# Influence of Majorana-parity degeneracy on the boundary Berezinskii-
Kosterlitz-Thouless quantum phase transition of a dissipative resonant level

Gu Zhang<EMAIL_ADDRESS>Beijing Academy of Quantum Information Sciences,
Beijing 100193, China  Zhan Cao  Beijing Academy of Quantum Information
Sciences, Beijing 100193, China  Dong E. Liu<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional Quantum Physics, Department of
Physics, Tsinghua University, Beijing, 100084, China  Beijing Academy of
Quantum Information Sciences, Beijing 100193, China  Frontier Science Center
for Quantum Information, Beijing 100184, China  Hefei National Laboratory,
Hefei 230088, China

(August 29, 2024)

###### Abstract

The interplay between a topological degeneracy and the residue degeneracy
(also known as the residue entropy) of quantum criticality remains as an
important but not thoroughly understood topic. We find that the degeneracy of
topological parity, provided by a Majorana zero mode pair, relaxes the
otherwise strictly requested symmetry requirement, to observe the boundary
Berezinskii-Kosterlitz-Thouless (BKT) quantum phase transition (QPT) of a
dissipative resonant level. Our work indicates that the topological-parity
degeneracy can be potentially viewed as an auxiliary symmetry that realizes a
robust boundary QPT. The relaxation of the symmetry requirement extends the
transition from a point to a finite area, thus greatly reducing the difficulty
to experimentally observe the QPT. This topology-involved exotic BKT phase
diagram, on the other hand, provides a piece of convincing evidence for a
Majorana zero mode.

††preprint: APS/123-QED

_Introduction_ —Boundary quantum phase transitions (QPTs) Vojta (2006);
Affleck (2008) are special QPTs triggered by the change of boundary
conditions. Similar as the bulk ones, boundary QPTs normally involve
symmetries. For instance, in multi-channel Kondo Emery and Kivelson (1992);
Affleck and Ludwig (1991a, 1992); Affleck  _et al._ (1992); Mitchell  _et al._
(2016); Lopes  _et al._ (2020a), Luttinger liquid resonant Komnik and Gogolin
(2003), and dissipative resonant level Mebrahtu  _et al._ (2012, 2013); Liu
_et al._ (2014) models, the observation of quantum critical point (QCP), i.e.
the point that separates two zero-temperature phases Sachdev (2011), requires
the impurity (a quantum dot or magnetic impurity) to symmetrically couple to
involved leads. Actually, the symmetric coupling that triggers quantum
frustration at QCP produces a non-trivial residue degeneracy, which is another
important feature of QPT Affleck and Ludwig (1991a); Vojta (2006). This
residue degeneracy, heavily focused on recently as a QPT-induced “anyon”
Cornfeld  _et al._ (2019); Lopes  _et al._ (2020b); Komijani (2020); Lotem
_et al._ (2022); Gabay  _et al._ (2022), has been proposed as an amplifier of
topological non-triviality Zhang and Spånslätt (2020). As another example, a
Berezinskii-Kosterlitz-Thouless (BKT) transition is predicted in a dissipative
resonant level model (DRLM) Chung  _et al._ (2009); Liu  _et al._ (2014),
between the ballistic-transmission and isolated-impurity phases. Its
observation, importantly, also requires perfectly symmetric lead-dot
couplings, since its ballistic-transmission phase consists of solely a line of
QCPs. This fine-tuning prerequisite of symmetry is among the difficulties to
observe this BKT QPT. To the best of our knowledge, a BKT-type boundary QPT,
has not yet been reported experimentally.

On the other hand, a pair of Majorana zero modes (MZMs), predicted to exist at
end points of a nanowire hetero-structure Kitaev (2001); Lutchyn  _et al._
(2010); Oreg  _et al._ (2010) and vortices of a topological superconductor
Read and Green (2000); Fu and Kane (2008), is known to provide a non-locally
defined topological degeneracy, given a long enough inter-MZM distance. This
topological degeneracy, distinct from a trivially local one, is among the key
elements of topological quantum computation Nayak  _et al._ (2008). Exploring
the intricate influences of topological degeneracy on the residue degeneracy
in QPTs, where symmetry is a prerequisite, constitutes a compelling avenue of
investigation. A crucial line of inquiry involves investigating the potential
role of topological degeneracy as an auxiliary symmetry capable of reinstating
boundary QPTs that have been undermined by asymmetry.

Summary of main results: In this work, we visit this topic by investigating
the influence of topological-parity degeneracy on the boundary BKT QPT in a
DRLM. Without the topological-parity degeneracy, it is known that the BKT
transition of a DRLM requires a perfect symmetry between two involved leads.
Surprisingly, as among our central results, this strict symmetry requirement
(to observe the BKT QPT) is remarkably relaxed, by coupling an MZM to the dot
of the DRLM [see Fig. 2 that is distinct from Fig. 1(b)]. With Coulomb gas
renormalization group (RG) equations and g-theorem Affleck and Ludwig (1991b,
1993), we further prove that the relaxation of symmetry requirement is a
direct consequence of the topological-parity degeneracy, with which the left-
right asymmetry becomes irrelevant near the strong-tunneling fixed point. Our
work, as far as we know, is pioneering in studying the interplay between
topological-parity degeneracy and boundary QPTs where symmetries are otherwise
strictly requested. In addition, the broadened ballistic-transmission phase
space (Fig. 2) greatly reduces the experimental difficulty to observe the BKT
QPT of a DRLM. Remarkably, the phase diagram featuring MZMs, which greatly
contrasts that of a regular DRLM, provides a clear and unequivocal indicator
for detecting the presence of MZMs.

Figure 1: (a) The model of the system. The MZM (the red dot) couples to the
dot of the DRLM, with the amplitude $t_{\text{M}}$. The dot couples to two
leads with corresponding dimensionless tunneling amplitudes $V_{L}\tau_{c}$
and $V_{R}\tau_{c}$, with $\tau_{c}$ the cutoff in imaginary time. The leads
are dissipative, with the total impedance $\mathcal{R}$. When
$t_{\text{M}}=0$, the system reduces to the DRLM, with the zero-temperature
equilibrium conductance in (b) (when $r=2.5$). The marked-out phases A, B, C
and D are illustrated in (c), with red crosses indicating vanishing lead-dot
communications.

_Model_ — The system we consider [with the Hamiltonian
$H=H_{\text{leads}}+H_{\text{impurity}}+H_{\text{T}}$, see Fig. 1(a)] contains
two dissipative leads ($H_{\text{leads}}$) that couple to ($H_{\text{T}}$) a
Majorana-contained impurity part
$H_{\text{impurity}}=\epsilon_{d}d^{\dagger}d+it_{\text{M}}(d+d^{\dagger})\gamma_{1}/\sqrt{2}$,
where $d$ indicates the dot that couples to one MZM $\gamma_{1}$. The other
MZM $\gamma_{2}$ [not shown in Fig. 1(a)] decouples from $\gamma_{1}$, to
enforce the degeneracy between topological parities. For later convenience, we
define an auxiliary fermionic operator
$d_{\text{MZM}}\equiv(\gamma_{1}-i\gamma_{2})/\sqrt{2}$ with MZMs. The lead
Hamiltonian $H_{\text{leads}}$ contains two free fermionic leads, and bosonic
environmental modes. With those modes included, the lead-impurity tunneling
Hamiltonian
$H_{\text{T}}=V_{L}\psi^{\dagger}_{L}de^{i\varphi_{L}}+V_{R}\psi^{\dagger}_{R}de^{i\varphi_{R}}+h.c.$
contains the dynamical phase $\varphi_{\alpha}$ that is conjugate to the
charge number $N_{\alpha}=\int dx\psi^{\dagger}_{\alpha}(x)\psi_{\alpha}(x)$,
in the corresponding lead $\alpha=L,R$. After integrating out the
environmental modes, $\varphi_{L}=-\varphi_{R}=\varphi$, with the long-time
dynamical feature $\langle\varphi(\tau)\varphi(0)\rangle\sim 2r\ln\tau$ Ingold
and Nazarov (1992), where $r=\mathcal{R}e^{2}/h$ is the ratio between the
Ohmic impedance $\mathcal{R}$ of the dissipative leads, and the resistance
quanta $h/e^{2}$. The required dissipation has been realized in real
experiments Parmentier  _et al._ (2011); Mebrahtu  _et al._ (2012, 2013);
Jezouin  _et al._ (2013). Finally, two leads couple to the dot, with
hybridization parameters $V_{L}$ and $V_{R}$.

Dissipative modes introduce effective many-body interaction to the system
Nazarov (1989); Safi and Saleur (2004); Florens  _et al._ (2007). We then
bosonize lead fermions after extending semi-infinite leads into chiral full-1D
ones Kane and Fisher (1992); Giamarchi (2004)
$\psi_{\alpha}(x)=F_{\alpha}\exp[i\phi_{\alpha}(x)]/\sqrt{2\pi a}$, where $a$
is the short-distance cutoff, and $F_{\alpha}$ is the Klein factor that
enforces the fermionic anti-commutator. Finally, the bosonic phase
$\phi_{\alpha}$ satisfies the commutation relation
$[\phi_{\alpha}(x),\phi_{\alpha^{\prime}}(x^{\prime})]=i\pi\Theta(x-x^{\prime})\delta_{\alpha,\alpha^{\prime}}$.
These fields are related to the charge numbers in two leads
$N_{L,R}=e[\phi_{L,R}(\infty)-\phi_{L,R}(-\infty)]/(2\pi)$. To proceed, we
define the common and difference fields with the rotation
$\phi_{c,f}=(\phi_{L}\pm\phi_{R})/\sqrt{2}$. The post-rotation fields
$\phi_{c}$ and $\phi_{f}$ reflect the total charge number $N_{L}+N_{R}$, and
the number difference $N_{L}-N_{R}$, respectively. With bosonization and the
rotation, the dissipative phase can be combined with the difference field
$\phi_{f}^{\prime}\equiv(\phi_{f}+\varphi/\sqrt{2})/\sqrt{1+r}$. The addressed
dissipative phase
$\varphi^{\prime}\equiv(\sqrt{r}\phi_{f}-\varphi/\sqrt{2r})/\sqrt{1+r}$
decouples, reducing the system degrees of freedom.

_BKT QPT of the DRLM_ — When the MZM decouples ($t_{\text{M}}=0$), the system
reduces to the regular DRLM, where a boundary BKT QPT is predicted Liu  _et
al._ (2014) when $2<r<3$ and $V_{L}=V_{R}$ Foo  (see an alternative QPT
interpretation in Ref. Chung  _et al._ (2009)). When $V_{L}=V_{R}$ is larger
than a critical value Sup , the system flows to the ballistic-transmission
phase with conductance $e^{2}/h$ [Line B in the phase diagram of Fig. 1(b)].
Notice that this conductance is different from that of Ref. Liu and Baranger
(2011); Vernek  _et al._ (2014), due to the dissipative and interacting effect
Safi and Schulz (1995); Maslov and Stone (1995); Ponomarenko (1995); Kawabata
(1996); de C. Chamon and Fradkin (1997); Mebrahtu  _et al._ (2012); Hu and
Kane (2016). By contrast, with couplings smaller than the critical value, the
system instead flows to the isolated-impurity phase [A of Fig. 1(b)] with zero
conductance. The transition (between phases A and B) is of BKT-type, due to
the absence of QCP. Indeed, it shares equivalent scaling equations as that of
an XY-model, the pioneering model of topological transition Berezinskii
(1971); Kosterlitz (1974); Kosterlitz and Thouless (2018). More specifically,
when $r>2$, the lead-dot coupling is originally RG irrelevant [i.e., with
scaling dimension larger than one Altland and Simons (2010), see Eq. (2)]. The
couplings then, initially, decrease during the RG flow. If $V_{L}=V_{R}$ are
initially large enough, the dot charge number gradually becomes fixed at half-
filling Kane and Fisher (1992); Liu  _et al._ (2014). The common field,
reflecting the charge number fluctuation, then loses its scaling dimension
after the RG, leading to a relevant lead-dot coupling if $r<3$. Briefly, the
change of scaling dimension (upon the RG flow) leads to the BKT QPT: the
system can enter the ballistic-transmission phase, only if $V_{L}$ and $V_{R}$
become relevant before they vanish; otherwise the flow terminates in the
isolated-impurity phase.

Importantly, this BKT QPT requires a strict left-right symmetry $V_{L}=V_{R}$.
Indeed, if $V_{L}\neq V_{R}$, the system flows to other two zero-conductance
phases [C and D of Fig. 1(b)]. The transition
$\text{C}\leftrightarrow\text{B}\leftrightarrow\text{D}$, with state B as the
QCP, is not of BKT type. As our main result, we show that with a finite
Majorana-dot coupling $t_{\text{M}}$, the strict left-right symmetry to
observe the BKT QPT is relaxed. Indeed, as shown in Fig. 2, the ballistic-
transmission phase (area B) has extended from a 1D line [Fig.1(b)] to a 2D
space. The specific phase diagram changes under the different values of
$t_{\text{M}}$: when $t_{\text{M}}$ decreases [i.e. increasing cutoff $l_{C}$,
from Fig. 2(a) to Fig. 2(b)], the ballistic-transmission phase requires a
smaller lead-dot tunnelings, but has a worse tolerance against left-right
asymmetry.

Figure 2:  Phase diagrams (indicated by the corresponding zero-temperature
equilibrium conductance) and RG flow curves when $r=2.5$. (a), (b) are the
phase diagrams with $l_{C}=2$ and $l_{C}=3$, respectively, with the
corresponding $t_{\text{M}}$ ratio $1:0.37$. (c) High-temperature RG flows
curves of $K_{\text{charge}}$ and $K_{\text{asy}}$, for the symmetric (red
curves) and asymmetric (blue curves) situations. The red and blue diamonds
marked out points that belong to the ballistic-transmission
($K_{\text{charge}}<K_{C}$) and isolated-impurity ($K_{\text{charge}}>K_{C}$)
phases, respectively, if high-temperature RG flow terminates at their
corresponding $l$. (d) Low-temperature RG flow of $K_{\text{charge}}$ and
$K_{\text{asy}}$, with their initial values given by the corresponding
diamond.

_Coulomb gas RG_ — In this work, we obtain the phase diagram of Fig. 2 (a,b)
with the method of Coulomb gas RG Anderson  _et al._ (1970); Kane and Fisher
(1992). Briefly, within the Coulomb gas representation, one works with the
partition function

$\displaystyle\mathcal{Z}$
$\displaystyle=\sum_{n}\int_{0}^{1/T}\\!d\tau_{1}\\!\cdots\int_{0}^{\tau_{i}-\tau_{c}}\\!d\tau_{i+1}\\!\cdots\\!\\!\int_{0}^{\tau_{n-1}-\tau_{c}}\\!d\tau_{n}$
(1) $\displaystyle\big{\langle}H_{\text{T}}(\tau_{1})\cdots
H_{\text{T}}(\tau_{i+1})\cdots
H_{\text{T}}(\tau_{n})\big{\rangle}_{H_{\text{leads}}+H_{\text{impurity}}},$

where $n$ refers to the number of lead-impurity tunneling events during
imaginary time $1/T$, and $\tau_{c}$ is the cutoff: for any $i\in[1,n-1]$,
$\tau_{i}-\tau_{i+1}>\tau_{c}$ is enforced. Expansions of Eq. (1) are carried
out within the interaction picture Bruus and Flensberg (2004), over lead-
impurity tunnelings $H_{\text{T}}$. One can perform RG calculations Kane and
Fisher (1992) as follows. During each RG step, one starts with the cuttoff
$\tau_{c}$, and increases it to $\tau_{c}+d\tau_{c}$ (indicating the
temperature decreasing). The pairs of tunneling events at
$\\{\tau_{i},\tau_{i+1}\\}$ with condition
$\tau_{c}<|\tau_{i}-\tau_{i+1}|<\tau_{c}+d\tau_{c}$ should be integrated out.
RG flow equations are then obtained, after rescaling the cutoff, and enforcing
the invariance of $\mathcal{Z}$. In Eq. (1), the correlator of $H_{\text{T}}$
equals the product of the free-lead correlation (with the environmental modes
included) and the impurity correlation. The free-lead contribution can be
straightforwardly obtained Kane and Fisher (1992); Liu  _et al._ (2014).

The impurity correlator can be obtained by solving the Hamiltonian
$H_{\text{impurity}}$, leading to four impurity eigenstates: two even-parity
ones $|\psi_{1,3}\rangle=\frac{1}{\sqrt{2}}(\mp i|1,1\rangle+|0,0\rangle)$,
and two odd-parity ones $|\psi_{2,4}\rangle=\frac{1}{\sqrt{2}}(\mp
i|1,0\rangle+|0,1\rangle)$, where $|n,m\rangle$ indicates the state with $n$
and $m$ particle (either 0 or 1) in the dot and the auxiliary dot
$d_{\text{MZM}}$, respectively. Assuming $t_{\text{M}}>0$, states
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ become the degenerate ground states:
they have the energy $-t_{\text{M}}$, lower than that ($t_{\text{M}}$) of
$|\psi_{3}\rangle$ and $|\psi_{4}\rangle$.

Without lead-dot tunneling, the impurity part stays in two possible ground
states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$. After one lead-dot
tunneling, they become
$d|\psi_{1}\rangle\\!=\\!-\\!\frac{i}{2}(|\psi_{2}\rangle\\!+\\!|\psi_{4}\rangle),d^{\dagger}|\psi_{1}\rangle\\!=\\!\frac{i}{2}(|\psi_{2}\rangle\\!-\\!|\psi_{4}\rangle),d|\psi_{2}\rangle\\!=\\!-\\!\frac{i}{2}(|\psi_{1}\rangle\\!+\\!|\psi_{3}\rangle),$
and
$d^{\dagger}|\psi_{2}\rangle\\!=\\!\frac{i}{2}(|\psi_{1}\rangle\\!-\\!|\psi_{3}\rangle)$,
resulting in (non-eigen) final states with impurity parities different from
the initial ones. The impurity state before the next lead-impurity tunneling
depends on the evolution time Sup . For the low-temperature limit $T\ll
t_{\text{M}}$, tunnelings are rare, the impurity state evolves back to the
ground states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ before the next
tunneling. By contrast, for the high-temperature situation $T\gg
t_{\text{M}}$, lead-dot tunnelings are frequent, and the impurity state stays
in the non-eigenstate before the next lead-impurity tunneling. We thus have to
treat the low-temperature and high-temperature situations separately.

_The high-temperature RG flow_ — We start by visiting the high-temperature
regime $T\gg t_{\text{M}}$, where all four impurity states are energetically
allowed. Of this case, the impurity-lead tunneling histories with alternating
creation and annihilation operators dominate the partition function Sup . The
effect of the MZM-dot coupling is then negligible, leading to RG equations
that approximately equal that of a normal DRLM (following Ref. Liu  _et al._
(2014), or reducing Luttinger liquid spinful results of Ref. Schiller and
Ingersent (1997) to the spinless case),

$\displaystyle
dV_{L,R}/\\!dl\\!=\\!V_{L,R}\left[1\\!-\\!\frac{1+r}{4}(1\\!+\\!K_{\text{charge}}\\!\pm\\!2K_{\text{asy}})\right],$
(2) $\displaystyle
dK_{\text{charge}}/\\!dl\\!=\\!-4\tau_{c}^{2}[K_{\text{charge}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!K_{\text{asy}}(V_{L}^{2}\\!-\\!V_{R}^{2})],$
$\displaystyle
dK_{\text{asy}}/\\!dl\\!=\\!-2\tau_{c}^{2}[K_{\text{asy}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!(V_{L}^{2}\\!-\\!V_{R}^{2})],$

where parameter $l$ indicates the RG flow progress: it increases when
temperature $T$ decreases $l\sim\ln(T_{0}/T)$, with $T_{0}$ the initial
temperature (a device-dependent parameter that can only be characterized
experimentally). $K_{\text{charge}}$, with its initial value $1/(1+r)$, refers
to the scaling dimension of the common field $\phi_{c}$. $K_{\text{asy}}$,
which equals zero initially, refers to the scaling dimension from the left-
right asymmetry Kane and Fisher (1992); Liu  _et al._ (2014). For the
symmetric case $V_{L}=V_{R}$, $K_{\text{asy}}$ is fixed at zero [red dashed
line of Fig. 2(c)]. The scaling dimension of lead-dot tunnelings becomes
smaller than one (thus RG relevant) when $K_{\text{charge}}$ has become
smaller than $K_{C}=4/(1+r)-1$. For asymmetric situations, by contrast,
$|K_{\text{asy}}|$ increases during the RG flow [blue dashed line of Fig.
2(c)], after which the larger lead-dot coupling becomes more relevant than the
weaker one, leading to an even more significant asymmetry Sup . The high-
temperature flow terminates when $T\sim t_{\text{M}}$.

_Low-temperature RG, and the phase diagram_ — For the low-temperature case
$T\ll t_{\text{M}}$, only ground states are allowed as real states in the
impurity, and after one lead-dot tunneling, the impurity system prefers to
evolve to one of the ground states ($|\psi_{1}\rangle$ or $|\psi_{2}\rangle$),
before the next lead-dot tunneling event. Neighbouring dot operators can be
thus the same: a consequence of the presence of topological degeneracy. Low-
temperature RG equations of an MZM-involved system become Sup

$\displaystyle
dV_{L,R}/dl\\!=\\!V_{L,R}\left[1\\!-\\!\frac{1+r}{4}(1\\!+\\!K_{\text{charge}}\\!\pm\\!2K_{\text{asy}})\right],$
(3) $\displaystyle dK_{\text{charge}}/dl=0,\ \
dK_{\text{asy}}/dl=-2\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{asy}},$

with initial values of $K_{\text{charge}}$ and $K_{\text{asy}}$, importantly,
inherited from the RG-flow of the previous stage. The low-temperature flow Eq.
(3) contains two major Majorana-induced modifications: (i)
$K_{\text{charge}}$, which initially decreases in the high-temperature
situation [solid lines of Fig. 2(c)], now stops flowing [solid lines of Fig.
2(d)], and, more importantly, (ii) $K_{\text{asy}}$, which increases in the
asymmetric case of the high-temperature limit [dashed red line of Fig. 2(c)],
now instead begins to decrease to zero [dashed lines of Fig. 2(d)].
Modification (ii), remarkably, is the key element that leads to the
irrelevance of a weak asymmetry. Indeed, when $K_{\text{asy}}$ has vanished,
$dV_{L,R}/dl\\!=\\!V_{L,R}\left[1\\!-\\!(1+r)(1\\!+\\!K_{\text{charge}})/4\right]$,
where $V_{L}$ and $V_{R}$ simultaneously increase to perfect or decrease to
zero, disregarding a weak asymmetry. Noticeably, the irrelevance of asymmetry
also removes phases C and D in Fig. 1(c), where only one lead-impurity
coupling (the smaller one of $V_{L}$ or $V_{R}$) flows to zero. As the
consequence of (i), the final phase is determined by the value of
$K_{\text{charge}}$, larger or smaller than $K_{C}$, at the end of high-
temperature flow [as shown in Fig. 2(d)].

Figure 3: Lead-impurity tunneling events. (a) In a regular DRLM, the dot state
alternates in time. (b) In great contrast, with tunneling into the MZM
degeneracy (the grey arrow), two neighbouring dot-lead tunneling events (shown
by the blue and red arrows) can both contain a creation operator in the dot.

One can physically understand these two modifications with Fig. 3. Briefly, in
DRLM [with corresponding RG equations (2)] the dot state alternates in time
[Fig. 3(a)], where $N_{L}+N_{R}$ has only two possible (integer) options.
Field $\phi_{c}$, reflecting fluctuations of $N_{L}+N_{R}$, then gradually
loses its scaling dimension [reflected by the decreasing $K_{\text{charge}}$
of Eq. (2)] when $N_{L}+N_{R}$ is fixed by a half-filling dot Kane and Fisher
(1992). In great contrast, with Majorana involved, the parity of the impurity
part, including the dot and a coupled MZM, alternates in time [Fig. 3(b)]. The
corresponding common field $\phi_{c}$ now becomes a free 1D field. Its scaling
dimension $K_{\text{charge}}$ thus remains invariant during RG flow of Eq.
(3). We emphasize, importantly, the necessity of topological degeneracy in the
picture above. Indeed, if the DRLM dot couples to a zero-energy Andreev bound
state, the impurity part has a unique ground state (instead of two degenerate
ground states of the MZM situation). This case reduces to the DRLM with a
detuned dot Sup : a system with only the isolated-impurity phase Mebrahtu  _et
al._ (2012, 2013); Liu  _et al._ (2014).

Modifications above also explain features of Fig. 2(a, b). More specifically,
with a larger $t_{\text{M}}$ of Fig. 2(a), (i) for the symmetric case, a
larger value of $V_{L}=V_{R}$ is required to enter the ballistic-transmission
phase, and (ii) for the asymmetric case, the ballistic-transmission phase
tolerates a larger value of $V_{L}-V_{R}$. Feature (i) is a consequence of an
earlier termination, due to a larger $t_{\text{M}}$, of the high-temperature
flow. Feature (ii) is related to the non-monotonous curve of
$K_{\text{charge}}$ [blue solid line of Fig. 2(c)].

Figure 4: The effect of left-right asymmetry (assuming $V_{L}<V_{R}$) in a
regular [(a) and (b)] and Majorana-involved DRLM [(c) and (d)]. The red and
blue dots refer to the MZM and the residue degeneracy ($g=\sqrt{1+r}$) of the
ballistic-transmission phase, respectively. The grey dot instead indicates the
hybridized dot degree of freedom. RG flow prefers the fixed point with a
smaller degeneracy.

_Discussions_ — In this paper, we show that the MZM-introduced topological
degeneracy can relax the strict symmetry requirement $V_{L}=V_{R}$ for a BKT
QPT in a DRLM. This result can be understood with the g-theorem Affleck and
Ludwig (1991b, 1993), by visiting the effect of a weak asymmetry in the strong
tunneling regime. Briefly, the g-theorem claims that in a boundary QPT, the
phase with a smaller degeneracy is more stable. Indeed, in a regular
asymmetry-present DRLM, the fixed point where the dot is fully hybridized by
the stronger lead has a smaller degeneracy [degeneracy $g=1$, in Fig. 4(b)]
than that of the ballistic-transmission fixed point [$g=\sqrt{1+r}$ Zheng  _et
al._ (2014), Fig. 4(a)]. The zero-conductance phase is thus more stable, in a
regular DRLM. With the Majorana-parity provided, the fixed point where dot is
hybridized by the stronger lead has the degeneracy $\sqrt{2}$ [Fig. 4(d)], the
degeneracy of a single MZM: larger than that [Fig. 4(b)] of the MZM absent
situation. For the ballistic transmission case, the Majorana couples to the
residued dot-degeneracy (i.e., $\sqrt{1+r}$). The degeneracy of the impurity
part, obtained after the MZM-residue fusion Ginsparg (1988); Francesco  _et
al._ (1997) (see e.g., Ref. Affleck  _et al._ (1995) as a pioneering work that
applies fusion rules to boundary QPT studies), is generally hard to obtain.
Indeed, an arbitrary $r$ does not necessarily correspond to a discrete
Virasoro algebra with central charge $c<1$ Ginsparg (1988). We can however
evaluate the possible range of degeneracy, with fusion between primary fields
of the Ising model ($c=1/2$). Indeed, if $r=1$, we have the fusion
$[\sqrt{1+r}]\times[\sqrt{2}]=[\sqrt{2}]\times[\sqrt{2}]=[1]+[2]$, with
numbers indicating the degeneracy. For $r=3$, instead
$[\sqrt{1+r}]\times[\sqrt{2}]=[2]\times[\sqrt{2}]=[\sqrt{2}]+[2\sqrt{2}]$.
Taking the smaller degeneracy of both cases, the Majorana-involved ballistic-
transmission phase thus has a smaller degeneracy [$1<g<\sqrt{2}$, Fig. 4(c)]
than that ($\sqrt{2}$) of the zero-conductance one, indicating a stabilized
ballistic-transmission phase against a weak asymmetry. Importantly, similar
g-theorem analysis applies to situations with a finite $\epsilon_{d}$: the BKT
QPT thus tolerates also a small dot detuning Sup .

Before closure, we emphasize that the predicted significant MZM-induced
modification of the BKT QPT phase diagram can also signify the Majorana
existence, in a much more convincing way than the quantized plateau, as our
proposal detects the non-trivial degeneracy of an MZM. Encouraged by
persistent progress of Majorana hunting in nano-deveices Mourik  _et al._
(2012); Deng  _et al._ (2016); Albrecht  _et al._ (2016); Fornieri  _et al._
(2019); Ren  _et al._ (2019); Vaitiekėnas  _et al._ (2021); Song  _et al._
(2022); Wang  _et al._ (2022a); Aghaee  _et al._ (2022); Dvir  _et al._
(2023), especially for the cases with dissipative environments Liu (2013); Liu
_et al._ (2022); Zhang  _et al._ (2022a); Wang  _et al._ (2022b); Zhang  _et
al._ (2022b), we anticipate the experimental capability to observe our
predicted phenomenon. Prospectively, QPTs are also potentially capable to
identify Majorana non-localities Semenoff and Sodano (2006); Fu (2010); Béri
and Cooper (2012); Altland and Egger (2013); Béri (2013); Hao  _et al._
(2022), another crucial elements in topological quantum device.

_Acknowledgements_.— The authors thank Hao Zhang for valuable discussions.
This work was supported by the Innovation Program for Quantum Science and
Technology (Grant No. 2021ZD0302400), the National Natural Science Foundation
of China (Grants No. 11974198 and No. 12004040).

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Supplementary Information for “Influence of Majorana-parity degeneracy on the
boundary Berezinskii-Kosterlitz-Thouless quantum phase transition of a
dissipative resonant level”
Gu Zhang, Zhan Cao, and Dong E. Liu

(Dated: March 20, 2023)

In this Supplementary Information, we provide details on: (i) RG equations and
descriptions of BKT QPT in a regular DRLM, (ii) Correlation function of the
impurity part, (iii) derivations of RG equations in the low-temperature
regime, (iv) RG flow equations in the presence of a finite detuning of the dot
($\epsilon_{d}$), and (v) the situation that the DRLM dot couples to a
topologically trivial Andreev bound state.

##  I BKT transition in DRLM

In this section, we provide more details to present the existence of a BKT QPT
in DRLM. We start with the DRLM RG flow equation, Eq. (5) of the main text,

$\displaystyle\frac{dV_{L,R}}{dl}=V_{L,R}\left[1\\!-\\!\frac{1+r}{4}(1\\!+\\!K_{\text{charge}}\\!\pm\\!2K_{\text{asy}})\right],$
(S1)
$\displaystyle\frac{dK_{\text{charge}}}{dl}\\!=\\!-4\tau_{c}^{2}[K_{\text{charge}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!K_{\text{asy}}(V_{L}^{2}\\!-\\!V_{R}^{2})],$
$\displaystyle\frac{dK_{\text{asy}}}{dl}\\!=\\!-2\tau_{c}^{2}[K_{\text{asy}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!(V_{L}^{2}\\!-\\!V_{R}^{2})].$

Without the Majorana, the flow of Eq. (S1) continues until arriving at the
cutoff: either temperature of bias. For the symmetric case $V_{L}=V_{R}$, we
obtain the flow of $K_{\text{charge}}$ in Fig. S1(a). Here the dashed and
solid red lines have $V_{L}=V_{R}=0.095/\tau_{c}$ and
$V_{L}=V_{R}=0.1/\tau_{c}$, respectively. The dashed grey line indicates the
critical value [$K_{C}=4/(1+r)-1$] of $K_{\text{charge}}$: lead-dot tunnelings
become relevant when $K_{\text{charge}}<K_{C}$, and irrelevant given the
opposite situation, i.e., $K_{\text{charge}}>K_{C}$. If
$K_{\text{charge}}>K_{C}$ during the entire RG flow [the red dashed line of
Fig. S1(a)], couplings of the dot to both leads are always irrelevant, and the
system flows to the isolated-impurity phase with zero equilibrium conductance.
By contrast, if $K_{\text{charge}}<K_{C}$ after the RG flow [the red solid
line of Fig. S1(a)], system flows instead to the ballistic-transmission phase
with perfect conductance $e^{2}/h$. Analysis above leads to the RG flow
diagram of Ref. Liu  _et al._ (2014) (one can see Ref. Chung  _et al._ (2009)
for an alternative interpretation of this BKT QPT), shown in Fig. S1(b).

Figure S1: RG flows of a DRLM, when $r=2.5$. (a) The flow of
$K_{\text{charge}}$, when $V_{L}=V_{R}=0.095\tau_{c}$ (the dashed line) and
$V_{L}=V_{R}=0.1\tau_{c}$ (the solid line), respectively. Of the latter case,
$K_{\text{charge}}$ flows beneath the critical value $K_{C}=-1+4/(1+r)$,
leading to the ballistic-transmission phase. By contrast, of the former case,
$K_{\text{charge}}$ saturates to a value larger than $K_{C}$, leading to the
isolated-impurity phase. (b) The corresponding RG flow diagram Liu  _et al._
(2014). It greatly mimics that of a Kondo model, where a BKT QPT is predicted
between the fully screened and the isolated-impurity phases Hewson (1993). (c)
The flow of $V_{L}$ and $V_{R}$ if they start with different initial values.
$V_{L}$ becomes relevant (i.e., increases with an increasing $l$) after the RG
flow. The weaker one $V_{R}$ however remains irrelevant. (d) The flow of
$K_{\text{charge}}$ and $K_{\text{asy}}$, with the same parameters in (c).
$V_{L}$ becomes relevant when $K_{\text{charge}}>K_{C}$, due to a finite
$K_{\text{asy}}$.

Now we move to discuss the influence of asymmetry. Figs. S1(c) and S1(d)
present the flow of dot-lead couplings and scaling parameters
($K_{\text{charge}}$ and $K_{\text{asy}}$), respectively. As the stronger
lead-dot tunneling, $V_{L}$ becomes RG relevant (i.e., increases with an
increasing $l$) after the flow. The weaker one however remains irrelevant. In
(d), we further see that $V_{L}$ becomes relevant when
$K_{\text{charge}}>K_{C}$, due to a finite $K_{\text{asy}}$.

##  II Correlation function of the impurity part

In this section we provide detailed discussions on the eigenstates of the
impurity part, and the evolution of impurity states after lead-impurity
tunnelings. As a reminder, in this work we consider the impurity Hamiltonian

$H_{\text{impurity}}=\epsilon_{d}d^{\dagger}d+it_{\text{M}}(d+d^{\dagger})\gamma_{1}/\sqrt{2},$
(S2)

with $d$ and $d^{\dagger}$ the dot operators, and $\gamma_{1}$ representing
for the Majorana. As has been shown in the main text, in this work we take the
Coulomb gas RG method, and work in the interaction picture that contains
expansions over lead-impurity tunnelings. The corresponding correlation
functions then contain the free-lead part and the isolated impurity part. The
former one is straightforward within the Luttinger liquid formalism Kane and
Fisher (1992), while the latter one equals

$\displaystyle\big{\langle}\Pi_{i=1}^{n}D(\varepsilon_{i},\tau_{i})\big{\rangle}=2\Pi_{i=1}^{n-1}g_{\text{dot}}(\tau_{i}-\tau_{i+1},\varepsilon_{i},\varepsilon_{i+1}),$
(S3)

where we have, for convenience, defined another dot operator
$D(\varepsilon_{i},\tau_{i})$: it equals $d^{\dagger}(\tau_{i})$ and
$d(\tau_{i})$ if $\varepsilon=\pm 1$, respectively. The green’s function
$g_{\text{dot}}(\tau,\varepsilon,\varepsilon^{\prime})=[\exp(t_{\text{M}}\tau)-\varepsilon\varepsilon^{\prime}\exp(-t_{\text{M}}\tau)]/2$:
it equals $\sinh(t_{\text{M}}\tau)$ if dot operators of two involved
neighbouring kinks are both creation or annihilation operators (i.e.,
$\varepsilon_{i}\varepsilon_{i+1}=1$); otherwise
($\varepsilon_{i}\varepsilon_{i+1}=-1$) it equals $\cosh(t_{\text{M}}\tau)$.
One can understand RG flows under different temperatures by visiting the
corresponding features of Eq. (S3). Briefly, if temperature is large,
$\tau<1/T\ll 1/t_{\text{M}}$,
$\sinh(t_{\text{M}}\tau)\ll\cosh(t_{\text{M}}\tau)$, and lead-tunneling
histories with alternating dot states (i.e., with
$\varepsilon_{i}\varepsilon_{i+1}=-1$) are preferred. The correlation function
of the impurity part then greatly mimics that of a regular DRLM. By contrast,
for a small temperature, $\tau\gg 1/t_{\text{M}}$ in general,
$\sinh(t_{\text{M}}\tau)\approx\cosh(t_{\text{M}}\tau)$, and now the parity of
the entire impurity part alternates in time, leading to RG flows that are
distinct from that of a regular DRLM.

One can similarly understand the difference between high-temperature and low-
temperature situations, by looking into the time-dependent evolution of the
impurity states. Without loss of generality, we assume the state
$|\psi_{2}\rangle=\frac{1}{\sqrt{2}}(-i|1,0\rangle+|0,1\rangle)$ of the
impurity at a moment. After receiving one electron via a lead-impurity
tunneling, impurity state becomes $|1,1\rangle$. As a non-eigen state,
$|1,1\rangle$ evolves in time, becoming
$\exp(-H_{\text{impurity}}\tau)|1,1\rangle=\cosh(t_{\text{M}}\tau)|1,1\rangle+i\sinh(t_{\text{M}}\tau)|0,0\rangle$,
after $\tau$ of the tunneling. For the high-temperature limit, tunnelings are
frequent, $t_{\text{M}}\tau\ll 1$ is a small quantity. The impurity state
$\cosh(t_{\text{M}}\tau)|1,1\rangle+i\sinh(t_{\text{M}}\tau)|0,0\rangle\approx|1,1\rangle$
approximately equals that right after the lead-impurity tunneling. By
contrast, for the low-temperature limit, tunnelings are rare,
$t_{\text{M}}\tau\gg 1$ is normally large. Of this case,
$\cosh(t_{\text{M}}\tau)\approx\sinh(t_{\text{M}}\tau)$, and the impurity
state approaches the ground state $|\psi_{1}\rangle$, which has an opposite
parity from the initial one $|\psi_{2}\rangle$, before the next lead-dot
tunneling. The RG flow is then distinct from that of a regular DRLM.

##  III Low-temperature RG flow equations

In the section we provide detailed derivations of the low-temperature RG flow
equations, Eq. (3) of the main text. Briefly, within the Coulomb-gas RG
formalizm, one deals with the partition function Kane and Fisher (1992)

$Z=\sum_{\left\\{q_{j}\right\\}}\sum_{\left\\{d_{j}\right\\}}\prod_{j}t_{j}\Big{\langle}\Pi_{j}e^{i\frac{1}{\sqrt{2}}[q_{j}\phi_{c}(\tau_{j})+d_{j}\phi_{f}(\tau_{j})]}\Big{\rangle}_{H_{\text{leads}}}\Big{\langle}\Pi_{j}D(\varepsilon_{j},\tau_{j})\Big{\rangle}_{H_{\text{impurity}}}$
(S4)

for a two-lead system, where $t_{j}=V_{L},V_{R}$ refers to the amplitude of
the $j$th tunneling event that occurs at time $\tau_{j}$. This tunneling event
contains the impurity and lead operators $D(\varepsilon_{j},\tau_{j})$ and
$\exp\left\\{i\frac{1}{\sqrt{2}}[q_{j}\phi_{c}(\tau_{j})+d_{j}\phi_{f}(\tau_{j})]\right\\}$,
respectively. Here $q_{j}$ and $d_{j}$ reflect the change of total lead charge
($N_{L}+N_{R}$ of the main text), and the charge difference in two leads
($N_{L}-N_{R}$ of the main text), respectively: they are charges considered in
a regular DRLM Liu  _et al._ (2014). The dot operator $D$ equals
$d^{\dagger}(\tau_{i})$ and $d(\tau_{i})$ if $\varepsilon=\pm 1$,
respectively.

Figure S2: Four possible lead-impurity tunneling processes, of the $i$th
tunneling. The cyan and purple dot refer to two different impurity ground
states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, respectively. The red small
dot refers to a tunneling electron. In each figure, the process at the right
side is the inverse process of that at the left side. Charges $q_{i}$ and
$d_{i}$ refer to the change of the total charge number, and the charge number
differences in two leads, respectively. The third charge $p_{i}$ refers to the
change of parity of the impurity part.

For the low-temperature situation, $\sinh(\tau t_{\text{M}})\approx\cosh(\tau
t_{\text{M}})\approx\exp(|\tau t_{\text{M}}|)$. In addition, in this regime,
the parity of the impurity part (instead of the dot occupation number of the
high) alternates in time. We thus introduce another charge number, the parity
charge $p_{j}$, to indicate the change of parity of the impurity part, before
the $j$th tunneling event. The partition function then becomes

$\mathcal{Z}\\!=\\!\sum_{N}\sum^{\prime}_{\left\\{q_{j}\right\\}}\sum^{\prime}_{\left\\{d_{j}\right\\}}\sum^{\prime}_{\left\\{p_{j}\right\\}}\prod_{j=1}^{N}t_{j}\int_{0}^{\beta-\tau_{c}}\\!d\tau_{N}\\!\cdot\cdot\cdot\\!\int_{0}^{\tau_{j+1}-\tau_{c}}\\!d\tau_{j}\\!\int_{0}^{\tau_{j}-\tau_{c}}\\!d\tau_{j-1}\\!\cdot\cdot\cdot\int_{0}^{\tau_{2}-\tau_{c}}\\!d\tau_{1}e^{-\sum_{i<j}\mathcal{V}_{ij}}\prod_{j=1}^{N-1}e^{|\tau_{j}-\tau_{j+1}|t_{\text{M}}}$
(S5)

where prime indicates $\sum_{j}q_{j}=\sum_{j}d_{j}=\sum_{j}p_{j}=0$, and the
“potential interaction” equals

$\mathcal{V}_{ij}=\frac{1+r}{2}\left[\frac{1}{1+r}q_{i}q_{j}+d_{i}d_{j}+K_{\text{parity}}p_{i}p_{j}+K_{\text{asy}}(p_{i}d_{j}+p_{j}d_{i})+\frac{1}{1+r}K_{\text{asy}}^{\prime}(p_{i}q_{j}+p_{j}q_{i})\right]\ln(\tau_{i}-\tau_{j})/\tau_{c},$
(S6)

where $K_{\text{parity}}$ is the scaling factor of parity charge. Its initial
value equals zero, since the lead fermion, after bosonization, contains only
the common and different fields. Due to the alternating parity, another
asymmetric scaling $K_{\text{asy}}^{\prime}$ is introduced, refering to the
asymmetric scaling of the charge sector. It also has the initial value zero.

Following Eq. (S5), two neighbouring kinks are not allowed to occur within the
time cutoff $\tau_{c}$. At the starting of each RG step, the cutoff increases
to $\tau_{c}+d\tau_{c}$. Neighbouring kinks then have the chance to occur
within the new cutoff, i.e.,
$\tau_{c}<|\tau_{i}-\tau_{i+1}|<\tau_{c}+d\tau_{c}$. There are two different
close kink pairs, as shown in Fig. S3. In Fig. S3(a), the state $S_{i-1}$ and
$S_{i+1}$ are different, called as the non-neutral pair. Of this case, we can
simply combine two kinks at $\tau_{i}$ and $\tau_{i+1}$. Of the second case,
in Fig. S3(b), these two states equal, known as the neutral pair. Of this
case, both kinks need to be integrated out. In this work the BKT QPT occurs
when transmissions are small. We thus ignore the non-neutral pairs. The
inclusion of neutral pairs changes only the critical value of the coupling
constant $V_{C}$, but not the qualitative feature.

Figure S3: Two different pairs to deal with in Coulomb gas RG. (a) Of the non-
neutral pair situation, the system state $S_{i-1}$ before the close kink pairs
(that occur at $\tau_{i}$ and $\tau_{i+1}$, respectively) are different from
that $S_{i+1}$ after the pairs. One needs to combine non-neutral kinks at
$\tau_{i}$ and $\tau_{i+1}$ into a single one. (b) When the pre and post-kink-
pair states equal $S_{i-1}=S_{i+1}$, kinks at both $\tau_{i}$ and $\tau_{i+1}$
should be removed.

The integral over neutral pair leads to the extra potential interaction
$\delta\mathcal{V}_{ij}$

$\delta\mathcal{V}_{ij}\\!=\\!\sum_{\alpha}t_{i}^{2}\frac{1\\!+\\!r}{2}\left[\frac{1}{1\\!+\\!r}q_{i}q_{j}\\!+\\!d_{i}d_{j}\\!+\\!K_{\text{parity}}p_{i}p_{j}\\!+\\!K_{\text{asy}}(p_{i}d_{j}\\!+\\!p_{j}d_{i})\\!+\\!\frac{1}{1\\!+\\!r}K_{\text{asy}}^{\prime}(p_{i}q_{j}\\!+\\!p_{j}q_{i})\right]\\!\tau_{c}\partial_{\tau}\ln(\tau=\tau_{j})/\tau_{c},$
(S7)

after summation over all possible kink choices shown in Fig. S2. Notice that
since the parity charge alternates in time, $p=p_{i}$ is enforced, for all
possible neutral pairs. One can then express other two charges of the neutral
pair in terms of the parity charge.

After summing over all possible kinks, we arrive at extra potential generated
from the neutral pair

$\delta\mathcal{V}_{j}=-2(1+r)\tau_{c}(V_{L}^{2}+V_{R}^{2})\left[K_{\text{parity}}pp_{j}+K_{\text{asy}}pd_{j}+K_{\text{asy}}^{\prime}pq_{j}\right]d\tau_{c}\ln(\tau_{i}-\tau).$
(S8)

The extra contribution above can be combined with other kinks, leading to the
renormalization of scaling parameters

$\frac{dK_{\text{parity}}}{dl}=-4\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{parity}},\
\frac{dK_{\text{asy}}}{dl}=-2\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{asy}},\
\frac{dK_{\text{asy}}^{\prime}}{dl}=-2\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{asy}}^{\prime}.$
(S9)

From Eq. (S9), we see that $K_{\text{parity}}$ and $K_{\text{asy}}^{\prime}$,
with vanishing initial values, are both fixed at zero: the parity charge will
not introduce extra asymmetric scaling parameters. In addition, although
$K_{\text{asy}}$ is finite at the end of the high-temperature flow, it
gradually decreases to zero in the low-temperature flow. The asymmetry is thus
irrelevant in the low-temperature regime.

##  IV RG flow of a finite dot energy $\epsilon_{d}$

In the main text, we take $\epsilon_{d}=0$, to focus on the effect of a left-
right asymmetry $V_{L}-V_{R}$. In this section, we take a finite
$\epsilon_{d}$, to study whether the topological-parity degeneracy can also
“restore” the detuning, or “particle-hole” asymmetry at the dot.

For the high-temperature situation, Majorana-coupling $t_{\text{M}}$ is
negligible, and RG flow mimics that of a regular DRLM. RG flow equations with
a finite $\epsilon_{d}$ then become (after reducing the spinful equations of
Ref. Schiller and Ingersent (1997) to the spinless case)

$\displaystyle\frac{dV_{L,R}}{dl}=V_{L,R}\left[1\\!-\\!\frac{1+r}{4}(1\\!+\\!K_{\text{charge}}\\!\pm\\!2K_{\text{asy}})\right],$
(S10)
$\displaystyle\frac{dK_{\text{charge}}}{dl}\\!=\\!-2\tau_{c}^{2}[K_{\text{charge}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!K_{\text{asy}}(V_{L}^{2}\\!-\\!V_{R}^{2})]\left(e^{\epsilon_{d}\tau_{c}}+e^{-\epsilon_{d}\tau_{c}}\right),$
$\displaystyle\frac{dK_{\text{asy}}}{dl}\\!=\\!-\tau_{c}^{2}[K_{\text{asy}}(V_{L}^{2}\\!+\\!V_{R}^{2})\\!+\\!(V_{L}^{2}\\!-\\!V_{R}^{2})]\left(e^{\epsilon_{d}\tau_{c}}+e^{-\epsilon_{d}\tau_{c}}\right),$
$\displaystyle\frac{d\epsilon_{d}}{dl}=\epsilon_{d}+(V_{L}^{2}+V_{R}^{2})\tau_{c}\left(e^{-\epsilon_{d}\tau_{c}}-e^{\epsilon_{d}\tau_{c}}\right).$

Clearly a finite $\epsilon_{d}$ is RG relevant, and increases when decreasing
the temperature. When $t_{\text{M}}=0$, the lead-dot tunnelings are then
turned off at zero temperature, where the dot is either fully unoccupied (when
$\epsilon_{d}<0$), or empty (when $\epsilon_{d}>0$). In addition, a finite
$\epsilon_{d}$, either positive or negative, accelerates the flow of scaling
factors $K_{\text{asy}}$ and $K_{\text{charge}}$. Following Eq. (S10), in the
high-temperature regime, a finite dot detuning $\epsilon_{d}$ increases the
system asymmetry, and tend to sabotage the BKT transition.

In the low-temperature regime, RG flow features above are however greatly
modified. Indeed, in this regime, the dot operator has the following influence
on the ground states

$\displaystyle
d|\psi_{1}\rangle=-\frac{i}{2}(\mathcal{A}|\psi_{2}\rangle+\mathcal{B}|\psi_{4}\rangle),\
\ $ $\displaystyle
d^{\dagger}|\psi_{1}\rangle=\frac{i}{2}(\mathcal{B}|\psi_{2}\rangle-\mathcal{A}|\psi_{4}\rangle),$
(S11) $\displaystyle
d|\psi_{2}\rangle=-\frac{i}{2}(\mathcal{A}|\psi_{1}\rangle+\mathcal{B}|\psi_{3}\rangle),\
\ $ $\displaystyle
d^{\dagger}|\psi_{1}\rangle=\frac{i}{2}(\mathcal{B}|\psi_{1}\rangle-\mathcal{A}|\psi_{3}\rangle),$

with two $\epsilon_{d}$-dependent positive parameters

$\mathcal{A}=\sqrt{1+\frac{\epsilon_{d}}{\sqrt{\epsilon_{d}^{2}+4t_{\text{M}}}}},\
\
\mathcal{B}=\sqrt{\frac{4t_{\text{M}}^{2}}{4t_{\text{M}}^{2}+\epsilon_{d}(\epsilon_{d}+\sqrt{\epsilon_{d}^{2}+4t_{\text{M}}^{2}})}}.$
(S12)

As the consequence, we obtain the low-temperature RG equations

$\displaystyle\frac{dV_{L,R}}{dl}\\!=\\!V_{L,R}\left[1\\!-\\!\frac{1+r}{4}(1\\!+\\!K_{\text{charge}}\\!\pm\\!2K_{\text{asy}})\right],\frac{dK_{\text{parity}}}{dl}=-4\mathcal{A}\mathcal{B}\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{parity}},$
(S13)
$\displaystyle\frac{dK_{\text{asy}}}{dl}=-2\mathcal{A}\mathcal{B}\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{asy}},\
\frac{dK_{\text{asy}}^{\prime}}{dl}=-2\mathcal{A}\mathcal{B}\tau_{c}^{2}(V_{L}^{2}+V_{R}^{2})K_{\text{asy}}^{\prime},\
\frac{dK_{\text{charge}}}{dl}=0,$

where the $\epsilon_{d}$-dependent factors $\mathcal{A}$ and $\mathcal{B}$
influence the flow of all scaling parameters. However, since
$\mathcal{A}\mathcal{B}>0$, the flow patterns of all scaling parameters
qualitatively agree with that of the fine-tuned situation, i.e., Eq. (S9), and
Eq. (3) of the main text. A finite $\epsilon_{d}$ simply modifies the
prefactors of RG flow equations. These modifications change only the critical
value of $V_{L}$, $V_{R}$ at the boundary of the transition. The qualitative
feature of the phase diagram however remains invariant.

##  V Coupling the DRLM dot to an Andreev bound state

In the main text, we have shown that by coupling the dot of a DRLM to an MZM,
the BKT QPT hosted by the DRLM will be greatly modified. It is also
illustrated and highlighted that the modification of the BKT phase diagram is
due to the topological degeneracy provided by the Majorana. To better
understand the importance of the topological degeneracy, we consider another
example where the dot instead couples to an Andreev bound state (ABS, with the
state operator $d_{\text{ABS}}$), with has instead a trivial local degeneracy.
Of this case, the impurity part has the Hamiltonian

$H_{\text{impurity}}=\epsilon_{\text{dot}}d^{\dagger}d+\epsilon_{\text{ABS}}d_{\text{ABS}}^{\dagger}d_{\text{ABS}}+t_{1}(d_{\text{dot}}^{\dagger}d_{\text{ABS}}+d_{\text{ABS}}^{\dagger}d_{\text{dot}})+t_{2}(d_{\text{dot}}^{\dagger}d^{\dagger}_{\text{ABS}}+d_{\text{ABS}}d_{\text{dot}}),$
(S14)

where $t_{1}=t_{2}$ for an MZM. For simplicity we have assumed $t_{1}$ and
$t_{2}$ to be both real numbers. The impurity Hamiltonian has four
eigenvalues,

$\displaystyle\epsilon_{1}$
$\displaystyle=\frac{1}{2}\left[\epsilon_{\text{dot}}+\epsilon_{\text{ABS}}-\sqrt{(\epsilon_{\text{dot}}+\epsilon_{\text{ABS}})^{2}+4t_{1}^{2}}\right],\
\
\epsilon_{2}=\frac{1}{2}\left[\epsilon_{\text{dot}}+\epsilon_{\text{ABS}}-\sqrt{(\epsilon_{\text{dot}}-\epsilon_{\text{ABS}})^{2}+4t_{2}^{2}}\right],$
(S15) $\displaystyle\epsilon_{3}$
$\displaystyle=\frac{1}{2}\left[\epsilon_{\text{dot}}+\epsilon_{\text{ABS}}+\sqrt{(\epsilon_{\text{dot}}+\epsilon_{\text{ABS}})^{2}+4t_{1}^{2}}\right],\
\
\epsilon_{4}=\frac{1}{2}\left[\epsilon_{\text{dot}}+\epsilon_{\text{ABS}}+\sqrt{(\epsilon_{\text{dot}}-\epsilon_{\text{ABS}})^{2}+4t_{2}^{2}}\right].$

Apparently, energies $\epsilon_{1}$ and $\epsilon_{2}$ are candidates of the
ground-state energy. If either $\epsilon_{\text{dot}}$ or
$\epsilon_{\text{ABS}}$ (corresponding to an accidentally fine-tuned ABS) is
zero, $\epsilon_{1}\neq\epsilon_{2}$ and the dot-ABS state reduces an
effectively detuned dot: of this case, the BKT QPT does not exist, in great
contrast to the enhanced BKT phase diagram of the MZM situation.

Specially, these two energies only equal when
$\epsilon_{\text{dot}}\epsilon_{\text{ABS}}+t_{1}^{2}=t_{2}^{2}$, leading to
an accidental degenerate state. We emphasize, however, that this accidental
degenerate situation will not sabotage the MZM evidence in real experiments.
Indeed, in real experiments, the tuning of $V_{L}$ and $V_{R}$, with which the
phase diagram is detected, will effectively tune the dot energy
$\epsilon_{\text{dot}}$ Mebrahtu  _et al._ (2012) (, and may also influence
$\epsilon_{\text{ABS}}$). As the consequence, we expect that the accidental
degenerate situation only leads to scattered full-transmission points in phase
space, being distinct from the real MZM situation.

Following analysis above, we clearly see that the Majorana-induced
modification of the low-temperature RG flow, physically explained in Fig. 3,
does require the presence of a topological degeneracy. A fine-tuned trivial
ABS ($\epsilon_{\text{ABS}}=0$) instead leads to always the isolated impurity
phase.

## References

  * Liu  _et al._ (2014) Dong E. Liu, Huaixiu Zheng, Gleb Finkelstein, and Harold U. Baranger, “Tunable quantum phase transitions in a resonant level coupled to two dissipative baths,” Phys. Rev. B 89, 085116 (2014).
  * Chung  _et al._ (2009) Chung-Hou Chung, Karyn Le Hur, Matthias Vojta, and Peter Wölfle, “Nonequilibrium transport at a dissipative quantum phase transition,” Phys. Rev. Lett. 102, 216803 (2009).
  * Hewson (1993) A. C. Hewson,  _The Kondo Problem to Heavy Fermions_ (Cambridge Univ. Press, Cambridge UK, 1993).
  * Kane and Fisher (1992) C. L. Kane and Matthew P. A. Fisher, “Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas,” Phys. Rev. B 46, 15233–15262 (1992).
  * Schiller and Ingersent (1997) A. Schiller and K. Ingersent, “Renormalization-group study of a magnetic impurity in a luttinger liquid,” Europhysics Letters 39, 645 (1997).
  * Mebrahtu  _et al._ (2012) H. T. Mebrahtu, I. V. Borzenets, Dong E. Liu, Huaixiu Zheng, Yu. V. Bomze, A. I. Smirnov, H. U. Baranger, and G. Finkelstein, “Quantum phase transition in a resonant level coupled to interacting leads,” Nature 488, 61–64 (2012).


3.0pt\raise
0.0pt\hbox{$\textstyle{T_{\tilde{q}}\tilde{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
133.62537pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
133.62537pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q}{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
182.72552pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
182.72552pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

As the off-diagonal term in (5.12) can be turned off in a one-parameter
family, one has a canonical orientation-preserving isomorphism

$\det T_{\tilde{q}}\tilde{\mathcal{M}}\cong\det
T_{\tilde{q}}\tilde{\mathcal{M}}^{\rm vert}\otimes\det T_{q}{\mathcal{M}}.$

Furthermore, considering the differential of the evaluation maps which are
isomorphisms onto corresponding normal spaces to the constraints. There is a
commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{T_{\tilde{q}}\tilde{\mathcal{M}}^{\rm
vert}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
86.3927pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.80647pt\raise-20.19444pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.94722pt\hbox{$\scriptstyle{D_{\tilde{q}}{\rm ev}^{\rm
vert}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
45.94635pt\raise-28.44444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
86.3927pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{\tilde{q}}\tilde{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
133.62537pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
98.00903pt\raise-30.16667pt\hbox{\hbox{\kern 0.0pt\raise
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146.17545pt\raise-20.19444pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
146.17545pt\raise-30.55559pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
182.72552pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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182.72552pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-40.3889pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
35.33856pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.33856pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\tilde{N}^{\rm
vert}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
92.23125pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
92.23125pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\tilde{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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182.72552pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
182.72552pt\raise-40.3889pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

By taking determinants, one obtains a commutative diagram

---
$\textstyle{\det
T_{\tilde{q}}\tilde{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det(D_{\tilde{q}})}$$\textstyle{\det\tilde{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\det
T_{\tilde{q}}\tilde{\mathcal{M}}^{\rm vert}\otimes\det
T_{q}{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det(D_{\tilde{q}}^{\rm
vert})\otimes\det(D_{q})}$$\textstyle{\det\tilde{N}^{\rm vert}\otimes\det N}$

where vertical arrows are orientation preserving. Now the top horizontal arrow
has a sign ${\rm sign}(\tilde{q})$ contributing to the GW invariant of $P$,
the bottom arrow has a sign ${\rm sign}(\tilde{q}^{\rm vert})\cdot{\rm
sign}(q)$ with the factors contributing to the graph GW invariant and the GW
invariant of the base. It follows that

${\rm sign}(\tilde{q})={\rm sign}(\tilde{q}^{\rm vert})\cdot{\rm sign}(q).$

##  6\. A fibration which does not split

The cohomological splitting stated in Theorem A and Corollary 1.2 potentially
fails for certain characteristics, if we drop the the non-divisibility
assumption on the Gromov–Witten invariant. Here we provide such an example.

######  Theorem 6.1.

There exists a smooth Fano variety $B$ (hence rationally connected) and a
smooth projective family $P\to B$ such that

$H^{*}(P;{\mathbb{F}}_{2})\neq H^{*}(B;{\mathbb{F}}_{2})\otimes
H^{*}(M;{\mathbb{F}}_{2}).$

We first collect some relevant topological arguments from [Eke09].

######  Definition 6.2.

Let $G$ be a complex reductive Lie group. A topological space $X_{n}$ is
called an $n$-th cohomological approximation of the classifying space $BG$ if
there is a map $X_{n}\to BG$ inducing an isomorphism

$H^{j}(BG;{\mathbb{Z}})\to H^{j}(X_{n};{\mathbb{Z}})$

for all $j\leq n$.

Following the argument of [Eke09, Proposition 2.1], one has the following non-
splitting result.

######  Lemma 6.3.

Let $G_{\rm Borel}\subset G$ be a Borel subgroup and $F:=G/G_{\rm Borel}$ the
flag variety. Suppose $H^{j}(BG;{\mathbb{Z}})$ has $p$-torsion for some $j$.
Then for $n$ sufficiently large and any $n$-th cohomological approximation
$X_{n}$ of $BG$, if $P_{n}\to X_{n}$ is the pullback $G$-bundle and
$Y_{n}:=P_{n}\times_{G}F$, then

$H^{*}(Y_{n};{\mathbb{F}}_{p})\neq H^{*}(F;{\mathbb{F}}_{p})\otimes
H^{*}(X_{n};{\mathbb{F}}_{p}).$

###### Proof.

Consider the universal $F$-fibration $Y_{\infty}:=EG\times_{G}F$. When $n$ is
sufficiently large, the natural map $Y_{n}\to Y_{\infty}$ induces an
isomorphism

$H^{j}(Y_{\infty};{\mathbb{Z}})\cong H^{j}(Y_{n};{\mathbb{Z}}).$

On the other hand, let $T\subset G_{\rm Borel}$ be the maximal torus. One has

$Y_{\infty}=EG\times_{G}F\cong EG/G_{\rm Borel}\simeq BG_{\rm Borel}\simeq BT$

whose cohomology is a polynomial algebra and has no torsion. However,
$H^{j}(X_{n};{\mathbb{Z}})\cong H^{j}(BG;{\mathbb{Z}})$ has $p$-torsion. Hence

$H^{*}(Y_{n};{\mathbb{Z}})\neq H^{*}(F;{\mathbb{Z}})\otimes
H^{*}(X_{n};{\mathbb{Z}})$

as the right hand side has $p$-torsion. The conclusion follows by tensoring
both sides with ${\mathbb{F}}_{p}$. ∎

_Proof of Theorem 6.1:_ Ottem–Rennemo [OR24] constructed a sequence of
cohomological approximations of the classifying spaces for a reductive group
$G=GO(4)^{\circ}$ by a sequence of smooth projective varieties $X_{n}$ (see
[OR24, Proposition 3.5]) with algebraic $G$-bundles $P_{n}\to X_{n}$. Their
computation ([OR24, Corollary 3.6]) shows
$H^{3}(BG;{\mathbb{Z}})={\mathbb{Z}}_{2}$. On the other hand, by [OR24,
Theorem 4.1], $X_{n}$ is Fano. Forming the associated
$F=G/G_{\operatorname{Borel}}$ bundle gives a fibration $Y_{n}\to X_{n}$ with
fiber the flag variety $F$. To see that that $Y_{n}$ is projective, choose a
$G$-equivariant embedding $F\hookrightarrow\mathbf{P}(\mathbb{V})$, where
$\mathbb{V}$ is some linear representation of $G$. The variety
$Z_{n}:=\mathbf{P}(\mathbb{V})\times_{G}P_{n}$ is projective, since it is the
projectivization of the vector bundle $\mathbb{V}\times_{G}P_{n}$ over $X_{n}$
and the projectivization of any vector bundle over $X_{n}$ is projective (over
$X_{n}$ and hence over $\mathbb{C}$) [Sta24, Tag 01W7]. The variety $Y_{n}$ is
a closed subvariety of $Z_{n}$ and is thus also projective. Hence, this
provides the example required by the statement of Theorem 6.1.

######  Remark 6.4.

We sketch an alternative construction of a fibration $P\to B$ satisfying the
hypotheses of Theorem 6.1. The base $B$ will be again be one of the smooth
Fano varieties $X_{n}$ from [OR24]. There is a non-trivial
$\mathbb{P}^{1}$-bundle over $X_{n}$ whose Brauer class represents the non-
trivial element $b\in H^{3}(X_{n},\mathbb{Z})$ (see [OR24, A.8]). Let $h\in
H^{2}(\mathbb{P}^{1},\mathbb{Z})$ be the hyperplane class. It follows from the
general theory of Brauer classes that on the $E^{3}$ page of the Leray
spectral sequence, we have

$d^{3}(h)=b.$

In particular, the spectral sequence fails to degenerate. This construction
applies more generally to smooth rationally connected projective varieties
with torsion in $H^{3}(X_{n},\mathbb{Z})$.

##  Appendix A Pseudocycles in characteristic $p$

For semi-positive symplectic manifolds, Gromov–Witten invariants are defined
via intersections of pseudocycles (see [RT95][MS04]). One can show that
pseudocycles up to cobordism are in one-to-one correspondence with integral
homology classes (see [Zin08]). Moreover, as intersection numbers are
cobordism invariant, the Gromov–Witten invariants are functions of integral
homology classes (modulo torsion).

We would like to consider the analogue of Zinger’s theorem in finite
characteristic. The results are well-known to experts (cf. [Wil24]), and we
include the discussion for completeness.

######  Definition A.1.

Let $p$ be a prime number. An oriented $k$-dimensional $p$-pseudocycle in a
smooth manifold $X$ is a smooth map $f:W\to X$ from a $k$-dimensional oriented
manifold with boundary $W$ to $X$ satisfying the following property:

  1. (1)

$f(W)$ is precompact in $X$.

  2. (2)

There is a $p$-fold oriented covering $\partial W\to V$ and a smooth map
$g:V\to X$ such that $f|_{\partial W}$ is the pullback of $g$.

  3. (3)

The frontier is small. More precisely, the $\Omega$-set of $f$ is

$\Omega_{f}:=\bigcap_{K\subset W\ {\rm compact}}\overline{f(W\setminus K)}.$

We require that it has dimension at most $k-2$.

In particular, a pseudocycle is a $p$-pseudocycle with $\partial W=\emptyset$.

Let $\tilde{\mathcal{H}}_{k}^{(p)}(X)$ be the set of all oriented
$k$-dimensional $p$-pseudocycles. It has the structure of an abelian group
where the sum of two $p$-pseudocycles is their disjoint union and the inverse
of a $p$-pseudocycle is the same map with domain orientation reversed.

######  Definition A.2.

Two oriented $k$-dimensional $p$-pseudocycles $f_{0}:W_{0}\to X$,
$f_{1}:W_{1}\to X$ are said to be $p$-cobordant if there exists another smooth
map $g:V\to X$ from an oriented $k$-dimensional manifold $V$ without boundary,
a smooth map $\tilde{f}:\tilde{W}\to X$ from an oriented $k+1$-dimensional
manifold $\tilde{W}$ with boundary with

$\tilde{f}(\tilde{W})\ {\rm is\ precompact},\ {\rm dim}\Omega_{\tilde{f}}\leq
k-1,$

an oriented diffeomorphism

$\partial\tilde{W}\cong-{\rm Int}W_{0}\sqcup{\rm Int}W_{1}\sqcup W^{\prime}$

where $W^{\prime}$ is a $k$-dimensional manifold without boundary (having the
induced orientation) such that the restriction of $\tilde{f}$ to ${\rm
Int}W_{0}$ resp. ${\rm Int}W_{1}$ coincides with $f_{0}$ resp. $f_{1}$, and an
oriented $p$-fold covering $W^{\prime}\to V$ such that
$\tilde{f}|_{W^{\prime}}$ is the pullback of $g$.

One needs to do some simple modifications to make certain naive operations
satisfy the above definition. For example, a $p$-pseudocycle $f:W\to X$ is
$p$-cobordant to itself. However, the naive map

$\tilde{f}:W\times[0,1]\to X,\ \tilde{f}(x,t)=f(x)$

is a $p$-cobordism only after removing the corner $\partial W\times\\{0,1\\}$.

It is clear that $p$-cobordant is an equivalence relation and respect the
additive structure. Let ${\mathcal{H}}_{k}^{(p)}(X)$ be the set of
$p$-cobordant classes of $p$-pseudocycles, which is an abelian group. As $p$
times of any $p$-pseudocycle is $p$-cobordant to the empty set via the empty
$p$-cobordism, ${\mathcal{H}}_{k}^{(p)}(X)$ is indeed an
${\mathbb{F}}_{p}$-vector space.

Before we discuss the relation between pseudocycles and homology classes, we
define the intersection pairing between pseudocycles in finite
characteristics.

Let

$f_{1}:W_{1}\to X,\ f_{2}:W_{2}\to X$

be two oriented $p$-pseudocycles in $X$ of complimentary dimensions. When they
intersect transversely, meaning that
$\overline{f_{1}(W_{1})}\cap\overline{f_{2}(W_{2})}=f_{1}({\rm Int}W_{1})\cap
f_{2}({\rm Int}W_{2})$ and the intersection is transverse, the signed count of
intersection points, modulo $p$, is defined to be the intersection number. One
can see easily that this intersection number is invariant under $p$-cobordism,
provided that the $p$-cobordism is in general position.

In addition, if $f_{1}:W_{1}\to X$ is $p$-cobordant to the empty set via a
$p$-cobordism $\tilde{f}_{1}:\tilde{W}_{1}\to X$, and if $\tilde{f}_{1}$ and
$f_{2}$ are transverse, then $\tilde{f}_{1}(\tilde{W}_{1})\cap f_{2}(W_{2})$
is a compact oriented 1-dimensional manifold with boundary being

$\Big{(}\tilde{f}_{1}(\partial\tilde{W}_{1})\cap f_{2}({\rm
Int}W_{2})\Big{)}\sqcup\Big{(}\tilde{f}_{1}({\rm Int}\tilde{W}_{1})\cap
f_{2}(\partial W_{2})\Big{)}.$

Besides the intersection $f_{1}({\rm Int}W_{1})\cap f_{2}({\rm Int}W_{2})$,
other boundary intersections contribute to a multiple of $p$.

In general, if relevant intersections are not transverse, then one can perturb
via ambient diffeomorphisms to achieve transversality. This allows us to
define intersection numbers between any pair of $p$-pseudocycles of
complementary dimensions and prove the independence of the choice of
perturbations, as all nearby diffeomorphisms are homotopic. Therefore, we have
defined a bilinear pairing

${\mathcal{H}}^{(p)}(X)\otimes{\mathcal{H}}^{(p)}(X)\to{\mathbb{F}}_{p}$

Next we will define the map from homology to pseudocycles. Recall that one
also has an abelian group ${\mathcal{H}}_{k}(X)$ of genuine $k$-dimensinoal
pseudocycles up to genuine cobordism. There is an obvious group homomorphism

${\mathcal{H}}_{*}(X)\to{\mathcal{H}}_{*}^{(p)}(X).$

Zinger [Zin08] constructed a natural isomorphism

$\Phi_{*}:H_{*}(X;{\mathbb{Z}})\cong{\mathcal{H}}_{*}(X).$

######  Theorem A.3.

There is a natural map

$\Phi^{(p)}_{*}:H_{*}(X;{\mathbb{F}}_{p})\to{\mathcal{H}}_{*}^{(p)}(X)$

satisfying the following conditions.

  1. (1)

The following diagram commutes.

$\textstyle{H_{*}(X;{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{*}}$$\textstyle{{\mathcal{H}}_{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}(X;{\mathbb{F}}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{*}^{(p)}}$$\textstyle{{\mathcal{H}}_{*}^{(p)}(X)}$
(A.1)

  2. (2)

Suppose a homology class $a\in H_{*}(X;{\mathbb{F}}_{p})$ is represented by a
smooth cycle $f:W\to X$ where $W$ is a compact oriented manifold with boundary
such that $f|_{\partial W}$ is a $p$-fold oriented covering of a map $g:V\to
X$ from a compact oriented manifold $V$ without boundary; in particular, $f$
is a $p$-pseudocycle. Then $\Phi_{*}^{(p)}(a)$ is represented by $f$.

  3. (3)

The map $\Phi_{*}^{(p)}$ intertwines the Poincaré pairing on
$H_{*}(X;{\mathbb{F}}_{p})$ with the intersection pairing on
${\mathcal{H}}_{*}^{(p)}(X)$.

###### Proof.

For the purpose of this paper, we do not need to show that $\Phi_{*}^{(p)}$ is
invertible. We just need to consider the case that $p>2$ as it is known that
for $p=2$ any homology class can be represented by a closed submanifold (see,
e.g., [BH81, Theorem B])

We follow the same approach as Zinger [Zin08]. Consider the complex of
singular chains with ${\mathbb{F}}_{p}$-coefficients. A homology class in
${\mathbb{F}}_{p}$ coefficients is represented by a singular cycle

$C=\sum_{i=1}^{N}a_{i}h_{i}$

where $a_{i}\in{\mathbb{F}}_{p}$ and $h_{i}:\Delta_{k}\to X$ is a continuous
map from the $k$-simplex $\Delta_{k}$. We can always choose the representative
such that each $h_{i}$ is smooth. Let $\delta^{j}(h_{i})$ be the $j$-th face
of $h_{i}$, which is a $k-1$-simplex. Moreover, let
$\tilde{a}_{i}\in\\{1,\ldots,p-1\\}$ be a lift of $a_{i}$ in ${\mathbb{Z}}$.
For any smooth map $g:\Delta_{k-1}\to X$ which may appear as a face of
$h_{i}$, consider

$\Delta_{i,g}^{\pm}(C)=\Big{\\{}j\ |\delta^{j}(h_{i})=\pm g\Big{\\}}$

and

$\Delta_{g}^{\pm}(C)=\bigcup_{i=1}^{N}\left(\underbrace{\Delta_{i,g}^{\pm}(C)\sqcup\cdots\sqcup\Delta_{i,g}^{\pm}(C)}_{\tilde{a}_{i}}\right)$

Then since $C$ is a cycle, one has

$\\#\Delta_{g}^{+}(C)-\\#\Delta_{g}^{-}(C)\in p{\mathbb{Z}}.$

Without loss of generality, assume
$\\#\Delta_{g}^{+}(C)\geq\\#\Delta_{g}^{-}(C)$. Then choose an injection
$\Delta_{g}^{-}(C)\hookrightarrow\Delta_{g}^{+}(C)$ and glue the corresponding
(interior of) faces, and removing all codimension two or higher facets from
the simplexes, one obtains a topological $k$-manifold with boundary $N(C)$
together with a map $f:N(C)\to X$. Notice that the boundary of $N(C)$ can be
identified as $p$ copies of a manifold. If we fix a certain standard way of
gluing standard $k$-simplexes along a face, $N(C)$ is then equipped with a
canonical smooth structure. One can perturb $f:N(C)\to X$ to a smooth map, and
hence a smooth $p$-pseudocycle. The cobordism class of the pseudocycle $f$ is
independent of the choice of the perturbation.

On the other hand, by using the same method as Zinger [Zin08, Lemma 3.3], one
can show that two homologous cycles induce cobordant $p$-pseudocycles. The
details are left to the reader. Hence we have constructed the map

$\Phi_{*}^{(p)}:H_{*}(X;{\mathbb{F}}_{p})\to{\mathcal{H}}_{*}^{(p)}(X).$

It is easy to see that this is an ${\mathbb{F}}_{p}$-linear map making the
diagram (A.1) commutative and satisfying (2) of Theorem A.3. Moreover, by
comparing the definition of the Poincaré pairing (which is essentially
counting transverse intersections of cycles) and the intersection pairing, (3)
of Theorem A.3 is also true. ∎

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