# Action formalism for geometric phases from self-closing quantum trajectories

Dominic Shea  Department of Physics, Lancaster University, Lancaster LA1 4YB,
United Kingdom.  Alessandro Romito  Department of Physics, Lancaster
University, Lancaster LA1 4YB, United Kingdom.

(August 28, 2024)

###### Abstract

When subject to measurements, quantum systems evolve along stochastic quantum
trajectories that can be naturally equipped with a geometric phase observable
via a post-selection in a final projective measurement. When post-selecting
the trajectories to form a close loop, the geometric phase undergoes a
topological transition driven by the measurement strength. Here, we study the
geometric phase of a subset of self-closing trajectories induced by a
continuous Gaussian measurement of a single qubit system. We utilize a
stochastic path integral that enables the analysis of rare self-closing events
using action methods and develop the formalism to incorporate the measurement-
induced geometric phase therein. We show that the geometric phase of the most
likely trajectories undergoes a topological transition for self-closing
trajectories as a function of the measurement strength parameter. Moreover,
the inclusion of Gaussian corrections in the vicinity of the most probable
self-closing trajectory quantitatively changes the transition point in
agreement with results from numerical simulations of the full set of quantum
trajectories.

††preprint: APS/123-QED

##  I Introduction

The global phase of a system’s quantum state is an unmeasurable $U(1)$ gauge
freedom. However, when a system is driven adiabatically in a closed cycle, the
accumulated phase difference becomes gauge invariant and, therefore,
experimentally accessible [1, 2]. This observable phase difference consists of
two components: a dynamical (or local) phase and a Berry (or geometric) phase
[3, 4], which is solely dependent on the path taken through the Hamiltonian’s
parameter space. Geometric phases are key to identifying topological phases of
matter like quantum Hall phases, topological insulators and superconductors
[5, 6, 7, 8], and have been exploited in the design of high-fidelity quantum
gates that are resilient to random noise [9, 10].

Geometric phases can be generalised to continuous evolution in a system’s
Hilbert space via the Aharonov-Anandan phase, which can be defined as a
holonomy element of a fibre bundle over the projective Hilbert space of the
system [5, 11, 12]. Avoiding the use of Hamiltonian parameter space, this
approach places no restrictions on the system dynamics, so that a geometric
phase can be associated with continuous non-unitary dynamics due to quantum
measurement back-action [13, 14]. Importantly for measurement-induced effects,
the dynamics acquire stochastic properties. The difference between the average
state and the full distribution of states along quantum trajectories leads to
new features in single-qubit dynamics [15, 16] and new out-of-equilibrium
many-body states [17, 18, 19, 20, 21, 22, 23, 24]. Much research has focused
on associating some generalized version of the geometric phase to the averaged
(mixed) state described by a density matrix, via the Uhlmann phase [25, 26,
27], or introducing an interferometric geometric phase [28, 29, 30]. It is
only recently that the statistical properties of the geometric phase along
individual stochastic realizations (aka quantum trajectories) have begun to be
investigated [31, 32, 33, 32, 34, 35]. Recently it has been shown that
geometric phases arising from a particular measurement post-selection exhibit
a topological transition as a function of the measurement strength [31, 33,
20]. This transition differs from the topological order transition induced by
measurements in many-body systems [36, 37, 38, 39, 40], and was originally
predicted for a qubit subject to a sequence of cyclically rotating
measurements. Further work has shown that the topological nature of the
transition is a more generic feature of single qubit measurement-induced
dynamics including different measurement procedures and protocols, as well as
Hamiltonian dynamics and dephasing [32, 34, 35]. The transition is also robust
against several perturbations, including additional adiabatic dynamics and
dephasing [41, 42]. This, in turn, has facilitated the experimental
observation in superconducting and optical architectures [42, 43].

Notwithstanding the various generalizations, the observation of measurement-
induced geometric phases requires quantum trajectories that form a closed
loop. For a generic quantum evolution, this may be achieved with a final post-
selection of a projective measurement onto the initial state. While this
procedure provides a way to associate a geometric phase to a generic quantum
trajectory (open geometric phase hereafter), it introduces a discontinuous
jump in the quantum evolution. Alternatively, one can define a post-selection
procedure of the subset of self-closing quantum trajectories originating from
the quantum dynamics, which gives rise to continuous quantum trajectories.

Here we take advantage of this continuous evolution to develop a path integral
formulation of the measurement-induced closed geometric phases, associated
with the set of self-closing quantum trajectories. In particular, we will
follow the approach developed by Chantasri-Dressel-Jordan (CDJ) for continuous
Gaussian measurements [44, 45, 46], which is a well-established technique for
investigating continuous measurement dynamics [47, 48, 49]. By incorporating a
phase variable in the CDJ action for continuous Gaussian measurements we
obtain a path-integral formulation for open and closed geometric phases. We
systematically investigate the most likely open and self-closing quantum
trajectories and their associated geometric phase. We find that the open
geometric phase transition remains topological for a variety of post-selection
conditions of the readout record and initial state preparations. A suitable
protocol can be designed to investigate the topological properties of closed-
geometric phases. In this case, a transition as a function of the measurement
strength exists and maintains its topological properties, although the
underlying mechanism of competing minima of the probability distribution
differs from its counterpart in the open-geometric phases. Finally, we develop
a Gaussian approximation around the optimal trajectory, which is shown to
wapture the statistics of geometric phases resulting from the full
distribution of quantum trajectories obtained from numerical simulations.

##  II Geometric Phases Induced by Gaussian Measurements

(a)

(b)

Figure 1: Cyclically Rotating Qubit Measurement Protocol and corresponding
quantum trajectories. Panel a) A continuous sequence of Gaussian measurements
of operators $\mathbf{\sigma}\cdot\mathbf{n}(t)$ at constant latitude
$\Theta$, represented by $\mathbf{n}(t)$ tracing out a path on the Bloch
Sphere (dotted lines). Panel b) Examples of quantum Trajectories on the Bloch
sphere generated by the measurement sequence in panel (a) for self-closing
(blue) and open (black) boundary conditions. Dotted lines indicate the length
minimising geodesics closing the open path trajectories.

###  II.1 Cyclically rotating Gaussian measurements

We consider the measurement-induced time evolution of a qubit, with a generic
state given by the density matrix
$\rho=\frac{1}{2}\left(\mathbb{I}+\boldsymbol{x}\cdot\boldsymbol{\sigma}\right)$.
The vector, $\boldsymbol{\sigma}$, consists of Pauli Matrices
$(\sigma_{X},\sigma_{Y},\sigma_{Z})$ and $\boldsymbol{x}$ is a unit vector on
the Bloch sphere of the system parameterised by latitude $\theta$ and
longitude $\phi$. The qubit is subject to a time-dependent sequence of weak
measurements over a fixed time, $T$. We define a continuous process by
dividing $T$ into $N$ time steps of length $\delta t$, where $t_{k}=k\delta t$
and $k\in\\{n\in\mathbb{N}\cup\\{0\\}\ |\ n<N\\}$. During each time interval,
we measure the operator $A(k)=\boldsymbol{n}(k)\cdot\boldsymbol{\sigma}$,
where $\boldsymbol{n}$ is a unit vector specified by spherical coordinates
$(\Theta,\Phi)$. We further specify $\Theta(k)=\Theta$ and $\Phi(k)=2\pi
t_{k}/T$, so that the target observables are constrained to closed loops of
constant latitude on the Bloch sphere (see Fig. 1(a)). At each time step, the
measurement of the observable $A(k)$ is performed as a Gaussian measurement,
with measurement outcomes distributed according to a Gaussian probability
distribution [13, 14].

For a system in a state $\rho_{k}$, the Gaussian measurement of $A_{k}$
entails a measurement outcome $r_{k}$ drawn from the probability distribution
$P(r_{k})=\mathrm{Tr}[E(r_{k})^{\dagger}E(r_{k})\rho_{k}]$ and a corresponding
state update $\rho_{k+1}$ given by

$P(r_{k})=\mathrm{Tr}[E(r_{k},k)^{\dagger}E(r_{k},k)\rho_{k}],$ (1)
$\rho_{k+1}=E(r_{k},k)\rho_{k}E(r_{k},k)^{\dagger}/P(r_{k}).$ (2)

The entire process is controlled by the set of Kraus operators $E(r_{k},k)$.
For the specific protocol at hand, with measurements constrained at a fixed
latitude, we have

$E(r_{k},k)=R(\mathbf{n}(k))^{-1}M_{\delta t}(r_{k})R(\mathbf{n}(k)),$ (3)

where

$M_{\delta t}(r_{k})=\sqrt[4]{\frac{\delta
t}{2\pi\tau}}\exp\left(-\frac{\delta t}{4\tau}(r_{k}-\sigma_{z})^{2}\right),$
(4)

and $R(\boldsymbol{n}(k))$ is a rotation operator
$R(\boldsymbol{n}(k))=e^{-\frac{1}{2}i\phi_{k}}\text{exp}\left[\frac{1}{2}i(\theta_{k}\sigma_{2}+\phi_{k}\sigma_{3})\right]$
that takes the Block sphere state $(\theta,\phi)$ to $|0\rangle$. Here,
$\boldsymbol{n}(k)$ denotes the measurement axis at time $t_{k}$.

This measurement process is understood by noting that $M_{\delta t}$
corresponds to a Gaussian measurement of the operator $\sigma_{Z}$. The
rotational component of $E(r_{k},k)$ ensures that the measured operator is
$\boldsymbol{\sigma}\cdot\boldsymbol{n}$. The parameter $\tau$ sets the
characteristic measurement time for each individual Kraus operator, i.e. the
time scale after which the state is close to an eigenvalue of the measured
observable, or, equivalently, the inverse measurement strength. The continuous
limit $\delta t\to 0$, $N\to\infty$ with $\delta tN=T$ transforms the
sequences of measurement readouts $r_{k}$ and the corresponding qubit state
variables $\phi_{k}$ and $\theta_{k}$, which are parameterizations of the
qubit state in the spherical Bloch sphere, into continuous functions of time,
$r(t)$, $\phi(t)$, $\theta(t)$ defining a continuous stochastic process which
we study throughout this paper (see Fig. 1(b)).

This measurement protocol is akin to the one presented in Ref. [31], in which
a topological transition in the measurement-induced geometric phase was
originally identified. However, a key difference is the use of the Gaussian
measurement protocol. The original protocol involves a quasi-continuous
sequence of measurements with binary outcomes $r(t)=r^{j}=j$ for
$j\in\\{1,0\\}$ and corresponding Kraus operators $E_{j}$ defined via Eq. 3
with $M_{\delta t}(r_{k})$ replaced by $M_{j}$,

$M_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-\frac{4c\delta
t}{T}}\end{array}\right)\text{and }M_{0}=\left(\begin{array}[]{cc}0&0\\\
0&\sqrt{\frac{4c\delta t}{T}}\end{array}\right),$ (5)

where $c$ is a dimensionless measurement strength parameter. This choice of
Kraus operators replaces the continuous set of Kraus operators parametrized by
$r$ in Eq. 3. In the continuous limit $\delta t\to 0$, The Gaussian
measurement backaction $E(r_{k},\mathbf{n}(k))$ continuously approaches the
identity operator $\mathbb{I}$, while Eq. 5 leads to discontinuous jumps
associated with the outcome $0$, though with vanishing probability for $\delta
t\to 0$. A quasi-continuous evolution from Eq. 5 is then distilled by
performing a post-selection for ‘Null’ type readout $j=1$.

###  II.2 Geometric Phase of the monitored qubit

Any of the measurement readout sequences $r(t)$, has an associated trajectory
of states on the Bloch sphere $|\psi(t)\rangle$. Each such trajectory is in
turn associated with a unique geometric phase via the functional,

$\chi^{\text{g}}[\psi(t)]=\text{arg}\langle\psi(0)|\psi(T)\rangle+i\int_{0}^{T}\langle\psi(t)|\dot{\psi}(t)\rangle
dt,$ (6)

where $|\psi(t)\rangle$ is a lift of the curve in projective space into
Hilbert space that satisfies an initial condition $|\psi(0)\rangle$ written in
some arbitrarily chosen gauge. While geometric phases are typically associated
with closed paths in Hilbert space, we note that Eq. 6 is valid for paths in
projective space that do not close [10]. In these cases, the geometric phase
is defined as the geometric phase of the given curve in projective space
concatenated with a length-minimizing geodesic that closes the trajectory. We
refer to the geometric phase as the open geometric phase, as opposed to the
closed geometric phase associated with self-closing trajectories. In practical
terms, the open geometric phase corresponds to the closed geometric phase of a
post-selected trajectory including an additional projective measurement onto
$|\psi(0)\rangle\langle\psi(0)|$.

An important feature of Gaussian measurement is that it parallel transports
the state of the system, so there is no dynamical phase contribution to
subtract from the global phase. Adopting a pure qubit state parameterisation
in spherical coordinates ($\mathbf{q}=(\phi,\theta,\chi)$) that includes a
gauge-dependent global phase $\chi$, the systems state is,

$|\psi(\mathbf{q}(t))\rangle=e^{i\chi(t)}\Bigg{(}\begin{array}[]{c}\cos\left(\frac{\theta(t)}{2}\right)\\\
e^{i\phi(t)}\sin\left(\frac{\theta(t)}{2}\right)\\\ \end{array}\Bigg{)}.$ (7)

The parallel transport condition $\langle\psi(t+\delta t)|\psi(t)\rangle>0$,
holds across each time step, simplifying the geometric phase from Eq. 6 to

$\displaystyle\chi^{\text{g}}$
$\displaystyle=\text{arg}\Bigg{[}e^{-i(\phi(0)-\chi(T))}\big{(}\sin\frac{\theta(0)}{2}e^{i\phi(T)}\sin\frac{\theta(T)}{2}$
$\displaystyle+e^{i\phi(0)}\cos\frac{\theta(0)}{2}\cos\frac{\theta(T)}{2}\big{)}\Bigg{]},$
(8)

where we have used gauge freedom to choose $\chi(0)=0$. When $\phi(0)=\phi(T)$
and $\theta(0)=\theta(T)$ then $\chi^{\text{g}}=\chi(T)$.

##  III Path Integral: incorporating phase data

###  III.1  Chantasri-Dressel-Jordan path Integral with Phase tracking

To study the geometric phases associated with quantum trajectories from
Gaussian measurements, we formulate a path integral for the probability
distribution of the induced quantum trajectories that explicitly incorporates
the phase of the monitored state. We do so by incorporating phase information
in the CDJ path integral formulation for a Gaussian-monitored qubit [44],
using the state parameterisation in Eq. 7. Since this phase is parallel
transported, the global phase is directly equivalent to the geometric phase on
a closed path. The path integral is constructed from the joint conditional
probability of finding a state $\mathbf{q}(T)$ and readout $r(T)$, given some
initialization $\mathbf{q}_{i}$, under the evolution in Eq. 2. This
probability, in the continuum limit, can be expressed as a product of
sequential conditional probabilities so

$\displaystyle\mathcal{P}(\mathbf{q}(T),r(T)|\mathbf{q}_{i})=\mathcal{F}(t_{0},t_{N})\times$
(9) $\displaystyle\lim_{\delta t\to
0,N\to\infty}\prod_{k=0}^{N-1}P(\mathbf{q}(k+1)|\mathbf{q}(k),r(k))P(r(k)|\mathbf{q}(k)),$

where
$\mathcal{F}(t_{0},t_{N})=\delta^{3}(\mathbf{q}(N)-\mathbf{q}_{f})\delta^{3}(\mathbf{q}(0)-\mathbf{q}_{i})$
sets the initial and final states to be $\mathbf{q}_{i}$ and $\mathbf{q}_{f}$.
The readout probability distribution function, $P(r|\mathbf{q})$, can be
obtained directly from Eq. 2 and 3, so

$P(r|q)\approx\sqrt{\frac{\delta t}{2\pi\tau}}\exp\left(-\frac{\delta
t}{2\tau}(r^{2}-2ra+1)\right),$ (10)
$a(\Theta,\Phi,\theta,\phi)=\cos(\theta)\cos(\Theta)+\sin(\theta)\sin(\Theta)\cos(\phi-\Phi).$
(11)

Here $\langle r\rangle=\int rP(r|\mathbb{q})dr=a$ is the mean measurement
record function, and corresponds to the most probable readout at each time
step. The remaining term in Eq. III.1 is the state update, which is
deterministic and can be expressed as a delta function

$\displaystyle P(\mathbf{q}(t+\delta t)|\mathbf{q}(t),r(t))$
$\displaystyle=\delta^{d}(\mathbf{q}(t+\delta t)|\mathbf{q}(t),r(t))$ (12)
$\displaystyle=\Big{(}\frac{1}{2\pi
i}\Big{)}^{d}\int^{i\infty}_{-i\infty}d^{d}\mathbf{p}e^{-\mathbf{p}\cdot(\mathbf{q}(t+\delta
t))},$

where by expressing the Dirac-delta functions in Fourier form each $q$ gains a
conjugate momentum $p_{q}$, so $\mathbf{p}=(p_{\phi},p_{\theta},p_{\chi})$.
The deterministic update associated with $\delta^{d}(\mathbf{q}(t+\delta
t)|\mathbf{q}(t),r(t))$ is expressed in the differential equations,

$\displaystyle\dot{\phi}$
$\displaystyle=-\frac{r}{\tau}f_{\Theta,\Phi}(\theta,\phi)$ (13)
$\displaystyle\dot{\theta}$
$\displaystyle=\frac{r}{\tau}g_{\Theta,\Phi}(\theta,\phi,)$ (14)
$\displaystyle\dot{\chi}$
$\displaystyle=\frac{r}{2\tau}h_{\Theta,\Phi}(\theta,\phi,).$ (15)

with $f_{\Theta,\Phi}(\theta,\phi)=\csc(\theta)\sin(\Theta)\sin(\phi-\Phi)$,
$g_{\Theta,\Phi}(\theta,\phi)=(\cos(\theta)\sin(\Theta)\cos(\phi-\Phi)-\sin(\theta)\cos(\Theta))$
and
$h_{\Theta,\Phi}(\theta,\phi)=\tan\left(\frac{\theta}{2}\right)\sin(\Theta)\sin(\phi-\Phi)$.
Following the usual procedure for constructing path integrals [50, 51, 44, 45,
52], we express $\mathcal{P}$ in terms of an action principle,

$\mathcal{P}\propto\int
D\mathbf{q}D\mathbf{p}Dr\exp\left(-\int_{0}^{T}S[\mathbf{q},\mathbf{p},r]dt\right),$
(16) $\displaystyle
S[\mathbf{q},\mathbf{p},r]=-p_{\theta}\left(\dot{\theta}-\frac{r}{\tau}g_{\Theta,\Phi}\right)-p_{\phi}\left(\frac{r}{\tau}f_{\Theta,\Phi}+\dot{\phi}\right)$
(17) $\displaystyle-
p_{\chi}\left(\dot{\chi}-\frac{r}{2\tau}h_{\Theta,\Phi}\right)+\frac{r(2a-r)-1}{2\tau}.$

The action $S[\mathbf{q},\mathbf{p},r]$, is a generalization of the
probability density of a quantum trajectory, encoding all the statistical
details of the measurement process. Compared to the CDJ action in Ref. [44],
Eq. 17 now includes the variable $\chi$ and the time dependence of the
measurement operators. This formalism allows us to identify the most probable
trajectories (with a given initial and final state) by variational methods.
This results in a system of differential equations for quantum trajectories
that are either the most-likely path to traverse between the boundary points
or a minimum/saddle point solution, given by Eq. 13,14 and15 in combination
with the time derivatives of the momentum variables, $\dot{p_{\chi}}=0$,

$\displaystyle\dot{p_{\phi}}=\frac{r}{\tau}p_{\phi}\frac{\partial
f_{\Theta,\Phi}}{\partial\phi}-\frac{r}{\tau}p_{\theta}\frac{\partial
g_{\Theta,\Phi}}{\partial\phi}-\frac{r}{2\tau}p_{\chi}\frac{\partial
h_{\Theta,\Phi}}{\partial\phi}-\frac{r}{\tau}\frac{\partial a}{\partial\phi},$
$\displaystyle\dot{p_{\theta}}=\frac{r}{\tau}p_{\phi}\frac{\partial
f_{\Theta,\Phi}}{\partial\theta}-\frac{r}{\tau}p_{\theta}\frac{\partial
g_{\Theta,\Phi}}{\partial\theta}-\frac{r}{2\tau}p_{\chi}\frac{\partial
h_{\Theta,\Phi}}{\partial\theta}-\frac{r}{\tau}\frac{\partial
a}{\partial\theta},$ (18)

and a constraint on the measurement record function

$\displaystyle r$
$\displaystyle=\frac{1}{2}p_{\chi}h(\theta,\Theta,\phi,\Phi)-p_{\phi}f(\theta,\Theta,\phi,\Phi)$
(19)
$\displaystyle+p_{\theta}g(\theta,\Theta,\phi,\Phi)+a(\theta,\Theta,\phi,\Phi).$

We observe that by introducing the variable $\chi$, the calculation of the
optimal geometric phase becomes more straightforward, as its value can be
determined concurrently with the optimal quantum trajectory.

###  III.2 Corotating coordinates

It will be useful to consider the action in Eq. LABEL:action rewritten in a
spherical coordinate system that co-rotates with the measurement axis defined
by new polar and azimuthal coordinates $(\tilde{\theta},\tilde{\phi})$. Such a
coordinate transformation is implemented by the change-of-basis matrix
$B=\exp(i\Phi\sigma_{z}/2)$. In this coordinate system, the dynamics are
described by the Kraus operator $\tilde{E}$ and an effective Hamiltonian
$\tilde{H}$. Here $\tilde{E}=R^{-1}(\tilde{\boldsymbol{n}})M_{\delta
t}R(\tilde{\boldsymbol{n}})$, and is now time-independent since
$\tilde{\boldsymbol{n}}=(\sin(\Theta),0,\cos(\Theta))$ is fixed. While
$\tilde{H}=i\dot{B}B^{\dagger}=-\frac{1}{2}\dot{\Phi}\sigma_{z}$, and acts
unitarily on the system state. The state update is given by $\rho(t+\delta
t)=\frac{e^{-i\tilde{H}}\tilde{E}\rho\tilde{E}^{\dagger}e^{i\tilde{H}}}{Tr{e^{-i\tilde{H}}\tilde{E}\rho\tilde{E}^{\dagger}e^{i\tilde{H}}}}.$
In terms of the action, this point transformation acts directly on the Bloch
sphere coordinates as,

$\displaystyle\tilde{\phi}$
$\displaystyle=\Phi-\phi(t),\quad\tilde{\theta}=\theta,$
$\displaystyle\dot{\phi}$
$\displaystyle=\frac{\partial\tilde{\phi}(\phi,t)}{\partial\phi}\dot{\tilde{\phi}}+\frac{\partial\tilde{\phi}(\phi,t)}{\partial
t}=-\dot{\tilde{\phi}}+\dot{\Phi},\quad\dot{\tilde{\theta}}=\dot{\theta},$
(20)

and also induces a contra-variant transformation on the conjugate momentum,

$p_{\phi}=\frac{\partial\tilde{\phi}(\phi,t)}{\partial\phi}\tilde{p}_{\phi}=-\tilde{p}_{\phi},\quad
p_{\phi}=\tilde{p_{\phi}}.$ (21)

These coordinate transformations applied to Eq.17 (omitting $\chi$ and
$p_{\chi}$), lead to the CDJ action in rotating coordinates,

$\displaystyle\tilde{S}[\tilde{\theta},\tilde{\phi},p_{\tilde{\theta}},p_{\tilde{\phi}}]=\frac{1}{2\tau}\Bigg{[}+2r\tilde{a}-r^{2}-1$
$\displaystyle+2\tilde{p}_{\theta}\Big{(}r\cos(\tilde{\theta})\sin(\Theta)\cos(\tilde{\phi})-r\sin(\tilde{\theta})\cos(\Theta)-\tau\dot{\tilde{\theta}}\Big{)}$
$\displaystyle+2\tilde{p}_{\phi}\left(\tau(\dot{\Phi}-\dot{\tilde{\phi}})-r\csc(\tilde{\theta})\sin(\Theta)\sin(\tilde{\phi})\right)\Bigg{]}.$
(22)

with
$\tilde{a}=\sin(\tilde{\theta})\sin(\Theta)\cos(\tilde{\phi})+\cos(\tilde{\theta})\cos(\Theta)$.
From this reformulation, it is evident that the rotating measurement protocol
equivalently captures the physics of the Zeno effect, i.e. the competition
between measurement and unitary evolution in a qubit.

###  III.3 Lagrangian Formulation

Since the path integral in Eq. 16 is Gaussian in $r$ it is possible to
integrate out the measurement record. The ensuing action is quadratic in the
momentum variables, which can then also be integrated out to give a
configuration space path integral,

$\displaystyle\mathcal{P}$ $\displaystyle\propto\int
D\mathbf{q}D\mathbf{p}Dre(-S[\mathbf{q},\mathbf{p},r])$ (23)
$\displaystyle=\int D\mathbf{q}\mathfrak{p}[\mathbf{q}]=\int
D\mathbf{q}\mu(\theta,\phi)e^{\int_{0}^{T}\mathcal{L}[\theta,\phi]}.$

This alternate formulation is characterized by a probability measure denoted
as $\mathfrak{p}[\theta,\phi]$ which consists of two components: a singular
[53] Lagrangian $\mathcal{L}[\theta,\phi]$, and a path-dependent functional
measure, $\mu(\theta,\phi)$, given by

$\displaystyle\mu(\theta,\phi)=\text{Det}[\frac{\csc^{2}(\theta)\sin^{2}(\Theta)\sin^{2}(\phi-\Phi)}{2\tau}]^{-\frac{1}{2}},$
(24)
$\displaystyle\mathcal{L}(\theta,\phi)=-\frac{1}{2}\tau\sin^{2}(\theta)\csc^{2}(\Theta)\dot{\phi}^{2}\csc^{2}(\phi-\Phi)-\frac{1}{2\tau}$
$\displaystyle-\frac{\dot{\phi}}{2}\sin(2\theta)\cot(\Theta)\csc(\phi-\Phi)-\sin^{2}(\theta)\cot(\phi-\Phi)\dot{\phi}.$
(25)

The path dependence of $\mu$ can be attributed to the multiplicative nature of
the underlying stochastic process which acts like a curvature effect in the
time axis [54]. The Lagrangian is characterised by a non-invertible Hessian
matrix. This particularity results from the imposition of semi-holonomic
constraints within the configuration space, as specified by the equations,

$\displaystyle\dot{\theta}=$
$\displaystyle\frac{\dot{\phi}}{2}\Big{(}(2\sin^{2}(\theta)\cot(\theta_{0})\csc(\phi-\phi_{0})$
(26) $\displaystyle-\sin(2\theta)\cot(\phi-\phi_{0})\Big{)},$
$\dot{\chi}=\frac{\dot{\phi}}{2}(\cos(\theta)-1).$ (27)

These constraints naturally appear during the process of functional
integration, wherein terms exhibiting, at most, linear dependence on momentum
play the role of Legendre multipliers. These multipliers, in turn, enforce Eq.
26 and 27.

##  IV Topological features of the open geometric phase

The action (Eq. 17) and its associated extremization in Eq. III.1, allow us to
determine the properties of the optimal geometric phases induced by Gaussian
measurements for any required boundary conditions. However, before exploring
new subsets of geometric phases, like optimal self-closing ones, we first
determine the properties of the open geometric phases under Gaussian
measurements, in particular, their topological features.

For rotating null-type measurements defined in Eq. 2 and 5, preparing the
qubit state along the initial measurement axis, ($\theta(0)=\Theta$,
$\phi(0)=\Phi(0)$), implementing a closed loop of measurements,
$\boldsymbol{n}(t)$, and post selecting the measurement readouts $r=1$
associates the path followed by the measurement with a unique trajectory on
the Bloch sphere. This map can then be applied to the family of trajectories
spanned by the initial conditions $\Theta\in[0,\pi]$. In this way the sphere
spanned by the direction of measurement operators, $\boldsymbol{n}(t)$, is
mapped onto a submanifold of the Bloch sphere, via Kraus operator $M_{1}$.
This mapping undergoes a topological transition [31] as a function of the
measurement strength $c$ from a phase in which the image of the trajectories
covers the Bloch sphere, to one in which it fails to cover the entire sphere.
These two topological regimes are distinguished by a Chern number

$C\equiv\frac{1}{2\pi}\int_{0}^{\pi}d\Theta\int_{0}^{1}Bdt=\frac{1}{2\pi}(\chi(\pi)-\chi(0)),$
(28)

with
$B(\Theta,t)=\text{Im}\\{\partial_{t}\langle\psi|\partial_{\Theta}|\psi\rangle-\partial_{\Theta}\langle\psi|\partial_{t}|\psi\rangle\\}$.
The Chern number can take discrete values, $C=-1$ or $C=0$, which are directly
related to the dependence of the geometric phase as shown in the last step of
Eq. 28. The topological transition of this mapping is then manifest in the
measurement-induced geometric phase as a function of measurement latitude,
$\chi(\Theta)$, bifurcating the function into monotonic and non-monotonic
regimes. The mapping $[0,\pi]\ni\Theta\to\chi\in[0,2\pi]$ is in fact a mapping
$S_{1}\to S_{1}$ with winding number $w=0,1$. The latter is equivalently
determined by $w=[\chi(\pi/2)-\chi(0)]/\pi\in\\{0,1\\}$

Gaussian measurements have been involved in researching this type of
topological transition [46]. Dual readout Gaussian measurements on a qutrit
were used to reproduce the effect of the null-type measurements on a two-level
sub-system. Here we are concerned with Gaussian measurements in their own
right, specifically in the protocol outlined in section II.1. In this case, a
continuous mapping between the two spheres is no longer constrained to the
null-type post-selection and more general post-selections can be imposed on
the measurement readout. Quite generally, we expect that a rotating
measurement protocol with ($\theta_{i}=\Theta,\phi_{i}=0$) features a
topological transition for post-selected record function $r(t)>0$, which
ensures that the continuity of the trajectory at the initial state in the
strong-measurement limit (i.e. the system is driven towards the positive
eigenstate of the measured operator at all times during the measurement
induced dynamics). For the special case when $\Theta=\frac{\pi}{2}$; the Bloch
Sphere equator corresponds to an invariant subspace for every Kraus operator
in the measurement sequence: states initialized on the equator remain therein.
This feature of the systems accessible trajectories manifests in the available
geometric phases $\\{\frac{\pi}{2}n|n\in\mathbb{Z}\\}$; each is associated
with a definite winding number. This applies irrespective of the state
preparation used.

(a)

(b)

(c)

Figure 2: Global Optimal quantum trajectories and their geometric phases. (a)
Geometric phase $\chi^{g}(\Theta)$ for a range of measurement strengths below
(blue) and above (black) the inverse critical measurement strength
$\tau_{c}/T\approx 0.1$. Quantum trajectories on the Bloch sphere with closing
geodesics (dotted) for $\tau/T=0.05$ (c) and $\tau/T=0.2$ (b). The family of
trajectories covers the Bloch sphere for measurements stronger than the
critical value (b) and does not otherwise (c).

As a first example, it is possible to ascertain the topological transition for
the family of most likely trajectories spanned by initial states that coincide
with the measurement axis. These most-probable post-selected optimal
trajectories are obtained from the observation that at each time-step, the
most probable outcome is given by $\langle r\rangle=a(\Theta,t)$ (cf. Eq. 10).
Hence, substituting the time continuous version, $r(t)=a(t)$ from Eq. 11 into
Eq. 13-15, with the initial conditions $\theta(0)=\Theta$ and
$\phi(0)=\Phi(0)$, produces the required optimum quantum trajectory. A
topological transition is observed in this case, as illustrated in Fig. 2,
where the family of quantum trajectories — including the closing geodesic— for
strong measurements (small $\tau/T$) wrap the Bloch sphere (Fig. 4(b), while
for weak measurements (large $\tau/T$) they do not (Fig. 4(c). The transition
measurement strength is determined numerically from the dependence
$\chi^{g}(\Theta)$, which is reported in Fig 2(a). For $\tau<\tau_{c}$ the
geometric phase evolve continuously from $0$ to $-\pi$, while for
$\tau>\tau_{c}$, $\chi(\pi/2)=\chi(0)$. We estimate that the transition occurs
for $\tau_{c}/T\approx 0.1$, The transition is further confirmed by a direct
numerical evaluation of the Chern number using Eq. 28.

(a)

(b)

(c)

Figure 3: Equilibrium state. Flow of Hamilton’s equations in the CDJ Phase
Space at $\theta=\pi/2$ (Eq. III.1) in the co-rotating coordinate system for
$\tau=0.1$, $\Theta=\frac{\pi}{2}$ (panel a) and $\tau=0.3$,
$\Theta=\frac{\pi}{2}$ (panel b). The equilibrium point is indicated by the
black dot. Panel c): Dependence of the equilibrium points
($\theta_{e},\phi_{e}$) on $\Theta$ and $\tau$. For each value of $\Theta$
($\frac{\pi}{6}$ (blue),$\frac{2\pi}{7}$ (green),$\frac{2\pi}{5}$
(orange),$\frac{\pi}{2}$(red), $\frac{3\pi}{5}$(orange),$\frac{5\pi}{7}$
(green)) darker shades corresponding to weaker measurements from $\tau/T=0.1$
to $\tau/T=10$.

A second kind of variation of the protocol concerns the state preparation.
From numerical simulation of a range of cases, it appears that the
initialization of the system does not affect the topological nature of the
transition and the associated phenomenology of the geometric phase provided
the state initialization $\theta_{i}(\Theta(0))$ spans the entire range of the
polar angle, is monotonic, and satisfies $\theta_{i}(0)=0$,
$\theta_{i}(\frac{\pi}{2})=\frac{\pi}{2}$, and $\theta(\pi)=\pi$ with
$\lim_{\tau\to 0}\phi_{i}=\Phi(0)$. Note we are explicitly allowing for the
possibility of a measurement strength-dependent state preparation. We are
particularly interested in using this freedom to choose a new state
preparation that, similar to the choice $\theta_{i}=\Theta$
$\phi_{i}=\Phi(0)$, will also continuously recover the projective measurement
limit - where states are initialized along the measurement axis and
meticulously follow the axis for their entire evolution. This then requires a
state initialization that gives $\theta_{i}\to\Theta(0)$ and
$\phi_{i}\to\Phi(0)$ in the strong measurement limit.

A natural case, which will be relevant later on for closed geometric phases,
has the initial state chosen to coincide with a fixed point (so
$\theta_{i}=\theta_{e}$ and $\phi_{i}=\phi_{e}$) of the optimal trajectory
dynamics in the co-rotating coordinate frame. This equilibrium point in the
co-rotating dynamics ($\theta_{e},\phi_{e}$) is defined by
$\dot{\tilde{\phi}}=0$, $\dot{\tilde{\theta}}=0$, $\dot{\tilde{p}}_{\phi}=0$,
$\dot{\tilde{p}}_{\theta}=0$. Hamilton’s Equations for the action (Eq. III.2)
can be solved to determine a closed-form expression,

$\displaystyle\theta_{e}$
$\displaystyle=\tan^{-1}\left(\frac{\tan(\Theta)}{\sqrt{4\pi^{2}\tau^{2}+1}}\right),$
(29) $\displaystyle\phi_{e}$ $\displaystyle=-\tan^{-1}(2\pi\tau).$ (30)

Fig. 3, which reports the phase space flow diagram from Hamilton’s equations
for the action in Eq. III.2 for different values of the measurement strength
[panels (a) and (b)] at $\theta=\Theta_{0}=\pi/2$. For increasingly strong
measurements $\phi_{e}$ tends toward the measurement axis. The position of
equilibrium points for generic latitudes on the Bloch sphere is reported in
Fig. 3(c), showing that this same limiting behaviour continues across the
entire Bloch Sphere. The quantum trajectories for this modified protocol (with
a generic choice of post-selection $r(t)=1$) are reported in Fig. (4[b-c]),
where small $\tau/T$ (panel b) led to trajectories wrapping the Bloch sphere
and large $\tau/T$ (panel c) don’t. Similarly to the case reported in Fig. 2,
we can identify a topological transition from the behaviour of $\chi$ as a
function of $\Theta$ [cf. Fig. 4(a)], which gives a critical measurement
strength $\tau_{c}/T\approx 0.22$. From these examples, it emerges that,
despite variations in the exact value of $\tau_{c}$, the fundamental
characteristics of the transition are unchanged for a wide range of state
preparation protocols.

(a)

(b)

(c)

Figure 4: Quantum trajectories and geometric phases with equilibrium state
initialization. Panel a) Geometric phase $\chi(\Theta)$ for a range of
measurement strengths below (blue) and above (black) the inverse critical
measurement strength $\tau_{c}\approx 0.22$. Quantum trajectories on the Bloch
Sphere with closing geodesics (dotted) for $\tau=0.15$ (b) $\tau=0.3$ (c). The
family of trajectories covers the Bloch sphere for measurements stronger than
the critical value (b) and does not otherwise (c).

##  V Topological Transition in the optimal closed geometric phase

We now use the developed action formalism to investigate the topological
features of the closed geometric phase. This involves considering the set of
self-closing trajectories generated solely by continuous monitoring, without a
projective measurement step. Since self-closing trajectories are not generally
achievable by conditioning on any single measurement record function, we post-
selected most likely self-closing trajectory, corresponding to the most-
probable closed geometric phase attained during measurement. We denote the
optimum phase as $\chi^{\text{opt}}$. On the Bloch Sphere equator, the
behaviour of $\chi^{\text{opt}}$ is fundamentally tied to the topology of
$S^{1}$. Attaining only two possible values corresponding to the winding
number of the associated quantum trajectory, either $-\pi$ or $0$.

Care must be taken to find a state preparation that will produce
$\chi^{\text{opt}}$ that recovers the projective measurement limit,

$\displaystyle\lim_{\tau\to
0}\chi^{\text{opt}}\approx-2\pi\sin^{2}\left(\frac{1}{2}\Theta\right),\lim_{\tau\to\infty}\chi^{\text{opt}}\approx
0.$ (31)

The state preparation $\theta(0)=\Theta$ and $\phi(0)=\Phi$, applied to Eq.
III.1, with self-closing boundary conditions produce a variety of candidate
optimal geometric phases that satisfy Eq. 31. In the regime of strong
measurements, one might anticipate an increased likelihood of candidate
solutions with $\lim_{\tau\to
0}\chi^{\text{opt}}\approx-2\pi\sin^{2}\left(\frac{1}{2}\Theta\right)$.
However, numerical assessment of the stochastic action reveals that these
solutions remain vanishing improbable even in the strong measurement limit.
This phenomenon can be understood upon examining the characteristics of these
candidate optima: spending the majority of their lifetime (approximately $T$)
following the equilibrium trajectory (Eq. 29), punctuated with rapid
transitions to and from the measurement axis at the beginning and end of the
protocol. It is these rapid transitions that suppress the likelihood of the
winding solution. Consequently, solutions associated with a vanishing winding
number dominate. This scenario leads to a $\chi^{\text{opt}}$ that does not
satisfy the prescribed condition. Instead, a suitable state preparation is
provided by the equilibrium point introduced in Eq. 29. The equilibrium points
span the entire range of latitudes, with $\theta\in[0,\pi]$. As we shall later
demonstrate, with this particular initialization, the value of
$\chi^{\text{opt}}$ adheres to the required limits given in Eq. 31. This
establishes the mapping between measurement parameters $(\Theta,\tau)$ and the
optimal self closing quantum trajectory and it’s associated closed geometric
phase $\chi^{\text{opt}}$ that we use to establish a new topological
transition.

###  V.1 Topological Transition

As discussed for the case of the open geometric phase, the topological
properties of the mapping of $\boldsymbol{n}(t)\to q(t)$ is dictated by the
fixed points at $\Theta=0$ and $\Theta=\pi$. When $\Theta=\frac{\pi}{2}$, the
simplest topological features can be investigated since here the accessible
trajectories are each associated with a definite winding number $n$ indexing
the available closed geometric phases $\\{\frac{\pi}{2}n\ |\
n\in\mathbb{Z}\\}$. Hamilton’s equations for the Stochastic Action, Eq. III.1,
restricted to the Bloch Sphere equator, have multiple solutions after imposing
boundary conditions $\phi(0)=\phi_{e}$ and $\phi(T)=\phi_{e}+2\pi n$,
generating a set of candidate most-likely self-closing quantum trajectories
and phases. A single candidate solution is generated for each value of $n$
corresponding to a local minimum of the action (or equivalently a local
maximum of the probability density, $\mathfrak{p}$ over the set of quantum
trajectories).

To determine which candidate solution occurs with a higher probability, we
evaluate and compare their actions. The solution corresponding to $n=1$,
equivalent to the equilibrium quantum trajectory in co-rotating coordinates,
is $\phi(t)^{n=1}_{\Theta=\frac{\pi}{2}}=2\pi t-\arctan{2\pi\tau}$. By
substituting this solution into Eq. 17, we find the associated probability
density is given by
$\mathfrak{p}^{n=1}_{\Theta=\frac{\pi}{2}}=e^{-2\pi^{2}\tau}.$ For $n=0$, a
numerical solution to Eq. III.1 is used to find
$\phi(t)^{n=0}_{\Theta=\frac{\pi}{2}}$, which is then substituted directly
into the stochastic action to evaluate
$\mathfrak{p}^{n=0}_{\Theta=\frac{\pi}{2}}$. Both of these probability
densities are plotted in Fig. 5(a). Solutions for other values of $n$ are
found to have strictly lower probabilities. We, therefore, focus our analysis
on the competition between the candidate optimum trajectories indexed by $n=0$
and $n=1$. The value of $\chi^{\text{opt}}$ is determined by the competition
between the two most prominent candidate optimums. As evidenced by Fig. 5(a),
there is a measurement strength, $\tau_{c}/T\approx 0.11$, at which the
optimum geometric phase jumps discontinuously from $0$ to $-\pi$.

(a)

(b)

Figure 5: Stochastic properties of winding and non-winding quantum
trajectories on the Bloch Sphere Equator. Panel a) Probability measure’s
$\mathfrak{p^{n=1}}$(solid black) and $\mathfrak{p^{n=0}}$ (black triangles)
as functions of $\tau$. Normalised self-closing conditional probability
$\mathcal{P}$ (state initialization
($\theta_{i}=\theta_{e},\phi_{i}=\phi_{e})$) for non-winding trajectories (red
triangles) and winding trajectories (red). $\mathcal{P}$ is obtained from
numerical simulation of Eq. 2 and 3 with 100 time-steps and 500 quantum
trajectories with bin size $\Delta\phi=0.1$. Panel b) The $\tau$ dependence of
the ratios $\mathfrak{R}$ and $R$ (c.f. section VI) calculated using numerical
data in panel a and the results of Gaussian corrections of section VI.

We now investigate if a transition of topological number in the optimal
geometric phase occurs across the whole Bloch sphere. To address this
question, we study the subset of solutions for Eq. III.1 that generate optimal
self-closing quantum trajectories (with $\theta(0)=\theta_{e}$,
$\phi(0)=\phi_{e}$ and $\chi(0)=0$) for all measurement latitudes. We identify
one family of solutions ($\phi_{eq}$) smoothly connected to
$\phi(t)^{n=1}_{\Theta=\frac{\pi}{2}}$, specified by $\phi_{eq}=\phi_{e}+2\pi
t$ and $\theta(t)=\theta_{e}$. This closed-form solution may be substituted
back into Eq. III.1 to find the corresponding geometric phase,

$\chi^{n=1}=-2\pi\sin^{2}\left(\frac{1}{2}\tan^{-1}\left(\frac{\tan(\Theta)}{\sqrt{4\pi^{2}\tau^{2}+1}}\right)\right),$
(32)

which is proportional to the solid angle of the spherical cap defined at the
latitude $\theta_{e}$. We note that $\lim_{\tau\to
0}\chi^{n=1}=-2\pi\sin^{2}\left(\frac{1}{2}\Theta\right)$, and that
$\chi^{n=1}$ decreases monotonically from 0 to $-\pi$ with increasing
$\Theta$. Similarly, we evaluate $S$ on this family of solutions, finding the
probability density

$\mathfrak{p}^{n=1}=\exp\left(-\frac{2\pi^{2}\tau\sin^{2}(\Theta)}{2\pi^{2}\tau^{2}\cos(2\Theta)+2\pi^{2}\tau^{2}+1}\right).$
(33)

Note that $\mathfrak{p}^{n=1}\to 1$ for $\tau\to 0$. These sets of solutions
for different $\Theta$ form a sub-manifold of the Bloch sphere with $C=1$,
independent of the measurement strength (see Fig. 6(a)). We may identify a
second family of quantum trajectories: the most likely solutions excluding
$\phi_{eq}$111As for the solutions at the equator, multiple solutions of the
Euler-Lagrange equations exist for given boundary conditions, corresponding to
local minima of the actions (see Fig. 6(b)). This set of solutions,
$\phi(t)^{n=0}$, is determined by numerically computing the stochastic action
in Eq. 17 and has no closed-form expression for either the accrued geometric
phase or the associated probability density. The mapping between
$(\Theta(t),\Phi(t))$ and $(\theta(t),\phi(t))$ in this case is always
characterized by $C=0$. The optimal (most likely) value of the geometric
phase, $\chi^{\text{opt}}$, is then determined by the competition between
these two families of solutions.

(a)

(b)

(c)

Figure 6:  Optimal self-closing quantum trajectories. Equilibrium quantum
trajectories $\phi_{eq}$ [panel (a)] and $\phi(t)^{n=0}$ [panel (b)]for
various choices of $\Theta$, with $\tau/T=0.2$. The shaded region in panel (b)
is a guide to the eyes highlighting the $C=0$ submanifold on the Bloch sphere.
The trajectories with $C=1$ cover the whole sphere as shown in panel (a).
Panel c) Candidate optimum geometric phases, $\chi_{eq}$ and $\chi_{2}$ and
corresponding probabilities, $P(\chi_{eq})$ and the competing optimum
$P(\chi_{2})$, at $\Theta=0.9<\Theta_{c}$ as a function of $\tau$. The two
candidate solutions merge into a single q-trajectory before the value of
$\tau_{c}$.

For strong measurements ($\tau\ll T$), solutions $\phi(t)^{n=0}$ are less
probable than $\phi_{eq}$, so the optimal geometric phase is given by Eq. 33.
In the weak measurement regime ($\tau\gg T$), we have the converse, solutions
$\phi(t)^{n=0}$ are more likely. We define a $\tau$ dependent function,
$\Theta_{\textrm{jump}}(\tau)$, that separates two types of behaviour in
$\chi^{\text{opt}}$ as it approaches the strong measurement limit. At
$\Theta_{\textrm{jump}}(\tau)$, $\chi^{\text{opt}}$ jumps discontinuously to
the value determined by $\phi_{eq}$, with $\tau_{c}$ marking the smallest
value of $\tau$ for which this discontinuous jump occurs. This jump in the
geometric phase is shown in Fig. 7. We name the largest value of
$\Theta_{\textrm{jump}}$, $\Theta_{C}$, and it is determined to be
$\Theta_{C}\approx 0.95$.

Below $\Theta_{C}$, $\chi^{\text{opt}}$ tends smoothly towards the geometric
phase associated with $\phi_{eq}$. Numerical evidence suggests that the family
of quantum trajectories $\phi(t)^{n=0}$ merge smoothly with the associated
trajectory $\phi_{eq}$. This behaviour is illustrated in Fig. 6(b), where the
blue-coloured quantum trajectories denote the region in which the family of
trajectories $\phi(t)^{n=0}$ merges into $\phi_{eq}$. This behaviour is
quantified in Fig. 6(c), which shows how, at $\Theta<\Theta_{C}$, the
geometric phase from the $\phi(t)^{n=0}$ family of trajectories coincides with
that from $\phi_{eq}$ at $\tau<\tau_{c}$. Crucially, this merger occurs above
the critical measurement strength $\tau_{c}$, suggesting that the value of
$\tau_{c}$ as determined on the Bloch sphere equator does correspond to a
topological transition across the entire Bloch Sphere. This transition is
manifest in the behaviour of $\chi^{\text{opt}}$ as a function of $\Theta$
(see Fig. 7), where $\tau_{c}$ separates phases that are continuous and
monotonically decreasing (from 0 to $\pi$) and those which are non-monotonic
(0 to 0). The transition in the topological number for optimal self-closing
trajectories and the corresponding discontinuity in the geometric phase are
distinct from the open geometric phase transition. While the latter is
associated with a vanishing post-selection probability at the critical point
[31], for self-closing trajectories, two trajectories from distinct families
become equally likely at the transition point.

Figure 7: Optimal geometric phases, $\chi^{\text{opt}}$, as a function of
$\Theta$ for a range of measurement strengths. The critical measurement
strength $\tau_{c}$ distinguishes the behaviour
$\chi^{\text{opt}}(\Theta=\pi/2)=0$ and
$\chi^{\text{opt}}(\Theta=\pi/2)=-\pi$. for $\Theta>\Theta_{C}$, the geometric
phase can exhibit a jump for $\Theta>\Theta_{C}$.

This topological transition is well defined in terms of the most likely
trajectories belonging to either of the two families identified above.
However, any experiment would not be able to access the most likely
trajectories directly. The set of trajectories to be compared must necessarily
include an ensemble of trajectories that are equivalent, up to the precision
of the experiment. The averaged geometric phase is expected to display a
crossover as opposed to a sharp transition at $\tau_{c}$. In a scenario with
finite experimental precision, the value of $\tau_{c}$ will therefore be
smeared out. However, a transition can still be identified in terms of which
of the ensembles (each associated with a distinct optimal geometric phase) is
most probable. We calculate the effect this has on the value of $\tau_{c}$ in
the next section.

##  VI Gaussian Corrections

To incorporate multiple trajectories in the picture we include Gaussian
corrections in the analyses of optimal self-closing trajectories. This method
allows us to account for the effect of solutions that deviate slightly from
the optimal solutions while still satisfying the required boundary conditions
(and hence can be associated with distinct geometric phases). Operationally,
the identification of $\tau_{c}$ when including extra trajectories is achieved
by replacing a direct comparison of the measure of two individual quantum
trajectories, with a comparison of the state transition probabilities,

$\mathfrak{R}=\frac{\mathfrak{p}(\phi^{n=1}_{\Theta=\frac{\pi}{2}})}{{\mathfrak{p}(\phi^{n=0}_{\Theta=\frac{\pi}{2}})}}\to
R=\frac{\mathcal{P}^{\text{eq}}(2\pi+\phi_{e}|\phi_{e})}{\mathcal{P}^{n=0}(\phi_{e}|\phi_{e})},$
(34)

so that $\mathcal{P}^{eq}(2\pi+\phi_{e}|\phi_{e})$ is the self-closing
transition probability evaluated for trajectories close to $\phi_{eq}$ with
geometric phases $-\pi$. $\mathcal{P}^{n=0}(\phi_{e}|\phi_{e})$ is the self-
closing transition probability evaluated for trajectories close to
$\phi^{n=0}$ which are associated with a vanishing geometric phase. Here, we
limit our analysis of Gaussian corrections to states initialized with
$\Theta=\pi/2$ since the transition and its critical value are determined by
the equatorial dynamics. The dynamics are then fully constrained to one
dimension, parametrized by $\phi$.

$\mathcal{P}^{\text{eq}}$ and $\mathcal{P}^{n=0}$ can be evaluated using a
saddle point approximation around their associated candidate optimum quantum
trajectory. Calculating a saddle point approximation is a standard procedure
[51] that consists in expanding the action around a chosen optimum solution by
rewriting $\mathbb{q}=\mathbb{q}^{*}+\delta\mathbb{q}$ and
$\mathbb{p}=\mathbb{p}^{*}+\delta\mathbb{p}$, where $\delta\mathbb{q}$ and
$\delta\mathbb{p}$ are small deviations from the optimal values
$\mathbb{q}^{*}$ and $\mathbb{p}^{*}$. The Gaussian terms in the path integral
may then be evaluated, neglecting higher-order corrections. The result is an
approximated probability,

$\mathcal{P}=\mathcal{N}\int\mathcal{D}\textbf{q}\mathcal{D}\textbf{p}\
e^{S[\textbf{q},\textbf{p}]}\approx\mathcal{N}\sqrt{\frac{(2\pi)^{n}}{\det(\mathbb{A}|_{\mathbb{q}^{*},\mathbb{p}^{*}})}}e^{S[\mathbb{q}^{*},\mathbb{p}^{*}],}$
(35)

where $\mathbb{A}|_{\mathbb{q}^{*},\mathbb{p}^{*}}$ is the functional hessian
of the CDJ action evaluated on the optimum path with zero Dirichlet boundary
conditions and $\mathcal{N}$ is an overall normalisation factor.

Our analysis is simplified by expressing the stochastic path integral in the
Lagrangian formulation (Eq. 16), despite the presence of the state-dependent
functional measure in Eq. 24. Simplifying Eq. 16, 24 and III.3 with
$\theta(t)=\frac{\pi}{2}$ and $\Theta=\frac{\pi}{2}$, and recalling the value
of $\theta_{e}$, the path integral we required for
$\mathcal{P}^{\text{eq}}(2\pi+\phi_{e}|\phi_{e})$ is,

$\displaystyle\mathcal{P}(2\pi+\phi_{e}|\phi_{e})\propto$ (36)
$\displaystyle\int\mathcal{D}\phi\sqrt{\frac{\tau}{2\pi\sin^{2}(2\pi
t-\phi)}}e^{\int\big{(}-\frac{1}{2}\tau\dot{\phi}^{2}\csc^{2}(2\pi
t-\phi)+\dot{\phi}\cot(2\pi t-\phi)\big{)}dt}$ (37) $\displaystyle\approx
e^{S[\phi_{eq}]}\int\mathcal{D}\delta\phi\sqrt{\frac{\tau}{2\pi\sin^{2}(2\pi
t-\phi_{eq})}}e^{\int\delta\phi\mathbb{A}|_{\phi_{eq}}\delta\phi dt}.$

Following the method described in [54], we apply a time reparameterization
around the equilibrium trajectory, so the new time variable $u$ is determined
by $u(t)=\frac{1}{\tau}\int_{0}^{t}\sin(2\pi
t^{\prime}-\phi_{eq}(t^{\prime}))dt^{\prime}$ with $u(0)=0$. This acts as a
local-scale transformation that eliminates the state-dependent functional
measure, simplifying the path integral to,

$\displaystyle\mathcal{P}(2\pi+\phi_{e}|\phi_{e})\approx\mathcal{N}e^{S[\phi_{eq}]}\int\mathcal{D}\delta\phi
e^{\int\delta\phi\mathbb{\Sigma}|_{\phi_{eq}}\delta\phi du}.$ (38)

This may then be evaluated as,

$\displaystyle e^{S[\phi_{eq}]}$ $\displaystyle\int\mathcal{D}\delta\phi
e^{\int\delta\phi\mathbb{\Sigma}|_{\phi_{eq}}\delta\phi
du}=e^{S[\phi_{eq}]}\Big{(}\det\mathbb{\Sigma}|_{\phi_{eq}}\Big{)}^{-\frac{1}{2}}$
(39)
$\displaystyle=e^{S[\phi_{eq}]}\Big{(}\frac{\det\big{[}\Sigma|_{\phi_{eq}}\big{]}}{\det\big{[}\frac{d^{2}}{du^{2}}\big{]}}\Big{)}^{-\frac{1}{2}}\Big{(}\det_{\zeta}\big{[}\frac{d^{2}}{du^{2}}\big{]}\Big{)}^{-\frac{1}{2}},$

where the final functional determinant is Riemann-Zeta regularized [51]. This
regularized determinant would normally be absorbed in the functional measure,
however, given the time parametrization we employed depends on the value of
the state parameter we must include the state-dependent part of the Riemann
Zeta regularised contribution of this term which can be expressed as [54]

$\text{det}_{\zeta}\frac{d^{2}}{du^{2}}=\prod\pi^{2}n^{2}\prod\frac{1}{u(T)^{2}}=|u(T)|=\frac{4\pi^{2}\tau}{4\pi^{2}\tau^{2}+1}.$
(40)

The ratio of functional determinants may be calculated by the Gelfand-Yaglom
method [56] giving,

$\frac{\det\big{[}\Sigma|_{\phi_{eq}}\big{]}}{\det\big{[}\frac{d^{2}}{du^{2}}\big{]}}=\frac{f(u|{t=1})}{f^{0}(u|{t=1})}=\frac{\tau\sinh\left(\frac{\sqrt{4\pi^{2}\tau^{2}+1}}{\tau}\right)}{\sqrt{4\pi^{2}\tau^{2}+1}},$
(41)

where $\Sigma|_{\phi_{eq}}f(u)=\lambda f(u)\text{ and
}\frac{d^{2}}{du^{2}}f^{0}(u)=\lambda^{0}_{i}f^{0}(u)$ with initial conditions
$f^{(0)}(0)=0$ and $\dot{f^{(0)}}(0)=1$. For the winding trajectory,
$\phi_{eq}=2\pi t-\arctan(2\pi\tau)$, we find a closed form expression for the
probability density,

$P(2\pi+\phi_{e}|\phi_{e})\propto
e^{-2\pi^{2}\tau}\left(\frac{4\pi^{2}\tau^{2}\sinh\left(\frac{\sqrt{4\pi^{2}\tau^{2}+1}}{\tau}\right)}{(4\pi^{2}\tau^{2}+1)^{\frac{3}{2}}}\right)^{-\frac{1}{2}}.$
(42)

For $\mathcal{P}^{n=0}(\phi_{e}|\phi_{e})$, we employ the same method,
however, finding no closed form solution for the saddle point, we resort to a
numerical approximation of each of the three contributing factors (equivalent
to Eq. 39 evaluated around $\phi^{n=0}$) directly evaluating
$e^{S[\phi^{n=0}]}$ and $|u(T)|$ numerically while approximating the ratio of
the two functional determinants using the smallest $N$ eigenvalues,

$\frac{\det\big{[}\Sigma|_{\phi^{n=0}}\big{]}}{\det\big{[}\frac{d^{2}}{du^{2}}\big{]}}\approx\frac{\prod_{i}^{N}\lambda_{i}}{\prod_{i}^{N}\lambda^{0}_{i}}.$
(43)

Using these results, the ratio $R$ is plotted in Fig. 5(b), where $R$ is
compared to $\mathfrak{R}$ and to the value of $R$ computed using numerical
simulations. we find that the effective value of $\tau^{\text{eff}}_{c}\approx
0.045$ is in excellent agreement with the results from trajectory simulations,
substantiating the validity of the Gaussian approximation. This shows that the
Gaussian action is a valid approximation to capture the whole statistics of
quantum self-closing trajectories, and this might be valuable in any
realization in which identifying the most probable trajectories might not be
feasible.

##  VII Conclusions

In this work, we have developed an action formalism that incorporates an
overall phase for the dynamics of a single qubit subject to Gaussian
measurements based on the formalism developed by Chantasri–Dressel–Jordan. We
have utilized this formalism to define a suitable Lagrangian density with
associated holonomic constraints. The inclusion of a phase degree of freedom
allows us to determine the statistical properties of measurement-induced
geometric phases for both trajectories with open boundary conditions (open
geometric phase) and sets of self-closing trajectories (closed geometric
phase). We determined the topological properties of the open geometric phase
for most likely trajectories and a variety of initial conditions, showing that
transition in a topological number is a general feature, although the
quantitative values of the critical measurement are protocol-dependent. Most
importantly, the formalism allows us to study self-closing geometric phases,
for which we show that a transition in the topological number is present for
the entire Bloch sphere, although the underlying mechanism differed from that
of open-geometric phases: in the former, the transition is dictated by the
competition of local most-likely trajectories, while in the latter, the
geometric phase is unobservable (occurring with vanishing probabilities) at
the transition point. Furthermore, we have shown that including multiple
trajectories around the optimal one via Gaussian approximation is an excellent
approximation of the full distribution simulated numerically, and leads to
modifications of the transition critical point.

The formalism developed in this work can be the basis for the efficient study
of measurement-induced dynamics in more complex systems, including multiple
qubits and their entanglement dynamics. It can also be exploited for a more
direct connection of self-closed geometric phases to experiments, where the
statistics of multiple trajectories are unavoidable.

## Acknowledgments

We would like to thank Y. Gefen and J. Ruostekoski for helpful discussions.
A.R. acknowledges support from the Royal Society, grant no. IECR2212041

## References

  * [1] Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin, Takuya Kitagawa, Eugene Demler, and Immanuel Bloch.  Direct measurement of the zak phase in topological bloch bands.  Nature Physics, 9(12):795–800, nov 2013.
  * [2] Michael Schüler, Umberto De Giovannini, Hannes Hübener, Angel Rubio, Michael A. Sentef, and Philipp Werner.  Local berry curvature signatures in dichroic angle-resolved photoelectron spectroscopy from two-dimensional materials.  Science Advances, 6(9):eaay2730, 2020.
  * [3] Michael Victor Berry.  Quantal phase factors accompanying adiabatic changes.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802):45–57, 1984.
  * [4] Barry Simon.  Holonomy, the quantum adiabatic theorem, and berry’s phase.  Phys. Rev. Lett., 51:2167–2170, Dec 1983.
  * [5] D. Chruscinski and A. Jamiolkowski.  Geometric Phases in Classical and Quantum Mechanics.  Progress in Mathematical Physics. Birkhäuser Boston, 2012.
  * [6] Di Xiao, Ming-Che Chang, and Qian Niu.  Berry phase effects on electronic properties.  Rev. Mod. Phys., 82:1959–2007, Jul 2010.
  * [7] Já nos K. Asbóth, László Oroszlány, and András Pályi.  A Short Course on Topological Insulators.  Springer International Publishing, 2016.
  * [8] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma.  Non-abelian anyons and topological quantum computation.  Rev. Mod. Phys., 80:1083–1159, Sep 2008.
  * [9] Shi-Liang Zhu and Paolo Zanardi.  Geometric quantum gates that are robust against stochastic control errors.  Phys. Rev. A, 72:020301, Aug 2005.
  * [10] Erik Sjöqvist.  Geometric phases in quantum information.  International Journal of Quantum Chemistry, 115(19):1311–1326, may 2015.
  * [11] Y. Aharonov and J. Anandan.  Phase change during a cyclic quantum evolution.  Phys. Rev. Lett., 58:1593–1596, Apr 1987.
  * [12] Young-Wook Cho, Yosep Kim, Yeon-Ho Choi, Yong-Su Kim, Sang-Wook Han, Sang-Yun Lee, Sung Moon, and Yoon-Ho Kim.  Emergence of the geometric phase from quantum measurement back-action.  Nature Physics, 15(7):665–670, April 2019.
  * [13] Kurt Jacobs.  Quantum Measurement Theory and its Applications.  Cambridge University Press, 2014.
  * [14] Kurt Jacobs and Daniel A. Steck.  A straightforward introduction to continuous quantum measurement.  Contemporary Physics, 47(5):279–303, sep 2006.
  * [15] Kyrylo Snizhko, Parveen Kumar, and Alessandro Romito.  Quantum zeno effect appears in stages.  Phys. Rev. Res., 2:033512, Sep 2020.
  * [16] Parveen Kumar, Alessandro Romito, and Kyrylo Snizhko.  Quantum zeno effect with partial measurement and noisy dynamics.  Phys. Rev. Res., 2:043420, Dec 2020.
  * [17] Alessandro Romito.  Quantum hardware measures up to the challenge.  Nature Physics, 19:1234–1235, 2023.
  * [18] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher.  Quantum zeno effect and the many-body entanglement transition.  Phys. Rev. B, 98:205136, Nov 2018.
  * [19] Amos Chan, Rahul M. Nandkishore, Michael Pretko, and Graeme Smith.  Unitary-projective entanglement dynamics.  Phys. Rev. B, 99:224307, Jun 2019.
  * [20] Brian Skinner, Jonathan Ruhman, and Adam Nahum.  Measurement-induced phase transitions in the dynamics of entanglement.  Phys. Rev. X, 9:031009, Jul 2019.
  * [21] M. Szyniszewski, A. Romito, and H. Schomerus.  Entanglement transition from variable-strength weak measurements.  Phys. Rev. B, 100:064204, Aug 2019.
  * [22] Jin Ming Koh, Shi-Ning Sun, Mario Motta, and Austin J. Minnich.  Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout.  Nature Physics, 19(9):1314–1319, June 2023.
  * [23] Google AI and Collaborators.  Measurement-induced entanglement and teleportation on a noisy quantum processor.  Nature, 622(7983):481–486, October 2023.
  * [24] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay.  Random quantum circuits.  Annual Review of Condensed Matter Physics, 14(1):335–379, March 2023.
  * [25] Armin Uhlmann.  Parallel transport and “quantum holonomy” along density operators.  Reports on Mathematical Physics, 24(2):229–240, 1986.
  * [26] Armin Uhlmann.  On berry phases along mixtures of states.  Annalen der Physik, 501(1):63–69, 1989.
  * [27] Armin Uhlmann.  A gauge field governing parallel transport along mixed states.  letters in mathematical physics, 21:229–236, 1991.
  * [28] Erik Sjöqvist, Arun K. Pati, Artur Ekert, Jeeva S. Anandan, Marie Ericsson, Daniel K. L. Oi, and Vlatko Vedral.  Geometric phases for mixed states in interferometry.  Phys. Rev. Lett., 85:2845–2849, Oct 2000.
  * [29] A. Carollo, I. Fuentes-Guridi, M. Fran ça Santos, and V. Vedral.  Geometric phase in open systems.  Phys. Rev. Lett., 90:160402, Apr 2003.
  * [30] Marie Ericsson, Erik Sjöqvist, Johan Brännlund, Daniel K. L. Oi, and Arun K. Pati.  Generalization of the geometric phase to completely positive maps.  Phys. Rev. A, 67:020101, Feb 2003.
  * [31] Valentin Gebhart.  Measurement-induced geometric phases.  PhD thesis, Masterarbeit, Universität Freiburg, 2017.
  * [32] Ludmila Viotti, Ana Laura Gramajo, Paula I. Villar, Fernando C. Lombardo, and Rosario Fazio.  Geometric phases along quantum trajectories, 2023.
  * [33] Kyrylo Snizhko, Nihal Rao, Parveen Kumar, and Yuval Gefen.  Weak-measurement-induced phases and dephasing: Broken symmetry of the geometric phase.  Phys. Rev. Res., 3:043045, Oct 2021.
  * [34] Kyrylo Snizhko, Nihal Rao, Parveen Kumar, and Yuval Gefen.  Weak-measurement-induced phases and dephasing: Broken symmetry of the geometric phase.  Physical Review Research, 3(4), oct 2021.
  * [35] Kyrylo Snizhko, Parveen Kumar, Nihal Rao, and Yuval Gefen.  Weak-measurement-induced asymmetric dephasing: Manifestation of intrinsic measurement chirality.  Physical Review Letters, 127(17), oct 2021.
  * [36] Graham Kells, Dganit Meidan, and Alessandro Romito.  Topological transitions in weakly monitored free fermions.  SciPost Phys., 14:031, 2023.
  * [37] Sthitadhi Roy, J. T. Chalker, I. V. Gornyi, and Yuval Gefen.  Measurement-induced steering of quantum systems.  Phys. Rev. Res., 2:033347, Sep 2020.
  * [38] Nicolai Lang and Hans Peter Büchler.  Entanglement transition in the projective transverse field ising model.  Phys. Rev. B, 102:094204, Sep 2020.
  * [39] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli.  Measurement-induced topological entanglement transitions in symmetric random quantum circuits.  Nature Physics, 17(3):342–347, January 2021.
  * [40] Yunzhao Wang, Kyrylo Snizhko, Alessandro Romito, Yuval Gefen, and Kater Murch.  Dissipative preparation and stabilization of many-body quantum states in a superconducting qutrit array.  Physical Review A, 108(1), July 2023.
  * [41] Kyrylo Snizhko, Parveen Kumar, Nihal Rao, and Yuval Gefen.  Weak-measurement-induced asymmetric dephasing: Manifestation of intrinsic measurement chirality.  Phys. Rev. Lett., 127:170401, Oct 2021.
  * [42] Yunzhao Wang, Kyrylo Snizhko, Alessandro Romito, Yuval Gefen, and Kater Murch.  Observing a topological transition in weak-measurement-induced geometric phases.  Phys. Rev. Res., 4:023179, Jun 2022.
  * [43] Manuel F. Ferrer-Garcia, Kyrylo Snizhko, Alessio D’Errico, Alessandro Romito, Yuval Gefen, and Ebrahim Karimi.  Topological transitions of the generalized pancharatnam-berry phase, 2022.
  * [44] A. Chantasri, J. Dressel, and A. N. Jordan.  Action principle for continuous quantum measurement.  Physical Review A, 88(4), oct 2013.
  * [45] Areeya Chantasri and Andrew N. Jordan.  Stochastic path-integral formalism for continuous quantum measurement.  Physical Review A, 92(3), sep 2015.
  * [46] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi.  Observing single quantum trajectories of a superconducting quantum bit.  Nature, 502(7470):211–214, oct 2013.
  * [47] Philippe Lewalle, Areeya Chantasri, and Andrew N. Jordan.  Prediction and characterization of multiple extremal paths in continuously monitored qubits.  Physical Review A, 95(4), apr 2017.
  * [48] Philippe Lewalle, John Steinmetz, and Andrew N. Jordan.  Chaos in continuously monitored quantum systems: An optimal-path approach.  Physical Review A, 98(1), jul 2018.
  * [49] Areeya Chantasri, Juan Atalaya, Shay Hacohen-Gourgy, Leigh S. Martin, Irfan Siddiqi, and Andrew N. Jordan.  Simultaneous continuous measurement of noncommuting observables: Quantum state correlations.  Physical Review A, 97(1), jan 2018.
  * [50] H. Kleinert.  Path Integrals In Quantum Mechanics, Statistics, Polymer Physics, And Financial Markets (5th Edition).  World Scientific Publishing Company, 2009.
  * [51] Alexander Altland and Ben D. Simons.  Condensed Matter Field Theory.  Cambridge University Press, 2 edition, 2010.
  * [52] Areeya Chantasri, Mollie E. Kimchi-Schwartz, Nicolas Roch, Irfan Siddiqi, and Andrew N. Jordan.  Quantum trajectories and their statistics for remotely entangled quantum bits.  Physical Review X, 6(4), dec 2016.
  * [53] H.J. Rothe and K.D. Rothe.  Classical and Quantum Dynamics of Constrained Hamiltonian Systems.  World Scientific lecture notes in physics. World Scientific, 2010.
  * [54] Miguel V. Moreno, Daniel G. Barci, and Zochil González Arenas.  Conditional probabilities in multiplicative noise processes.  Phys. Rev. E, 99:032125, Mar 2019.
  * [55] As for the solutions at the equator, multiple solutions of the Euler-Lagrange equations exist for given boundary conditions, corresponding to local minima of the actions.
  * [56] Klaus Kirsten and Alan J McKane.  Functional determinants for general sturm–liouville problems.  Journal of Physics A: Mathematical and General, 37(16):4649–4670, apr 2004.


# Unusual upper critical fields of the topological nodal-line semimetal
candidate SnxNbSe2-δ

Riffat Munir  K A M Hasan Siddiquee  Charuni Dissanayake  Department of
Physics, University of Central Florida, Orlando, Florida 32816, USA  Xinzhe Hu
Yasumasa Takano  Department of Physics, University of Florida, Gainesville,
Florida 32611, USA  Eun Sang Choi  National High Magnetic Field Laboratory,
Florida State University, Tallahassee, Florida 32816, USA  Yasuyuki Nakajima∗
Department of Physics, University of Central Florida, Orlando, Florida 32816,
USA

###### Abstract

We report superconductivity in SnxNbSe2-δ, a topological nodal-line semimetal
candidate with a noncentrosymmetric crystal structure. The superconducting
transition temperature $T_{c}$ of this compound is extremely sensitive to Sn
concentration $x$ and Se deficiency $\delta$, 5.0 K for Sn0.13NbSe1.70 and 8.6
K for Sn0.14NbSe1.71 and Sn0.15NbSe1.69. In all samples, the temperature
dependence of the upper critical field $H_{c2}(T)$ differs from the prediction
of the Werthamer-Helfand-Hohenberg theory. While the zero-temperature value of
the in-plane upper critical field of SnxNbSe2-δ with the higher $T_{c}$ is
lower than the Pauli paramagnetic limit $H_{P}$, that of the lower $T_{c}$
sample exceeds $H_{P}$ by a factor of $\sim$2\. Our observations suggest that
odd-parity contribution dominates the superconducting gap function of
SnxNbSe2-δ, and it can be fine-tuned by the Sn concentration and Se
deficiency.

Topological superconductors, characterized by a topologically nontrivial
gapped state in the bulk with gapless surface states Sato and Ando (2017),
have attracted great interest because of potential applications to
topologically protected quantum computing Kitaev (2003); Nayak et al. (2008).
Such nontrivial gapped states can be stabilized by odd-parity Cooper pairing,
occasionally realized in noncentrosymmetric superconductors with strong spin-
orbit coupling. The lack of inversion symmetry in the crystal structures
allows supersosition of spin singlet (even parity) and spin triplet (odd
parity) in superconducting gap functions. This mixing of parity can be fine-
tuned by the spin-orbit coupling strength, as observed in the crossover from
even to odd-parity pairing states in Li2(Pd,Pt)3Bi as the Pt concentration is
varied Harada et al. (2012). Hence, the search for noncentrosymmetric
superconductors with strong spin-orbit coupling serves as one of the routes
toward the realization of topological superconductivity.

The noncentrosymmetric ABSe2 (A=Sn or Pb and B=Nb or Ta) family is a promising
candidate for a topological superconductor. Recent experimental studies have
revealed superconductivity in PbTaSe2 at $T_{c}=3.8$ K Ali et al. (2014); Long
et al. (2016), with a fully opened superconducting gap probed by various
measurements, including specific heat Zhang et al. (2016), thermal
conductivity Wang et al. (2016), and penetration depth measurements Pang et
al. (2016). Moreover, angle-resolved photoemission spectroscopy has identified
this materials as a topological nodal-line semimetal with drumhead surface
states Bian et al. (2016); Chang et al. (2016). Although the parity of the
superconducting pairing state is yet to be determined, the lack of inversion
symmetry that along with the nontrivial topological band structure can induce
topological superconductivity in this system in PbTaSe2.

In addition to this compound, PbNbSe2, SnNbSe2, and PbTaSe2 are predicted by
ab-initio calculations to be superconductors with nontrivial topological nodal
lines in the band structure Chen et al. (2016). According to the calculations,
the superconducting transition temperatures, $T_{c}$, of these materials are
expected to be higher than that of PbTaSe2, and notably, the predicted $T_{c}$
of SnNbSe2 is 7 K, suggestive of a possible realization of topological
superconductivity in this material with the highest $T_{c}$, compared with
known bulk topological superconductor candidates, such as half Heusler
compounds YPtBi ($T_{c}$ = 0.6 K) Butch et al. (2011) and YPdBi ($T_{c}$ = 1.6
K) Nakajima et al. (2015), In-doped SnTe ($T_{c}$ = 1.2 K) Sasaki et al.
(2012), metal-intercalated Bi2Se3 ($T_{c}$ = 3.2 K) Hor et al. (2010); Kriener
et al. (2011); Liu et al. (2015), and $\beta$-Bi2Pd ($T_{c}$ = 5.4 K) Imai et
al. (2012); Liu et al. (2020). The higher $T_{c}$ can be advantageous to
detect Majorana bound sates with the energy spacing $\Delta^{2}/E_{F}$, where
$\Delta$ and $E_{F}$ are the superconducting gap and Fermi energy,
respectively, and can eliminate one bottleneck in the exploration for
topological superconductivity Beenakker (2015).

We here report unusual superconductivity in SnxNbSe2-δ, which retains the
noncentrosymmetric crystal structure of stoichiometric ABSe2. The
superconducting transition temperature of SnxNbSe2-δ varies with Sn
concentration and Se deficiency—up to 8.6 K, the highest among bulk
topological superconductor candidates known to date. The measured upper
critical fields of SnxNbSe2-δ cannot be described by the Werthamer-Helfand-
Hohenberg (WHH) theory for conventional type-II superconductors Werthamer et
al. (1966), and the zero-temperature value of the upper critical field for
Sn0.13NbSe1.70 exceeds the Pauli paramagnetic limit by a factor of $\sim$2\.
The unusual temperature dependence of the upper critical fields and the
enhancement beyond the Pauli paramagnetic limit suggest that an odd-parity
component dominates the superconducting gap function of SnxNbSe2-δ, indicating
that this compound is a promising candidate for realizing topological
superconductivity.

Single crystals of SnxNbSe2-δ were grown by using a self-flux method. A
mixture of NbSe2 and Sn in the ratio of NbSe2:Sn=1:2$-$6 was sealed in a
quartz tube, heated up to 900∘C, kept for 2$-$3 days, and slowly cooled down
to 600∘C. The excess of molten Sn flux was removed by centrifuging. The
typical size of obtained flake-like crystals is about 1$\times$1$\times$0.3
mm3.

Figure 1: Crystal structure and x-ray diffraction pattern of SnxNbSe2-δ. (a)
Top view (left panel) and side view (right panel) of the crystal structure of
SnNbSe2 with the space group $P\bar{6}$m2 Momma and Izumi (2011). (b) X-ray
diffraction pattern for single crystal Sn0.15NbSe1.69 measured with Cu Kα
radiation. We only observe (00$\ell$) Bragg peaks from the $ab$ plane. The
obtained lattice constant $c$ is 9.2976(12) Å. Inset: Optical image of a
SnxNbSe2-δ single crystal. The typical crystal size is $\sim 1\times 1\times
0.3$ mm3.

The noncentrosymmetric crystal structure of SnxNbSe2-δ with the space group
$P\bar{6}$m was confirmed by single-crystal x-ray diffraction. As shown in
fig.1b, we observed only (00$\ell$) reflections, which yielded the lattice
constant $c=$ 9.2976(12) Å close to the reported value of 9.30 Å for
polycrystalline SnNbSe2 Gentile et al. (1979). The atomic ratio of the
crystals was found to be Sn:Nb:Se = 0.11$-$0.15:1:1.60$-$1.74 with x-ray
fluorescence spectroscopy, suggestive of slightly intercalated Sn and
deficiency of Se. Although the Sn concentrations in our samples are only
11%$-$15% of the stoichiometric value, the identified crystal structure is the
same as that of ABSe2 and different from the 2H-NbSe2-type structure of Sn-
intercalated NbSe2 with $x$ up to 0.04 Naik et al. (2019).

We reveal superconductivity in SnxNbSe2-δ with resistivity measurements from
300 K down to 2 K. The resistivity at $T=$ 300 K is about 700 $\mu\Omega$ cm.
Upon cooling, the resistivity shows metallic behavior, followed by a
superconducting transition at low temperatures (fig.1). The superconducting
transition temperatures $T_{c}$, determined by the midpoint of resistive
transitions, are 5.0 K for Sn0.13NbSe1.70 and 8.6 K for Sn0.14NbSe1.71 and
Sn0.15NbSe1.69 (fig.1 inset). In contrast, no superconductivity has been
observed down to 1.5 K in polycrystalline Sn2/3NbSe2 Karnezos et al. (1975).
The normal-state resistivity of SnxNbSe2-δ right above $T_{c}$ is much higher
than that of PbTaSe2. Because of this high residual resistivity, the residual
resistivity ratio RRR = $\rho$(300 K)/$\rho$($T_{c}$) of SnxNbSe2-δ is
2.3$-$3.0, 30$-$50 times as small as that of single-crystal PbTaSe2 Zhang et
al. (2016); Long et al. (2016); Sankar et al. (2017). Interestingly, despite
the high residual resistivity, or the low RRR, the $T_{c}$ of 8.6 K not only
surpasses the theoretical prediction of 7 K Chen et al. (2016) but is to date
the highest superconducting transition temperature among known topological
superconductor candidates.

Figure 2: Temperature dependence of the resistivities of SnxNbSe2-δ. Overall
temperature dependence of the resistivity of Sn0.15NbSe1.69. Preceded by
metallic behavior, SnxNbSe2-δ undergoes a superconducting transition at low
temperatures. The residual resistivity ratio $RRR$ = $\rho$(300
K)/$\rho$($T_{c}$) is 2.3$-$3.0, suggestive of abundant disorder. Inset: Low
temperature resistivities of SnxNbSe2-δ, indicating superconducting
transitions at 5.0 K for Sn0.13NbSe1.70 and 8.6 K for Sn0.14NbSe1.71 and
Sn0.15NbSe1.69.

To determine the upper critical fields of SnxNbSe2-δ, we measured the
resistivity in a 35 T resistive magnet equipped with a 3He cryostat at the
National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida. We
show magnetoresistance of SnxNbSe2-δ in two different magnetic field
configurations: $H\parallel I\parallel ab$ (in-plane configuration) and
$H\perp ab,I\parallel ab$ (out-of-plane configuration). For both field
configurations and in both Sn0.13NbSe1.70 and Sn0.14NbSe1.71, the
magnetoresistance is indiscernible even at an applied magnetic field of 35 T
(figs.3a$-$d), indicative of short mean free paths of charge carriers due to
disorder caused most likely by slightly intercalated Sn and deficiency of Se.
On the other hand, we find a notable difference in the anisotropy of upper
critical fields, $\Gamma=H_{c2}^{ab}/H_{c2}^{c}$, between these two samples.
In the out-of-plane configuration for Sn0.13NbSe1.70 (fig.3b), the upper
critical field parallel to the $c$ axis, $H_{c2}^{c}$, defined by the
midpoints of sharp resistive transitions, is 5 T at 0.3 K. In the in-plane
configuration for Sn0.13NbSe1.70 (fig.3a), the transition widths are broad,
and the upper critical field parallel to the $ab$ plane, $H_{c2}^{ab}$, is
extremely high, extracted to be 15 T at 0.3 K. These values of $H_{c2}$ yield
a large anisotropy of the upper critical field, $\Gamma=3$, for
Sn0.13NbSe1.70. In contrast, we observe similar transition widths with very
similar $H_{c2}$ in both field configurations for Sn0.14NbSe1.71, suggestive
of a nearly isotropic upper critical field with $\Gamma=1.1$ (figs.3c and d).
In the isostructural compound PbTaSe2, the anisotropy of $H_{c2}$ is measured
to be 11.6 Long et al. (2016), much larger than the observed anisotropy in
SnxNbSe2-δ, although the two compounds share similar electronic band
structures Chen et al. (2016).

Figure 3: Magnetotransport in SnxNbSe2-δ. Resistivity of Sn0.13NbSe1.70 as a
function of magnetic fields parallel to (a) the $ab$ plane and (b) the $c$
axis at several temperatures. Electrical current is applied in the $ab$ plane.
While the resistive transitions for $H\parallel c$ are sharp, the transitions
for $H\parallel ab$ are broad. The upper critical fields obtained by the
midpoints of the resistive transitions show high anisotropy
$\Gamma=H_{c2}^{ab}/H_{c2}^{c}=$ 3 in Sn0.13NbSe1.70. Resistivity of
Sn0.14NbSe1.71 as a function of magnetic fields parallel to (c) the $ab$ plane
and (d) the $c$ axis at various temperatures. Despite the layered structure of
the compound, the upper critical field is nearly isotropic, $\Gamma=$ 1.1, in
Sn0.14NbSe1.71. Figure 4: Temperature dependence and anisotropy of the upper
critical field of SnxNbSe2-δ. (a) Temperature dependence of upper critical
field determined by the midpoint of resistive transitions in figs.3 and 6 for
Sn0.13NbSe1.70. Regardless of the field orientation, the upper critical field
increases linearly with decreasing temperature down to 0.3 K. The zero-
temperature values of in-plane upper critical fields $\mu_{0}H_{c2}^{ab}(0)$
for $H\parallel I$ and $H\perp I$ exceed the Pauli paramagnetic limit
$\mu_{0}H_{P}=1.84T_{c}=$ 9.3 T, indicated by a blue dashed line. The zero-
temperature value of out-of-plane $\mu_{0}H_{c2}^{c}(0)$ is lower than
$\mu_{0}H_{P}$. Inset: Normalized upper critical field
$h=H_{c2}/(-T_{c}dH_{c2}/dT|_{T=Tc})$ as a function of normalized temperature
$t=T/T_{c}$ for Sn0.13NbSe1.70. For all the field orientations, the normalized
upper critical field clearly deviates from the WHH model at low temperatures
and is close to the calculation for the polar $p$-wave state Scharnberg and
Klemm (1980). (b) Temperature dependence of the upper critical field
determined by the midpoint of resistive transitions in fig.3 for
Sn0.14NbSe1.71. The upper critical fields for $H\parallel ab$ and $H\parallel
c$ exhibit linear-in-temperature behavior down to 0.3 K. The zero-temperature
values of in-plane upper critical fields $\mu_{0}H_{c2}^{ab}(0)$ for
$H\parallel I$ and $H\perp I$ are lower than the Pauli paramagnetic limit
$\mu_{0}H_{P}=$ 15.6 T, indicated by a blue dashed line. Inset: Normalized
upper critical field $h$ as a function of $t$ for Sn0.14NbSe1.71. The
normalized upper critical fields for both $H\parallel ab$ and $H\parallel c$
depart from the WHH model at low temperatures and are close to the calculation
for the polar $p$-wave state Scharnberg and Klemm (1980).

The upper critical fields of SnxNbSe2-δ exhibit quite unusual behavior, as
shown in fig.4. With decreasing temperature starting from $T_{c}$, $H_{c2}$
increases linearly in all the field configurations for both Sn0.13NbSe1.70 and
Sn0.14NbSe1.71. The initial slopes of the upper critical fields parallel to
the $ab$ plane for $H\parallel I$, $dH_{c2}/dT|_{T=T_{c}}$, are $-$3.2 T/K for
Sn0.13NbSe1.70 and $-$1.5 T/K for Sn0.14NbSe1.71, extremely larger than
$-$0.19 K/T for PbTaSe2 for $H\parallel ab$ Zhang et al. (2016). The observed
temperature dependence of $H_{c2}$ for SnxNbSe2-δ completely differs from
those of conventional type-II superconductors. In conventional type-II
superconductors, upper critical fields due to the orbital effect are
explicated by the Werthamer-Helfand-Hohenberg (WHH) theory Werthamer et al.
(1966). In the WHH theory, the zero-temperature value of upper critical field
can be written as $H_{c2}(0)=-\alpha T_{c}dH_{c2}/dT|_{T=T_{c}}$, where
$\alpha$ is 0.69 for the dirty limit and 0.72 for the clean limit Werthamer et
al. (1966). However, as shown in the insets to figs.4a and b, the normalized
upper critical field $h=H_{c2}/(-T_{c}dH_{c2}/dT|_{T=T_{c}})$ strikingly
deviates from the WHH values and is close to the calculations for the polar
$p$-wave state Scharnberg and Klemm (1980). The extracted $\alpha$ for
SnxNbSe2-δ is $\sim 1.0$.

Equally unusual in SnxNbSe2-δ are the magnitudes of the upper critical fields.
The zero temperature value of the in-plane upper critical field
$\mu_{0}H_{c2}^{ab}(0)$ for Sn0.13NbSe1.70 is $\sim$16 T, exceeding the Pauli
paramagnetic limit $\mu_{0}H_{P}=1.84T_{c}=$ 9.3 T obtained from a simple
estimation within the weak-coupling BCS theory with $2\Delta$ =
3.5$k_{B}T_{c}$ (fig.4a) Clogston (1962). The persistence of superconductivity
beyond the Pauli paramagnetic limit suggests dominant odd-parity contribution
in the superconducting gap function,as found in noncentrosymmetric heavy
fermions superconductor CePt2Si Bauer et al. (2004) and metal-intercalated
Bi2Se2 Bay et al. (2012).

In Sn0.14NbSe1.71, $\mu_{0}H_{c2}^{ab}(0)$ is 13 T, comparable to but slightly
lower than $\mu_{0}H_{P}=$ 15.6 T enhanced by the higher $T_{c}$ of 8.6 K
(fig.4b). The zero temperature values of out-of-plane upper critical fields
$\mu_{0}H_{c2}^{c}(0)$ of SnxNbSe2-δ are lower than the Pauli paramagnetic
limit. Utilizing the Ginzburg–Landau relation,
$\mu_{0}H_{c2}^{c}(0)=\phi_{0}/2\pi\xi_{ab}^{2}$ and
$\mu_{0}H_{c2}^{c}(0)=\phi_{0}/2\pi\xi_{c}\xi_{ab}$, where $\phi_{0}$ is the
magnetic flux quantum, we can estimate the coherence lengths: $\xi_{ab}=$ 15
nm and $\xi_{c}=$ 4.5 nm for Sn0.13NbSe1.70 and $\xi_{ab}=$ 5.0 nm and
$\xi_{c}=$ 6.0 nm for Sn0.14NbSe1.71.

Although our observations suggest the presence of considerable odd-parity
contribution to the pairing states at least in Sn0.13NbSe1.70 with $T_{c}$ of
5.0 K, its superconductivity is robust against scattering due to nonmagnetic
disorder. In general, nonmagnetic scattering strongly suppresses
unconventional superconductivity Mackenzie et al. (1998), including odd-parity
pairing states, as described by the Abrikosov-Gorkov model Abrikosov and
Gor’kov (1961). The Abrikosov-Gorkov model shows that $T_{c}$ of
unconventional superconductors are dramatically suppressed to zero by
nonmagnetic scattering, as the mean free path $\ell$ becomes comparable to the
coherence length, $\xi\sim\ell$. To estimate the mean free paths in
SnxNbSe2-δ, we exploit the Hall effect, along with electrical resistivity.
Hall resistivity $\rho_{yx}$ of Sn0.11NbSe1.60 is positive and perfectly
linear in magnetic fields in the normal state above $T_{c}$ (fig.5),
indicative of a dominant contribution to the charge transport from a large
cylindrical hole band, around the $\Gamma$ point, predicted by theoretical
calculations Chen et al. (2016); Xu et al. (2019). The slope of the Hall
resistivity gives the Hall coefficient $R_{H}=5.3\times$ 10-3 cm3/C, hence the
carrier density $n=1.2\times 10^{21}$ cm-3. Assuming a single cylindrical hole
band, we can extract mean free paths by using $\ell=(\hbar/\rho
e^{2})\sqrt{2\pi c/n}$, where $c$ is the lattice constant along the $c$ axis.
The extracted mean free paths at $T_{c}$ are $\ell$ = 5.9 nm for
Sn0.13NbSe1.70 and 3.1 nm for Sn0.14NbSe1.71. Surprisingly, these values are
much smaller than the coherence lengths in the $ab$ plane, intuitively
inconsistent with the possible odd-parity pairing state.

Figure 5: Field dependence of the Hall resistivity of Sn0.11NbSe1.60 at 7 K
($H\parallel c,I\parallel ab$). The Hall resistivity $\rho_{yx}$ is linear in
$H$ with the positive sign indicating that the hole band dominates the
transport in SnxNbSe2-δ. The extracted Hall coefficient $R_{H}$ from the slope
of $\rho_{yx}$ is 5.3 $\times$ 10-3 cm3/C, yielding a carrier density of
$1.2\times 10^{21}$ cm-3.

However, there is a precedence in which odd-parity superconductivity was found
to be invulnerble to nonmagnetic disorder—metal-intercalated Bi2Se3 Sasaki et
al. (2011); Kriener et al. (2011); Bay et al. (2012). In this topological
superconductor, the robustness of the odd-parity superconductivity is thought
to be due to strong spin-orbit coupling Michaeli and Fu (2012); Nagai et al.
(2014). Similarly, nonnegligible spin-orbit coupling due to Sn, a relatively
heavy element, may be protecting superconductivity in SnxNbSe2-δ against
strong nonmagnetic scattering. The shared features of superconductivity in the
two compounds—the upper critical fields beyond the Pauli paramagnetic limit
and the robust superconductivity against disorder—strongly suggest that the
odd-parity superconductivity is realized in SnxNbSe2-δ.

Figure 6: Transverse magnetoresistance of Sn0.13NbSe1.70 in magnetic fields
parallel to the $ab$ plane. Resistivity of Sn0.13NbSe1.70 as a function of in-
plane magnetic field perpendicular to the applied current in the $ab$ plane.
The resistive transitions are broad, and show kinks indicated by arrows,
associated with vortex motion due to the Lorentz force. The extracted
$H_{c2}^{ab}$ for $H\perp I$ is slightly lower than $H_{c2}^{ab}$ for
$H\parallel I$, as shown in fig.4a.

We briefly comment on a possible nematic superconducting state in the PbTaSe2
family. Nematic superconductivity, accompanied by a rotational symmetry
breaking in the superconducting state, is observed in metal-intercalated
Bi2Se3, evidencing topological superconductivity in the system Matano et al.
(2016); Pan et al. (2016); Smylie et al. (2017); Yonezawa et al. (2017).
Similarly, in PbTaSe2, soft point-contact spectroscopy Le et al. (2020)
elucidates the rotational symmetry breaking in superconducting properties,
possibly associated with nematic superconductivity. In Sn0.13NbSe1.70, we
observe clear anisotropy in in-plane upper critical fields for two different
configurations, $H\parallel I$ and $H\perp I$, as shown in fig.4a. However,
the measured anisotropy can be attributed to the presence/absence of flux flow
due to the Lorentz force as observed in MgB2 Shi et al. (2003), masking
intrinsic twofold symmetry associated with electronic nematicity in this
system, if present. Indeed, we observe kinks in the resistive transition
associated with vortex motion due to the Lorentz force only in the $H\perp I$
configuration as shown in fig.6 Zhu et al. (2010). To determine the intrinsic
in-plane anisotropy, further experimental investigations will be required.

In summary, we have grown single crystals of SnxNbSe2-δ with a
noncentrosymmetric crystal structure, and found superconductivity with the
highest $T_{c}$, 8.6 K among known topological superconductor candidates. The
upper critical field of Sn0.13NbSe1.70, with $T_{c}=$ 5 K, exceeds the Pauli
paramagnetic limit, suggestive of a sizable contribution of odd-parity paring
in the superconductivity. The possible odd-parity pairing component, a
prerequisite for topological superconductivity, makes this system a promising
material for further study.

###### Acknowledgements.

This work was supported by the start-up fund from the University of Central
Florida. H.S and Y.N. are supported by NSF CAREER DMR-1944975, and X.H. and
Y.T. by the NHMFL UCGP program. The NHMFL is supported by the National Science
Foundation through NSF/DMR-1644779 and the State of Florida.

## References

  * Sato and Ando (2017) M. Sato and Y. Ando, Reports on Progress in Physics 80, 076501 (2017).
  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003).
  * Nayak et al. (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * Harada et al. (2012) S. Harada, J. J. Zhou, Y. G. Yao, Y. Inada, and G.-q. Zheng, Phys. Rev. B 86, 220502 (2012).
  * Ali et al. (2014) M. N. Ali, Q. D. Gibson, T. Klimczuk, and R. J. Cava, Phys. Rev. B 89, 020505 (2014).
  * Long et al. (2016) Y.-J. Long, L.-X. Zhao, P.-P. Wang, H.-X. Yang, J.-Q. Li, H. Zi, Z.-A. Ren, C. Ren, and G.-F. Chen, Chinese Physics Letters 33, 037401 (2016).
  * Zhang et al. (2016) C.-L. Zhang, Z. Yuan, G. Bian, S.-Y. Xu, X. Zhang, M. Z. Hasan, and S. Jia, Phys. Rev. B 93, 054520 (2016).
  * Wang et al. (2016) M. X. Wang, Y. Xu, L. P. He, J. Zhang, X. C. Hong, P. L. Cai, Z. B. Wang, J. K. Dong, and S. Y. Li, Phys. Rev. B 93, 020503 (2016).
  * Pang et al. (2016) G. M. Pang, M. Smidman, L. X. Zhao, Y. F. Wang, Z. F. Weng, L. Q. Che, Y. Chen, X. Lu, G. F. Chen, and H. Q. Yuan, Phys. Rev. B 93, 060506 (2016).
  * Bian et al. (2016) G. Bian, T.-R. Chang, R. Sankar, S.-Y. Xu, H. Zheng, T. Neupert, C.-K. Chiu, S.-M. Huang, G. Chang, I. Belopolski, et al., Nature Communications 7, 10556 EP (2016).
  * Chang et al. (2016) T.-R. Chang, P.-J. Chen, G. Bian, S.-M. Huang, H. Zheng, T. Neupert, R. Sankar, S.-Y. Xu, I. Belopolski, G. Chang, et al., Phys. Rev. B 93, 245130 (2016).
  * Chen et al. (2016) P.-J. Chen, T.-R. Chang, and H.-T. Jeng, Phys. Rev. B 94, 165148 (2016).
  * Butch et al. (2011) N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, and J. Paglione, Phys. Rev. B 84, 220504 (2011).
  * Nakajima et al. (2015) Y. Nakajima, R. Hu, K. Kirshenbaum, A. Hughes, P. Syers, X. Wang, K. Wang, R. Wang, S. R. Saha, D. Pratt, et al., Science Advances 1 (2015).
  * Sasaki et al. (2012) S. Sasaki, Z. Ren, A. A. Taskin, K. Segawa, L. Fu, and Y. Ando, Phys. Rev. Lett. 109, 217004 (2012).
  * Hor et al. (2010) Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett. 104, 057001 (2010).
  * Kriener et al. (2011) M. Kriener, K. Segawa, Z. Ren, S. Sasaki, and Y. Ando, Phys. Rev. Lett. 106, 127004 (2011).
  * Liu et al. (2015) Z. Liu, X. Yao, J. Shao, M. Zuo, L. Pi, S. Tan, C. Zhang, and Y. Zhang, Journal of the American Chemical Society 137, 10512 (2015).
  * Imai et al. (2012) Y. Imai, F. Nabeshima, T. Yoshinaka, K. Miyatani, R. Kondo, S. Komiya, I. Tsukada, and A. Maeda, Journal of the Physical Society of Japan 81, 113708 (2012).
  * Liu et al. (2020) P.-F. Liu, J. Li, X.-H. Tu, H. Yin, B. Sa, J. Zhang, D. J. Singh, and B.-T. Wang, Phys. Rev. B 102, 155406 (2020).
  * Beenakker (2015) C. W. J. Beenakker, Rev. Mod. Phys. 87, 1037 (2015).
  * Werthamer et al. (1966) N. Werthamer, E. Helfand, and P. Hohenberg, Phys. Rev. 147, 295 (1966).
  * Momma and Izumi (2011) K. Momma and F. Izumi, Journal of Applied Crystallography 44, 1272 (2011).
  * Gentile et al. (1979) P. Gentile, D. Driscoll, and A. Hockman, Inorganica Chimica Acta 35, 249 (1979).
  * Naik et al. (2019) S. Naik, G. K. Pradhan, S. G. Bhat, B. C. Behera, P. A. Kumar, S. L. Samal, and D. Samal, Physica C: Superconductivity and its Applications 561, 18 (2019).
  * Karnezos et al. (1975) N. Karnezos, L. B. Welsh, and M. W. Shafer, Phys. Rev. B 11, 1808 (1975).
  * Sankar et al. (2017) R. Sankar, G. N. Rao, I. P. Muthuselvam, T.-R. Chang, H. T. Jeng, G. S. Murugan, W.-L. Lee, and F. C. Chou, Journal of Physics: Condensed Matter 29, 095601 (2017).
  * Scharnberg and Klemm (1980) K. Scharnberg and R. A. Klemm, Phys. Rev. B 22, 5233 (1980).
  * Clogston (1962) A. Clogston, Phys. Rev. Lett. 9, 266 (1962).
  * Bauer et al. (2004) E. Bauer, G. Hilscher, H. Michor, C. Paul, E. W. Scheidt, A. Gribanov, Y. Seropegin, H. Noël, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004).
  * Bay et al. (2012) T. V. Bay, T. Naka, Y. K. Huang, H. Luigjes, M. S. Golden, and A. de Visser, Phys. Rev. Lett. 108, 057001 (2012).
  * Mackenzie et al. (1998) A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, Phys. Rev. Lett. 80, 161 (1998).
  * Abrikosov and Gor’kov (1961) A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 12, 1243 (1961).
  * Xu et al. (2019) X. Xu, Z. Kang, T.-R. Chang, H. Lin, G. Bian, Z. Yuan, Z. Qu, J. Zhang, and S. Jia, Phys. Rev. B 99, 104516 (2019).
  * Sasaki et al. (2011) S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Phys. Rev. Lett. 107, 217001 (2011).
  * Michaeli and Fu (2012) K. Michaeli and L. Fu, Phys. Rev. Lett. 109, 187003 (2012).
  * Nagai et al. (2014) Y. Nagai, Y. Ota, and M. Machida, Phys. Rev. B 89, 214506 (2014).
  * Matano et al. (2016) K. Matano, M. Kriener, K. Segawa, Y. Ando, and G.-q. Zheng, Nat Phys 12, 852 (2016).
  * Pan et al. (2016) Y. Pan, A. M. Nikitin, G. K. Araizi, Y. K. Huang, Y. Matsushita, T. Naka, and A. de Visser, Scientific Reports 6, 28632 EP (2016).
  * Smylie et al. (2017) M. P. Smylie, K. Willa, H. Claus, A. Snezhko, I. Martin, W.-K. Kwok, Y. Qiu, Y. S. Hor, E. Bokari, P. Niraula, et al., Phys. Rev. B 96, 115145 (2017).
  * Yonezawa et al. (2017) S. Yonezawa, K. Tajiri, S. Nakata, Y. Nagai, Z. Wang, K. Segawa, Y. Ando, and Y. Maeno, Nat Phys 13, 123 (2017).
  * Le et al. (2020) T. Le, Y. Sun, H.-K. Jin, L. Che, L. Yin, J. Li, G. Pang, C. Xu, L. Zhao, S. Kittaka, et al., Science Bulletin 65, 1349 (2020).
  * Shi et al. (2003) Z. X. Shi, M. Tokunaga, T. Tamegai, Y. Takano, K. Togano, H. Kito, and H. Ihara, Phys. Rev. B 68, 104513 (2003).
  * Zhu et al. (2010) X. D. Zhu, J. C. Lu, Y. P. Sun, L. Pi, Z. Qu, L. S. Ling, Z. R. Yang, and Y. H. Zhang, Journal of Physics: Condensed Matter 22, 505704 (2010).