# A Solvable Model for Discrete Time Crystal Enforced by Nonsymmorphic
Dynamical Symmetry

Zi-Ang Hu  Department of Physics, The University of Hong Kong, Pokfulam Road,
Hong Kong, China  Bo Fu  Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong, China  Xiao Li  Department of Physics, City
University of Hong Kong, Kowloon, Hong Kong, China  City University of Hong
Kong Shenzhen Research Institute, Shenzhen 518057, Guangdong, China  Shun-Qing
Shen<EMAIL_ADDRESS>Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong, China

(August 28, 2024)

###### Abstract

Discrete time crystal is a class of nonequilibrium quantum systems exhibiting
subharmonic responses to external periodic driving. Here we propose a class of
discrete time crystals enforced by nonsymmorphic dynamical symmetry. We start
with a system with nonsymmorphic dynamical symmetry, in which the
instantaneous eigenstates become Möbius twisted, hence doubling the period of
the instantaneous state. The exact solution of the time-dependent Schrödinger
equation shows that the system spontaneously exhibits a period extension
without undergoing quantum superposition states for a series of specific
evolution frequencies or in the limit of long evolution period. Moreover, in
such case the system gains a $\pi$ Berry phase after two periods’ evolution.
Finally, we show that the subharmonic response is stable even when many-body
interactions are introduced, indicating a DTC phase in the thermodynamic
limit.

## Introduction

Recently, the spontaneous breaking of time translation symmetry has attracted
tremendous attention and led to the idea of time crystal [1]. Although it has
been shown that no system can spontaneously break the continuous time
translation symmetry [2], it is possible to break the discrete time
translation symmetry in Floquet quantum many-body systems [3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14]. This new nonequilibrium phase has since been verified
in a series of experiments [15, 16, 17, 18, 19, 20, 21, 22]. Subsequent
studies in nonequilibrium Floquet quantum systems further generalize to
incorporate both spatial and temporal dimensions, giving rise to a wide range
of new phenomena, including the space-time crystals [23, 24, 25, 26, 27]. In
particular, it has been shown that a nonsymmorphic symmetry can enforce energy
band crossings and create a Möbius twist as well as additional topological
properties [28, 29, 30, 31, 32, 33, 27]. However, it remains unclear what will
happen when a system hosts nonsymmorphic dynamical symmetry.

Here we propose a class of discrete time crystals (DTCs) enforced by the
nonsymmorphic dynamical symmetry. We reveal the dynamical behavior of such
systems by presenting an exact solution of the time-dependent evolution of a
two-level system. The instantaneous eigenstates of the system are Möbius
twisted when the nonsymmorphic dynamical symmetry is present. The exact time-
dependent solution shows that the quantum states of the system exhibit
evolution along the twisted instantaneous state so that period of the states
is doubled to that of the system Hamiltonian when the ratio between the width
of the instantaneous energy band to the evolution frequency of the system’s
Hamiltonian equals a set of discrete values, or when the system evolves in the
long-period limit. Besides, we find that the system has a nontrivial topology
with a half-integer winding number and a $\pi$ Berry phase in two cycles of
evolution. Also, we explore the system’s stability in the presence of many-
body interactions and show the robustness of this class of DTCs. The exact
solution provides an explicit evidence to support the spontaneous breaking of
the time translation symmetry in discrete time crystals.

## Model with a Möbius twist in instantaneous state

Figure 1:  Illustration of Möbius twist of the instantaneous eigenstate and
evolution process of DTC. The red and blue lines label different eigenstates
of the symmetry operator $G$. The vertical direction shows the relativity
energy of two states. The two eigenstates will swap after one period of
evolution and go back to themselves after two periods. The green ball
represent the quantum states evolution which will go into a superposition
state of instantaneous states (translucent balls) and go to another
instantaneous state after one period evolution.

The nonsymmorphic dynamical symmetry operation represents a unique category of
symmetry operations that integrates both spatial transformations and non-
trivial time translation operations, which cannot be performed independently.
When nonsymmorphic symmetry is incorporated, the resulting extension of the
symmetry group exhibits topologically non-trivial properties, leading to
intriguing characteristics in the group representations. In the realm of group
representation theory, the instantaneous eigenstate of the system adheres to
the irreducible representation of the symmetry group, while the Hamiltonian
aligns with the induced representation of the nonsymmorphic symmetry. When the
system exhibits nonsymmorphic symmetry, the period of the irreducible
representation does not consistently align with the period of the induced
representation. Consequently, this inconsistency generates a mismatch between
the periods of the instantaneous state and the Hamiltonian.

Specifically, we start with a time-dependent two-level Hamiltonian with the
dynamical glide symmetry [29],

$H_{0}(t)=\frac{1}{2}\hbar\Omega\sin(\omega
t)\sigma_{x}+\hbar\Omega\sin^{2}(\frac{\omega t}{2})\sigma_{y},$ (1)

where $\sigma_{i}$ ($i=x,y,z$) are the Pauli matrices. The Hamiltonian has a
period of $T=\frac{2\pi}{\omega}$. At time $t$, the instantaneous eigenstates
are $\phi_{\chi}(t)=\frac{1}{\sqrt{2}}[\chi e^{-i\omega t/2},1]^{\mathsf{T}}$
($\mathsf{T}$ indicating the transpose), with $\chi=\pm 1$. The corresponding
energy eigenvalues are $E_{\chi}(t)=\chi\hbar\Omega\sin(\omega t/2)$. The
model possesses a nonsymmorphic dynamical symmetry $\tilde{g}$, the square of
which will be a time translation symmetry with one period. The matrix
representation of $\tilde{g}$ in the basis of the two-level system is the
reduced representation of the whole group and has the following form

$G(t)=\begin{pmatrix}0&e^{-i\omega t}\\\ 1&0\end{pmatrix}.$ (2)

As a result, the Hamiltonian commutes with $G(t)$, $[H(t),G(t)]=0$. Because
$G(t)$ is the induced representation, it has the same period $T$ as the
Hamiltonian, $G(t+T)=G(t)$. Furthermore, $G^{2}(t)=e^{-i\omega t}$ is the
irreducible representation of the time translation symmetry with a period $T$.
Consequently, the eigenvalues of $G(t)$ are $g_{\chi}(t)=\chi e^{-i\omega
t/2}$. The corresponding eigenstates are the instantaneous eigenstates of the
system $\phi_{\chi}(t)$. The eigenvalues $g_{\chi}(t)$ are in the irreducible
representations of the extended symmetry group whose period is twice the
system’s period. In addition, it has the swapping property within one period
as $g_{\chi}(t+T)=g_{-\chi}(t)$ and $g_{\chi}(t+2T)=g_{\chi}(t)$. The
corresponding eigenstates $\phi_{\chi}(t)$ share the same properties. The
swapping property of the eigenvalues and eigenstates is a manifestation of the
group monodromy. It directly illustrates the periodic extension by introducing
the nonsymmorphic symmetry. Another consequence of this period mismatch is
that the instantaneous eigenstates of the system automatically become Möbius
twisted. Figure 1 illustrates how the Möbius twist happens. The two colored
lines represent the two eigenstates of the symmetry operator $G(t)$, which are
also the instantaneous eigenstates of the Hamiltonian. After one cycle of
evolution, the two eigenstates will swap. Since the instantaneous eigenstates
of the Hamiltonian still host the discrete time translation symmetry, it is
forced to undergo a symmetry-enforced crossing in the evolution process. The
two eigenstates are Möbius twisted at $t=T$ such that they only come back to
themselves after a two-period evolution, matching the period of the
irreducible representation of the symmetry group.

Besides period doubling in the two-level system with a dynamical glide
symmetry, systems with other nonsymmorphic symmetries can also have other
forms of period extension. For example, a dynamical screw symmetry in multi-
level systems allows the period extension of more than two [34].

## Exact solution and DTC

The exact solution $\Psi(t)$ of the time-dependent Hamiltonian in Eq. (1) can
be obtained by solving the time-dependent Schrödinger equation,
$i\hbar\partial_{t}\Psi(t)=H_{0}(t)\Psi(t)$. In the basis of the instantaneous
eigenstates $\phi_{\chi}(t)$, an ansatz for the time-dependent solution is
$\Psi(t)=\sum_{\chi}c_{\chi}(t)\exp\left[i\frac{\omega
t}{4}-\frac{i}{\hbar}\int_{0}^{t}E_{\chi}(t^{\prime})dt^{\prime}\right]\phi_{\chi}(t),$
where $c_{\chi}(t)$ are two time-dependent coefficients. Substituting the wave
function into the time-dependent Schrödinger equation, we find that
$c_{\chi}(t)$ satisfies

$\partial_{x}^{2}c_{\chi}-i\chi\alpha\sin(2x)\partial_{x}c_{\chi}+c_{\chi}=0,$
(3)

where $x=\frac{\omega t}{4}$ and $\alpha=\frac{8\Omega}{\omega}.$ Note that
$c_{\chi}(t)$ depend on $\chi$, $\alpha$, and $x$. We denote the initial state
as $\Psi(0)=\sum_{\chi}c_{\chi}(0)\phi_{\chi}(0)$ with
$c(x)\equiv(c_{+}(x),c_{-}(x))^{\mathsf{T}}$. The solutions for $c_{\chi}(t)$
are found as follows:

$c(x)=\begin{pmatrix}H_{c}^{+}(\alpha,x)&-i\sin(x)H_{c}^{-}(\alpha,x)\\\
-i\sin(x)H_{c}^{-}(\alpha,x)^{*}&H_{c}^{+}(\alpha,x)^{*}\end{pmatrix}c(0).$
(4)

Here we define

$H_{c}^{\chi}(\alpha,x)\equiv
H_{c}(i\alpha,-\frac{\chi}{2},-\frac{1}{2},-\frac{i\alpha}{2},\frac{1}{8}+\frac{i\alpha}{4};\sin^{2}x),$
(5)

where $H_{c}$ is the confluent Heun function [35, 36], which satisfies the
boundary conditions $H_{c}^{\chi}(\alpha,x=0)=1$ and
$\frac{d}{dx}H_{c}^{\chi}(\alpha,x)\big{|}_{x=0}=0$. The complex conjugate of
Eq. (3) requires that $H_{c}^{\chi}(-\alpha,x)=H_{c}^{\chi}(\alpha,x)^{*}$.
Assuming the initial state is the eigenstate of $\sigma_{x}$ with eigenvalue
$\chi=+1$, i.e., $c_{+}(0)=1$ and $c_{-}(0)=0$, we have
$c_{+}(x)=H_{c}^{+}(\alpha,x)$ and $c_{-}(x)=-i\sin xH_{c}^{-}(-\alpha,x)$.
The wave function normalization further requires that
$|H_{c}^{+}(\alpha,x)|^{2}+\sin^{2}(x)|H_{c}^{-}(\alpha,x)^{2}=1$.

Figure 2:  The remaining probability $\rho(T)$ after one period of evolution
with different value of $\alpha$. The red line plots
$H_{c}^{-}(\alpha,\frac{\pi}{2})$, while the green line plots
$\frac{\pi}{2}J_{0}(\frac{\alpha}{2})$ which is an asymptotic approximation of
the former. Finally, the blue dots are the value of numerical methods.

The solution in Eq. (4) reveals a subharmonic behavior of dynamical evolution
to the $H(t)$ at a series of the specific ratio $\alpha$, i.e.,

$\Psi(t+2T)=-\Psi(t),$ (6)

which is a hallmark feature of a DTC. Although the instantaneous eigenstates
are Möbius twisted and only return to themselves after $2T$, the quench
dynamics starting from a generic initial state usually does not follow the
same behavior. To analyze this question, we introduce the time evolution
operator $U(t)$, which satisfies $\Psi(t)=U(t)\Psi(0)$. By comparing the
initial condition $c_{\chi}(0)=\phi_{\chi}^{\dagger}(0)\Psi(0)$ with the exact
solution in Eq. (4), we can obtain the general expression for the time
evolution operator $U(t)$ [34]. Consider the initial state $\phi_{+}(0)$.
After one period $t=T$ or $x\equiv\frac{\omega T}{4}=\frac{\pi}{2}$, the
probability that the system still stays in the initial state is given by

$\rho(T)=|\phi_{+}^{\dagger}(0)U(T)\phi_{+}(0)|^{2}=|H_{c}^{-}(\alpha_{n},\frac{\pi}{2})|^{2}.$

Specifically, $H_{c}^{-}(\alpha,\frac{\pi}{2})$ decays oscillatorilly with
$\alpha$, and vanishes for several $\alpha=\alpha_{n}$,
$H_{c}^{-}(\alpha_{n},\frac{\pi}{2})=0$ as shown in Fig. 2. According to the
normalization condition, the other Heun function satisfies
$\left|H_{c}^{+}(\alpha_{n},\frac{\pi}{2})\right|=1$. Hence, we can write
$H_{c}^{+}(\alpha_{n},\frac{\pi}{2})\equiv\exp[i\theta_{n}]$. The first ten
$\alpha_{n}$ and $\theta_{n}$ are listed in Table 1. For a large $n$,
$\alpha_{n}\simeq 2n\pi-\frac{\pi}{2}$ and
$\theta_{n}\simeq\frac{\pi}{\alpha_{n}}$. Consequently, we have
$c_{\pm}(\frac{\pi}{2})=\exp[\pm i\theta_{n}]c_{\pm}(0)$, indicating that the
coefficients gain extra phases in one period of evolution. This result shows
that the state does not return to the initial one after one period. In this
case, the time evolution operator has the form of

$\displaystyle U_{n}(T)$
$\displaystyle=-i\sigma_{z}e^{-i\sigma_{x}(\frac{\alpha_{n}}{2}-\theta_{n})}$
(7)

for specific values of $\alpha=\alpha_{n}$. For $t=2T$, it follows that
$U_{n}(2T)=-1$. Thus, after a two-period evolution, the quantum state will
return to its initial state, and gains a $\pi$ Berry phase as shown in Eq.
(6). This manifests the presence of the DTC.

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
$\alpha_{n}$ | 4.21 | 10.73 | 17.11 | 23.44 | 29.75 | 36.07 | 42.36
$\theta_{n}$ | 0.391 | 0.199 | 0.138 | 0.108 | 0.0889 | 0.0756 | 0.0664
$n$ | 8 | 9 | 10 | 11 | 12 | 13 | 14
$\alpha_{n}$ | 48.66 | 54.95 | 61.24 | 67.46 | 73.76 | 80.07 | 86.34
$\theta_{n}$ | 0.0592 | 0.0535 | 0.0483 | 0.0459 | 0.0424 | 0.0396 | 0.0371
Table 1:  The numerical value of first ten $\alpha_{n}$ of the Heun function
and the phase factor $\theta_{n}.$ With $n\to\infty$, they have asymptotic
behavior as $\alpha_{n}\simeq 2\pi n-\frac{\pi}{2}$ and
$\theta_{n}\simeq\frac{\pi}{\alpha_{n}}.$

When the period is long ($\alpha\gg 1$), we can approximate the confluent Huen
function by the zeroth Bessel function, $H_{c}^{-}(\alpha,\frac{\pi}{2})\simeq
e^{i\frac{\alpha}{2}}\frac{\pi}{2}J_{0}(\frac{\alpha}{2})$, as shown in Fig.
2. The time evolution operator then has the form,

$U(T)\simeq\left[-i\sigma_{z}\sqrt{1-\frac{\pi^{2}}{4}J_{0}^{2}(\frac{\alpha}{2})}+e^{i\frac{\alpha}{2}}\frac{\pi}{2}J_{0}(\frac{\alpha}{2})\right]e^{-i\sigma_{x}(\frac{\alpha}{2}-\frac{\pi}{\alpha})}.$
(8)

If the initial state is one of the eigenstates of $\sigma_{x}$, say
$\phi_{+}(0)$, the state will mainly evolve into the other eigenstate
$\phi_{-}(0)$, and the probability to stay in the state $\phi_{+}(0)$ is just
$\rho(T)$. Interestingly, the energy crossing here invalidates the adiabatic
theorem. Furthermore, after a two-period evolution, we have $U(2T)\simeq-1$.
In this limit, the system is still approximately a DTC even if $\alpha$ is not
equal to $\alpha_{n}$.

To illustrate the periodicity of the evolution of the quantum state, we
project the state $\Psi(t=nT)\equiv U(nT)\Psi(0)$ onto the two basis states
$\phi_{\chi}(0)$, where $n$ is a positive integer. We denote the two resulting
projection parameters as $a_{\pm}(n)=|\langle\phi_{\pm}(0)|\Psi(nT)\rangle|$,
which satisfies $a_{+}^{2}(n)+a_{-}^{2}(n)=1$ because of the wave function
normalization. Consequently, the points $(a_{+}(n),a_{-}(n))$ should all
locate on a quarter circle with a unit radius. The results for different
values of $\alpha$ and the initial states are plotted in Fig. 3. For general
$\alpha$ and initial states, the time evolution of the state does not exhibit
any periodicity since all quantum states will be visited as $n$ increases. The
points $(a_{+}(n),a_{-}(n))$ are expected to fill the entire quarter circle,
indicating ergodicity for an arbitrary initial state as shown in Fig. 3(a). We
also find that there are limitations to the ergodicity because a portion of
states near the axis will not be visited for some specific initial states and
the value of $\alpha$ after a large number $n$ periods of evolution, as shown
in Fig. 3(b). However, for $\alpha=\alpha_{n}$, we note that $\Psi(t=nT)$ just
have two states: $\Psi(t=nT)=\pm\Psi(0)$ for an even $n$, and $\Psi(t=nT)=\pm
U(T)\Psi(0)$ for an odd $n$. The state $\Psi(t=nT)$ just bounces back and
forth between $\Psi(0)$ and $U(T)\Psi(0)$, as shown in Fig. 3(c). The period
of the state becomes $2T$, indicating the formation of a DTC. If the initial
state happens to be one of the eigenstates of $U(T)$, we have
$U(T)\Psi(0)\varpropto\Psi(0)$. The evolution period is just the same as that
of the Hamiltonian. In this case, the time translation symmetry is respected.
For $\alpha$ slightly deviating from the $\alpha_{n}$ or $\alpha\gg 1$, as
shown in Fig. 3(d), the system approximately keeps the two-period oscillation
for a relatively long time.

H

Figure 3:  The stroboscopic projection values in the first 300 periods. (a)
The projections values at $\alpha=8$ (not one of $\alpha_{n}$) for the initial
state $(1,0)^{\mathsf{T}}$, indicating no sign of periodicity in the time
evolution. (b) The projection values at $\alpha=8$ for
$(0.6,0.8)^{\mathsf{T}}$. (c) The projection values at
$\alpha=\alpha_{3}=17.11$. The different color paired points indicate for the
different initial states, $(1,0)^{\mathsf{T}}$ (red),$(0.6,0.8)^{\mathsf{T}}$
(purple), $(0.90,0.436)^{T}$(blue), and
$(\sqrt{2}/2,\sqrt{2}/2)^{\mathsf{T}}$. (d) The projection values at
$\alpha=80$ (large, but not one of $\alpha_{n}$). The initial state:
$(1,0)^{\mathsf{T}}$(blue), and $[0.6,0.8]^{\mathsf{T}}$(pink).

## Stability of the DTC

Upon deviating from the $\alpha_{n}$, the quantum states no longer return to
their initial state after two cycles of evolution, causing the off-diagonal
term of $\sigma_{x}$ to dominate and inevitably leading to the dynamical chaos
of relevant observables. As illustrated in Fig. 4(a), we depict
$\langle\sigma_{x}(t)\rangle=\langle\Psi(t)|\sigma_{x}|\Psi(t)\rangle$ for the
noninteracting model when $\alpha$ diverges from $\alpha_{n}.$ Notably, the
off-diagonal term of $\sigma_{x}$ in the instantaneous eigenstate basis can be
represented as

$\langle\phi_{+1}|\sigma_{x}|\phi_{-1}(t)\rangle=e^{-i\alpha\sin^{2}\frac{\omega
t}{4}}\sin\frac{\omega t}{2}.$ (9)

With a large $\alpha\gg 1$, the dynamical phase factor
$e^{-i\alpha\sin^{2}\frac{\omega t}{4}}$ prompts rapid oscillation in the
observable $\langle\sigma_{x}(t)\rangle$. Furthermore, multiple peaks emerge
in the Fourier spectrum of $\langle\sigma_{x}(t)\rangle$. Thus no time
translation symmetry is respected in this case.

We demonstrate that incorporating many-body interactions can stabilize the
subharmonic response, thereby achieving a prethermal DTC. This prethermal DTC
is expected to endure in the thermodynamic limit and exhibit robustness
against perturbations and imperfect single-spin driving fields. Specifically,
we consider a spin chain by comprising multiple identical Floquet two-level
spin system with uniform nearest-neighbor Ising couplings. The Hamiltonian can
be expressed as

$H(t)=H_{0}(t)+J\sum_{i}\sigma_{\mu}^{i}\sigma_{\mu}^{i+1},$ (10)

with $H_{0}(t)$ given by Eq. (1). Here $J$ represents the Ising interaction
between nearest-neighbor sites, $\sigma_{\mu}^{i}$ denotes the spin operator
of the $i$-th site, and $\mu$ takes the values of $x,y,$or $z$. We assume that
the interaction strength $J$ is considerably smaller than the driving term and
system’s frequencies, such that $|J|\ll\hbar\omega$. This interaction induces
decoherence and thermalization of the Floquet system [37, 38]. If the system
is initialized in one of the instantaneous eigenstates of $H_{0}(t)$,
decoherence among subsystems suppresses the rapid oscillation caused by off-
diagonal terms, resulting in a smoother oscillation pattern. To demonstrate
the interaction’s impact, we assess the average value of $\sigma_{x}^{i}$ for
all sites, i.e., $\langle
m_{x}(t)\rangle=\frac{1}{N}\sum_{i}\langle\sigma_{x}^{i}(t)\rangle,$ and
display the oscillation of $\langle m_{x}(t)\rangle$ over the initial $40$
periods. Considering Ising interactionsin the $\mu=x$ direction, we find that
the model exhibits a persistent oscillation with a $2T$ period, even when
$\alpha$ deviates from $\alpha_{n}$. The DTC reinforced by nonsymmorphic
symmetry is resilient to various initial states and site-resolved measurements
[34]. The oscillation’s Fourier spectrum also presents an isolated peak at
$\frac{\omega}{2}$ in Fig. 4(b). This substantiates explicitly the existence
of the DTC phase in the presence of the interaction. To further verify that
the system constitutes a prethermal DTC, we carry out a finite-size scaling of
$\langle m_{x}(t)\rangle$ at stroboscopic time $t=nT$ ($n$ is an integer),
defined as $\langle m_{x}(nT)\rangle=(-1)^{n}Z(n)$, with the initial state
polarized in the x-direction and present the results for different systems’
size in Fig. 4(c). The calculated $Z(n)$ show that it drops to $0.8\sim 0.9$
and maintains for a while before starting to drop drastically to around zero,
which allows us to define the lifetime $\tau=n_{c}T$ for the DTC with
different parameters. The inset of Fig. 4(c) illustrates that the lifetime
$\tau$ grows exponentially with the system size $L$ , with $\tau\sim e^{bL}$,
where $b$ is a phenomenological parameter that depends on the system’s
parameters. This characteristic is a key feature of a prethermal DTC with
spontaneously broken symmetry [3]. Remarkably, the interaction preserves the
DTC phase even when the symmetry is slightly broken by a small perturbation
term. For comparison, we also present the lifetime and finite-size scaling for
$\alpha=\alpha_{13}$ in Fig. 4(d). It is noted that the stability of $\langle
m_{x}(t)\rangle$ and the lifetime of the DTC are enhanced drastically. Thus
the interacting multiple Floquet spin system is a prethermal DTC for
$\alpha=\alpha_{n}$ and even for $\alpha$ deviating slightly from $\alpha_{n}$
in the case of interaction.

Figure 4:  (a) $\langle\sigma_{x}(t)\rangle$ and its Fourier spectrum for the
non-interacting system for the first 40 periods. (b) $\langle m_{x}(t)\rangle$
and its Fourier spectrum for the interacting DTC with an initial state
polarized in the $x$-direction with length $L=10$. (c) $Z(n)$ at the
stroboscopic time in systems with varying sizes. The inset shows the lifetime
as a function of the size $L$ of the system, $\tau\propto\exp(bL),$ with
$b=1.56$ for $\alpha=81.60$ (slightly deviated from $\alpha_{13}=80.07$). (d)
$Z(n)$ at the stroboscopic time for $\alpha=\alpha_{13}$ with corresponding
$b=2.49$. Here we set the interaction $J=0.2\hbar\omega$.

## Discussion

In the previous section, we demonstrated that nonsymmorphic symmetry enforces
the instantaneous eigenstate into a Möbius twist, resulting in the period-
doubled evolution. This period extension is attributed to the realization of a
DTC. The Hamiltonian in Eq. (1) possesses an additional chiral symmetry,
defined as $\Gamma=\sigma_{z},$ such that $\\{\Gamma,H_{0}\\}=0$. The
topological property of the system can be characterized by the winding number,
defined as
$\nu=\int\frac{dk}{2\pi}(h_{y}\partial_{t}h_{x}-h_{x}\partial_{t}h_{y})$, if
we decompose the Hamiltonian into Pauli matrices as $H_{0}=\bm{h}\cdot\sigma.$
When the instantaneous energy bands exhibit a single crossing, the winding
number becomes a half-integer [29]. This unique winding number is intimately
connected to the system’s period extension.

In contrast to gapped systems where the adiabatic theorem applies and the
quantum state will always stay in instantaneous eigenstate in evolution
process, the quantum state of the DTC with band crossing returns to its
initial state only after two periods, acquiring an extra $\pi$ Berry phase
when the frequency equals to certain values or in the long-period limit.
Experimentally, quantum simulation of our DTC proposal can be performed using
qubits in a quantum computer, such as quantum superconducting circuits and
nuclear magnetic resonance systems [39, 40, 41, 22]. These systems enable spin
manipulation for arbitrary Hamiltonians. The period extension can be directly
observed by monitoring corresponding observable quantities like spin
polarization. Additionally, the unique Berry phase in our model can also be
measured in these systems. By selecting an initial state polarized in the
$x$-direction, the $\pi$ Berry phase can be measured from the $2T$ periodic
oscillation of $\langle\sigma_{x}\rangle$ when $\alpha=\alpha_{n}$ or in the
long-period limit.

## Acknowledgements

The authors would like to thank Dr. Chenjie Wang and Dr. Huanwen Wang for
useful discussions. This work was supported by the Research Grants Council,
University Grants Committee, Hong Kong under Grant Nos. C7012-21GF, 17301220,
21304720, and 11300421.

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# Superconducting fluctuations in a thin NbN film probed by the Hall effect

Daniel Destraz<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Konstantin Ilin
Institut für Mikro- und Nanoelektronische Systeme (IMS), Karlsruher Institut
für Technologie, Germany  Michael Siegel  Institut für Mikro- und
Nanoelektronische Systeme (IMS), Karlsruher Institut für Technologie, Germany
Andreas Schilling  Physik-Institut, Universität Zürich, Winterthurerstrasse
190, CH-8057 Zürich, Switzerland  Johan Chang  Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

###### Abstract

We present a comprehensive study of how superconducting fluctuations in the
normal state contribute to the conductivity tensor in a thin
($119\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) film of NbN. It is shown how
these fluctuations drive a sign change in the Hall coefficient
$R_{\mathrm{H}}$ for low magnetic fields near the superconducting transition.
The scaling behaviours as a function of distance to the transition
$\epsilon=\ln(T/T_{\mathrm{c}})$ of the longitudinal ($\sigma_{\mathrm{xx}}$)
and transverse ($\sigma_{\mathrm{xy}}$) conductivity is found to be consistent
with Gaussian fluctuation theory. Moreover, excellent quantitative agreement
between theory and experiment is obtained without any adjustable parameters.
Our experimental results thus provide a case study of the conductivity tensor
originating from short-lived Cooper pairs.

##  I Introduction

Superconducting fluctuations in the normal state have long been known to
influence bulk properties such as conductivity and magnetization. Generally, a
stronger response to fluctuations is expected for lower dimensions Ussishkin
_et al._ (2002). Many studies have therefore been carried out on thin films or
layered compounds such as the high-temperature cuprate superconductors Emery
and Kivelson (1995). Of particular interest are systems that host a
Bardeen–Cooper–Schrieffer (BCS) regime to Bose–Einstein condensate (BEC)
cross-overs Randeria (2010); Gantmakher and Dolgopolov (2010). The BCS regime
is characterized by conventional Gaussian fluctuations Ussishkin  _et al._
(2002); Michaeli and Finkel’stein (2009); Serbyn  _et al._ (2009) whereas
phase fluctuations are expected in the BEC regime. In the cuprates
paraconductivity, torque magnetization, and Nernst effect experiments – inside
the pseudogap phase – have been interpreted as evidence for phase fluctuations
of superconductivity Xu  _et al._ (2000); Wang  _et al._ (2006); Li  _et al._
(2010); Baity  _et al._ (2016). The topic, however, remains controversial as
the same techniques have also produced results consistent with Gaussian
fluctuation theory Chang  _et al._ (2012); Pourret  _et al._ (2006); Cabo  _et
al._ (2007); Rullier-Albenque  _et al._ (2011); Leridon  _et al._ (2007);
Perfetti  _et al._ (2015); Behnia and Aubin (2016). To make progress, one way
forward is to study superconducting fluctuations of related systems. Recently,
a pseudogap phase has been identified in disordered films of NbN and TiN and
it has been conjectured that it stems from phase fluctuating superconductivity
Sacépé  _et al._ (2010); Mondal  _et al._ (2011a); Chand  _et al._ (2012). In
this context, careful studies of the normal state fluctuations are called for.
Recently, the sister compound TaN, for which no pseudogap has been identified,
has been studied and it was demonstrated that superconducting fluctuations
manifest themselves in the Hall coefficient Breznay  _et al._ (2012); Breznay
and Kapitulnik (2013) – consistent with predictions of Gaussian fluctuation
theory Michaeli  _et al._ (2012).

Due to its promising potential for applications such as single-photon
detection and hot-electron bolometers, NbN is one of the best characterized
superconducting films. Both the normal state metallic and the superconducting
properties have been widely studied Chockalingam  _et al._ (2008); Chand  _et
al._ (2009, 2012); Semenov  _et al._ (2009). Perhaps for this reason, it has
served as a model system for studies of out-of-equilibrium dynamics of
superconductivity Beck  _et al._ (2011); Matsunaga and Shimano (2012); Beck
_et al._ (2013); Matsunaga  _et al._ (2014). Here, we use a NbN film just
outside the pseudogap regime to carry out a combined paraconductivity and Hall
effect study of the normal state superconducting fluctuations. The sign of the
contribution from superconducting fluctuations to the Hall conductivity is
defined by the derivative
$\kappa=-\mathrm{d}\ln(T_{\mathrm{c}})/\mathrm{d}\mu$ where $\mu$ is the
chemical potential Aronov  _et al._ (1995); Michaeli  _et al._ (2012). For
most conventional superconductors, including NbN, $\kappa<0$ and hence the
Hall coefficient due to Gaussian fluctuations is expected to be positive
[$R_{\mathrm{H}}(\mathrm{SC})>0$]. It is also known that charge transport in
NbN films is governed by electron-like carriers Chockalingam  _et al._ (2008),
which generate a negative normal state quasiparticle (QP) Hall coefficient
[$R_{\mathrm{H}}(\mathrm{QP})<0$]. In NbN short lived Cooper pairs and
quasiparticles are thus expected to contribute with opposite sign to the Hall
effect. Near $T_{\mathrm{c}}$, but still within the normal state
($T>T_{\mathrm{c}}$), we indeed find a sign reversal of the Hall effect
response. This sign reversal facilitates the disentanglement of the Hall
signal from quasiparticles and short lived Cooper pairs and hence enables us
to study the response from superconducting (SC) fluctuations to the Hall
effect. Although the Hall conductivity $\Delta\sigma_{\mathrm{xy}}$ generated
by SC fluctuations is generally highly non-linear, it does scale with magnetic
field $B$ in the limit $B\rightarrow 0$. Consistent with Gaussian fluctuation
theory, $\Delta\sigma_{\mathrm{xy}}/B\propto\epsilon^{-2}$ scales with
$\epsilon=\ln(T/T_{\mathrm{c}})$ that for $\epsilon\ll 1$ is a measure of the
distance to the superconducting transition
$(T-T_{\mathrm{c}})/T_{\mathrm{c}}$. Furthermore, from the normal state Hall
conductivity isotherms we extract a ghost critical field
$B^{*}\propto\epsilon$. This combined with a paraconductivity that scales as
$\Delta\sigma_{\mathrm{xx}}\propto\epsilon^{-1}$ makes a convincing case for
Gaussian fluctuations in NbN. Furthermore, excellent quantitative agreement
between Gaussian fluctuation theory and the experiment is found without any
adjustable parameters. Our study therefore provides an experimental
demonstration of how Gaussian fluctuations of superconductivity contribute to
the conductivity tensor.

##  II Methods

A thin film of NbN ($T_{\mathrm{c}}=$ $14.96\text{\,}\mathrm{K}$) was grown on
a sapphire substrate using dc reactive magnetron sputtering of a pure Nb
target in a mixture of Ar and N2 gasses. The average thickness $d=$
$119(2)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was measured with a stylus
profiler. Six gold contacts with Hall bar geometry were deposited onto the
film. Resistivity and Hall effect experiments were carried out – using a
commercial Quantum Design PPMS – in magnetic fields up to 9 Tesla. Magnetic
field and temperature were stabilized before measuring. Reversal of the field
direction was used to eliminate contributions originating from contact
misalignment.

##  III Results

Figure 1:  In-plane resistivity of a
$119\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ thin NbN film as a function of
temperature for magnetic fields in steps of $1\text{\,}\mathrm{T}$. The
magnetic field is applied perpendicular to the film plane. Solid lines are
guides to the eye. The upper critical field $B_{\mathrm{c2}}(T)$ (shown in the
inset) is defined by the point with the steepest slope on the respective
transitions. The red line in the inset is a linear fit used to evaluate
$B_{\mathrm{c2}}(0)$ – see main text for further explanation.

In Fig. 1 the longitudinal resistivity $\rho_{\mathrm{xx}}$ is shown as a
function of temperature and magnetic field perpendicular to the film. The
zero-field resistivity curve yields $T_{\mathrm{c}}=$
$14.96\text{\,}\mathrm{K}$, defined by the temperature with the largest
derivative $\mathrm{d}\rho_{\mathrm{xx}}/\mathrm{d}T$. Notice that the
sharpness of the transition allows determination of $T_{\mathrm{c}}$ with
$20\text{\,}\mathrm{mK}$ precision. When a magnetic field $B$ is applied
perpendicular to the film, the transition temperature is gradually suppressed
as indicated in the inset of Fig. 1.

Raw Hall resistivity ($\rho_{\mathrm{xy}}$) isotherms are shown in Fig. 2a.
Well above the superconducting transition, the negative Hall response
$\rho_{\mathrm{xy}}$ scales linearly with magnetic field strength. Essentially
no magnetoresistance is observed in $\rho_{\mathrm{xx}}$ and
$R_{\mathrm{H}}=\rho_{\mathrm{xy}}/B$. This is consistent with a single band
picture where the Hall coefficient is given by $R_{\mathrm{H}}=-1/(ne)$. Our
film has a carrier density $n=$
$4.2\text{\times}{10}^{23}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ and hence
$k_{\mathrm{F}}=(3\pi^{2}n)^{1/3}$= 2.3 Å-1. The electronic mean free path
$\ell=\hbar k_{\mathrm{F}}/ne^{2}\rho=$ 2.0 Å confirms that our NbN film
belongs to the dirty regime with a Ioffe-Regel parameter
$k_{\mathrm{F}}\ell=4.6$. Being in the dirty regime, we use the Werthamer-
Helfand-Hohenberg relation Werthamer  _et al._ (1966),
$B_{\mathrm{c2}}(0)=-0.69T_{\mathrm{c}}\frac{\mathrm{d}B_{\mathrm{c2}}}{\mathrm{d}T}$,
to estimate the zero-temperature upper critical field
$B_{\mathrm{c2}}(0)\approx$ $18\text{\,}\mathrm{T}$ (see inset of Fig. 1).
This implies a zero-temperature coherence length $\xi_{0}=(\Phi_{0}/[2\pi
B_{\mathrm{c2}}(0)])^{1/2}=$ $43\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
where $\Phi_{0}$ is the flux quantum. As our measurements are taken near the
superconducting transition temperature where $\xi$ and the penetration depth
Mondal  _et al._ (2011b) diverge, the superconducting length scales are
generally larger than the film thickness. Our system thus displays two-
dimensional superconductivity whereas the electrons sense a three-dimensional
environment due to their short mean-free path.

Figure 2: (a) Hall resistivity $\rho_{\mathrm{xy}}$ isotherms for $T\approx
T_{\mathrm{c}}$, 15.3 and $20.0\text{\,}\mathrm{K}$. The inset displays the
high-field linear field dependence of $\rho_{\mathrm{xy}}$. (b) The non-linear
Hall resistance $\Delta\rho_{\mathrm{xy}}$ obtained by subtracting the linear
high-field dependence for each of the respective isotherms. Solid lines are
guides to the eye. Figure 3: Hall conductivity due to superconducting
fluctuations. The subtraction of the normal state response is described in the
text. Isotherms of $\Delta\sigma_{\mathrm{xy}}$ shown in (a) are compared to
Gaussian fluctuation theory (solid lines) explained in the text. The same data
represented as $\nu_{\mathrm{H}}=\Delta\sigma_{\mathrm{xy}}/B$ versus B are
shown in (b). For magnetic fields lower than $B^{*}$ (indicated by arrows),
the isotherms of $\nu_{\mathrm{H}}$ become constant at values
$\nu_{\mathrm{H,0}}$. For the isotherm at $15.0\text{\,}\mathrm{K}$ the
constant plateau is not reached at the lowest measurable fields and hence the
flat line indicates a lower bound. The grey bars below the arrows show the
estimated uncertainty of $B^{*}$ and all solid lines are guides to the eye.
Figure 4: (a) The low-field value
$\nu_{\mathrm{H,0}}=\Delta\sigma_{\mathrm{xy}}/B$ for ($B\rightarrow 0$) (left
axis) and the paraconductivity $\Delta\sigma_{\mathrm{xx}}$ (right axis) as a
function of $\epsilon$. Dashed lines are the predicted dependencies from
Gaussian fluctuation theory without any adjustable parameters – see text for a
detailed explanation. (b) The ghost critical field as a function of
$\epsilon$, obtained from the isotherms shown in Fig 3. The dashed line
corresponds to $B^{*}=B_{\mathrm{c2}}\epsilon/2$ with $B_{\mathrm{c2}}=18$ T
obtained from the resistivity data shown in Fig. 1. The dotted line
corresponds to the mirror image of the red line in the inset of Fig. 1.
Vertical error bars correspond to the grey bars below the arrows in Fig. 3.
Horizontal error bars in (a) and (b) correspond to an uncertainty in
$T_{\mathrm{c}}$ of $20\text{\,}\mathrm{m}\mathrm{K}$.

Hall effect isotherms taken near the superconducting transition
$T_{\mathrm{c}}$, display a sign change from negative to positive values at
low magnetic fields (Fig. 2a). This sign change is observed in a narrow
temperature window of 0.3 K above $T_{\mathrm{c}}$. Deviations from linear
low-field dependence is, however, observed up to $\sim 1$ K above the
superconducting transition. We thus analyse the isotherms in term of a
negative normal state contribution $\rho_{\mathrm{xy}}\propto B$ and a
positive response with a non-linear field dependence. To investigate the
positive response, the negative linear normal state component is subtracted,
i.e.
$\Delta\rho_{\mathrm{xy}}=\rho_{\mathrm{xy}}-\rho_{\mathrm{xy}}^{\mathrm{n}}$.
As shown in Fig. 2b, the positive Hall effect response
$\Delta\rho_{\mathrm{xy}}$ decreases rapidly with increasing temperature. In
fact, it vanishes below the detection limit about 1 Kelvin above
$T_{\mathrm{c}}$.

Next, to compare with theoretical predictions, the contribution from
superconductivity to the conductivity tensor is being isolated. As NbN
displays essentially no magnetoresistance and
$\rho_{\mathrm{xx}}\gg\rho_{\mathrm{xy}}$, the normal state conductivity
$\sigma_{\mathrm{xx}}^{\mathrm{n}}$ is – in the temperature regime of interest
– given by
$\sigma_{\mathrm{xx}}^{\mathrm{n}}=1/\rho_{\mathrm{xx}}($9\text{\,}\mathrm{T}$)$.
The paraconductivity – shown in Fig. 4a – is then given by
$\Delta\sigma_{\mathrm{xx}}=\sigma_{\mathrm{xx}}-\sigma_{\mathrm{xx}}^{\mathrm{n}}$
where $\sigma_{\mathrm{xx}}=1/\rho_{\mathrm{xx}}($0\text{\,}\mathrm{T}$)$. The
Hall conductivity due to superconductivity is extracted in a similar fashion:
$\Delta\sigma_{\mathrm{xy}}=\sigma_{\mathrm{xy}}-\sigma_{\mathrm{xy}}^{\mathrm{n}}$
where
$\sigma_{\mathrm{xy}}^{\mathrm{n}}=-\rho_{\mathrm{xy}}^{\mathrm{n}}/[(\rho_{\mathrm{xx}}^{\mathrm{n}})^{2}+(\rho_{\mathrm{xy}}^{\mathrm{n}})^{2}]$
and
$\sigma_{\mathrm{xy}}=-\rho_{\mathrm{xy}}/[\rho_{\mathrm{xx}}^{2}+\rho_{\mathrm{xy}}^{2}]$.
In Fig. 3 isotherms of $-\Delta\sigma_{\mathrm{xy}}$ and
$\nu_{\mathrm{H}}=\Delta\sigma_{\mathrm{xy}}/B$ for temperatures just above
the superconducting transition $T_{\mathrm{c}}$ are shown. In the limit
$B\rightarrow 0$, $\nu_{\mathrm{H}}(B)$ saturates at $\nu_{\mathrm{H,0}}$ and
becomes essentially independent of magnetic field. The amplitude of the
plateau ($\nu_{\mathrm{H,0}}$) is strongly temperature dependent. As shown in
Fig. 4a, $\nu_{\mathrm{H,0}}$ drops almost two orders of magnitude by heating
just half a Kelvin above the superconducting transition. The onset of this
low-field plateau defines a field scale $B^{*}$ that scales with
$\epsilon=\ln(T/T_{\mathrm{c}})$ (Fig. 4b).

##  IV Discussion

We now discuss $\Delta\sigma_{\mathrm{xx}}$, $\Delta\sigma_{\mathrm{xy}}$, and
the ghost critical field $B^{*}$. Generally, the paraconductivity
$\Delta\sigma_{\mathrm{xx}}$ scales with the correlation length $\xi^{2}(T)$
that diverges as $T\rightarrow T_{\mathrm{c}}$. Gaussian fluctuations lead to
a power-law divergence of the correlation length
$\xi(T)\propto\epsilon^{-1/2}$. By contrast, phase fluctuations are expected
to have an exponentially diverging correlation length $\xi(T)$. To bridge the
two regimes, Halperin and Nelson Halperin and Nelson (1979); Mondal  _et al._
(2011b) proposed a phenomenological function
$\Delta\sigma_{\mathrm{xx}}\propto\sinh^{2}{(\sqrt{b\tau_{\mathrm{c}}/\tau})}$
– where $b$ is a constant,
$\tau_{\mathrm{c}}=(T_{\mathrm{c}}-T_{\mathrm{KT}})/T_{\mathrm{KT}}$, and
$\tau=(T-T_{\mathrm{KT}})/T_{\mathrm{KT}}$. Phase fluctuating
superconductivity would exist in between $T_{\mathrm{KT}}$ – the Kosterlitz-
Thouless transition temperature – and $T_{\mathrm{c}}$. For
$b\tau_{\mathrm{c}}/\tau\gg 1$, an exponential divergence
$\Delta\sigma_{\mathrm{xx}}\propto\exp(2\sqrt{b\tau_{\mathrm{c}}/\tau})$ is
dominant whereas for $b\tau_{\mathrm{c}}/\tau\ll 1$ the Gaussian power-law
dependence is recovered. It has been demonstrated theoretically that the
dimensionless constant $b$ scales with $\sqrt{\tau_{\mathrm{c}}}$ Benfatto
_et al._ (2009). As the superconducting transition of our film is very sharp,
it is clear that $\tau_{\mathrm{c}}$ and hence $b$ are vanishing small. The
Kosterlitz-Thouless regime is thus not relevant for our system and we
therefore discuss the superconducting fluctuations within the Gaussian
paradigm.

A central prediction is that the normal state superconducting fluctuations
should display a characteristic field scale $B^{*}$ that in essence marks the
cross-over between coherence length $\xi(T)=\xi/\sqrt{\epsilon}$ and the
magnetic length scale $\ell_{B}=\sqrt{\hbar/\alpha eB}$. Different numerical
values of $\alpha=1,2$, and 4 have previously been used Breznay  _et al._
(2012); Breznay and Kapitulnik (2013); Pourret  _et al._ (2007, 2006, 2009).
The cross-over condition ($\xi(T)=\ell_{B}$) implies that
$B^{*}=2B_{\mathrm{c2}}(0)\epsilon/\alpha$ and since $B^{*}$ scales with
$B_{\mathrm{c2}}(0)$, it is often referred to as the ghost critical field
scale. For Nernst effect isotherms ($N$ versus $B$), this field scale is
commonly defined by a maximum in $N$ Tafti  _et al._ (2014); Pourret  _et al._
(2006); Chang  _et al._ (2012). However, this maximum is not expected to
strictly vanish for $T\rightarrow T_{\mathrm{c}}$ and indeed Nernst effect
experiments typically find a saturation of $B^{*}$ near $T_{\mathrm{c}}$ Chang
_et al._ (2012); Yamashita  _et al._ (2015). For this reason, it makes sense
to define the ghost critical field $B^{*}$ as the field scale below which
$\nu=N/B$ or $\nu_{\mathrm{H}}=\Delta\sigma_{\mathrm{xy}}/B$ is constant (see
Fig. 3). Within error bars the extracted ghost critical field scales with
$\epsilon$ as shown in Fig. 4b. Moreover, using $B_{\mathrm{c2}}(0)\approx$
$18\text{\,}\mathrm{T}$ (derived from the resistivity curves – shown in Fig.
1) good agreement with the experiment is obtained with $\alpha=4$. This
implies, as has previously been observed in TaN and cuprate superconductors
Tafti  _et al._ (2014); Breznay  _et al._ (2012); Breznay and Kapitulnik
(2013), that the ghost critical field $B^{*}$ is generally lower than the
upper critical field $B_{\mathrm{c2}}(0)$.

Gaussian / amplitude fluctuations of superconductivity constitute short-lived
Cooper pairs that open a new channel for charge transport. In two dimensions,
this yields the following contributions to the conductivity tensor Tikhonov
_et al._ (2012); Michaeli  _et al._ (2012):

$\Delta\sigma_{\mathrm{xx}}^{(1)}=\frac{e^{2}}{16\hbar
d}\frac{1}{\epsilon}\quad\textrm{and}\quad\frac{\Delta\sigma_{\mathrm{xy}}^{(1)}}{B}=\frac{|e|D\kappa}{3}\frac{e^{2}}{16\hbar
d}\frac{1}{\epsilon^{2}}$ (1)

where $\Delta\sigma_{\mathrm{xx}}$ (the Aslamazov-Larkin (AL) term Aslamazov
and Larkin (1968)) is independent of material properties and
$\Delta\sigma_{\mathrm{xy}}$ depends only on the diffusion constant $D$ and
$\kappa$. The AL-term is expected valid in the regime $Gi<\epsilon\ll 1$
Larkin and Varlamov (2005), where $Gi=(e^{2}\rho^{n}_{\mathrm{xx}})/(16\hbar
d)\approx 0.001$ is the Ginzburg-Levanyuk parameter here defined by the
condition $\Delta\sigma_{\mathrm{xx}}^{(1)}=\sigma^{n}_{xx}$ Larkin and
Varlamov (2005). Scattering of electrons on the fluctuating superconductivity
is described by the so-called Maki-Thompson terms Maki (1968); Thompson
(1970); Lang  _et al._ (1994):

$\Delta\sigma_{\mathrm{xx}}^{(2)}=\frac{2\epsilon\Delta\sigma_{\mathrm{xx}}^{(1)}}{\epsilon-\delta}\ln\left(\frac{\epsilon}{\delta}\right)\quad\textrm{and}\quad\frac{\Delta\sigma_{\mathrm{xy}}^{(2)}}{B}=-2\mu_{\mathrm{H}}\Delta\sigma_{\mathrm{xx}}^{(2)}$
(2)

where $\delta=\pi\tau_{0}/(8\tau_{\phi})$, $\tau_{0}=\hbar/k_{\mathrm{B}}T$,
$\tau_{\phi}$ is the dephasing time Reizer (1992), and $\mu_{\mathrm{H}}$ is
the electron mobility. We have no direct experimental measure of the dephasing
time $\tau_{\phi}$ in NbN. For the optimally doped high-temperature
superconductor YBCO, $\tau_{\phi}=$ $86\text{\,}\mathrm{fs}$ has previously
been reported Lang  _et al._ (1994). Assuming a similar or shorter dephasing
time for NbN, implies that
$\Delta\sigma_{\mathrm{xx}}^{(1)}\gg\Delta\sigma_{\mathrm{xx}}^{(2)}$ and
$\Delta\sigma_{\mathrm{xy}}^{(1)}\gg\Delta\sigma_{\mathrm{xy}}^{(2)}$. The
absence of a $\ln(\epsilon/\delta)$ dependence of $\Delta\sigma_{\mathrm{xx}}$
(Fig. 4a) lends experimental support for this ansatz. Finally, density-of-
state (DOS) corrections are predicted to be subleading contributions Tikhonov
_et al._ (2012). We therefore discuss the experimentally observed
paraconductivity by setting $\Delta\sigma_{i}=\Delta\sigma_{i}^{(1)}$ with
$i\in\\{\mathrm{xx},\mathrm{xy}\\}$.

The exact experimental $T$-dependence of $\Delta\sigma_{i}$ in the limit
$\epsilon\rightarrow 0$ is sensitive to how $T_{\mathrm{c}}$ is defined
Leridon  _et al._ (2007); Breznay  _et al._ (2012). When defining
$T_{\mathrm{c}}=$ $14.96\text{\,}\mathrm{K}$ by the steepest point of the
superconducting transition, we find that $\Delta\sigma_{\mathrm{xx}}$ scales
perfectly with $\epsilon^{-1}$ and $-\nu_{\mathrm{H,0}}$ is consistent with a
$\epsilon^{-2}$ dependence (Fig. 4a). Gaussian fluctuation theory thus
provides an excellent description of the temperature dependence of
$\Delta\sigma_{\mathrm{xx}}$, $\Delta\sigma_{\mathrm{xy}}$ and $B^{*}$ without
any adjustment of $T_{\mathrm{c}}$.

The absolute magnitude of $\Delta\sigma_{\mathrm{xx}}$ and
$\Delta\sigma_{\mathrm{xy}}$ is also well described by Gaussian fluctuation
theory. Using the film thickness $d=119$ Å, the predicted magnitude of
$\Delta\sigma_{\mathrm{xx}}$ is within 20$\%$ in agreement with the experiment
(Fig. 4a). The amplitude of $\Delta\sigma_{\mathrm{xy}}$ can be evaluated by
estimating the product $|e|D\kappa$. Notice that
$\kappa=\beta/\epsilon_{\mathrm{F}}$ where $\epsilon_{\mathrm{F}}$ is the
Fermi energy and
$\beta=-0.5\lambda\cdot\mathrm{d}\ln(T_{\mathrm{c}})/\mathrm{d}\lambda$ is
depending on the superconducting coupling constant $\lambda$ Michaeli  _et
al._ (2012). In the weak coupling limit ($\lambda\ll 1$), where
$T_{\mathrm{c}}\propto\exp(-1/\lambda)$ Bardeen  _et al._ (1957) one finds
$\beta=-1/(2\lambda)$. NbN, however, belongs to the strongly coupled
$\lambda\sim 1$ limit Chockalingam  _et al._ (2008); Beck  _et al._ (2011)
where
$T_{\mathrm{c}}\propto\exp([-1.04(1+\lambda)]/[\lambda-\mu^{*}(1+0.62\lambda)])$
with $\mu^{*}=0.13$ being the screening potential McMillan (1968);
Chockalingam  _et al._ (2008). A recent experimental study found
$\lambda=1.1\pm 0.1$ Beck  _et al._ (2011) implying that $\beta=-0.77$.
Finally, using $D=v_{\mathrm{F}}^{2}\tau/3$ where $v_{\mathrm{F}}$ is the
Fermi velocity and $\tau$ is the mean free life time, it is found that
$|e|D\kappa=2\mu_{\mathrm{H}}\beta/3$ where $\mu_{\mathrm{H}}=|e|\tau/m^{*}$
is the Hall mobility and $m^{*}$ is the quasiparticle mass. The Hall mobility
can be derived directly from the experiment
$\mu_{\mathrm{H}}=|R_{\mathrm{H}}/\rho_{\mathrm{xx}}^{\mathrm{n}}|=$$1.3\text{\times}{10}^{-5}\text{\,}\mathrm{T}^{-1}$.
We find, without any adjustable parameters, that the theoretical prediction of
$\Delta\sigma_{\mathrm{xy}}$ (solid line in Fig. 4a) is in excellent agreement
with the experiment.

Finally, it is also possible – as done in Ref. Breznay  _et al._ , 2012 – to
analyze a more extended region of magnetic fields by fitting the isotherms.
Gaussian theory predicts the isotherms to be described by

$\Delta\sigma_{\mathrm{xy}}^{(1)}(\omega\rightarrow
0)=\frac{e^{2}k_{\mathrm{B}}T\kappa}{\pi\hbar
d}\mathrm{sgn}(B)\sum_{j=0}^{\infty}\frac{(j+1)(\zeta_{j+1}-\zeta_{j})^{3}}{\zeta_{j}\zeta_{j+1}(\zeta_{j+1}+\zeta_{j})^{2}}$
(3)

where

$\zeta_{j}=\epsilon+\frac{\kappa\omega}{2}+\Psi\left(\frac{1}{2}+\frac{(j+1/2)4D|eB|-i\omega}{4\pi
k_{\mathrm{B}}T}\right)-\Psi\left(\frac{1}{2}\right)$ (4)

and $\Psi$ is the digamma function. By using $\kappa$ and $D$ as fit
parameters it is possible, as shown in Fig. 3a, to describe reasonably well
the isotherms of $\sigma_{\mathrm{xy}}$. The obtained values of $\kappa$ and
$D$ are in good agreement with the $B\rightarrow 0$ analysis presented in Fig.
4a. As a concluding remark, we stress the advantage of the $B\rightarrow 0$
analysis. As the product $\kappa D$ is proportional to the Hall mobility, the
Gaussian theory can be tested without knowing or fitting $\kappa$ and $D$.

##  V Conclusions

In summary, we presented a systematic study of the conductivity response
generated by superconducting fluctuations in the normal state of a
$119\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ thin NbN film. It is shown how
these fluctuations drive a sign change in the Hall coefficient. Isolating the
longitudinal $\Delta\sigma_{\mathrm{xx}}$ and transverse
$\Delta\sigma_{\mathrm{xy}}$ conductivity due to superconducting fluctuations
allowed direct comparison to Gaussian fluctuation theory. We found that these
transport quantities are scaling with the distance to the superconducting
transition $\epsilon=\ln(T/T_{\mathrm{c}})$ as predicted
($\Delta\sigma_{\mathrm{xx}}\propto\epsilon^{-1}$ and
$\Delta\sigma_{\mathrm{xy}}\propto\epsilon^{-2}$). Furthermore, excellent
quantitative agreement between Gaussian fluctuation theory and the experiment
was obtained. The presented study thus demonstrates experimentally how
Gaussian fluctuations of superconductivity contribute to the conductivity
tensor.

###### Acknowledgements.

The authors thank X. Zhang and K. Michaeli for useful discussions. D. D. and
J. C. acknowledge support by the Swiss National Science Foundation under grant
PP00P2 _150573.

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