangle$
$\displaystyle\langle C_{r}\rangle$ $\displaystyle=\left\langle-\frac{\partial
H}{\partial r}\right\rangle$ (B.4)

To progress further it is important to note that the scattering quantities
$\ell$ and $r$ are functions of the bare quantities $\nu_{0}$ and $g$ in the
Hamiltonian, see Eq. (5). Evaluating Eq. (B.4), one can find the following
microscopic definitions for the contact operators:

$\displaystyle C_{\ell}$
$\displaystyle=\int_{-\infty}^{\infty}dx\frac{g^{2}}{4}\phi^{\dagger}(x)\phi(x)$
$\displaystyle C_{r}$
$\displaystyle=\int_{-\infty}^{\infty}dx\frac{g^{2}}{4}\phi^{\dagger}(x,t)\left(i\partial_{t}+\frac{\partial_{x}^{2}}{4}+2\mu\right)\phi(x,t)$
(B.5)

In order to obtain Eq. (B.5), I have used the Heisenberg equation of motion
for the molecular field: $\partial_{t}\phi(x,t)=i\left[H,\phi(x,t)\right]$, to
integrate out the fermionic degrees of freedom in the definition of $C_{r}$.

##  Appendix C Imaginary Part of the T-Matrix

In this appendix I calculate the imaginary part of the T-matrix to leading
order in the virial expansion. There are two contributions, the first is due
to the positive frequency scattering states, and the second is due to the
bound state.

To begin, the molecular propagator to leading order in the virial expansion is
given by:

$\displaystyle\left(\frac{g^{2}D(Q,z-i\delta)}{4}\right)^{-1}=\frac{1}{\ell}+\left(z-\frac{Q^{2}}{4}+2\mu\right)r$
$\displaystyle+\int_{-\infty}^{\infty}\frac{dk}{\pi}\
k^{2}\left[\frac{1}{k^{2}-z+\frac{Q^{2}}{4}-2\mu+i\delta}-\frac{1}{k^{2}}\right]$
$\displaystyle=\frac{1}{\ell}+\left(z-\frac{Q^{2}}{4}+2\mu\right)-\sqrt{-z+\frac{Q^{2}}{4}-2\mu+i\delta}$
(C.1)

which is simply the molecular propagator in the presence of the vacuum. From
Eq. (C.1) one can absorb the $Q$ and $\mu$ dependence by shifting variables to
$z^{\prime}=z-Q^{2}/4+2\mu$. Hence I will ignore the dependence of the
$T$-matrix on $Q$ and $\mu$. For positive frequencies, $z>0$, one can show
that the imaginary part of the $T$-matrix is:

$\Im\left[\frac{g^{2}D(0,z-i\delta)}{4}\right]=\frac{\sqrt{z}}{\left(\frac{1}{\ell}+zr\right)^{2}+z}$
(C.2)

At negative frequencies, $z<0$, the imaginary part of the $T$-matrix comes
from the two-body bound state. The two-body bound state for this system is
given by the pole of the $T$-matrix, or equivalently the pole of Eq. (C.1).
This occurs when:

$\left(\frac{g^{2}D(0,-z_{B})}{4}\right)^{-1}=0=\frac{1}{\ell}-z_{B}r-\sqrt{z_{B}}$
(C.3)

As one can see from Eq. (C.3), the low-energy bound state exists for $\ell>0$.
For small effective range, $r^{2}z_{B}\ll 1$, one obtains:

$z_{B}=\frac{1}{\ell^{2}}\left(1-2\frac{r}{\ell}\right)$ (C.4)

which is perturbative in the effective range, $r$. This is evidence for the
statement that the effective range is an irrelevant quantity in understanding
the energetics and dynamics.

In principle Eq. (C.3) also has a pole that is approximately given by:
$z_{B}\approx 1/r^{2}$. This term must be discarded as it is beyond the low-
energy approximation which requires: $z_{B}\ll r^{-2}$.

To evaluate the imaginary part of the $T$-matrix for negative frequencies,
expand the denominator of the $T$-matrix around $-z_{B}$:

$\left[\frac{g^{2}D(0,z-i\delta)}{4}\right]^{-1}\approx\left(\frac{1}{2\sqrt{z_{B}}}+r\right)(z+z_{B})-i\delta$
(C.5)

The imaginary part of the $T$-matrix at negative frequencies is then:

$\Im\left[\frac{g^{2}D(Q,z-i\delta)}{4}\right]=\left(\frac{1}{2\sqrt{z_{B}}}+r\right)^{-1}\pi\delta(z+z_{B})\theta(\ell)$
(C.6)

In terms of the contacts, the bound state contributions to Eq. (7) are equal
to:

$\displaystyle\left.\frac{\langle C_{\ell}\rangle}{L}\right|_{\text{m}ol.}$
$\displaystyle=\frac{e^{2\beta\mu}}{\sqrt{\pi\beta}}e^{\beta
z_{B}}\frac{2\sqrt{z_{B}}}{1+2\sqrt{z_{B}}r}\theta(\ell)$
$\displaystyle\left.\frac{\langle C_{r}\rangle}{L}\right|_{\text{m}ol.}$
$\displaystyle=-\frac{e^{2\beta\mu}}{\sqrt{\pi\beta}}\ e^{\beta
z_{B}}z_{B}\frac{2\sqrt{z_{B}}}{1+2\sqrt{z_{B}}r}\theta(\ell)$ (C.7)

From Eq. (C.7), one can see that the molecular contribution vanishes at
resonance, while it appears to become quite large in the weakly interacting
limit. Near resonance, and for $\ell>0$ the bound state energy is small but
finite. In this limit the contribution due to the closed channel molecules is
important. Near resonance there can be a large population of molecules since
the wavefunctions of the closed channel molecules have substantial overlap
with the scattering states. In the weakly interacting limit, however, the
bound state is exceptionally deep. For a thermal gas in the upper branch [3],
the relaxation of scattering atoms into the closed channel molecules is
exceptionally slow. For this reason it is safe to neglect the molecular
contribution to the contacts.

##  Appendix D Local Density Approximation at High-Temperatures

In this appendix I discuss how to evaluate the contacts for a trapped 1D Fermi
gas in the local density approximation at high temperatures. At high-
temperatures, the Fermi-Dirac distribution can be replaced by a Boltzmann
distribution:

$f(k,x)=\exp\left[-\beta\left(\frac{k^{2}}{2}+\frac{1}{2}\omega^{2}x^{2}-\mu\right)\right]$
(D.1)

where I have set $\hbar$ and $m$ to be unity for the time being and $\mu$ is
the chemical potential at the center of the trap. The density is then given
by:

$n(x)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}f(k,x)=\frac{1}{\sqrt{2\pi\beta}}\exp\left[-\beta\left(\frac{1}{2}\omega^{2}x^{2}-\mu\right)\right]$
(D.2)

normalizing with respect to the total number of particles, $N$ gives the
following condition for the chemical potential:

$N=\frac{e^{\beta\mu}}{\beta\omega}$ (D.3)

or equivalently the density is given by:

$n(r)=\frac{\beta\hbar\omega N}{\lambda_{th}}e^{-\frac{1}{2}\beta
m\omega^{2}x^{2}}$ (D.4)

where I have restored the factors of $\hbar$ and $m$. Note here that the
virial expansion is equivalent to the statement $N\beta\hbar\omega\ll 1$.

From Eq. (D.4) the initial moment of inertia can then be calculated:

$\displaystyle\langle x^{2}\rangle(0)$
$\displaystyle=\int_{-\infty}^{\infty}dxx^{2}n(x)=\frac{N}{m\beta\omega^{2}}$
(D.5)

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# An AI approach to measuring financial risk ††thanks: Financial support from
the Deutsche Forschungsgemeinschaft (DFG) via SFB 649 “Ökonomisches Risiko”,
IRTG 1792 “High-Dimensional Non-Stationary Times Series” and Sim Kee Boon
Institute for Financial Economics, Singapore Management University,as well as
the Czech Science Foundation under grant no. 19-28231X, the Yushan Scholar
Program and the European Union’s Horizon 2020 research and innovation program
”FIN-TECH: A Financial supervision and Technology compliance training
programme” under the grant agreement No 825215 (Topic: ICT-35-2018, Type of
action: CSA), Humboldt-Universität zu Berlin, is gratefully acknowledged.

Lining Yu Research associate at Ladislaus von Bortkiewicz Chair of Statistics,
C.A.S.E. - Center for Applied Statistics and Econometrics, IRTG 1792,
Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.
Email<EMAIL_ADDRESS>Wolfgang Karl Härdle Ladislaus von Bortkiewicz
Professor of Statistics, C.A.S.E. - Center for Applied Statistics and
Econometrics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
Berlin, Germany; Wang Yanan Institute for Studies in Economics, Xiamen
University, 422 Siming Road, Xiamen 361005, China; Sim Kee Boon Institute for
Financial Economics, Singapore Management University, 90 Stamford Road,
Singapore 178903, Singapore; Department of Mathematics and Physics, Charles
University Prague, Ke Karlovu 2027/3, 12116 Praha 2, Czech. Email:
<EMAIL_ADDRESS>Lukas Borke Research associate at Ladislaus von
Bortkiewicz Chair of Statistics, C.A.S.E. - Center for Applied Statistics and
Econometrics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099
Berlin, Germany. Email<EMAIL_ADDRESS>Thijs Benschop Research associate at
Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. - Center for Applied
Statistics and Econometrics, Humboldt-Universität zu Berlin, Unter den Linden
6, 10099 Berlin, Germany. Email<EMAIL_ADDRESS>
( )

###### Abstract

AI artificial intelligence brings about new quantitative techniques to assess
the state of an economy. Here we describe a new measure for systemic risk: the
Financial Risk Meter (FRM). This measure is based on the penalization
parameter ($\lambda$) of a linear quantile lasso regression. The FRM is
calculated by taking the average of the penalization parameters over the 100
largest US publicly traded financial institutions. We demonstrate the
suitability of this AI based risk measure by comparing the proposed FRM to
other measures for systemic risk, such as VIX, SRISK and Google Trends. We
find that mutual Granger causality exists between the FRM and these measures,
which indicates the validity of the FRM as a systemic risk measure. The
implementation of this project is carried out using parallel computing, the
codes are published on www.quantlet.de with keyword  FRM. The R package
RiskAnalytics is another tool with the purpose of integrating and facilitating
the research, calculation and analysis methods around the FRM project. The
visualization and the up-to-date FRM can be found on hu.berlin/frm.

Keywords: Systemic Risk, Quantile Regression, Value at Risk, Lasso, Parallel
Computing, Financial Risk Meter

JEL: C21, C51, G01, G18, G32, G38.

This is a post-peer-review, pre-copyedit version of an article published in
the Singapore Economic Review. The final authenticated version is available
online at:
http://dx.doi.org/10.1142/S0217590819500668

##  1 Introduction

Systemic risk is hazardous for the stability of financial markets, since the
failure of one firm may impact the stability of the whole system. There are
various definitions of systemic risk. One of the most popular definitions is
introduced in Schwarcz (2008). He defined systemic risk as a trigger event,
such as an economic shock or institutional failure, causing a chain of bad
economic consequences, sometimes referred to as domino effect. This definition
indicates that interlinkages and interdependencies in a system or market are
very crucial for controlling systemic risk. The financial crisis in 2008 is an
example. After the bankruptcy of Lehman Brothers, several more financial
cooperations bankrupted as a result of their interlinkages with Lehman
Brothers. Consequently, there has been a surge in the interest in measuring
and controlling systemic risk since the 2008 crisis, which has led to an
increase in the research on this topic.

Several methodologies for measuring systemic risk have been proposed. Adrian
and Brunnermeier (2016) proposed CoVaR, the value at risk of financial
institutions conditional on the other institutions being under distress, which
uses two linear quantile regressions. Hautsch et al. (2015) refined this
algorithm by introducing linear quantile lasso regression with a fixed
penalization parameter $\lambda$ for each company to select the relevant risk
drivers. Fan et al. (2016) and Härdle et al. (2016) use a nonlinear Single
Index Model (SIM) combined with a variable selection technique to select risk
factors. In their application, they use data on 200 financial companies and 7
macro variables to estimate CoVaR. During the estimation procedure, a time-
varying penalization parameter $\lambda$ is generated. This series has a
striking pattern: higher values correspond to financial crisis times and lower
values correspond to stable periods. This observation has led to the idea to
use the penalization parameter $\lambda$ itself as a measure for systemic
risk. The time-varying feature is not observed in Hautsch et al. (2015), who
applied a fixed $\lambda$ for each firm, and not time varying.

Fan et al. (2016) provide the $\lambda$ series for single companies. In
contrast, we would like to see the overall behavior of $\lambda$. Härdle et
al. (2016) compare the linear quantile lasso model and SIM, and conclude that
SIM is more suited than the linear model, but that the linear quantile lasso
model is also valid in terms of backtesting. Indeed if one generates $\lambda$
series for 100 firms with more than $300$ observations each, then the
application of SIM is not realistic. Since linear quantile lasso is easier to
apply and time saving, we decided to use it to compute the FRM.

We use log return data from the 100 largest US publicly traded financial
institutions as well as 6 macro variables. Our model is based on daily log
returns of these financial institutions. The time period under consideration
runs from April 5, 2007 until September 23, 2016 and covers several documented
financial crises (2008, 2011). We observe that the pattern of this risk
measure is more informative on financial risk. The shape and volatility of the
series correspond to the market volatility and financial events with a large
impact on systemic risk are clearly visible. Therefore, this series of
averaged $\lambda$ may be called a Financial Risk Meter (FRM). Zbonakova et
al. (2017) apply linear quantile lasso regression to analyze the behavior of
the $\lambda$ series. They find that $\lambda$ is sensitive to the changes of
volatility, which provide the theoretical evidence for the FRM to be a
systemic risk measure, as high volatility indicates high risk.

We introduce the methodology of the FRM, describe the risk levels, the
computational implementation as well as possible visualizations. We compare
the FRM with other systemic risk measures, such as VIX (Hallett; 2009), SRISK
(see Brownlees and Engle; 2016) as well as the Google trends of key words
related to financial crises (see Preis et al.; 2013). We find that the FRM and
these risk measures mutually Granger cause, which indicates the validity of
the FRM as a systemic risk measure.

The remainder of this paper is organized as follows. In Section 2 the
methodology used to construct our FRM, which is quantile lasso modeling, is
presented. Section 3 presents the data, computational challenge and the
visualization of the results. Section 4 shows the validity of our FRM as a
measure for financial risk by comparing with other financial risk measures.
Section 5 concludes, the financial institutions applied in this paper is
listed in Section 6 Appendix. All the R programs for this paper can be found
on www.quantlet.de (Borke and Härdle; 2017a).

##  2 FRM methodology and estimation

In this section we describe the methodology and algorithm used to compute FRM.
Since the penalization parameters are computed based on an $L_{1}$-norm
(LASSO) quantile linear regression, this regression framework is introduced
first. Within this framework, the penalization parameter $\lambda$ is
exogenous. Since the FRM is distilled from the selected penalization
parameter, we subsequently discuss methods to select $\lambda$.

###  2.1 Linear Quantile Lasso Regression Model

Following Härdle et al. (2016), we introduce the quantile lasso regression
model. Let $m$ be the number of macro variables describing the state of the
economy, $k$ the number of firms under consideration, $j\in\\{1,\ldots,k\\}$.
Then $p=k+m-1$ represents the number of covariates. $t\in\\{1,\ldots,T\\}$ is
the time point with $T$ the total number of observations (days). $s$ is the
index of moving window, $s\in\\{1,\ldots,(T-(n-1))\\}$, where $n$ is the
length of window size. Then the quantile lasso regression is defined as:

$X^{s}_{j,t}=\alpha^{s}_{j}+A_{j,t}^{s,\top}\beta^{s}_{j}+\varepsilon^{s}_{j,t},$
(1)

where $A^{s}_{j,t}\stackrel{{\scriptstyle def}}{{=}}$
$\left[\begin{array}[]{c}M^{s}_{t-1}\\\ X^{s}_{-j,t}\\\ \end{array}\right]$,
$M^{s}_{t-1}$ the $m$ dimensional vector of macro variables, $X^{s}_{-j,t}$ is
the $p-m$ dimensional vector of log returns of all other firms except firm $j$
at time $t$ and in moving window $s$, $\alpha^{s}_{j}$ is a constant term and
$\beta^{s}_{j}$ is a $p\times 1$ vector defined for moving window $s$.

The regression is performed using $L_{1}$-norm quantile regression proposed by
Li and Zhu (2008):

$\displaystyle\min\limits_{\alpha^{s}_{j},\beta^{s}_{j}}\left\\{n^{-1}\sum\limits_{t=s}^{s+(n-1)}\rho_{\tau}\big{(}X^{s}_{j,t}-\alpha^{s}_{j}-A_{j,t}^{{s,\top}}\beta^{s}_{j}\big{)}+\lambda^{s}_{j}\parallel\beta^{s}_{j}\parallel_{1}\right\\},$
(2)

where $\lambda^{s}_{j}$ is the penalization parameter, and the check function
$\rho_{\tau}(u)$ is:

$\rho_{\tau}(u)=|u|^{c}|\mathbf{I}(u\leq 0)-\tau|,$

where $c=1$ corresponds to quantile regression. The $L_{1}$-norm quantile
linear regression can be used to select relevant covariates (other firms and
macro state variables) for each firm.

###  2.2 Penalization Parameter $\lambda$

Since Equation (2) has an $L_{1}$ loss function and an $L_{1}$-norm penalty
term, the optimization problem is an $L_{1}$-norm quantile regression
estimation problem. The choice of the penalization parameter $\lambda_{j}^{s}$
is crucial. There are several options to select $\lambda_{j}^{s}$, e.g. with
the Bayesian Information Criterion (BIC) or using the Generalized Approximate
Cross-Validation criterion (GACV). Yuan (2006) conducted simulations and
concluded that GACV outperforms BIC in terms of statistical efficiency.
Therefore, we determine $\lambda_{j}^{s}$ with the GACV criterion in the FRM
model and set $\lambda_{j}^{s}$ as the solution of the following minimization
problem:

$\min
GACV(\lambda_{j}^{s})=\min\frac{\sum_{t=s}^{s+(n-1)}\rho_{\tau}\big{(}X^{s}_{j,t}-\alpha^{s}_{j}-A_{j,t}^{s\top}\beta^{s}_{j}\big{)}}{n-df},$

where $df$ is a measure of the effective dimensionality of the fitted model.
The advantage of GACV is that it also works for $p>n$, which can be important
for the FRM if the moving window size is small.

To compute the FRM, we perform the regression analysis as described above and
select the $\lambda_{j}^{s,*}$ for each firm $j$ using GACV. The Financial
Risk Meter is defined as the average lambdas over the set of $k$ firms for all
windows:

$FRM\stackrel{{\scriptstyle
def}}{{=}}\frac{1}{k}\sum_{j=1}^{k}\lambda_{j}^{*}$

##  3 Computational challenges and visualization

###  3.1 Data

To compute the FRM, we use data from 100 US publicly traded financial
institutions as well as six macro variables. The selection of financial
companies is based on the NASDAQ company list111See the NASDAQ webpage:
http://www.nasdaq.com/screening/companies-by-industry.aspx?industry=Finance
and based on the market capitalization. The selected companies are the 100 US
publicly traded financial institutions with the largest market capitalization,
see Table LABEL:table:List_of_Firms in Appendix.

Figure 1: The $x$-axis represents the number of firms ordered by market
capitalization and the $y$-axis the percentage of total market capitalization.

FRM_per_cap

Initially, we used data on the 200 US publicly traded financial institutions
with the largest market capitalization to compute the FRM. However, the
smaller companies in this set change regularly over the time period under
consideration (2007-2016) due to, for instance, bankruptcies. This leads to
obvious data download issues and therefore we use only 100 firms. Figure 1
shows the cumulative market capitalization of US financial firms. The $x$-axis
represents the firms ordered by market capitalization and the $y$-axis the
cumulative market capitalization. We observe that the largest 100 firms cover
more than $85\%$ of the total market capitalization of all companies in the US
financial market and we therefore can restrict our analysis to 100 firms.
Furthermore, the results of estimating the FRM based on 100 or 200 firms are
very similar if the moving window size is the same. Figure 2 plots both FRM
series with the window size $n=126$, the shape and the trends of them are
similar.

Figure 2: FRM with 100 firms (black) and FRM with 200 firms (grey), moving
window size $n=126$.

FRM_compare_nf

Figure 3: FRM with different moving window size, $n=63$ (black) and $n=126$
(grey), both series are scaled into the interval [0,1], from July 6, 2007
until September 23, 2016.

FRM_compare_ws

We select six macro state variables to represent the general state of the
economy: 1) the implied volatility index, VIX from Yahoo Finance; (2) the
changes in the three-month Treasury bill rate from the Federal Reserve Bank of
St. Louis; (3) the changes in the slope of the yield curve corresponding to
the yield spread between the ten-year Treasury rate and the three-month bill
rate from the Federal Reserve Bank of St. Louis; (4) the changes in the credit
spread between BAA-rated bonds and the Treasury rate from the Federal Reserve
Bank of St. Louis; (5) the daily S&P500 index returns from Yahoo Finance, and
(6) the daily Dow Jones US Real Estate index returns from Yahoo Finance.

To compute the FRM we employ the tail parameter $\tau=0.05$. To find the a
stable window size, $n$, we have to make a trade-off. We find that the lasso
selection technique performs worse if the window size is too small. Since we
use daily data, the moving window size should be larger than $50$, so that the
estimation for each window is more precise. The results of using different
window sizes (we have considered window sizes $n=63$ (one quarter) and $n=126$
(half a year)) are shown in Figure 3. The larger the window size, the more
lagged, but also the smoother the plot is. Cross correlation can be used to
determine the time delay of a time series, which we apply here for the
estimate of the FRM with $n=63$ and the FRM with $n=126$. In Figure 4 and
Table 1, the largest autocorrelation between FRM with $n=63$ and the lagged
FRM with $n=126$ is $0.967$ from lag $-29$ to lag $-22$. We conclude that the
FRM with $n=63$ leads the FRM $n=126$ by at least 22 periods. From all the
preceding we set the moving window size to $n=63$.

Figure 4: Cross correlation between FRM with $n=63$ and FRM with $n=126$,
where the number of firms is 100.

FRM_compare_ws

For each firm we have $2,386$ daily observations and $105$ covariates (99
firms and 6 macro-state variables). The FRM is the average of the $\lambda$’s
computed from the 100 individual firms. The $\lambda$’s for the individual
firms are more volatile and less smooth than the average over 100 firms and
therefore more robust to reflect the impact from financial events on systemic
risk. Figure 5 illustrates this by plotting the $\lambda$ of firm Wells Fargo
(the largest firm by market capitalization) and the FRM.

Lag | -30 | -29 | -28 | -27 | -26 | -25 | -24 | -23 | -22 | -21
---|---|---|---|---|---|---|---|---|---|---
Cross correlation | 0.963 | 0.964 | 0.964 | 0.964 | 0.964 | 0.964 | 0.964 | 0.964 | 0.964 | 0.963

Table 1: Cross correlation between the estimates of the FRM with $n=63$ and
FRM with $n=126$.

Figure 5: FRM (black) and $\lambda$ of Wells Fargo (grey), both series are
scaled into interval [0,1], from April 5, 2007 until September 23, 2016.

FRM_compare_of

###  3.2 Computational challenges

We wrote a script to automatically download the data from Yahoo Finance and
Federal Reserve Bank of St. Louis. The R package _quantmod_ is used. More
details and the script are available from Quantnet ( FRM_download_data).

The $L_{1}$-norm quantile regression used to generate the $\lambda$ series is
computationally intensive and therefore time-consuming, if applied
sequentially for a large number of firms, see for instance the code from
Quantnet ( FRM_lambda_series). Therefore, we consider parallel computing in R
to reduce the computation time. R offers several algorithms for performance
computing, such as _lapply_ , _mclapply_ , _parLapply_ , _for_ and _foreach_
222The webpage http://www.parallelr.com/r-with-parallel-computing/ provides an
overview.. For our purposes the _foreach_ loops is the fastest solution, which
we use for implementation.

We use the _doParallel_ and _foreach_ packages in R as developed and proposed
by Calaway, Weston, Tenenbaum and Analytics (2015) and Calaway, Weston and
Analytics (2015), see also Kane et al. (2013). Since we have 100 financial
firms we use the _foreach_ loops twice: the first loop is for the 100
financial firms with the second loop nested in the first loop to perform the
moving window estimation. The speed of computation is increased considerably,
the script is available from Quantnet:  FRM_parallel_compute.

Without the use of parallel computing, i.e. using a processor with four cores
for each moving window, it requires around two minutes to generate the FRM
estimate for one day. The Research Data Center (RDC) of Humboldt-Universität
zu Berlin has provided access to their multi-core servers. Their servers have
respectively 24, 32, and 40 cores. By using these servers combined with
parallel computing, the average computation time is reduced approximately $12$
seconds to obtain a daily value for the FRM. The FRM webpage is updated
weekly, which takes only 1 minute to generate the FRM series for five working
days.

###  3.3 Visualization

To implement the visualization of the FRM, we use the JavaScript framework
D3.js (or just D3 for Data-Driven Documents), which is a JavaScript library
for producing dynamic, interactive data visualizations in web browsers. The
QuantNetXploRer is a good example of D3 in power. More information about the
D3 architecture, its various designs and the D3-based QuantNetXploRer can be
found in Bostock et al. (2011) and Borke and Härdle (2017b).

Figure 6: The graph of Financial Risk Meter (FRM).

FRM_parallel_compute

Figure 6 illustrates the D3-based FRM visualization, and more examples e.g.
for Asia, Europe are available on Althof et al. (2019).

####  3.3.1 Descriptive statistics

Figure 6 shows the FRM series from April 5, 2007 through September 23, 2016.
The FRM has no theoretical upper bound. In the time frame under consideration,
the maximum value is $0.075$, which occurred on December 15, 2008 and the mean
value is $0.021$. We observe several peaks in the FRM series, which correspond
to crises and other events in these periods. Two peaks correspond to the
financial crises in 2008 and 2010. The peak in the first quarter of 2009 is at
the height of the Great Recession: 800 thousand jobs were lost and the
unemployment rate rose to $7.8\%$ in the US, which was the highest since June
1992. Another peak around the fourth quarter of 2011 coincides with the
decline in stock markets in August 2011, which was due to fears of contagion
of the European sovereign debt crisis to Spain and Italy.

Therefore one may state that the peaks of FRM series identify financial events
and their impact on financial and systemic risk. The minimum of the FRM series
in the time period under consideration is observed in August 26, 2014, with a
value of $0.009$. This was a relatively stable period.

####  3.3.2 Risk levels

For convenience and following the color scheme of US homeland security office
we divide risk into five levels with different classifications and colors. The
levels of risk are defined as different intervals of ratios for the FRM. These
ratios are computed based on the past values of the FRM. As shown in Figure 7,
we have five levels of risk with five color codes. The current risk level is
determined by the ratio based on all past FRM observations into which the
current $\lambda$ falls. Table 2 presents the risk levels as well as the
colors, descriptions and ratios of the risk levels.

Figure 7: Risk levels of FRM Color | Risk level description | FRM ratio
---|---|---
Green | Low risk of crisis in the financial market. |  $<$20
| The incidence of a crisis is less likely than usual. |
Blue | General risk of crisis in the financial market. | 20-40
| There is no specific risk of a crisis. |
Yellow | Elevated risk of crisis in the financial market. | 40-60
| The incidence of a crisis is somewhat higher than usual. |
Orange | High risk of crisis in the financial market. | 60-80
| A crisis might occur very soon. |
Red | Severe risk of a crisis in the financial market. |  $>$80
| A financial crisis is imminent or happening right now. |
Table 2: Risk levels, color codes and ratios for FRM

As an example, on September 23, 2016 the value of FRM was $0.013$. Since the
maximum of FRM series up to that date was $0.075$, the ratio of this risk
measure on September 23, 2016 was $17.3\%$. Since this is less than the
20%-ratio, we classify the risk on that day as low risk of crisis in the
financial market with color green. On the website the current risk level is
marked with a cross as shown in Figure 7 for this example.

##  4 Causality of FRM and other systemic risk measures

Zbonakova et al. (2017) analyze the factors affecting the value of $\lambda$
and summarize that $\lambda$ depends on three major factors: the variance of
the error term, the correlation structure of the covariates and the number of
non-zero coefficients of the model. Since high volatility indicates high risk
in finance and the number of non-zero coefficients is related to the
connectedness of the financial firms, they provide more theoretical evidence
for the FRM as a risk measure. In their application, they find a co-
integration relationship between $\hat{\lambda}$ and other systemic risk
measures. We extend their idea and use Granger causality analysis to validate
FRM. We select three measures: VIX (see Hallett; 2009), SRISK (see Brownlees
and Engle; 2016) as well as the Google trends of the key word "financial
crisis" (see Preis et al.; 2013).

For the causality analysis we first need to introduce the Vector
Autoregression (VAR) model briefly. Lütkepohl (2005) proposes the VAR(P) model
as follows:

$y_{t}=\alpha+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots+A_{P}y_{t-P}+{\mathbf{u}_{t}},$
(3)

where $y_{t}\stackrel{{\scriptstyle def}}{{=}}(y_{1t},\ldots,y_{Kt})^{\top}$,
$A_{i}$ are fixed ($K\times K$) coefficient matrices, $\mathbf{u}_{t}$ is a
$K$ dimensional process. The coefficients could be estimated by applying
multivariate least squares estimation. In order to perform the Granger
causality test, the vector of endogenous variables $y_{t}$ is split into two
subvectors $y_{1t}$ and $y_{2t}$ with dimensions $(K_{1}\times 1)$ and
$(K_{2}\times 1)$ and $K=K_{1}+K_{2}$. Then the VAR(P) model can be rewritten
as follows:

$\displaystyle y_{t}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}y_{1t}\\\ y_{2t}\\\
\end{array}\right)=\left(\begin{array}[]{c}\alpha_{1}\\\ \alpha_{2}\\\
\end{array}\right)+\left(\begin{array}[]{cc}A_{11,1}&A_{12,1}\\\
A_{21,1}&A_{22,1}\\\ \end{array}\right)\left(\begin{array}[]{c}y_{1,{t-1}}\\\
y_{2,{t-1}}\\\ \end{array}\right)+\cdots$ (19) $\displaystyle\hskip
86.11084pt+\left(\begin{array}[]{cc}A_{11,P}&A_{12,P}\\\ A_{21,P}&A_{22,P}\\\
\end{array}\right)\left(\begin{array}[]{c}y_{1,{t-P}}\\\ y_{2,{t-P}}\\\
\end{array}\right)+\left(\begin{array}[]{c}u_{1t}\\\ u_{2t}\\\
\end{array}\right)$

The null hypothesis of the Granger causality test is that the subvector
$y_{1t}$ does not Granger-cause $y_{2t}$, which is defined as $A_{21,i}=0$ for
$i=1,2,\ldots,P$. The alternative hypothesis states that the subvector
$y_{1t}$ Granger-causes $y_{2t}$ and is defined as: $\exists\,A_{21,i}\neq 0$
for $i=1,2,\ldots,P$. The test statistic follows an $F$ distributions with
$PK_{1}K_{2}$ and $KJ-n^{*}$ degrees of freedom, where $J$ is the sample size
and $n^{*}$ equals the total number of parameters in the above VAR(P) model.

###  4.1 FRM versus VIX

The VIX series is often addressed as a “fear index” and can be interpreted as
a measure for systemic risk (Hallett; 2009). For reasons of comparability, we
standardize these two series by setting the lowest value in the sample to zero
and the highest to one. Figure 8 plots the standardized FRM series (thick
black line) and the VIX series (thin red line). The plot shows that both
indicators move in the same direction, with the VIX series a little more
volatile. We also get some evidence of some financial events by observing the
corresponding volatility levels of the FRM and VIX. For example, in the end of
2008 there is a sharp upward trend of FRM, whereas the upward trends dominates
VIX as well, which corresponds to the bankruptcy of Lehman Brothers on
September 15, 2008. Both FRM and VIX have higher values between 2008 and 2010,
which corresponds to the time period of the financial crises. After 2013 the
values of FRM are relative stable at a low level, while there is similar
pattern of VIX, which shows signs of the slow recovery of the global economy
from the recession.

Figure 8: Scaled FRM (thick black line) and VIX (thin red line)

FRM_VIX

Before we perform the Granger causality test, we test for stationarity of both
time series with the Augmented Dickey-Fuller (ADF) test. The results of the
test are shown in Table 3. For the FRM series, the $p$ value is larger than
$0.05$, so we cannot reject the null hypothesis, i.e. the FRM series may have
a unit root and may be non-stationary. We reject the null hypothesis for the
VIX series with a $p$ value smaller than $0.05$ and conclude that the VIX
series is stationary. We do not need to consider the co-integration problem,
since only if both series are non-stationary, we should take into account the
co-integration. There is a trade-off between using the original data and the
transformed (differenced) data to find the causality relationship. Sims (1980)
prefers to use the original data. He argues that VAR with non-stationary
variables may provide important insights, if one is interested in the nature
of relationships between variables. Brooks (2014) also states that
differencing will destroy information on any long-run relationships between
the series. However, other people argue that the original non-stationary data
might lead to untrusted estimation, see Yule (1926) and Granger and Newbold
(1974). In our case, we consider both the original data and transformed data.

Series |  $p$ values
---|---
FRM | $0.28$
VIX | 0.01
DFRM | $0.01$

Table 3: $p$ values of ADF test for stationarity

FRM_VIX

Model | AIC | HQ | SC | FPE
---|---|---|---|---
FRM and VIX | 20 | $3$ | $3$ | 20
DFRM and VIX | 19 | $8$ | $5$ | 19
Table 4: Suggested order for VAR process by different criteria

Model | Order VAR | PT (asymptotic) | PT (adjusted) | BG | ES
---|---|---|---|---|---
FRM and VIX | 3 | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $1.1\times 10^{-07}$ | $1.0\times 10^{-07}$
11 | $2.5\times 10^{-07}$ | $2.0\times 10^{-07}$ | $1.6\times 10^{-01}$ | $1.7\times 10^{-01}$
| 20 | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $3.1\times 10^{-08}$ | $4.1\times 10^{-08}$
DFRM and VIX | 5 | $2.2\times 10^{-16}$ | $2.2\times 10^{-16}$ | $3.2\times 10^{-08}$ | $3.1\times 10^{-08}$
8 | $6.7\times 10^{-12}$ | $4.9\times 10^{-12}$ | $1.4\times 10^{-06}$ | $1.5\times 10^{-06}$
11 | $2.3\times 10^{-09}$ | $1.8\times 10^{-09}$ | $1.5\times 10^{-03}$ | $1.7\times 10^{-03}$
19 | $1.7\times 10^{-03}$ | $1.6\times 10^{-03}$ | $5.5\times 10^{-08}$ | $7.2\times 10^{-08}$

Table 5: $p$ values of model selection tests

Cause | Effect |  $p$ values
---|---|---
FRM | VIX | $4.0\times 10^{-08}$
VIX | FRM | $6.1\times 10^{-11}$
DFRM | VIX | $6.6\times 10^{-11}$
VIX | DFRM | $8.7\times 10^{-13}$

Table 6: $p$ values of Granger causality test

FRM_VIX

Figure 9: Autoregression functions of FRM and VIX

FRM_VIX

Figure 10: Autoregression functions of DFRM and VIX

FRM_VIX

Firstly, we consider the original data. We choose the VAR order according to
four criteria: the Akaike information criterion (AIC), the Hannan-Quinn
information criterion

(HQ), the Schwarz criterion (SC) and the Prediction Error Criterion (FPE), see
Table 4. While HQ and SC suggest an order 3 VAR process, AIC and FPE suggest
an order 20 process. We fit both VAR models with order 3 and order 20. Next,
we check the autocorrelation of the residuals to decide the optimal order.
Four tests are carried out: the asymptotic Portmanteau Test, the adjusted
Portmanteau Test, the Breusch-Godfrey LM test and the Edgerton-Shukur F test.
The null hypothesis of these tests is that there is no first order
autocorrelation among residuals. Choosing order 3 and 20 leads to the
rejection of all these tests (cf. Table 5). Subsequently we try the other
orders and find that with order 11 both the Breusch-Godfrey LM test and the
Edgerton-Shukur F tests are passed. Therefore, we select order 11. The
autocorrelation function of the residuals is plotted in Figure 9. Table 6
shows the results of the Granger causality test. All $p$ values are smaller
than $0.05$ which indicates that the null hypothesis is rejected. Therefore,
FRM Granger causes VIX, and also VIX Granger causes FRM.

Next, we consider the transformed series. Since FRM is non-stationary, we take
the first difference. The transformed series is called as DFRM. In Table 3 we
see that DFRM is stationary. Then the same procedure as before is performed.
While HQ suggests an order 8 process, SC suggest an order 5, and AIC and FPE
both suggest an order 19 (cf. Table 4). After checking the four tests for
autocorrelation of the residuals, we conclude that the optimal order is 19.
Although it does not pass the autocorrelation test, the $p$ value is close to
the critical value $0.05$, and the autocorrelation function confirms this
result (cf. Table 5 and Figure 10). The result of the Granger causality test
is summarized in Table 6. We find that all $p$ values are significantly
smaller than $0.05$, which indicates that the null hypothesis is rejected.
Therefore we conclude that DFRM Granger causes VIX, and also VIX Granger
causes DFRM.

###  4.2 FRM versus SRISK

SRISK is a macro-finance measure of systemic risk (Acharya et al.; 2012;
Brownlees and Engle; 2016). Our data on SRISK for the US are obtained from
V-Lab 333See the Systemic Risk Analysis Welcome Page:
https://vlab.stern.nyu.edu/welcome/risk/. We also standardize SRISK, so that
both series are comparable on the same scale. Figure 11 plots the standardized
FRM series (thick black line) and the SRISK series (thin blue line). We see
that there is a peak in the first quarter of 2008 for SRISK, but afterwards
FRM and SRISK have similar patterns. Especially during the beginning of 2010
and the beginning of 2012, the two series have a similar shape.

Figure 11: Scaled FRM (thick black line) and SRISK (thin blue line)

FRM_SRISK

Variables | p-values
---|---
FRM | $0.48$
SRISK | 0.10
Table 7: $p$ values of ADF test for stationarity for FRM and SRISK

Explanatory (Cause) | Response (Effect) | Value of test-statistic | Critical value at 5%
---|---|---|---
FRM | SRISK | -3.1 | -1.95
SRISK | FRM | -2.7 | -1.95

Table 8: Results of Engle Granger 2-step co-integration test

FRM_SRISK

We perform the same procedure as in section 4.1. The results of the ADF test
for the SRISK series in Table 7 show that the series is non-stationary. Since
the FRM series is neither stationary, we consider the co-integration of them.
From Granger (1988) we know that if both series are co-integrated, then there
must be Granger causality between them in at least one way. We perform the
Engle Granger 2-step test for co-integration, which is suitable for bivariate
time series. In the first step, the linear regression of FRM on SRISK is
carried out, i.e. FRM is the explanatory variable and SRISK the response
variable. In the second step, we test the residuals of the aforementioned
linear regression. If these residuals are stationary, then there is co-
integration of FRM and SRISK. The null hypothesis of this test is that the
residuals are non-stationary. The result of this test are summarized in Table
8. We conclude that FRM and SRISK are co-integrated, in other words, FRM
Granger causes SRISK. If we regress SRISK on FRM, i.e. SRISK is the
explanatory variable and FRM the response variable, we also conclude that
SRISK and FRM are co-integrated, which indicates that SRISK Granger causes
FRM. We thus conclude that there is mutual causality between FRM and SRISK.

###  4.3 FRM versus Google Trends

Finally, we analyze the relationship between FRM and Google Trends (GT) for
the keyword "financial crisis". Google Trends provides data on the search
volume of particular words and phrases relative to the total search volume.
This can be disaggregated by countries. If a keyword is more frequently
searched for, this might indicate a particular interest. Preis et al. (2013)
analyzed the data related to finance from Google Trends, and find that Google
Trends data did not only reflect the current state of the stock markets, but
may have also been able to forecast certain future trends. We use Google
Trends for the keyword "financial crisis", assuming that more people will
search for this term if they feel the risk for a financial crisis is high. The
Google Trends data are weekly data. To allow for comparison with the FRM we
apply cubic interpolation to estimate daily data from the weekly Google Trends
series. This series is compared with the daily FRM series. Figure 12 plots
both the daily FRM series as well as the cubic interpolated Google Trends
daily series. Both series are standardized to the interval zero-one for
comparison. We observe some co-movement between both series, but no continuous
superiority of the Google Trends above FRM.

Figure 12: Scaled FRM (thick black line) and Google Trends (thin green line)

FRM_GT

The ADF test shows that the GT series is stationary (cf. Table 9). We perform
two tests for the relationship between the two series. Firstly, we consider
the original data of FRM, then we consider the transformed data. We perform
four criteria to find the optimal order of VAR model. As the results in Table
10 show, all the criteria suggest an order 20 VAR process. Therefore, we apply
an order 20 VAR model. Next, the autocorrelation of the residuals is tested.
Although none of the tests can be passed (cf. Table 11), we have no better
choice for the order than 20. The autocorrelation function of residuals are
plotted in Figure 13. Table 12 shows the results of the Granger causality
test. All $p$ values are significantly smaller than $0.05$, which indicates
that the null hypothesis is rejected. Therefore, FRM Granger causes GT, and GT
Granger causes FRM.

For the first differenced FRM, i.e. DFRM, the same procedure is used. In Table
10 all the criteria suggest an order 20 VAR process. The result of the
autocorrelation tests are presented in Table 10. Although none of the tests is
passed, we still use order 20. The autocorrelation function of the residuals
is shown in Figure 14. Table 12 shows the results of the Granger causality
test. All $p$ values are significantly smaller than $0.05$, which indicates
that the null hypothesis is rejected. Therefore, DFRM Granger causes GT, and
GT Granger causes DFRM.

Variables | p-values
---|---
FRM | $0.48$
GT | 0.01
DFRM | $0.01$
Table 9: $p$ values of ADF test for stationarity for FRM and GT Model | AIC | HQ | SC | FPE
---|---|---|---|---
FRM and GT | 20 | $20$ | $20$ | 20
DFRM and GT | 20 | $20$ | $20$ | 20
Table 10: Suggested order for VAR process by different criteria

Model | Order | PT (asymptotic) | PT (adjusted) | BG | ES
---|---|---|---|---|---
FRM and GT | 20 | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$
DFRM and GT | 20 | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$ | $<2.2\times 10^{-16}$

Table 11: $p$ values of model selection tests

Cause | Effect | p-values
---|---|---
FRM | GT | $1.1\times 10^{-10}$
GT | FRM | $2.1\times 10^{-12}$
DFRM | GT | $6.8\times 10^{-11}$
GT | DFRM | $4.1\times 10^{-10}$

Table 12: $p$ values of Granger causality test

FRM_GT

Figure 13: Autoregression functions of FRM and GT

FRM_GT

Figure 14: Autoregression functions of DFRM and GT

FRM_GT

##  5 Conclusion

In this paper we propose and develop an AI based measure for systemic risk in
financial markets: the Financial Risk Meter (FRM). The FRM is a measure for
systemic risk based on the penalty term $\lambda$ of the linear quantile lasso
regression, which is defined as the average of the $\lambda$ series over the
100 largest US publicly traded financial institutions. The implementation is
carried out by using parallel computing. The risk levels are classified by
five levels. The empirical result shows that our Financial Risk Meter can be a
good indicator for trends in systemic risk. Compared with other systemic risk
measures, such as VIX, SRISK, Google Trends with the keyword “financial
crisis”, we find that the FRM and VIX, FRM and SRISK, FRM and GT mutually
granger cause one another, which means that our FRM is a good measure of
systemic risk for the US financial market. All the codes of FRM are published
on www.quantlet.de with keyword  FRM. The R package RiskAnalytics (Borke;
2017b) is another tool with the purpose of integrating and facilitating the
research, calculation and analysis methods around the FRM project (Borke;
2017a). The up-to-date FRM can be found on hu.berlin/frm.

##  6 Appendix: Financial Institutions

Table 13: The list of 100 financial companies used to estimate FRM in our sample. WFC | Wells Fargo & Company | AON | Aon plc
---|---|---|---
JPM | J P Morgan Chase & Co | ALL | Allstate Corporation
BAC | Bank of America Corporation | BEN | Franklin Resources, Inc.
C | Citigroup Inc. | STI | SunTrust Banks, Inc.
AIG | American International Group, Inc. | MCO | Moody’s Corporation
GS | Goldman Sachs Group, Inc. | PGR | Progressive Corporation
USB | U.S. Bancorp | AMP | Ameriprise Financial Services, Inc.
AXP | American Express Company | AMTD | TD Ameritrade Holding Corporation
MS | Morgan Stanley | HIG | Hartford Financial Services Group, Inc.
BLK | BlackRock, Inc. | TROW | T. Rowe Price Group, Inc.
MET | MetLife, Inc. | NTRS | Northern Trust Corporation
PNC | PNC Financial Services Group, Inc. (The) | MTB | M&T Bank Corporation
BK | Bank Of New York Mellon Corporation (The) | FITB | Fifth Third Bancorp
SCHW | The Charles Schwab Corporation | IVZ | Invesco Plc
COF | Capital One Financial Corporation | L | Loews Corporation
PRU | Prudential Financial, Inc. | EFX | Equifax, Inc.
TRV | The Travelers Companies, Inc. | PFG | Principal Financial Group Inc
CME | CME Group Inc. | RF | Regions Financial Corporation
CB | Chubb Corporation | MKL | Markel Corporation
MMC | Marsh & McLennan Companies, Inc. | FNF | Fidelity National Financial, Inc.
BBT | BB&T Corporation | LNC | Lincoln National Corporation
ICE | Intercontinental Exchange Inc. | CBG | CBRE Group, Inc.
STT | State Street Corporation | KEY | KeyCorp
AFL | Aflac Incorporated | NDAQ | The NASDAQ OMX Group, Inc.
CINF | Cincinnati Financial Corporation | CACC | Credit Acceptance Corporation
CNA | CNA Financial Corporation | BRO | Brown & Brown, Inc.
HBAN | Huntington Bancshares Incorporated | ERIE | Erie Indemnity Company
SEIC | SEI Investments Company | OZRK | Bank of the Ozarks
ETFC | E*TRADE Financial Corporation | WTM | White Mountains Insurance Group, Ltd.
AMG | Affiliated Managers Group, Inc. | SNV | Synovus Financial Corp.
RJF | Raymond James Financial, Inc. | ISBC | Investors Bancorp, Inc.
UNM | Unum Group | MKTX | MarketAxess Holdings, Inc.
NYCB | New York Community Bancorp, Inc. | LM | Legg Mason, Inc.
Y | Alleghany Corporation | CBSH | Commerce Bancshares, Inc.
SBNY | Signature Bank | BOKF | BOK Financial Corporation
CMA | Comerica Incorporated | EEFT | Euronet Worldwide, Inc.
AJG | Arthur J. Gallagher & Co. | DNB | Dun & Bradstreet Corporation
TMK | Torchmark Corporation | WAL | Western Alliance Bancorporation
WRB | W.R. Berkley Corporation | EV | Eaton Vance Corporation
AFG | American Financial Group, Inc. | CFR | Cullen/Frost Bankers, Inc.
SIVB | SVB Financial Group | MORN | Morningstar, Inc.
EWBC | East West Bancorp, Inc. | THG | The Hanover Insurance Group, Inc.
ROL | Rollins, Inc. | UMPQ | Umpqua Holdings Corporation
ZION | Zions Bancorporation | CNO | CNO Financial Group, Inc.
AIZ | Assurant, Inc. | FHN | First Horizon National Corporation
PACW | PacWest Bancorp | WBS | Webster Financial Corporation
AFSI | AmTrust Financial Services, Inc. | PB | Prosperity Bancshares, Inc.
ORI | Old Republic International Corporation | PVTB | PrivateBancorp, Inc.
PBCT | People’s United Financial, Inc. | SEB | Seaboard Corporation
FCNCA | First Citizens BancShares, Inc. | MTG | MGIC Investment Corporation

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