σ$\sigma$ θ$\theta$ π$\pi$

# Topologically enabled superconductivity

Michael A. Rampp  Institute for Theory of Condensed Matter, Karlsruhe
Institute of Technology, 76131 Karlsruhe, Germany  Elio J. König  Max Planck
Institute for Solid State Research, D-70569 Stuttgart, Germany  Jörg Schmalian
Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology,
76131 Karlsruhe, Germany  Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

###### Abstract

Majorana zero modes are a much sought-after consequence of one-dimensional
topological superconductivity. Here we show that, in turn, zero modes
accompanying dynamical instanton events strongly enhance - in some cases even
enable - superconductivity. We find that the dynamics of a one-dimensional
topological triplet superconductor is governed by a $\theta$-term in the
action. For isotropic triplets, this term enables algebraic charge-$2e$
superconductivity, which is destroyed by fluctuations in non-topological
superconductors. For anisotropic triplets, zero modes suppress quantum phase
slips and stabilize superconductivity over a large region of the phase
diagram. We present predictions of correlation functions and thermodynamics
for states of topologically enhanced superconductivity.

††preprint: APS/123-QED

One-dimensional topological $p$-wave superconductors are widely proposed as
building blocks for quantum information processing [1, 2, 3, 4, 5, 6, 7, 8].
While proximity-induced superconductivity is discussed most frequently, the
prospect of intrinsic superconductivity in one-dimensional structures would
certainly allow for more versatile architectures. However, despite the obvious
challenge of identifying the right material, there seems to be a more
fundamental limitation to such an approach. In low dimensions the influence of
fluctuations is very strong. The impact of order-parameter fluctuations on
one-dimensional topological superconductivity has been extensively studied for
the spinless $p$-wave case [9, 10, 11, 12] and it has been demonstrated that
vacuum tunneling by $2\pi$ quantum phase slips (QPS) is suppressed in these
systems [10, 13], thus enlarging the superconducting domain in the phase
diagram. However, for isotropic triplet superconductivity the role of
fluctuations in the spin sector is ordinarily so strong as to completely
destroy $2e$ superconductivity [14, 15, 16, 17].

Figure 1:  Ground state phase diagram of the trivial (a), (c) and topological
(b), (d) superconductor. The top (bottom) row depicts the spin-isotropic
(strongly anisotropic) case. The axes correspond to the renormalized, i.e.,
experimentally relevant, charge ($K_{c}$) stiffness, and the bare
($K_{s}^{0}$) and renormalized ($K_{s}$) spin stiffness in the isotropic and
the anisotropic case, respectively. All superconducting states have algebraic,
quasi long-range order. While in the isotropic case only vestigial charge-$4e$
superconductivity is allowed for topologically trivial systems, a
$\theta$-term in the theory enables charge-$2e$ pairing of topological
superconductors, which occurs in a much larger domain of the phase diagram. At
finite spin anisotropy, disorder-inducing quantum phase slips are suppressed
by zero modes, stabilizing superconductivity.

In this paper we show that charge-$2e$ topological triplet superconductivity
in one-dimensional quantum wires becomes possible while it is not allowed for
non-topological systems. The latter can only undergo vestigial charge-$4e$
pairing in a much reduced regime of the phase diagram, see Fig. 1 a) and b).
Zero modes, primarily discussed as static Majorana bound states of topological
superconductivity, emerge in our analysis as dynamical events accompanying
instantons of the order parameter field. They are shown to suppress order-
parameter fluctuations via destructive interference due to a Berry phase,
enabling the charge-$2e$ superconducting state. This Berry phase leads to a
topological term in the field theory, a $\theta$-term [18, 19]. For the
topological angle we find $\theta=\pi$ for topological superconductors and
$\theta=0$ for non-topological ones. We obtain this result using non-Abelian
bosonization and, using a physically more transparent reasoning, by
demonstrating that dynamical zero modes give rise to a complex QPS fugacity.

These findings occur for systems that are isotropic in spin space where we
exploit a connection to Haldane’s conjecture [20] for spin chains. Including
an anisotropy in spin space, disorder-inducing configurations are suppressed
by zero modes that are more akin to what is known for spinless systems [10,
13].

_Model:_ We study a time reversal invariant $p$-wave spin triplet
superconductor in one dimension at zero temperature. The Bogoliubov-de Gennes
Hamiltonian for given space- and time-dependent order-parameter configuration
$\bm{\Delta}\left(x,\tau\right)$ in the Nambu spinor basis
$\Psi=(\psi_{\uparrow},\psi_{\downarrow},\bar{\psi}_{\downarrow},-\bar{\psi}_{\uparrow})^{T}$
is

${\cal
H}\left[\bm{\Delta}\right]=\left(\begin{array}[]{cc}-\frac{\partial_{x}^{2}}{2m}-\mu&\bm{\Delta}\left(x,\tau\right)\cdot\bm{\sigma}\partial_{x}\\\
-\partial_{x}\bm{\Delta}^{\dagger}\left(x,\tau\right)\cdot\bm{\sigma}&\frac{\partial_{x}^{2}}{2m}+\mu\end{array}\right),$
(1)

with $m$ the fermion mass and $\mu$ the chemical potential. The order
parameter takes the form of a real unit vector $\bm{n}$, that describes the
orientation of the Cooper pair spin, times a global phase $\vartheta$, _i.e._
$\bm{\Delta}=\lvert\Delta\rvert e^{i\vartheta}\bm{n}$, with the pairing
strength $\lvert\Delta\rvert$. $\mu<0$ corresponds to a trivial and $\mu>0$ to
a topological superconductor [21]. The order parameter manifold is
$\mathcal{M}=\left(S^{1}\times S^{2}\right)/\mathbb{Z}_{2}$, where the
$\mathbb{Z}_{2}$ quotient stems from the equivalence of field configurations
$(\vartheta,\bm{n})\sim(\vartheta+\pi,-\bm{n})$ [22, 23]. The fermionic
dynamics is then governed by the action

$S_{f}=\frac{1}{2}\int\mathrm{d}\tau\mathrm{d}x\Psi^{\dagger}\left({\cal\partial_{\tau}+{\cal
H}}\right)\Psi.$ (2)

In low-dimensional intrinsic superconductors order-parameter fluctuations are
important and are governed by the non-linear $\sigma$-model (NL$\sigma$M):

$\displaystyle S_{b}$ $\displaystyle=$
$\displaystyle\frac{K_{c}^{0}}{2\pi}\int\mathrm{d}\tau\mathrm{d}x\left(\frac{1}{v_{c}}\left(\partial_{\tau}\vartheta\right)^{2}+v_{c}\left(\partial_{x}\vartheta\right)^{2}\right)$
(3) $\displaystyle+$
$\displaystyle\frac{K_{s}^{0}}{2\pi}\int\mathrm{d}\tau\mathrm{d}x\left(\frac{1}{v_{s}}\left(\partial_{\tau}\bm{n}\right)^{2}+v_{s}\left(\partial_{x}\bm{n}\right)^{2}\right).$

Here $K_{c}^{0}$ and $K_{s}^{0}$ denote the bare stiffnesses of the charge and
spin sector respectively. $v_{c}$ and $v_{s}$ are the respective velocities.
We use $K_{c,s}$ for the renormalized stiffnesses.

This model is a generalization of previous descriptions for spinless fermions
[24, 25, 11].

_Topological field configurations:_ The fields $\vartheta,\bm{n}$ that
describe $U(1)$-phase and spin of a Cooper pair allow for several distinct
QPSs, shown in Fig. 2: we call vortices of $\vartheta$ with winding $\nu_{c}$
in space-time $2\pi\nu_{c}$ charge QPSs. Space-time skyrmions in $\bm{n}$ have
integer winding $Q$ which relies on $\pi_{2}(S^{2})=\mathbb{Z}$. The
evaluation of the partition function contains a sum over all possible
topological sectors, determined by a set of numbers $N=\\{\nu_{c},Q,\cdots\\}$
[19]

$Z=\sum_{N}\int\mathcal{D}\vartheta_{N}\mathcal{D}\bm{n}_{N}Z_{N}^{f}[\vartheta,\bm{n}]e^{-S_{b}[\vartheta,\bm{n}]},$
(4)

where $\int D\vartheta_{N}\mathcal{D}\bm{n}_{N}\dots$ denotes the integral
over smooth bosonic fluctuations on top of the topological field configuration
and $Z_{N}^{f}[\vartheta,\bm{n}]$ is the fermionic partition sum in a given
bosonic background of fixed $N$. Notice, all fluctuations around the trivial
state, including these instantons, must be taken into account as long as their
action is finite. Estimating the core size of QPS we obtain, following Ref.
[26], $r_{\rm QPS}\approx\left(K_{c}v/2\pi E_{\mathrm{cond}}\right)^{1/2}$,
and the core action $S_{\rm QPS}\approx K_{c}/2$.

_Topologically trivial superconductivity:_ To analyze the model it is tempting
to argue that fermions are gapped and should not change the universal behavior
of order-parameter fluctuations. Then, the NL$\sigma$M of Eq. (3) implies
$\langle\bm{\Delta}\rangle=0$ with exponentially decaying correlations, caused
by fluctuations in the spin sector. As last resort the system can still enter
a state of algebraic vestigial order characterized by the composite
$\bm{\Delta}\cdot\bm{\Delta}$ below a Berezinskii-Kosterlitz-Thouless (BKT)
transition [22, 23]. However, this state, where two spin triplets form a
charge-$4e$ spin singlet, can be destroyed by the proliferation of $\pi$, i.e.
fractional QPSs. In our units, the QPS configurations with smallest winding
number $\nu_{c}$ become relevant at a critical stiffness of
$K_{c}=2/\nu_{c}^{2}$. Hence, in order to stabilize $4e$ order with
$\nu_{c}=1/2$ fractional vortices, the charge stiffness has to be four times
larger compared to the usual BKT transition, which implies $K_{c}=8$; see the
left panel of Fig. 1. This behavior, deduced from the two-dimensional
classical model, is indeed correct in the topologically trivial phase. Yet, as
we will see next, it does not apply to topological superconductors. In the
topological phase order-parameter configurations with nontrivial topology in
space-time play a key role[27, 28, 29, 30, 31, 32, 33], requiring us to
carefully distinguish their effects and their interplay with the fermionic
degrees of freedom.

Figure 2: Selected topologically nontrivial order-parameter configurations in
space-time [34]. Left panel: skyrmion of topological charge $Q=1$. Middle and
right panel $Q=\pm\tfrac{1}{2}$ merons, i.e. spin vortices that differ by the
orientation of the spin inside the core.

_Integrating out fermions:_ In order to systematically integrate out fermions
in a one-dimensional system we employ non-Abelian bosonization [35], that
manifestly preserves the symmetry properties of interacting fermion theories
and is expressed in terms of the Wess-Zumino-Novikov-Witten (WZNW) action. We
employ this method to effectively eliminate the massive degrees of freedom
while retaining the effect of zero modes. The degrees of freedom of the WZNW
theory are group valued boson fields defined on the orthogonal group $O(4)$.
The matrix components of this field can be related to fermionic currents by
$g_{ij}=(-i/M)\chi_{\rm{R}}^{i}\chi_{\rm{L}}^{j}$, where
$\chi_{\rm{R}},\chi_{\rm{L}}$ are right- and left-moving Majorana fermions
that can be constructed from the microscopic fermions, and $M$ is a
regularization dependent mass-scale. The degrees of freedom described by the
BdG-Hamiltonian (1) deep in the topological phase are exactly four right-
moving and four left-moving Majorana fermions, which explains the manifold
$O(4)$. Effectively, the field $g$ encodes the fermionic fluctuations and the
opening of a gap corresponds to breaking the chiral symmetry, _i.e._
individual components of $g$ develop a finite expectation value. The action
with coupling constant $\gamma$ reads

$S_{W}=\frac{1}{4\gamma^{2}}\int\mathrm{d}\tau\mathrm{d}x{\rm
tr}\left[\partial_{\mu}g\partial^{\mu}g^{-1}\right]+\Gamma,$ (5)

where the Wess-Zumino term $\Gamma[g]$ is defined by extending the domain of
the group valued field $g\in O(4)$ to a hemisphere of the three-dimensional
unit sphere with $S^{2}$ as boundary

$\Gamma=\frac{\varepsilon^{\mu\nu\rho}}{24\pi
i}\int\mathrm{d}^{3}r\operatorname{tr}\left[g^{-1}(\partial_{\mu}g)g^{-1}(\partial_{\nu}g)g^{-1}(\partial_{\rho}g)\right].$
(6)

The pairing term is bosonized and introduces a $(\vartheta,\bm{n})$ dependent
mass term for $g$. In the adiabatic limit $g$ follows the variation of the
background fields $(\vartheta,\bm{n})$ and thereby traces the equator of
$O(4)$. The details of the subsequent analysis are given in the supplement
[36]. The first term in (5) yields terms identical to the NL$\sigma$M in (3),
i.e. it yields a renormalization of the stiffnesses due to fermions. More
interesting is the second term. When constrained to the equator, the Wess-
Zumino term (6), which measures the solid hyperangle on $O(4)$ normalized to
$2\pi$, may only take values of $\pi\,\text{mod}\,2\pi$. We find that

$\Gamma\rightarrow
S_{\theta}=i\frac{\theta}{4\pi}\int\mathrm{d}\tau\mathrm{d}x\bm{n}\cdot(\partial_{\tau}\bm{n}\times\partial_{x}\bm{n}),$
(7)

where $\theta=\pi$. For a topologically trivial superconductor we obtain
instead $\theta=0$. Eq. (7) has profound implications. According to Haldane’s
conjecture [20] where $\theta=\pi$ occurs in the same action for $\bm{n}$ for
half-integer spins [37], $S_{\theta}$ leads to a critical state described by a
$SU_{1}(2)$ WZNW theory [38, 39], rather than a state with finite correlation
length. Hence, algebraic charge-$2e$ superconductivity becomes possible for
topological superconductors while it is forbidden in topologically trivial
ones.

Before we discuss further implications of this finding, we offer an
alternative and physically more transparent derivation of the $\theta$-term.
We consider nontrivial skyrmion configurations but in this context it is
particularly convenient to introduce a soft easy-plane anisotropy [40]. This
anisotropy yields spin-vortex configurations but allows $\bm{n}$ to escape the
plane in a core region, whose size is determined by the strength of the
anisotropy, avoiding a singularity. There are two kinds of $2\pi$ spin QPSs
distinguished by the orientation of the vector in the core $(\pm n_{z})$,
called meron and antimeron with skyrmion winding number $Q=\pm\tfrac{1}{2}$;
see Fig. 2. The fermionic operator ${\cal K}\equiv\partial_{\tau}+{\cal H}$
acts in space and (imaginary) time. Since the Nambu spinor satisfies the
reality condition $\Psi^{T}=\Psi^{\dagger}C$ (where $C=\sigma^{y}\tau^{y}$
with $\sigma^{i}$ acting on spin and $\tau^{i}$ on Nambu degrees of freedom),
the fermionic partition function
$Z_{N}^{f}[\vartheta,\bm{n}]=\operatorname{Pf}(C\mathcal{K}[\vartheta,\bm{n}])$
follows directly from the Pfaffian of the fermionic kernel. Hence, in the case
that the fermionic kernel possesses a zero eigenvalue mode the fermionic
partition function vanishes. As summarized in [36], integrating out fermions
yields in the limit of large anisotropy that two zero modes are shifted to
finite eigenvalue in such a manner that

$Z_{N}^{f}[\vartheta,\bm{n}]\propto e^{-i\pi Q}.$ (8)

With this complex fugacity, the contribution of $2\pi$ spin QPS vanishes upon
summation over $Q$, the internal degree of freedom of the vortex [40]. Within
a field theoretical language, this behavior is precisely the effect that
follows from a $\theta$-term in the action, $S_{\theta}=i\theta Q$, where
$Q=\int\mathrm{d}\tau\mathrm{d}x\bm{n}\cdot(\partial_{\tau}\bm{n}\times\partial_{x}\bm{n})/(4\pi)$
is the associated topological charge [19]. It reveals that the $\theta$-term
yields destructive interference of disordering spin configurations.

In addition to nontrivial spin textures, we can also analyze charge vortices
or combinations of charge and spin vortices. We find that one can map this
dynamic problem onto an effective Hermitian single-particle Hamiltonian
$H_{\rm eff}$ in two dimensions $(x,\tau)$. $H_{\rm eff}$ can then be reduced
to the two-dimensional Fu-Kane Hamiltonian of a 3D topological insulator
surface state in contact with an s-wave superconductor [41]. Established
results for zero modes due to static vortex configurations of the Fu-Kane
Hamiltonian can now be used to obtain the number and character of zero modes
for ${\cal K}$. Since one coordinate of this two-dimensional problem
corresponds to (Euclidean) time, those are again dynamical instanton events.
The details of this rather powerful but straightforward analogy are summarized
in the supplementary material [36]. It yields for example two zero modes of
${\cal K}$ for a charge $2\pi$ QPS ($\nu_{c}=1$) and one zero mode for a
combined $2\pi$ half QPS ($\nu_{c}=\nu_{s}=\tfrac{1}{2}$), with winding number
of a planar spin vortex $\nu_{s}$. More generally we obtain a zero mode for
each odd $\nu_{c}\pm\nu_{s}$. These dynamical zero modes can be understood as
protected level crossings under the adiabatic variation of a parameter.
Perturbations that do not destroy the level crossing also do not lift the
dynamic zero modes. Hence $Z_{N}^{f}[\vartheta,\bm{n}]=0$ for these single-
defect configurations and the corresponding vortex fugacities vanish. More
importantly, vortex-antivortex pairs do not contribute to a BKT transition as
the exponentially small overlap between modes gives rise to a linear,
confining potential overruling the usual logarithmic interaction. Our mapping
to the Fu-Kane model can also be applied to spinless $p$-wave superconductors
where it agrees with past results on QPSs in this system [10, 13].

_Quantum Phase diagram._ If we combine the $\theta$-term and the presence of
zero modes due to charge vortices, we can determine the phase diagrams shown
in Fig. 1. In the topologically trivial phase $2e$ superconducting order is
destroyed because $\bm{n}$ fields are gapped but $4e$ superconducting order
may exist for $K_{c}>8$. In contrast, in the topological phase the
$\bm{n}$-fields are algebraically ordered and the charge-$2e$ superconductor
persists. A BKT transition driven by $\nu_{c}=2$, i.e. $4\pi$, QPS disorders
the phase sector for $K_{c}<2/\nu_{c}^{2}=1/2$. The resulting state possesses
vestigial order of the composites
$\Delta_{x}^{\dagger}\Delta_{y}+\Delta_{y}^{\dagger}\Delta_{x}$,
$\Delta_{x}^{\dagger}\Delta_{x}-\Delta_{y}^{\dagger}\Delta_{y}$, and
$\Delta_{z}^{\dagger}\Delta_{z}$. It is a charge insulator and a spin-nematic
state. Hence, the critical stiffness for the destruction of superconductivity
is $16$ times lower than in the trivial case; see Fig. 1 b.

_Correlators:_ Let us discuss experimentally and numerically measurable
consequences of the topological phases in terms of bosonic and fermionic
correlators. In topologically trivial superconductors, fermions are gapped and
fermion correlators decay exponentially. This is qualitatively different for
topological superconductors. The power-law behavior of the single-fermion
correlator is again caused by dynamical zero modes. We obtain

$\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle\sim\delta_{\sigma\sigma^{\prime}}{r}^{-\frac{1}{8K_{c}}-\frac{K_{c}}{2}-\frac{1}{2}},$
(9)

where ${r}=\sqrt{x^{2}+v_{c}^{2}\tau^{2}}$ denotes the Euclidean norm of a
point $(x,\tau)$ in space-time. For simplicity we assumed $v_{s}=v_{c}$.
Hence, the topological superconductor possesses gapless charged fermionic
excitations. This is in contrast to nodal excitations which can appear in
higher dimensions and are charge neutral. However, experiments which measure
the phase coherence, such as flux quantization or the a.c. Josephson effect
are expected to detect a condensate charge of $2e$.

Eq. (9) implies a power-law dependence on energy in the tunneling density of
states $\rho(E)$, which can be detected in scanning tunneling microscopy
measurements,

$\rho(E)\sim E^{\frac{1}{8K_{c}}+\frac{K_{c}}{2}-\frac{1}{2}},\quad E>0.$ (10)

Similarly, we expect power-law dependence on temperature of various transport
coefficients, such as the thermal conductivity [42].

Notice that nonetheless the system cannot be described as a Tomonaga-Luttinger
liquid of charge-$e$ fermions, since the behavior of two-particle correlators
differs. Zero modes also nontrivially affect two-particle correlators in the
topological state [36]. Specifically, we find the tunneling density of states
for tunneling a singlet and triplet Cooper pair into the wire

$\displaystyle\rho_{\rm{sg}}(E)$ $\displaystyle\sim$ $\displaystyle
E^{2K_{c}+\frac{1}{2K_{c}}-1},$ $\displaystyle\rho_{\rm{tr}}(E)$
$\displaystyle\sim$ $\displaystyle E^{\frac{1}{2K_{c}}}.$ (11)

The exponent for the singlet pair-correlator differs from the Luttinger liquid
result (extracting the Luttinger parameters from the one-particle correlator)
for general values of $K_{c}$, $\rho_{\rm{sg}}^{\rm{LL}}\sim
E^{1+\frac{1}{2K_{c}}}$. In the $2e$-ordered phase the singlet correlations
are less singular than in a conventional Luttinger liquid. At low energies the
amplitude for tunneling triplet pairs into the system is enhanced compared to
the one for singlet pairs. A detailed derivation is given in [36].

_Easy axis anisotropy:_ We already discussed an additional easy axis
anisotropy $S_{\mathrm{an}}=\lambda\int\mathrm{d}\tau\mathrm{d}xn_{z}^{2}$
when we offered an alternative derivation of the $\theta$-term in Eq. (8). The
fact that topological superconductivity is stabilized over the topologically
trivial case can also be seen at $\lambda>0$. It is however more similar to
what is known from spinless $p$-wave superconductors [10, 13], since the spin
sector is now also governed by a $U(1)$ order parameter and charge-$2e$
superconductivity becomes possible for both topological and non-topological
superconductors. However, at small stiffnesses algebraic order is now
destroyed by the proliferation of three kinds of space-time topological
defects: $2\pi$ charge, spin, or combined vortices; see also Refs. [43, 44,
45, 46]. This yields the four possible phases presented in Fig. 1 (c and d): a
completely ordered (all QPSs expelled), a completely disordered (all QPSs
proliferate), and two vestigial phases (only one kind of QPS proliferates).
Physically the resulting phases can be identified as: a spin nematic
charge-$2e$ superconductor, where the two gapless excitations are spin-$1$
Cooper pairs, a correlated insulator, a charge-$4e$ superconductor with
gapless excitations carrying charge $4e$ and spin $0$, and a spin-nematic
phase similar to the isotropic case. While thermodynamically the phase diagram
of topological and trivial limits are similar, in the former case $2\pi$
charge, spin, or combined QPSs are suppressed by dynamic zero modes. Hence,
the leading transitions are effected only by $4\pi$ QPSs, and as a result we
find again that the superconducting state is stabilized for the topological
phase. Similar to the isotropic case, zero modes lead to gapless fermion and
two-particle excitations in the $2e$ phase. In the vestigial charge-$4e$
phase, single fermions are gapped, but the singlet pair-correlator remains
gapless. There is another distinction between the topological and the trivial
vestigial phases: in the topological case the bosonic operator $\phi_{s}(x)$
($\phi(x)$), which is the dual field to the phase $\vartheta_{s}(x)$
($\vartheta(x)$), develops bona-fide long range $\mathbb{Z}_{2}$ order in the
vestigial charge (spin) phase on the top left (bottom right) of the phase
diagram. The reason is that the partial disorder is induced by $4\pi$ spin
(charge) QPSs that preserve a remnant $\mathbb{Z}_{2}$ order. Physically,
these order parameters correspond to the magnetization (charge) integrated up
to a point $x$ along the 1D system - thus they are non-local order parameters.

_Conclusion:_ We find that superconducting fluctuations in topological and
topologically trivial superconductors are qualitatively different, leading to
distinct phase boundaries, symmetry breaking, and excitation spectra. The
common theme is that superconductivity in topological systems is much more
robust against fluctuations. The reason is the crucial role of dynamic zero
modes. The most dramatic effect is the emergence of otherwise forbidden charge
$2e$ superconductivity for isotropic triplets caused by a topological term in
the action. It yields algebraic superconducting order for a stiffness 16 times
lower than charge $4e$-superconductivity in the non-topological counterpart.

_Acknowledgements:_ We are grateful to P. Coleman, R. M. Fernandes, L. I.
Glazman, P. Goswami, Y. Komijani, Y. Oreg, A. Schnyder, and A. Shnirman for
useful discussions. M. A. R. and J. S. acknowledge support by the Deutsche
Forschungsgemeinschaft (German Research Foundation) Project ER 463/14-1. The
work by E. J. K. was partly performed at the Aspen Center for Physics, which
is supported by National Science Foundation grant PHY-1607611.

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Supplementary materials on

Topologically enabled superconductivity

Michael A. Rampp1, Elio J. König2, Jörg Schmalian1,3

1Institut für Theorie der Kondensierten Materie, Karlsruher Institut für
Technologie, 76131 Karlsruhe, Germany

2Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany

3Institut für Quantenmaterialien und Technologien, Karlsruher Institut für
Technologie, 76021 Karlsruhe, Germany

##  S-I Computation of zero modes

###  S-I.1 Construction of the $(2+0)$-dimensional effective Hamiltonian

In the following it will be argued that the computation of the Pfaffian of the
fermionic kernel $\mathcal{K}$ (to be defined below) of a dynamical problem in
$(1+1)$-dimensions can be mapped to a time-independent Hamiltonian problem in
two spatial dimensions. Subsequently, the zero modes of this Hamiltonian will
be found for the topological background configurations introduced in the main
text. The kernel $\mathcal{K}$ is not Hermitian. This prohibits in general an
expansion in a complete set of eigenmodes. By multiplying $\mathcal{K}$ with a
matrix such that the resulting operator is Hermitian, this issue can be
circumvented. This procedure allows to determine the value of the Pfaffian up
to a phase, since $\operatorname{Pf}(AB)^{2}=\det(A)\det(B)$, which follows
from $\operatorname{Pf}(A)^{2}=\det(A)$ [47] and multiplicativity of the
determinant.

Let us first construct the kernel $\mathcal{K}$. Consider the fermions to be
deep in the topological superconducting phase $\mu\gg
2m\lvert\Delta\rvert^{2}$. Then the low-energy degrees of freedom are situated
near the Fermi points. Define the real and imaginary parts of the components
of the gap function as
$\Delta^{\prime}_{j}\equiv\operatorname{Re}\lvert\Delta\rvert
k_{\mathrm{F}}e^{i\vartheta}n_{j}$ and
$\Delta^{\prime\prime}_{j}\equiv\operatorname{Im}\lvert\Delta\rvert
k_{\mathrm{F}}e^{i\vartheta}n_{j}$. We expand close to the Fermi points $\pm
k_{\mathrm{F}}=\pm\sqrt{2m\mu}=\pm mv_{F}$. The Nambu spinors now have eight
components corresponding to spin, Nambu, and right/left moving components,
which we describe using the Pauli matrices $\sigma^{i}$, $\tau^{i}$, and
$\rho^{i}$, respectively. The reality condition then takes the form
$\Psi^{\dagger}=\Psi^{T}\sigma^{y}\tau^{y}\rho^{x}\equiv\Psi^{T}C$. (The
charge-conjugation operator $C$ also exchanges left- and right-movers). Then
for $\mathcal{K}\equiv(\partial_{\tau}+\mathcal{H})$, holds
$Z^{f}[\vartheta,\bm{n}]=\rm{Pf}(C\mathcal{K}[\vartheta,\bm{n}])$. Explicitly
we have

$\displaystyle C\mathcal{K}$ $\displaystyle=$
$\displaystyle\sigma^{y}\tau^{y}\rho^{x}(\partial_{\tau}+v_{\mathrm{F}}(-i\partial_{x})\tau^{z}\rho^{z}$
(S1)
$\displaystyle+\Delta^{\prime}_{j}\sigma^{j}\tau^{y}\rho^{z}+\Delta^{\prime\prime}_{j}\sigma^{j}\tau^{x}\rho^{z}).$

Note that $C$ and $\mathcal{K}$ have a different matrix structure than in the
main text, since the expansion around the Fermi points was not explicitly
performed there. The sign of the gap function at the left Fermi point is
reversed, due to the antisymmetry of the $p$-wave pairing in momentum space.
The term $\sim\partial_{x}(ue^{-i\vartheta}{\bm{n}})$ has been neglected,
because typical bosonic momenta are much smaller than $k_{\mathrm{F}}$.

In order to map this problem to an Hermitian operator, we multiply by the
matrix $\sigma^{y}\tau^{y}\rho^{z}$ and obtain

$\displaystyle H_{\mathrm{eff}}\equiv\sigma^{y}\tau^{y}\rho^{z}C\mathcal{K}$
$\displaystyle=$ $\displaystyle
i\rho^{y}\partial_{\tau}+i\tau^{z}\rho^{x}v_{\mathrm{F}}\partial_{x}$ (S2)
$\displaystyle-\Delta^{\prime}_{j}\sigma^{j}\tau^{y}\rho^{x}-\Delta^{\prime\prime}_{j}\sigma^{j}\tau^{x}\rho^{x}.$

This is an effective Hamiltonian of a $(2+0)$-dimensional system with
coordinates $(x,\tau)$. It has a chiral symmetry $\mathcal{C}=\rho^{z}$. The
kinetic term has the structure of a Dirac Hamiltonian with $\alpha$-matrices
$\alpha^{x}=-\tau^{z}\rho^{x}$ and $\alpha^{\tau}=-\rho^{y}$ that satisfy the
Clifford algebra $\\{\alpha^{i},\alpha^{j}\\}=2\delta_{ij}$. It takes the form

$H_{\mathrm{eff,kin}}=-iv_{\mathrm{F}}\alpha^{x}\partial_{x}-i\alpha^{\tau}\partial_{\tau}.$
(S3)

The components of the gap function act as coordinate dependent mass terms. It
is convenient to introduce polar coordinates in the $(x,\tau)$-plane by
$x=r\cos\phi,\,v_{\mathrm{F}}\tau=r\sin\phi$, and associated Dirac
$\alpha$-matrices
$\alpha^{r}=\cos\phi\alpha^{x}+\sin\phi\alpha^{\tau},\,\alpha^{\phi}=-\sin\phi\alpha^{x}+\cos\phi\alpha^{\tau}$.
In these coordinates the effective Hamiltonian takes the form

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle-
iv_{\mathrm{F}}\alpha^{r}\partial_{r}-i\frac{\alpha^{\phi}}{r}\partial_{\phi}-\Delta^{\prime}_{j}(r,\phi)\sigma^{j}\tau^{y}\rho^{x}$
(S4) $\displaystyle-$
$\displaystyle\Delta^{\prime\prime}_{j}(r,\phi)\sigma^{j}\tau^{x}\rho^{x}.$

###  S-I.2 Zero Modes on QPSs

Consider now a charge QPS with winding number $\nu_{c}\in\mathbb{Z}$ situated
at the origin. This corresponds to a configuration of the gap function where
$\Delta_{j}^{\prime}=\Delta_{0}(r)\cos(\nu_{c}\phi)n_{j}$ and
$\Delta_{j}^{\prime\prime}=\Delta_{0}(r)\sin(\nu_{c}\phi)n_{j}$. The real and
positive amplitude of the $\Delta_{0}(r)$ has to vanish at the origin for the
gap function to be well-defined. Far away from the core the amplitude
approaches the equilibrium value $\lvert\Delta\rvert k_{\mathrm{F}}$. The
$\bm{n}$-vector can be chosen to point in any constant direction, and we
choose the $z$-direction $\bm{n}=\bm{e}_{z}$. Hence, for this particular
configuration the effective Hamiltonian reads

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle-
iv_{\mathrm{F}}\alpha^{r}\partial_{r}-i\frac{\alpha^{\phi}}{r}\partial_{\phi}-\Delta_{0}(r)\cos(\nu_{c}\phi)\sigma^{z}\tau^{y}\rho^{x}$
(S5) $\displaystyle-$
$\displaystyle\Delta_{0}(r)\sin(\nu_{c}\phi)\sigma^{z}\tau^{x}\rho^{x}.$

This Hamiltonian is identical to a direct sum of two (2+0)-dimensional Fu-Kane
Hamiltonians of a 3D topological insulator surface state in contact with an
s-wave superconductor [41]. The decomposition is with respect to the two spin-
sectors that are defined by the choice of the constant vector $\bm{n}$ which
defines a quantization axis. It is known that the Fu-Kane Hamiltonian
possesses a single localized zero mode in a background field with odd winding
number and no zero mode in a background field with even winding number. We can
conclude that the Hamiltonian (S5) possesses two localized zero modes in the
field of a charge QPS of odd winding number. Moreover, there exists an anti-
unitary operator that commutes with (S5) and squares to $-1$, an effective
time-reversal symmetry, given by $\mathcal{T}=\sigma^{x}\tau^{y}\mathfrak{K}$.
This enforces the double-degeneracy of the spectrum, while the particle-hole
symmetry $\mathcal{P}$ defined by $\mathcal{C}=\mathcal{PT}$ protects the zero
modes.

The solutions can also be found explicitly for $\nu=\pm 1$ by looking for
angle-independent solutions to the zero mode equation. For $\nu_{c}=+1$ we
find, introducing
$g_{c}(r)=N\exp\left(-\int_{0}^{r}\mathrm{d}r^{\prime}\Delta_{0}(r^{\prime})/v_{\mathrm{F}}\right)$,

$\displaystyle\Psi_{1}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{c}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&1&0&0&0&0&0\end{pmatrix},$ (S6a)
$\displaystyle\Psi_{2}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{c}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0&1&0&0&0&0\end{pmatrix}.$ (S6b)

For the case of $\nu_{c}=-1$ we find analogously

$\displaystyle\Psi_{1}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{c}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0&1&0&1&0\end{pmatrix},$ (S7a)
$\displaystyle\Psi_{2}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{c}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0&0&1&0&1\end{pmatrix}.$ (S7b)

One easily realizes that the conclusions made about the zero modes do not rely
on our simplifying assumption that $v_{\mathrm{F}}=v_{c}$.

###  S-I.3 Zero mode on combined QPSs

Consider now a combined charge and spin QPS, that is a configuration in which
both the phase and the $\bm{n}$-vector wind around the origin with winding
numbers $\nu_{c}$ and $\nu_{s}$ respectively. Note that if both $\nu_{c}$ and
$\nu_{s}$ are half-integer the gap function is still well-defined. We choose
the $\bm{n}$-vector to lie in the $xy$-plane, thus the components of the gap
function take the form
$\Delta_{x}^{\prime}=\Delta_{0}(r)\cos(\nu_{c}\phi)\cos(\nu_{s}\phi)$,
$\Delta_{y}^{\prime}=\Delta_{0}(r)\cos(\nu_{c}\phi)\sin(\nu_{s}\phi)$,
$\Delta_{x}^{\prime\prime}=\Delta_{0}(r)\sin(\nu_{c}\phi)\cos(\nu_{s}\phi)$,
and
$\Delta_{y}^{\prime\prime}=\Delta_{0}(r)\sin(\nu_{c}\phi)\sin(\nu_{s}\phi)$.
The remaining components vanish. The resulting effective Hamiltonian reads

$\displaystyle
H_{\mathrm{eff}}=-iv_{\mathrm{F}}\alpha^{r}\partial_{r}-i\frac{\alpha^{\phi}}{r}\partial_{\phi}-\Delta_{0}(r)\begin{pmatrix}0&e^{-i\nu_{s}\phi}\\\
e^{i\nu_{s}\phi}&0\end{pmatrix}\otimes\begin{pmatrix}0&-ie^{i\nu_{c}\phi}\\\
ie^{-i\nu_{c}\phi}&0\end{pmatrix}\otimes\rho^{x}.$ (S8)

It can be seen that the Hamiltonian decomposes into two sectors where the
Cooper-pair background field is that of a QPS with winding number
$\nu_{c}\pm\nu_{s}$ respectively. If $\nu_{c}$ and $\nu_{s}$ are both half-
integer, then the field in one of the sectors has odd winding and the other
has even winding. Therefore there is a single zero mode on such a combined
QPS. This zero mode is protected by the particle-hole symmetry operator
$\mathcal{P}=\tau^{z}\rho^{x}\mathfrak{K}$.

###  S-I.4 Bound states on merons

At finite easy-plane anisotropy the relevant topological excitations that
determine the properties of the spin sector of the model [Eq. (3)] are merons.
As outlined in the main text a meron is essentially a vortex with a finite
core in which the vector (otherwise confined to lie in the plane) escapes the
plane and points in a direction perpendicular to the plane. This endows the
meron with an additional degree of freedom: the orientation of the core. In
other words, the meron carries two distinct topological charges, the vortex
winding number $\nu_{s}$, and the skyrmion charge $Q$. The latter can only
take the values $\pm\tfrac{1}{2}$.

Our goal is to find a physically more transparent derivation of the
$\theta$-term in the action. Since in the this case the configuration of the
superconducting phase is globally topologically trivial, it can be set to
$\vartheta=0$, and we can parametrize the meron texture
$\bm{n}=(\cos(\vartheta_{s})\sin(\beta),\sin(\vartheta_{s})\sin(\beta),\cos(\beta))$
by the in-plane angle $\vartheta_{s}$ and the out-of-plane angle $\beta$. We
choose the meron to wind around the origin in the $xy$-plane, i.e. we set
$\vartheta_{s}(r,\phi)=\nu_{s}\phi$. The lifting out of the plane is then
described by $\beta$, which has to take the value $\pi/2$ far from the core,
and $\pi(1+2Q)/2$ in the core, depending on the skyrmion charge $Q$. The
nonzero components of the gap function are now given by
$\Delta_{x}^{\prime}=\lvert\Delta\rvert
k_{\mathrm{F}}\sin(\beta(r))\cos(\nu_{s}\phi)$,
$\Delta_{y}^{\prime}=\lvert\Delta\rvert
k_{\mathrm{F}}\sin(\beta(r))\sin(\nu_{s}\phi)$, and
$\Delta_{z}^{\prime}=\lvert\Delta\rvert k_{\mathrm{F}}\cos(\beta(r))$. This
yields the effective Hamiltonian

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle-i\alpha^{r}v_{\mathrm{F}}\partial_{r}-i\frac{\alpha^{\phi}}{r}\partial_{\phi}-\lvert\Delta\rvert
k_{\mathrm{F}}\sin(\beta(r))(\cos(\nu_{s}\phi)\sigma^{x}$ (S9)
$\displaystyle+\sin(\nu_{s}\phi)\sigma^{y})\tau^{y}\rho^{x}-\lvert\Delta\rvert
k_{\mathrm{F}}\cos(\beta(r))\sigma^{z}\tau^{y}\rho^{x}.$

If the anisotropy is strong, _i.e._ the core size $L_{\rm
an}\sim\sqrt{v_{s}/\lambda}$ is small, then it is natural to decompose this
Hamiltonian into a unperturbed $H_{\mathrm{eff}}^{0}$ that contains the vortex
winding of the meron, and a perturbation $V$ due to the core region

$\displaystyle H_{\mathrm{eff}}^{0}$ $\displaystyle=$
$\displaystyle-i\alpha^{r}v_{\mathrm{F}}\partial_{r}-i\frac{\alpha^{\phi}}{r}\partial_{\phi}-\lvert\Delta\rvert
k_{\mathrm{F}}\sin(\beta(r))(\cos(\nu_{s}\phi)\sigma^{x}$ (S10)
$\displaystyle+\sin(\nu_{s}\phi)\sigma^{y})\tau^{y}\rho^{x},$ $\displaystyle
V$ $\displaystyle=$ $\displaystyle-\lvert\Delta\rvert
k_{\mathrm{F}}\cos(\beta(r))\sigma^{z}\tau^{y}\rho^{x}.$ (S11)

The unperturbed Hamiltonian $H_{\mathrm{eff}}^{0}$ can again be decomposed
into a direct sum of Fu-Kane Hamiltonians. Indeed it possesses a time-reversal
symmetry that squares to $-1$, given by
$\mathcal{T}=\sigma^{x}\rho^{y}\mathfrak{K}$. The perturbation explicitly
breaks this symmetry. Therefore, the unperturbed problem possesses two zero-
modes for odd winding number, and the splitting effected by the perturbation
must be symmetric (due to the still intact particle-hole symmetry).

For $\nu_{s}=+1$ the two (unperturbed) zero mode solutions read

$\displaystyle\Psi_{1}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{s}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0&1&0&0&1\end{pmatrix},$ (S12)
$\displaystyle\Psi_{2}^{T}(r)$ $\displaystyle=$ $\displaystyle
g_{s}(r)\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&1&0&0&0&0&0\end{pmatrix},$ (S13)

where
$g_{s}(r)=N\exp\left(-\int_{0}^{r}\mathrm{d}r^{\prime}\frac{\lvert\Delta\rvert
k_{\mathrm{F}}}{v_{\mathrm{F}}}\sin\beta(r)\right)$ is a real function that is
exponentially localized near $r=0$. In the space of the zero modes the
perturbation takes the form

$\displaystyle V_{zm}$ $\displaystyle=\pi\lvert\Delta\rvert
k_{\mathrm{F}}\int_{0}^{\infty}\mathrm{d}rrg_{s}(r)^{2}\cos\beta(r)\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}$ $\displaystyle\sim\begin{cases}Q(\lvert\Delta\rvert
k_{\mathrm{F}})(L_{\rm{an}}/\xi)^{2},&L_{\rm{an}}\ll\xi,\\\
Q(\lvert\Delta\rvert k_{\mathrm{F}}),&L_{\rm{an}}\gg\xi.\end{cases}$ (S14)

In first order perturbation theory the two zero modes are split symmetrically
around zero - the perturbative calculation is justified as long as the
splitting is small compared to higher Caroli-de Gennes-Matricon states, which
results in the condition $L_{\rm{an}}\ll\sqrt{\lvert\Delta\rvert/v_{\rm F}}$.
In this limit one may neglect the coupling to higher energy states which
implies that the kernel is block diagonal with one block consisting of the
zero mode subspace and the other of the finite energy states (the restriction
of $\mathcal{K}$ to this subspace is denoted $\mathcal{K}^{\prime}$). Then the
Pfaffian factorizes

$\operatorname{Pf}\begin{pmatrix}C\mathcal{K}^{\prime}&0\\\
0&V_{zm}\end{pmatrix}=\operatorname{Pf}\left(C\mathcal{K}^{\prime}\right)\operatorname{Pf}\left(V_{zm}\right).$
(S15)

The Pfaffian of $V_{zm}$, a $2\times 2$-matrix, is given by the upper right
off-diagonal element. Thus $\operatorname{Pf}V_{zm}=-iaQ$, where $a$ is a real
and positive number. It can be seen that Pfaffians of configurations with
opposite skyrmion charge $Q$ have opposite sign. Hence the contributions of
merons and antimerons to the effective fugacity cancel and can be written in
the form

$\operatorname{Pf}\left(C\mathcal{K}\right)\sim e^{-i\pi Q}.$ (S16)

The relative phase produced by the fermions of the complex fugacity of
configurations with different skyrmion charge $Q$ is therefore identical to
the phase produced by a $\theta$-term with topological angle $\theta=\pi$.

###  S-I.5 Numerical Results

We also verify the existence of zero modes by numerical computation. To this
end, the fermionic kernel is discretized on a lattice. Then the determinant
(which is the square of the Pfaffian) is computed with the background field
configuration of interest inserted. Due to the finite size of the lattice and
due to the finite numerical accuracy it is not possible to show that the
determinant in a certain background field vanishes exactly. However, we
demonstrate two effects that strongly suggest the presence of zero modes: (i)
The numerical value of the determinant in an appropriate topologically non-
trivial background field is multiple orders of magnitude smaller compared to
the determinant in a homogeneous configuration. (ii) In the background of a
QPS-anti-QPS pair the determinant scales exponentially with the separation
$\ell$ of the defects $\det(C\mathcal{K})\sim e^{-\ell/\xi}$. This is evidence
for the linear confinement of QPS-anti-QPS pairs, in distinction to the usual
logarithmic interaction.

Figure S1: Numerical results for the fermion determinant. The logarithm of the
determinant of the fermionic kernel is plotted against the separation between
two topological defects. As expected from the presence of zero modes, the
logarithm diminishes linearly with the separation. When a defect comes too
close to the edge of the system, it hybridizes with the zero mode on the edge,
and the linear behavior is altered. (a) QPS-anti-QPS pair with winding number
$n=1$. (b) spin QPS-anti-QPS pair with winding number $n=1$. (c) QPS-anti-QPS
pair with winding number $n=1$ subject to a phase bias of $2\pi$ along the
spatial direction.

Concretely, we consider the kernel $C\mathcal{K}$ deep in the topological
phase, Eq. (S1). It is placed on a space-time lattice of $n_{\tau}=100$ sites
along the imaginary-time direction and $n_{x}=300$ sites along the spatial
direction. A QPS-anti-QPS pair with winding number $n=1$ is used as a
background configuration and the separation between the defects is varied. For
an exemplary separation of $50$ lattice sites the determinant in the non-
trivial background is about $10^{58}$-times smaller compared to the
determinant in the homogeneous background. Furthermore, the expected linear
scaling of the logarithm of the determinant is observed, until the zero modes
begin to hybridize with the edge modes. The same scaling behavior is observed
for a pair of spin-QPS and a QPS pair subject to phase bias, Figs S1 (b),(c).

##  S-II Computation of fermion correlators

As outlined in the main text the presence of Fermi zero modes in an instanton
background leads to algebraic decay of fermion correlators. This result can
also be understood as analogous to the problem of chiral symmetry breaking in
gauge theories: The fermion parity symmetry (which is separately conserved in
the two spin sectors) does not allow gapless excitations with charge $e$
naively. But taking into account the instanton sectors renders this symmetry
anomalous. In the following sections one- and two-fermion correlators are
computed by considering contributions from zero modes in topologically
nontrivial background fields. For simplicity we assume that $v_{c}=v_{s}$ in
the following sections.

###  S-II.1 Inclusion of Gauge Fields

Before we proceed we briefly comment on the role of fluctuating
electromagnetic fields in charged systems. There are two effects, both of
which only affect the charge field $\vartheta$, but leave the theory in spin-
space unchanged: First, the logarithmic interaction of charge QPSs is formally
screened beyond a space-time scale $L$ [48] which is, however, exponentially
large in the square of ratio of the London penetration depth and some
microscopic scale. [49, 50] Second, the phase mode hybridizes with the 1D
plasmon mode and decays faster than powerlaw because of a logarithmic momentum
dependence of the stiffness [51]. However, any metallic gate, located at a
distance $d_{s}$ will screen the long-range interaction. While formally both
phenomena would destroy algebraic superconductivity, it persists in the
important and experimentally relevant regime $d_{s}\ll x\ll L$.

###  S-II.2 Single-fermion correlator

The fermion correlator is given by a functional integral over all possible
configurations of the fermions and the fluctuating order parameter weighted by
the action

$\displaystyle\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle$
(S17) $\displaystyle=$
$\displaystyle\sum_{N}\int\mathcal{D}\vartheta_{N}\mathcal{D}\bm{n}_{N}\mathcal{D}\psi\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)e^{-S[\psi,\vartheta,\bm{n}]}.$

A given configuration of the order parameter can be divided into a singular
vortex part and a regular fluctuation part
$\vartheta=\vartheta^{v}+\vartheta^{sw},\,\vartheta_{s}=\vartheta_{s}^{v}+\vartheta_{s}^{sw},\beta=\beta^{v}+\beta^{sw}$.
Let us focus on half QPS configurations, since they will generate the most
relevant contribution to the correlator. We first perform a gauge
transformation in order to separate the contribution of the fermions in the
QPS background from the fluctuations. Introduce fermion operators
$\chi_{\kappa}$ through

$\psi_{\sigma}(r)=e^{i\frac{\vartheta^{sw}(r)}{2}}U_{\sigma\kappa}(\bm{n}(r))\chi_{\kappa}(r),$
(S18)

where $U_{\sigma\kappa}(\bm{n})$ denotes the elements of an $SU(2)$ matrix
that satisfies

$U^{\dagger}\bm{n}(\vartheta_{s}^{v}+\vartheta_{s}^{sw},\beta^{v}+\beta^{sw})\cdot\bm{\sigma}U=\bm{n}(\vartheta_{s}^{v},\beta^{v})\cdot\bm{\sigma},$
(S19)

such that it transforms to a frame in which the fermions are subject only to
the topologically nontrivial configuration. The matrix $U$ can be given
explicitly as

$U=e^{-i\frac{\vartheta_{s}^{sw}}{2}\sigma^{z}}e^{-i\frac{\beta^{sw}}{2}(\sigma^{y}\cos\vartheta_{s}^{v}+\sigma^{x}\sin\vartheta_{s}^{v})}=\begin{pmatrix}e^{i\frac{\vartheta_{s}^{sw}}{2}}\cos\left(\frac{\beta^{sw}}{2}\right)&-e^{i\left(\frac{\vartheta_{s}^{sw}}{2}+\vartheta_{s}^{v}\right)}\sin\left(\frac{\beta^{sw}}{2}\right)\\\
e^{-i\left(\frac{\vartheta_{s}^{sw}}{2}+\vartheta_{s}^{v}\right)}\sin\left(\frac{\beta^{sw}}{2}\right)&e^{-i\frac{\vartheta_{s}^{sw}}{2}}\cos\left(\frac{\beta^{sw}}{2}\right)\end{pmatrix}$
(S20)

However, it cannot be expressed solely through the fluctuating angles, but it
depends explicitly on the underlying vortex configuration through
$\vartheta_{s}^{v}$. This has to be taken into account when the averages are
performed. We therefore find

$\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle=\langle
e^{i(\vartheta^{sw}(r)-\vartheta^{sw}(0))/2}\rangle\sum_{n,\bm{d}_{i}}\frac{1}{(n!)^{2}}\int\mathrm{d}X_{1}\dots\mathrm{d}X_{2n}\mathcal{D}\chi\langle
U_{\sigma\kappa}(\bm{n}(r))U^{\dagger}_{\kappa^{\prime}\sigma^{\prime}}(\bm{n}(0))\rangle\chi_{\kappa}(r)\chi_{\kappa^{\prime}}^{\dagger}(0)e^{-S[\chi,\vartheta^{v},\bm{n}^{v}]},$
(S21)

where the sum goes over $n$ QPS-anti-QPS pairs at the locations
$X_{1},\dots,X_{2n}$ with orientation $\bm{d}_{i}$. Hence it can be seen that
the correlator factorizes into a sum of fermion correlators in QPS-anti-QPS
background fields and fluctuation contributions from $\vartheta$ and $\bm{n}$.
(Of course even though the $2\pi$ combined QPSs are confined they still have
to be summed over in the functional integral.)

Let us now turn to the computation of the correlator of $\chi$-fermions. As
outlined above it involves a sum over singular (QPS) background field
configurations. We make the following approximation: we consider only the
contribution of a single QPS-anti-QPS pair (of charge-spin combined QPSs)
situated at distance $r$ (or more precisely, in a region of size $\xi$ around
$r$) and at $0$. Then there is a localized (in space-time) zero mode with spin
$\kappa$ at $r$ and a localized zero mode with spin $\kappa^{\prime}$ at $0$.
Taking into account configurations with higher numbers of QPS-anti-QPS pairs,
these zero modes will eventually broaden into a band [52, 53] with the Fermi
energy at half-filling, and the near-zero modes will become extended states.
Whether the approximation above is justified depends on which order-parameter
configurations dominate the path integral, dilute QPS-anti-QPS pair
configurations, or dense ones. In any case, since the contributions of the
extended zero modes to the correlator will also decay algebraically, they do
not change the general conclusion of the presence of gapless fermion
excitations.

Going back to the single QPS-anti-QPS pair, the expectation value
$\langle\chi_{\kappa}(r)\chi_{\kappa^{\prime}}^{\dagger}(0)\rangle$ in the
background field is a constant independent of $r$ because of the integration
over the (approximate, exact in the limit $r\rightarrow\infty$) Fermi zero
modes. This can be seen by noting that the functional integral is defined as
an integration over Grassmann numbers. Expanding the Nambu spinor fields
$\Psi$ and its transposed $\Psi^{T}$ in some basis with Grassmann numbers
$a_{n}$

$\displaystyle\Psi(r)$ $\displaystyle=\sum_{n}\eta_{R,n}(r)a_{n},$
$\displaystyle\Psi^{T}(r)$ $\displaystyle=\sum_{n}\eta_{L,n}^{T}(r)a_{n},$

where the left and right basis functions can be chosen to be mutually
orthonormal $\langle\eta_{L,n}|\eta_{R,m}\rangle=\delta_{n,m}$, the Majorana-
Grassmann integral takes the form

$\int\prod_{n}\mathrm{d}a_{n}\sum_{m,k}\eta_{R,m,1}(r)\eta_{L,k,1}^{T}(r)a_{m}a_{k}\prod_{l,s}\left(1-a_{l}\mathcal{K}_{ls}a_{s}\right),$
(S22)

where $\mathcal{K}_{ls}$ denotes the matrix elements of the kernel in the
introduced basis. From the preceding sections it is known that without the
additional insertions the Grassmann integral vanishes, since it is equal to
the Pfaffian $\operatorname{Pf}(\mathcal{K})$. From the point of view of
Grassmann integration this means that there are Grassmann numbers that do not
appear in the integrand. These are the zero modes. Hence the Grassmann
integral becomes non-zero only if there is an insertion for every zero mode.
From this it can be concluded that the expectation value above takes a finite
value in the limit $r\rightarrow\infty$. The long-distance behavior now comes
from the instanton action $\sim\exp\left(-\frac{K_{s}+K_{c}}{2}\log r\right)$
(this is just the interaction energy of the QPS and the AQPS). Of course
combined QPSs with higher winding number $\nu$ will also contribute to the
correlator, but their contribution will be subleading, because of the higher
interaction energy that leads to the exponent $\nu^{2}(K_{s}+K_{c})/2$.

The fluctuation contributions of the superconducting phase are given by
$\langle e^{i(\vartheta^{sw}(r)-\vartheta^{sw}(r^{\prime}))/2}\rangle\sim
r^{-1/8K_{c}}$. In a state without superconducting order this correlator
decays exponentially. Hence the loss of phase coherence renders the fermions
massive. Even though the zero modes are still present (locally) they do not
influence the long-range properties anymore.

The fluctuation contribution of the $\bm{n}$-vector is encoded in the
correlator $\langle
U_{\sigma\kappa}(\bm{n}(r))U^{\dagger}_{\kappa^{\prime}\sigma^{\prime}}(\bm{n}(0))\rangle$.
Using Eq. (S21) we can write

$\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle=\langle
e^{i(\vartheta^{sw}(r)-\vartheta^{sw}(0))/2}\rangle_{sw}\sum_{\kappa,\kappa^{\prime}}\left\langle\int\mathcal{D}\chi\chi_{\kappa}(r)\chi_{\kappa^{\prime}}^{\dagger}(0)\langle
U_{\sigma\kappa}(\bm{n}(r))U^{\dagger}_{\kappa^{\prime}\sigma^{\prime}}(\bm{n}(0))\rangle_{sw}\right\rangle_{v},$
(S23)

where $\langle\dots\rangle_{v(sw)}$ denotes the average over vortex
(fluctuation) configurations. Taking
$\sigma=\uparrow,\sigma^{\prime}=\uparrow$ we find that three of the four
terms in the sum depend on $\vartheta_{s}^{v}$. What happens to these terms
upon averaging? The relevant contributions of the Grassmann integral come from
close to the vortex cores. There the phase texture $\vartheta_{s}^{v}$ varies
rapidly. Hence all terms involving $\vartheta_{s}^{v}$ will be suppressed. The
remaining contribution is

$\displaystyle\langle\langle\chi_{\uparrow}(r)\chi_{\uparrow}^{\dagger}(0)\rangle\rangle_{v}$
(S24) $\displaystyle\times$ $\displaystyle\langle
e^{i\frac{\vartheta_{s}^{sw}(r)-\vartheta_{s}^{sw}(0))}{2}}\cos\frac{\beta^{sw}(r)}{2}\cos\frac{\beta^{sw}(0)}{2}\rangle_{sw}.$

In the anisotropic case this is now easy to evaluate and we find

$\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle\sim\delta_{\sigma\sigma^{\prime}}r^{-1/8K_{c}-1/8K_{s}-K_{c}/2-K_{s}/2}.$
(S25)

In the isotropic case $\vartheta_{s}^{sw}$ and $\beta^{sw}$ cannot be regarded
as independent fields. Their dynamics must be described by a $SU_{1}(2)$ WZNW
model. This can be effected by the replacement $K_{s}\rightarrow 1/2$. Hence

$\langle\psi_{\sigma}(r)\psi_{\sigma^{\prime}}^{\dagger}(0)\rangle\sim\delta_{\sigma\sigma^{\prime}}r^{-1/8K_{c}-K_{c}/2-1/2}.$
(S26)

###  S-II.3 Two-fermion correlators

As we have seen above, the summation over combined QPSs (that host a single
localized Fermi zero mode) leads to algebraic decay of single-fermion
correlators. Analogously, configurations with two zero modes localized at the
same point lead to algebraic decay of two-fermion correlators. This is an
effect that does not originate from the presence of gapless single fermion
excitations, which is why it persists in the vestigial phases. It has to be
distinguished from the possible effect of extended single zero modes which
appear for dense configurations of QPS-AQPS pairs. Essentially, if the single
Fermi zero modes hybridize and form a band of finite width, then two fermion
correlators are naturally gapless with a scaling dimension that is inherited
from the single-fermion correlator. In the opposite limit of a very dilute
QPS-AQPS pair configuration this is not implied. In this case the two-fermion
correlator can only become gapless through order-parameter configurations with
two zero modes localized at the same point.

Consider a $2\pi$ charge QPS. It possesses two localized zero modes of
opposite spin, because the tunneling event changes the fermion parity in both
the spin-up and -down sectors. This pair of zero modes forms a singlet, since
it is unchanged by rotations in spin space. (Therefore the corresponding
operator represents a type of order differing from the primary triplet order
$\bm{\Delta}$.) The annihilation operator with the same quantum numbers
(charge $2e$ and spin $0$) is given by $\psi_{\sigma}(r)\psi_{-\sigma}(r)$.
Hence in the approximation that only configurations containing a single QPS-
AQPS pair contribute, we obtain

$\displaystyle\langle(\psi_{\sigma}\psi_{-\sigma})(x)(\psi^{\dagger}_{-\sigma}\psi^{\dagger}_{\sigma})(0)\rangle$
(S27) $\displaystyle=$ $\displaystyle\langle
e^{i(\vartheta^{sw}(r)-\vartheta^{sw}(0))}\rangle\times\langle\chi_{\sigma}\chi_{-\sigma}(r)\chi^{\dagger}_{-\sigma}\chi^{\dagger}_{\sigma}(0)\langle\cos^{2}\left(\frac{\beta^{sw}(r)}{2}\right)\cos^{2}\left(\frac{\beta^{sw}(0)}{2}\right)\rangle_{sw}$
$\displaystyle-\chi_{\sigma}\chi_{-\sigma}(r)\chi^{\dagger}_{\sigma}\chi^{\dagger}_{-\sigma}(0)\langle\cos^{2}\left(\frac{\beta^{sw}(r)}{2}\right)\sin^{2}\left(\frac{\beta^{sw}(0)}{2}\right)\rangle_{sw}-\chi_{-\sigma}\chi_{\sigma}(r)\chi^{\dagger}_{-\sigma}\chi^{\dagger}_{\sigma}(0)\langle\sin^{2}\left(\frac{\beta^{sw}(r)}{2}\right)\cos^{2}\left(\frac{\beta^{sw}(0)}{2}\right)\rangle_{sw}$
$\displaystyle+\chi_{-\sigma}\chi_{\sigma}(r)\chi^{\dagger}_{\sigma}\chi^{\dagger}_{-\sigma}(0)\langle\sin^{2}\left(\frac{\beta^{sw}(r)}{2}\right)\sin^{2}\left(\frac{\beta^{sw}(0)}{2}\right)\rangle_{sw}\rangle_{v}.$

Using the anticommutativity of the Grassmann numbers it turns out that the
angular contributions cancel out. We find therefore

$\langle(\psi_{\sigma}\psi_{-\sigma})(r)(\psi^{\dagger}_{-\sigma}\psi^{\dagger}_{\sigma})(0)\rangle\sim
r^{-1/2K_{c}-2K_{c}}.$ (S28)

Because this correlator is independent of the variables in the spin sector, it
remains gapless in the spin disordered vestigial charge-$4e$ phase.

We can also consider a $2\pi$ spin QPS. It hosts two zero modes of opposite
spin, but now the phase is advanced in the opposite direction in both spin
sectors, which implies that the fermion parity changes in the opposite
direction. The corresponding operator thus has to be neutral and carry
spin-$1$ (the spin QPS like $\bm{n}$ under spin rotations). Such an operator
is given by $\psi_{\sigma}^{\dagger}\psi_{-\sigma}$. Note that the zero modes
are only present in the limit of infinite anisotropy, otherwise they are
gapped out. In this limit we find

$\displaystyle\langle(\psi_{\sigma}^{\dagger}\psi_{-\sigma})(r)(\psi_{-\sigma}^{\dagger}\psi_{\sigma})(0)\rangle$
$\displaystyle\sim$ $\displaystyle\langle e^{-\sigma
i(\vartheta_{s}^{sw}(r)-\vartheta_{s}^{sw}(0))}\rangle$ (S29)
$\displaystyle\sim$ $\displaystyle r^{-1/2K_{s}-2K_{s}}.$

It is independent of the charge variables. Hence at infinite anisotropy the
charge disordered vestigial spin-$2$ phase possesses gapless spin-$1$
excitations.

##  S-III Non-Abelian bosonization

In the following, a detailed derivation of the $\theta$-term using non-Abelian
bosonization [35] is presented. Non-Abelian bosonization is a technique by
which a $(1+1)$-dimensional system of fermions with a $O(N)\times O(N)$ (or
$U(N)\times U(N)$) chiral symmetry can be rewritten as a fixed point of a
bosonic theory, with the bosonic field being defined on the group manifold
$O(N)$ ($U(N)$). The advantage of the approach, compared to Abelian
bosonization, lies in the manifest preservation of the complete chiral
symmetry. The action governing the group-valued field $g$ is the WZNW action
of level $k=1$

$S_{W}[g]=\frac{1}{4\gamma^{2}}\int\mathrm{d}^{2}x\operatorname{tr}\left[\partial_{\mu}g\partial^{\mu}g^{-1}\right]+k\Gamma[g],$
(S30)

where the Wess-Zumino term is defined extending the domain of $g$ to a
hemisphere of the three-dimensional unit sphere with the original domain
$S^{2}$ as its boundary

$\Gamma[g]=-\frac{i}{24\pi}\int_{S^{3}_{\pm}}\mathrm{d}^{3}y\varepsilon^{\mu\nu\rho}\operatorname{tr}\left[g^{-1}\frac{\partial
g}{\partial y^{\mu}}g^{-1}\frac{\partial g}{\partial
y^{\nu}}g^{-1}\frac{\partial g}{\partial y^{\rho}}\right].$ (S31)

By virtue of Stokes’ theorem the Wess-Zumino term is independent of the
precise choice of the extension.

In the case discussed here, the Lagrangian of fermions with spin in the
background of a $p$-wave pairing field deep in the topological phase is
expressed as a Lagrangian of Majorana fermions with a $O(4)\times O(4)$ chiral
symmetry that is manifestly broken by the pairing terms that act as space-time
dependent Majorana masses. In this form the free theory can be represented by
a WZNW action of level $k=1$, and the pairing terms become mass terms for the
group valued bosonic field $g$ that effectively pin $g$ to a fixed value.
Under adiabatic variation of the gap function $g$ traces out a submanifold
$\simeq(S^{1}\times S^{2})/\mathbb{Z}_{2}$ of $O(4)$. We demonstrate that the
Wess-Zumino term (S31) measures the skyrmion charge $Q$ of the pairing field
configuration, $\Gamma[g(\vartheta,\bm{n})]=i\pi Q$, and is hence responsible
for the appearance a $\theta$-term with $\theta=\pi$ in the effective theory
describing the superconductor.

###  S-III.1 Majorana representation of the free fermion action

We begin by representing the free fermion action as a theory of massless
Majorana fermions. Define the right- and left-moving fermion fields $R$ and
$L$, and the right- and left-moving spinors as

$\phi_{\rm{R}}=\begin{pmatrix}R\\\
i\sigma^{y}L^{*}\end{pmatrix},\quad\phi_{\rm{L}}=\begin{pmatrix}L\\\
i\sigma^{y}R^{*}\end{pmatrix}.$ (S32)

After expansion around the Fermi points the BdG Hamiltonian [Eq. (1)] without
pairing terms takes the form

$\mathcal{H}=\frac{1}{2}\int\mathrm{d}x\left(\phi_{\rm{R}}^{\dagger}H_{\rm{R}}\phi_{\rm{R}}+\phi_{\rm{L}}^{\dagger}H_{\rm{L}}\phi_{\rm{L}}\right),$
(S33)

with $H_{\rm{R}}=v_{\rm{F}}(-i\partial_{x})\tau^{z}$ and
$H_{\rm{L}}=-H_{\rm{R}}$. The spinors are connected by the reality conditions
$\phi_{\rm{R}}^{\dagger}=\phi_{\rm{L}}^{T}C$ and
$\phi_{\rm{L}}^{\dagger}=\phi_{\rm{R}}^{T}C$, where $C=\sigma^{y}\tau^{y}$.
Hence the Hamiltonian has the same number of degrees of freedom as that of $8$
Majorana fermions, but expressed through $\phi_{\rm{R}}$ and $\phi_{\rm{L}}$
it does not yet have the appropriate form. The desired Nambu spinor should
satisfy the Majorana condition $\Phi^{\dagger}=\Phi^{T}$. This condition is
satisfied by the spinor
$\chi^{T}=(\chi_{1\rm{R}}^{T},\chi_{2\rm{R}}^{T},\chi_{1\rm{L}}^{T},\chi_{2\rm{L}}^{T})$
defined through

$\displaystyle R_{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\chi_{\sigma 1\rm{R}}+i\chi_{\sigma
2\rm{R}},\right)$ (S34a)
$\displaystyle\sigma^{y}_{\sigma\sigma^{\prime}}L_{\sigma^{\prime}}$
$\displaystyle=$ $\displaystyle-\frac{1}{\sqrt{2}}\left(\chi_{\sigma
1\rm{L}}-i\chi_{\sigma 2\rm{L}}\right).$ (S34b)

Under this transformation the kinetic part of the Hamiltonian becomes
$H_{\rm{kin}}=v_{\rm{F}}i\partial_{x}\rho^{z}$.

It remains to express the pairing terms in this representation. Before the
transformation they are given by
$H_{\rm{p,R}}=\Delta_{j}^{\prime}\sigma^{j}\tau^{y}+\Delta_{j}^{\prime\prime}\sigma^{j}\tau^{x}$
and $H_{\rm{p,L}}=-H_{\rm{p,R}}$. The result after the transformation is

$\displaystyle H_{\rm{p}}$ $\displaystyle=$
$\displaystyle\Delta_{x}^{\prime}\sigma^{x}\tau^{y}\rho^{x}+\Delta_{x}^{\prime\prime}\sigma^{x}\tau^{0}\rho^{y}-\Delta_{y}^{\prime}\sigma^{y}\tau^{0}\rho^{x}-\Delta_{y}^{\prime\prime}\sigma^{y}\tau^{y}\rho^{y}$
(S35)
$\displaystyle+\Delta_{z}^{\prime}\sigma^{z}\tau^{y}\rho^{x}+\Delta_{z}^{\prime\prime}\sigma^{z}\tau^{0}\rho^{y}.$

It can be seen that this BdG Hamiltonian only contains off-diagonal
$\rho$-matrices. Hence it only couples left- and right-moving fermions.
Therefore it effect is to generate a mass term linear in $g$ upon
bosonization.

###  S-III.2 Bosonization

Now, the group valued field $g$ is introduced. The correspondences between the
currents are given by

$\displaystyle\chi_{\rm{L}}^{i}\chi_{\rm{L}}^{j}$
$\displaystyle\longleftrightarrow$
$\displaystyle-\frac{1}{2\pi}\left(g^{-1}\partial_{+}g\right)^{ij},$ (S36a)
$\displaystyle\chi_{\rm{R}}^{i}\chi_{\rm{R}}^{j}$
$\displaystyle\longleftrightarrow$
$\displaystyle\frac{1}{2\pi}\left((\partial_{-}g)g^{-1}\right)^{ij},$ (S36b)
$\displaystyle\chi_{\rm{R}}^{i}\chi_{\rm{L}}^{j}$
$\displaystyle\longleftrightarrow$ $\displaystyle iMg^{ij},$ (S36c)

where $M$ is a quantity with dimensions of mass that depends on the
regularization procedure.

These formulas can now be used to represent the pairing terms through $g$.
Parametrize the order parameter as above. Consider e.g. the first contribution
to Eq. (S35). It holds

$\displaystyle\chi^{T}\tau^{y}\sigma^{x}\rho^{x}\chi$ $\displaystyle=$
$\displaystyle\chi_{\rm{R}}^{T}\tau^{y}\sigma^{x}\chi_{\rm{L}}+\chi_{\rm{L}}^{T}\tau^{y}\sigma^{x}\chi_{\rm{R}}$
(S37) $\displaystyle=$ $\displaystyle
iM\left(g^{ij}(\tau^{y}\sigma^{x})_{ij}-g^{ji}(\tau^{y}\sigma^{x})_{ij}\right)$
$\displaystyle=$ $\displaystyle-
iM\mathrm{tr}\left[g\left(\tau^{y}\sigma^{x}-(\tau^{y}\sigma^{x})^{T}\right)\right]$
$\displaystyle=$
$\displaystyle-2iM\mathrm{tr}\left[g\tau^{y}\sigma^{x}\right].$

Proceeding analogously for the remaining terms yields

$S_{\rm{p}}[g,\vartheta,\bm{n}]=-\lvert\Delta\rvert
k_{\mathrm{F}}M\operatorname{tr}\left[gO_{U}^{T}\right],$ (S38)

where $O_{U}$ defines an embedding of the order parameter manifold into the
target space of $g$

$\displaystyle(S^{1}\times S^{2})/\mathbb{Z}_{2}\rightarrow
O(4):\,\,(\vartheta,\bm{n})\longmapsto O_{U},$ $\displaystyle
O_{U}=-i\cos\vartheta\left(n_{x}\tau^{y}\sigma^{x}-n_{y}\tau^{0}\sigma^{y}+n_{z}\tau^{y}\sigma^{z}\right)$
$\displaystyle-\sin\vartheta\left(n_{x}\tau^{0}\sigma^{x}-n_{y}\tau^{y}\sigma^{y}+n_{z}\tau^{0}\sigma^{z}\right).$
(S39)

It can be checked that $O_{U}$ is indeed orthogonal. Moreover its determinant
is equal to one $\det(O_{U})=1$ which means that the mapping is into the
$SO(4)$ component of $O(4)$.

###  S-III.3 Mean-Field Solution

For fixed background fields $\bm{n},\vartheta$ the pairing term is minimized
by the configuration $g=O_{U}$. This can be seen by virtue of the following
argument: The matrix $-gO_{U}^{T}$ is an element of $O(4)$. Its trace is given
by the sum of its eigenvalues. Because of the orthogonality the possible
eigenvalues lie on the complex unit circle, and, if they are not real, come in
pairs of complex conjugates. Hence the minimal value of the trace is $-4$
which corresponds to $4$ eigenvalues of $-1$. Indeed a matrix with $4$
eigenvalues of $-1$ is equal to the negative of the identity matrix and hence
also unique.

The mean-field solution can be used as a starting point to find the effective
action. As discussed above it parametrizes the Goldstone manifold.
Fluctuations around this manifold are massive. Define $h=gO_{U}^{T}$ and use
the Wiegmann-Polyakov formula [54]

$\displaystyle S_{W}[hO_{U}]$ $\displaystyle=$ $\displaystyle
S_{W}[h]+S_{W}[O_{U}]$ (S40)
$\displaystyle+\frac{1}{2\pi}\int\mathrm{d}^{2}x\mathrm{tr}\left(h^{-1}\bar{\partial}h\,O_{U}\partial
O_{U}^{T}\right),$

where $\bar{\partial}=\frac{1}{2}(\partial_{x}+i\partial_{\tau})$ and
$\partial=\frac{1}{2}(\partial_{x}-i\partial_{\tau})$. $S_{W}[O_{U}]$
constitutes the leading order contribution to the effective action. The
kinetic part gives

$S_{\rm{kin}}=\frac{1}{\gamma^{2}}\int\mathrm{d}^{2}x\left(\left(\partial_{\mu}\vartheta\right)^{2}+\left(\partial_{\mu}\bm{n}\right)^{2}\right),$
(S41)

which is simply the action of the NL$\sigma$M corresponding to the order
parameter manifold. This constitutes a renormalization of the NL$\sigma$M due
to the contribution of the fermions (the starting point is already a
NL$\sigma$M with stiffnesses $K_{c,s}^{0}$). Its magnitude can be estimated by
using the fixed point value of the $O(4)\times O(4)$ symmetric model as a bare
value. Then $\gamma^{2}=4\pi$ and therefore the renormalization of the
stiffness is $K_{\mathrm{f}}=1/2$.

Let us now determine the effect of the Wess-Zumino (WZ) term. In order to
substitute the mean-field configuration into the WZ term it is necessary to
extend it to the three-sphere $S^{3}$. Following [19] parametrize $S^{3}$ by
the coordinates on $S^{2},\,(x_{1},x_{2})$ and an additional polar angle
$\xi\in[0,\pi]$. The equator corresponds to $\xi=\pi/2$. Hence the extension
$\tilde{g}$ has to satisfy
$\tilde{g}\left(x_{1},x_{2},\frac{\pi}{2}\right)=g(x_{1},x_{2})$ and the limit
$\tilde{g}(\xi\rightarrow 0)$ has to exist.

Since the only field that can obtain a nontrivial $\theta$-term from the WZNW
action is the vector $\bm{n}$ it is sufficient to set the superconducting
phase $\theta$ to a constant value. For simplicity $\theta=\pi/2$ is chosen.
Then the semi-classical configuration that is to be extended reads

$g=-\tau^{y}\bm{n}\cdot\bm{\Sigma},\quad\bm{\Sigma}\equiv\begin{pmatrix}\tau^{y}\sigma^{x}&-\sigma^{y}&\tau^{y}\sigma^{z}\end{pmatrix},$
(S42)

where the vector of matrices $\bm{\Sigma}$ has been introduced. It satisfies
the algebra

$\Sigma_{i}\Sigma_{j}=\delta_{ij}1-i\epsilon_{ijk}\Sigma_{k},$ (S43)

which is analogous to the algebra of Pauli matrices up to a sign. Now, a
concrete example of an extension is given by

$\tilde{g}=i\tau^{y}\exp\left(i\xi\bm{n}\cdot\bm{\Sigma}\right).$ (S44)

This can be seen by noting that

$\tilde{g}=i\tau^{y}\left(\cos\xi+i\bm{n}\cdot\bm{\Sigma}\sin\xi\right)=i\tau^{y}\cos\xi+g\sin\xi.$
(S45)

This coincides with $g$ for $\xi=\pi/2$ and is an element of $O(4)$ on the
whole three-sphere.

The expression (S44) can now be inserted into the WZ term. Defining
$\bm{\omega}_{i}\equiv\bm{n}\times\partial_{i}\bm{n}$ it can be expressed as

$\displaystyle\Gamma[g=O_{U}]$ $\displaystyle=$
$\displaystyle-\frac{6i}{24\pi}\int_{0}^{\frac{\pi}{2}}\mathrm{d}\xi\left(-i\sin^{2}\xi\right)\int\mathrm{d}^{2}x$
(S46)
$\displaystyle\times\mathrm{tr}[\bm{n}\cdot\bm{\Sigma}(\cos\left(\xi\right)\partial_{1}\bm{n}+\sin\left(\xi\right)\bm{\omega}_{1})\cdot\bm{\Sigma}$
$\displaystyle\times(\cos\left(\xi\right)\partial_{2}\bm{n}+\sin\left(\xi\right)\bm{\omega}_{2})\cdot\bm{\Sigma}].$

By repeatedly applying Eq. (S43) and using that the $\Sigma_{i}$ are traceless
this can be brought into the form

$\displaystyle-$
$\displaystyle\frac{6i}{24\pi}\int_{0}^{\frac{\pi}{2}}\mathrm{d}\xi(-4)\sin^{2}\xi\int\mathrm{d}^{2}x\cos^{2}\xi\,\bm{n}\cdot\left(\partial_{1}\bm{n}\times\partial_{2}\bm{n}\right)$
(S47) $\displaystyle+$
$\displaystyle\sin^{2}\xi\,\bm{n}\cdot\left(\bm{\omega}_{1}\times\bm{\omega}_{2}\right)$
$\displaystyle+$
$\displaystyle\sin\xi\cos\xi\,\bm{n}\cdot\left(\bm{\omega}_{1}\times\partial_{2}\bm{n}+\partial_{1}\bm{n}\times\bm{\omega}_{2}\right).$

The third term vanishes

$\bm{n}\cdot\left(\bm{\omega}_{1}\times\partial_{2}\bm{n}\right)=-\bm{\omega}_{1}\cdot\bm{\omega}_{2}=-\bm{n}\cdot\left(\partial_{1}\bm{n}\times\bm{\omega}_{2}\right),$
(S48)

and the first two terms combine,
$\bm{n}\cdot\left(\bm{\omega}_{1}\times\bm{\omega}_{2}\right)=\bm{n}\cdot\left(\partial_{1}\bm{n}\times\partial_{2}\bm{n}\right)$.
The integral over $\xi$ evaluates to $\pi/4$ and therefore the result is

$\Gamma[g=O_{U}]=\frac{6\pi
i}{24\pi}\int\mathrm{d}^{2}x\bm{n}\cdot\left(\partial_{1}\bm{n}\times\partial_{2}\bm{n}\right).$
(S49)

But this can be identified as the $\theta$-term of a NL$\sigma$M defined on
$S^{2}$ [19]

$S_{\theta}=\frac{i\theta}{4\pi}\int\mathrm{d}^{2}x\bm{n}\cdot\left(\partial_{1}\bm{n}\times\partial_{2}\bm{n}\right).$
(S50)

Comparison with the result from the WZNW model shows that the $\theta$-angle
takes the value $\pi$.

###  S-III.4 BCS interactions

In the following we demonstrate starting from a purely fermionic model with
BCS interactions in the $p$-wave channel, that a state characterized by a
superconducting order parameter of the form $\bm{\Delta}\sim
e^{i\vartheta}\bm{n}$ is favored. We show that the family of mean-field states
parametrized by Eq. (S39) arises as the ground-state manifold of the
interacting model.

Consider the local interaction operator

$S_{\rm{int}}=-V\int\mathrm{d}^{2}x\left(\psi^{\dagger}\bm{\sigma}i\sigma^{y}\partial_{x}\psi^{*}\right)\left(\psi^{T}i\sigma^{y}\bm{\sigma}\partial_{x}\psi\right).$
(S51)

For positive interaction constant $V>0$ this interaction is attractive and
drives a BCS instability in the $p$-wave channel. A mean-field decoupling
yields the BdG Hamiltonian [Eq. (1)]. In order to bosonize this term, we
expand the fermion operators close to the Fermi points and only consider
momentum-conserving terms. At half-filling Umklapp scattering becomes
relevant. This is neglected. There are two types of momentum-conserving
scattering events: Two right- (left-) moving Fermi fields can scatter into to
right- (left-) moving Fermi fields. After bosonization these terms will
contain derivatives of the group-valued field $g$ and hence lead to a
renormalization of the kinetic term. Hence they are not considered further. On
the other hand, a right- and a left-mover can scatter into a right- and a
left-mover. These processes lead to a potential term that reduces the gapless
degrees of freedom. We obtain

$\displaystyle S_{\rm{int}}$ $\displaystyle=$ $\displaystyle
Vk_{\mathrm{F}}^{2}\int\mathrm{d}^{2}x\left(R^{\dagger}\bm{\sigma}i\sigma^{y}L^{*}-L^{\dagger}\bm{\sigma}i\sigma^{y}R^{*}\right)$
(S52)
$\displaystyle\times\left(-R^{T}i\sigma^{y}\bm{\sigma}L+L^{T}i\sigma^{y}\bm{\sigma}R\right).$

Define $\tau^{\pm}\equiv\tau^{0}\pm\tau^{y}$. Transforming to the Majorana
representation (S34) yields

$\displaystyle S_{\rm{int}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}Vk_{\mathrm{F}}^{2}\int\mathrm{d}^{2}x\left(i\chi_{R}^{T}\bm{\sigma}\tau^{-}\chi_{L}+i\chi_{L}^{T}\sigma^{y}\bm{\sigma}\sigma^{y}\tau^{+}\right)$
(S53)
$\displaystyle\times\left(i\chi_{R}^{T}\sigma^{y}\bm{\sigma}\sigma^{y}\tau^{+}+i\chi_{L}^{T}\bm{\sigma}\tau^{-}\chi_{R}\right).$

In this form the interaction can be bosonized using Eqs (S36). We define the
effective interaction strength $\eta=Vk_{\mathrm{F}}^{2}M^{2}>0$ and find

$S_{\mathrm{int}}[g]=Vk_{\mathrm{F}}^{2}M^{2}\int\mathrm{d}^{2}x\operatorname{tr}\left[g\sigma^{y}\bm{\sigma}\sigma^{y}\tau^{+}\right]\operatorname{tr}\left[g\bm{\sigma}\tau^{-}\right].$
(S54)

In order to show that the family of solutions parametrized by Eq. (S39) indeed
minimizes Eq. (S54), we expand the action in fluctuations around this
solution. Therefor we introduce the imaginary, antisymmetric, and traceless
matrices $L_{ab}=-i(e_{ab}-e_{ba})$ for $a<b,\,a,b\in\\{1,2,3,4\\}$, where
$e_{ab}$ has entry $1$ at the position $(ab)$ and zero everywhere else. These
matrices form a basis of the Lie algebra $\mathfrak{so}(4)$. This allows to
write $g=O_{U}\exp\left(\tfrac{i}{\sqrt{2}}\phi_{ab}L_{ab}\right)$, where
$\phi_{ab}$ are fluctuating fields. Inserting this into the action (S54) and
retaining only terms up to second order in the fields $\phi_{ab}$ yields

$S_{\mathrm{int}}=\eta\left(-16+\frac{1}{2}C_{(ab)(cd)}(\vartheta,\bm{n})\phi_{ab}\phi_{cd}\right).$
(S55)

The mass matrix $C_{(ab)(cd)}$ possesses three zero eigenvalues and three
positive eigenvalues for all $(\vartheta,\bm{n})$. The three zero-eigenvalue
modes correspond to translations along the manifold, while the three positive
eigenvalues describe gapped fluctuations with mass $\sqrt{8\eta}$. Hence the
solutions $g=O_{U}(\vartheta,\bm{n})$ indeed constitute the ground-state
manifold of the interaction (S51), and it therefore favors an order parameter
of the form $\bm{\Delta}\sim e^{i\vartheta}\bm{n}$.

####  S-III.4.1 Hubbard-Stratonovich Transformation

The fermion-fermion interaction operator Eq. (S51) provides one specific
starting point to obtain the two-fluid model Eq. (1). A Hubbard-Stratonovich
decoupling yields the Hamiltonian [Eq. (1)] and integrating out amplitude
fluctuations produces the NL$\sigma$M. Our results do not rely on this
concrete derivation, since the two-fluid model is the generic low-energy
theory with the desired order-parameter symmetry. The NL$\sigma$M now encodes
the soft fluctuations of the Goldstone manifold. If this manifold is unstable
the model flows to strong coupling, indicating the absence of the ordered
phase. Let us make this explicit by performing the Hubbard-Stratonovich
decoupling for this concrete example. We introduce the order parameter
$\bm{\Delta}$ as a Hubbard-Stratonovich (HS) field and rewrite the exponential
of the interaction operator as

$\displaystyle
e^{-S_{\mathrm{int}}}\propto\int\mathcal{D}\bm{\Delta}^{\dagger}\mathcal{D}\bm{\Delta}\exp\Bigg{(}-\int\mathrm{d}^{2}x\bigg{[}\frac{\bm{\Delta}^{\dagger}\cdot\bm{\Delta}}{V}-\bm{\Delta}\cdot\left(\psi^{\dagger}\bm{\sigma}i\sigma^{y}\partial_{x}\psi^{*}\right)\bigg{.}\Bigg{.}-\Bigg{.}\bigg{.}\bm{\Delta}^{\dagger}\cdot\left(\psi^{T}i\sigma^{y}\bm{\sigma}\partial_{x}\psi\right)\bigg{]}\Bigg{)}.$
(S56)

The linear couplings between the HS field and the fermion bilinears generates
the pairing terms of the BdG description. The NL$\sigma$M formulation is
recovered upon fixing the amplitude of the HS field to a non-zero value. The
bare stiffnesses of this model are equal to zero. The coupling to the fermions
dynamically generates a finite renormalized stiffness. Performing a gradient
expansion one finds
$K_{c,s}^{\mathrm{ren}}=2F\left(\frac{\mu}{2m\lvert\Delta\rvert^{2}}\right)$
with $F(x)\rightarrow 1$ for $x\rightarrow\infty$, $F(x<0)=0$, and $F(x)\sim
x^{1/4}\log^{1/2}(1/x)$ for $x\rightarrow 0^{+}$. Hence $K_{c,s}=2$ deep in
the topological phase for this concrete model. On the mean-field level a self-
consistent solution with a finite amplitude exists if the ground-state energy
is lowered compared to the normal state. Consider the change in ground-state
energy density

$\Delta\mathcal{E}=\int\frac{\mathrm{d}k}{2\pi}\left(\xi_{k}-\sqrt{\xi_{k}^{2}+\lvert\bm{\Delta}\rvert^{2}k^{2}}\right)+\frac{\lvert\bm{\Delta}\rvert^{2}}{V}.$
(S57)

This quantity has to be negative for a nontrivial self-consistent solution to
exist. Deep in the weak-paired phase this is certainly the case: expanding
around the Fermi points and assuming a constant density of states yields the
familiar BCS criterion.