††thanks: Both authors contributed equally to this work††thanks: Both authors
contributed equally to this work††thanks: On leave from Dipartimento di Fisica
e Astronomia ’Ettore Majorana’, Università di Catania, Italy

#  Coherent phase slips in coupled matter-wave circuits

A. Pérez-Obiol  Barcelona Supercomputing Center, 08034 Barcelona, Spain
<EMAIL_ADDRESS>J. Polo  Quantum Research Centre, Technology Innovation
Institute, Abu Dhabi, UAE<EMAIL_ADDRESS>L. Amico  Quantum Research Centre,
Technology Innovation Institute, Abu Dhabi, UAE  INFN-Sezione di Catania, Via
S. Sofia 64, 95127 Catania, Italy  Centre for Quantum Technologies, National
University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore  LANEF
‘Chaire d’excellence’, Université Grenoble-Alpes & CNRS, F-38000 Grenoble,
France

###### Abstract

Quantum Phase slips are dual process of particle tunneling in coherent
networks. Besides to be of central interest for condensed matter physics,
quantum phase slips are resources that are sought to be manipulated in quantum
circuits. Here, we devise a specific matter-wave circuit enlightening quantum
phase slips. Specifically, we investigate the quantum many body dynamics of
two side-by-side ring-shaped neutral bosonic systems coupled through a weak
link. By imparting a suitable magnetic flux, persistent currents flow in each
ring with given winding numbers. We demonstrate that coherent phase slips
occur as winding number transfer among the two rings, with the populations in
each ring remaining nearly constant. Such a phenomenon occurs as a result of a
specific entanglement of circulating states, that, as such cannot be captured
by a mean field treatment of the system. Our work can be relevant for the
observation of quantum phase slips in cold atoms experiments and their
manipulation in matter-wave circuits. To make contact with the field, we show
that the phenomenon has clear signatures in the momentum distribution of the
system providing the time of flight image of the condensate.

## Introduction

Phase slips are jumps of the phase of the wave function. In coherent systems
as superconducting and cold atoms networks, they occur because of the
suppression of the amplitude of the superconducting/superfluid order parameter
making the phase unrestricted and able to jump by a discrete amount (in
multiples of $2\pi$) Arutyunov  _et al._ (2008); D’Errico  _et al._ (2017).
When such suppression is caused by thermal fluctuations, a thermal phase slip
occurs. Quantum Phase Slips (QPS) instead, are induced by quantum
fluctuations. A way to engineer QPS in mesoscopic physics is through Josephson
junctions, in which such events correspond to tunneling of the phase of the
order parameter Arutyunov  _et al._ (2008); Rastelli  _et al._ (2013).
Intriguingly, relying on a phase-charge duality, it was argued that such
tunneling events occur in proximity of the Coulomb blocked regime Mooij and
Nazarov (2006); Hriscu and Nazarov (2011).

In cold atoms settings, QPS have been investigated in different settings with
enhanced control and flexibility of the physical conditions Polkovnikov  _et
al._ (2002, 2005); Khlebnikov and Pryadko (2005); Danshita (2013); Roscilde
_et al._ (2016). Atomtronic circuits, in particular, define coherent networks
to study mesoscopic effects and quantum transport of ultracold atoms Amico
_et al._ (2017); Amico and et. al (2021); Amico  _et al._ (2021); Brantut  _et
al._ (2012); Burchianti  _et al._ (2018). In the interesting configuration of
bosonic condensates in a toroidal geometry, thermal and QPS have been
suggested as the main responsible for the dynamics of the atoms’ persistent
flow through the nucleation of vortex states and associated phonon emission,
or through the formation of dark solitons activated thermally or by the
stirring protocol Wright  _et al._ (2013); Ramanathan  _et al._ (2011); Eckel
_et al._ (2014); Yakimenko  _et al._ (2015); Polo  _et al._ (2018, 2019);
Pérez-Obiol and Cheon (2020); Pérez-Obiol  _et al._ (2020).

Although over the past three decades several phenomena in superconducting
networks have been related to QPS’ formation Lau  _et al._ (2001); Bezryadin
_et al._ (2000); Bollinger  _et al._ (2008); Altomare  _et al._ (2006); Masluk
_et al._ (2012); Manucharyan  _et al._ (2009); Weißl  _et al._ (2015); Pop
_et al._ (2010), a convincing experimental evidence of their occurrence in
solid state physics has been obtained only recently Astafiev  _et al._ (2012).
In cold atoms instead, QPS, as a coherent transfer of vortices or flux, have
not been observed yet.

Besides their interest in fundamental physics, QPS can be a resource for
quantum technology. Josephson junctions based quantum devices harnessing QPS
have been carried out Mooij and Harmans (2005); Belkin  _et al._ (2015); Pop
_et al._ (2010). For atomtronic ring circuits interrupted by weak links,
defining the atomic counterparts of SQUID devices, QPS localized at the weak
link play a crucial role for creating the superposition of the current states
that is expected to be especially important for quantum sensing Cominotti  _et
al._ (2014); Aghamalyan  _et al._ (2016); Amico  _et al._ (2015); Ryu  _et
al._ (2013); Polo  _et al._ (2019, 2021).

Figure 1: Schematic diagram of an optical lattice consisting of two sided
rings of five sites each. We represent the small link between rings, $t_{l}$,
with a small interfering pattern connecting the rings. Both rings have an
effective artificial gauge field denoted by $\Omega_{L}$ and $\Omega_{R}$.

Relying on the considerable know-how achieved in magneto-optic circuit design
and atom manipulation techniques, integrated atomtronic circuits define an
interesting direction of the field Amico and et. al (2021); Amico  _et al._
(2021). In this context, simple circuits of coupled rings and wave guides have
been considered Amico  _et al._ (2014); Safaei  _et al._ (2019); Mañas  _et
al._ (2021); Richaud and Penna (2017); Aghamalyan  _et al._ (2013); Polo  _et
al._ (2016); Pelegrí  _et al._ (2019); Bland  _et al._ (2020). Coupled ring
condensates, in particular, are prototype systems for the definition of
coherent cold atom networks in which matter wave flows are manipulated as a
resource. Even though several attempts have been done in this direction, the
analysis carried out so far show that independent winding numbers can coexist
in the two rings separately Pelegrí  _et al._ (2019); Bland  _et al._ (2020).
The transfer of winding numbers, though, has not been achieved.

Here, we solve the above bottleneck and demonstrate a coherent QPS in two
coupled rings arranged side-by-side as in Fig. 1. In the two rings, matter-
wave currents with different winding numbers are assumed to be imparted
through effective magnetic flux Dalibard  _et al._ (2011). We will argue that
to observe such effect it is important that the system works in the full-
fledged quantum regime. Indeed, the QPS we observe results from an oscillation
of entangled states of angular momentum states of the two rings.

## The model.

We consider a system of bosonic atoms trapped in two coupled coplanar rings
lattice, each of $N_{s}$ sites, and subjected to an effective magnetic flux
$\Omega_{\alpha}$, $\alpha=L,R$. See Fig. 1 for a schematic picture of the
system. The system’s Hamiltonian reads

$\displaystyle H=H_{L}+H_{R}+H_{I}(\tau)\;,$ $\displaystyle
H_{\alpha}\\!=\\!\\!\\!\sum_{i=0}^{N_{s}-1}\\!\\!\left[U(\hat{n}_{\alpha,i}^{2}-\hat{n}_{\alpha,i})-t\left(e^{-i\frac{2\pi\Omega_{\alpha}}{N_{s}}}\hat{a}_{\alpha,i}^{\dagger}\hat{a}_{\alpha,i+1}+h.c.\right)\right]\\!,$
$\displaystyle
H_{I}(\tau)=-t_{l}(\tau)\left(\hat{a}_{L,0}^{\dagger}\hat{a}_{R,0}+\hat{a}_{R,0}^{\dagger}\hat{a}_{L,0}\right).$
(1)

$\hat{a}_{\alpha,i}$ and $\hat{a}_{\alpha,i}^{\dagger}$ annihilate and create,
respectively, a boson in the site $i$ of the ring $\alpha$, satisfying
periodic conditions $\hat{a}_{\alpha,N_{s}}=\hat{a}_{\alpha,0}$,
$\hat{a}_{\alpha,N_{s}}^{\dagger}=\hat{a}_{\alpha,0}^{\dagger}$, and the
bosonic number operator is
$\hat{n}_{\alpha,i}=\hat{a}_{\alpha,i}^{\dagger}\hat{a}_{\alpha,i}$. The
parameters $t$ and $U\geq 0$ are the intra-ring hopping amplitude and
repulsive interaction; $t_{l}$ describes the tunneling among the two rings. We
set $t=1$.

The limits $U=0$ and $U\to\infty$ can be treated analytically for $t_{l}=0$.
The former reduces to the one particle problem, while the latter maps to
tight-binding hard-core bosonsWu and Yang (2006). We focus on the regime with
weak inter-ring coupling $t_{l}<t$, allowing us to work out both the numerical
simulations obtained by exact diagonalization and perturbative analysis in
$t_{l}/t$.

For uncoupled symmetric rings, $t_{l}=\Omega_{L}=\Omega_{R}=0$, the energy
spectrum is invariant under the exchange of left and right states, or under
the inversion of angular momentum directions in each separate ring, resulting
in specific degeneracies in the spectrum of (1).

For $\Omega_{L}=\Omega_{R}>0$, the rotational symmetry is broken, and
clockwise and counterclockwise current states split into separate energy
levels. In this regime, the eigenstates are found either non-degenerate or
two-fold degenerate. A finite $t_{l}$ splits the remaining degenerate energy
levels, and the new eigenvectors become a superposition of the uncoupled
degenerate states. We shall see that such level splitting is important for the
formation of specific entangled states enabling the QPS (see supplemental for
an example of such energy splittings).

Figure 2: Particle number’s expectation, standard deviation, and current
oscillations after quenching a system of two rings of fives sites and initial
fluxes $\Omega_{L}=1$, $\Omega_{R}=0$. The final velocities and interactions,
particle number and $t_{l}$ are (a) $\Omega_{L}=\Omega_{R}=0.05$, $U=0$,
$N_{p}=1$, $t_{l}=0.05$ and (b) $\Omega_{L}=\Omega_{R}=0.1$, $U=100$,
$N_{p}=2$, $t_{l}=0.1$ ($N_{p}>2$ is considered in Fig. 4).

## Quench Protocol.

We first construct a state with the same density of particles in each ring,
$\frac{N_{p}}{2}$, but with zero angular momentum on the right, $\langle
J_{R}\rangle=0$, and a non vanishing current on the left, $\langle
J_{L}\rangle\doteq J_{max}$. We denote such state as
$|L_{j};R_{0}\rangle\doteq|\frac{N_{p}}{2},J_{max};\frac{N_{p}}{2},0\rangle$,
resulting to be an eigenstate of the system when $\Omega_{L}=j$,
$\Omega_{R}=0$, with $j$ integer. Next, we perform the quench on the system to
$\Omega_{L}=\Omega_{R}\gtrsim 0$. Then, the state $|L_{j};R_{0}\rangle$ is a
superposition of quasi-degenerate eigenvectors, each with the same particle
number $\frac{N_{p}}{2}$ and total current $J_{max}$. After the quench, a time
evolution occurs on the expectation value of the number of particles
$\langle\hat{n}_{\alpha}\rangle$, its variance $\sigma^{2}_{\alpha}$, and the
current $\langle J_{\alpha}\rangle$, defined as

$\displaystyle\sigma^{2}_{\alpha}=\langle\hat{n}_{\alpha}^{2}\rangle-\langle\hat{n}_{\alpha}\rangle^{2}\,,$
(2) $\displaystyle\langle
J_{\alpha}\rangle=-i\sum_{k}\langle\hat{a}_{\alpha,k}^{\dagger}\hat{a}_{\alpha,k+1}-\hat{a}_{\alpha,k+1}^{\dagger}\hat{a}_{\alpha,k}\rangle\,,$
(3)

with $\hat{n}_{\alpha}=\sum_{i}\hat{n}_{\alpha,i}$.

Figure 3: Energy gap (such that $T=\frac{2\pi}{\Delta E}$, see supplemental
material) as a function of the interring coupling $t_{l}$ and
$\Omega_{L}=\Omega_{R}=0.01,0.05,0.1$ for: (a) one particle ($\Delta
E=\frac{2t_{l}}{N_{s}}$), (b) two particles and $U=100$ ($\Delta E\propto
t_{l}^{2}$).

## Results.

Current oscillations between left and right rings are found for small and
large interactions, $U\lesssim 0.01$ and $U\gtrsim 10$, when following the
quench. See Fig. 2 for the cases $U=0$ and $U=100$. The currents in each ring
oscillate completely out of phase, between a maximum value $J_{max}$ and zero:

$|L(\tau);R(\tau)\rangle=\cos\left(\omega\,\tau\right)|L_{j};R_{0}\rangle+\sin\left(\omega\,\tau\right)|L_{0};R_{j}\rangle,$
(4)

with $\omega=\frac{\Delta E}{2\pi}$, $\Delta E$ being the energy gap between
the involved states. Importantly, the expectation number
$\langle\hat{n}_{\alpha}\rangle$ in each of the two rings results to be nearly
constant at all times: no net transfer of particles between rings occur and
current oscillations happen due to the phase slipping through the weak link.
Note that the phase by itself does not carry any angular momentum or
direction, and for the current to change an arbitrary small flux
$\Omega_{L/R}=\Omega_{\alpha}$ is required to be applied in each ring.

The specific particle configuration $n_{\alpha}$ and $\sigma_{\alpha}^{2}$,
and the maximum current $J_{max}$ in $|L_{j};R_{0}\rangle$ and
$|L_{0};R_{j}\rangle$ depend on $U$ and $N_{p}$ (see Table 1 and supplemental
material). By using perturbation analysis, we find that $\omega$ depends
linearly in $t_{l}$ for $U=0$, and quadratically for large $U$ (except for odd
$N_{p}$, see Fig. 3 and Table 1). Important insights on the effect of the
interaction can be obtained by studying the limit $U\to\infty$. Although the
expected number of particles results to be barely affected by $U$, its
variance $\sigma_{\alpha}$ does. For $U=0$,
$\sigma_{\alpha}^{2}=\frac{N_{p}}{4}$. For large interactions, instead, any
measurement of the occupation would always find half the particles in each
ring, and therefore $\sigma_{\alpha}^{2}=0$ is found (or $\frac{N_{p}\pm
1}{2}$ particles in each ring and $\sigma_{\alpha}^{2}=\frac{1}{4}$ for
$N_{p}$ odd). We also remark that, because of the particle-hole symmetry
holding for large interactions, QPS for holes occur similarly to the particles
ones (see Fig. 4 and supplemental material). As for the maximum current, in
the case $U=0$, it results to scale linearly with the number of particles,
$J_{max}=2\,N_{p}\sin\left(\frac{2\pi}{N_{s}}\right)$, while for large
interactions we find
$J_{max}=4\cos\left(\frac{\pi}{N_{s}}\right)\sin\left(\frac{\pi
N_{p}}{N_{s}}\right)$. To transfer larger currents, one can simply start with
an integer flux $\Omega_{L}$ in $1<n<k_{\textrm{max}}$. Numerical tests for
various ring sizes and initial currents corroborate that QPS are still found
as $N_{p}$ and angular momenta are increased (see Fig. 4 and supplemental
material).

|  | $\langle\hat{n}\rangle$ | $\sigma^{2}$ | $J_{max}$ | $T$
---|---|---|---|---|---
$U=0$ |  | $\frac{N_{p}}{2}$ | $\frac{N_{p}}{4}$ | $2\,N_{p}\sin\left(\frac{2\pi}{N_{s}}\right)$ | $\frac{\pi N_{s}}{t_{l}}$
$U\to\infty$ |  $N_{p}$ even | $\frac{N_{p}}{2}$ | $0$ | $4\cos\left(\frac{\pi}{N_{s}}\right)\sin\left(\frac{\pi N_{p}}{N_{s}}\right)$ | $\propto\frac{1}{t_{l}^{2}}$
$N_{p}$ odd | $\frac{1}{4}$ | $\propto\frac{1}{t_{l}}$
Table 1:  Expected value of the occupation and variance in each ring,
amplitude and period of current oscillations, in the case of no interactions
and for $U\to\infty$.

By controlling $t_{l}$ in time, and relying on our condition of weak ring-ring
coupling, we note that different entangled states of angular momenta can be
engineered by our scheme. The transfer of angular momentum can be obtained by
manipulating $t_{l}$ on $\tau=\frac{T}{2}\times(2n+1)$. For
$\tau=\frac{T}{4}\times(2n+1)$, for example, the entangled state
$|L_{j};R_{0}\rangle+|L_{0};R_{j}\rangle$ can be achieved.

Figure 4: Each plot compares QPS where (excess) particles $N_{e}$ and holes
$2N_{s}-N_{e}$ are exchanged in the strongly interacting regime, $U=1000$, and
in rings of $N_{s}=3$ sites. Particle-hole symmetry is displayed for two
different commensurate fillings, such that the total number of particles is
$N_{p}=2N_{s}\times N_{b}+N_{e}$, with $N_{b}=0$ (left plots) and $N_{b}=1$
(right plots), and $N_{e}=\\{2,4\\}$ (top plots) and $N_{e}=\\{1,5\\}$ (bottom
plots). We quench from $\Omega_{L}=1$, $\Omega_{R}=0$, to $\Omega_{L}=0.01$,
$\Omega_{R}=0.01$, and from $t_{l}=10^{-5}$ to $t_{l}=0.01$. Note that the
period substantially changes for each commensurate filling $N_{b}=0,1$,
despite having the same number of particles and holes. These changes appear
because of the energy levels shift proportionally to $N_{b}$.

## Read-out of the QPS.

Matter wave currents can be detected through time-of-flight (TOF) measurements
Moulder  _et al._ (2012); Amico  _et al._ (2005). Such measurements in the far
field are directly related to the momentum distribution at the moment in which
the condensate is released from the ring trap. Therefore, the time evolution
of the persistent current is reflected in the time evolution of the momentum
distribution: $n({\bf k},t)=\sum_{i,j}\>e^{i\bf k\cdot({\bf R}_{i}-{\bf
R}_{j})}C_{i,j}(t)$ with
$C_{i,j}(t)=\langle\psi(t)|a^{\dagger}_{i}a_{j}|\psi(t)\rangle$ being the one-
body correlation function between different sites and $\textbf{R}_{j}$
denoting the position of the lattice sites of the ring. In each ring, we find
that $n({\bf k},t)$ evolves from a peak momentum distribution to the
characteristic circular-shaped one as soon as the system acquires one unit of
angular momentum. Such dynamics in the TOF provides the read-out of the
transfer of coherent phase slips between zero and one unit of angular
momentum. See Fig. 5.

## Discussion and Conclusions.

We have theoretically demonstrated QPS between two tunnel coupled rings of
interacting bosons: We prepare two different phase-states of the two separated
rings; after quenching the tunnel between the rings, we observe a coherent
oscillation between the phase states with nearly vanishing population
fluctuations (in each of the two rings). Once calibrated, the scheme can be
used to produce, transfer, and entangle current states by tuning on and off
the the weak link at specific times after the quenching protocol. We find that
the phase slips faster from one ring to the other for stronger inter-ring
couplings. Interactions reduce the maximum current in each ring, and make
phase slips slower (see supplemental material for the interplay between
interactions and period of oscillations). Indeed, such phenomenon occurs as a
direct consequence of the entangled state created between the phase states of
the two rings (macroscopic superposition of all particles rotating with
different angular momenta in each ring). The coherent oscillations of the QPS
are characterized by the simultaneous creation and destruction of current
states in each ring (see Sec. A of supplemental material for specific
examples). As such, QPS transfer is a genuine quantum effect that cannot be
captured by standard mean-field analysis such as Gross-Pitaevskii based
approach. In fact, the latter neglects entanglement and, to the best of our
knowledge, cannot describe coherent transfer of matter-wave without transfer
of population (which is an essential trait of our demonstration). We note that
the coupling between the rings is perturbative and as such, the corresponding
emergence of quasi-degenerate states involving a superposition of left and
right current states hold for large particle numbers. The scope of our results
can be further enlarged by resorting to a suitable particle-hole symmetry.
Therefore, our QPS are expected to occur also in systems with large particle
numbers. We studied the momentum distribution that is the standard method to
analyze neutral matter-wave currents in cold atoms experiments Moulder  _et
al._ (2012); Amico  _et al._ (2005); Greiner  _et al._ (2002); Gerbier  _et
al._ (2005); Kato  _et al._ (2008); Hoffmann and Pelster (2009)

Figure 5:  Time-of-flight expansion of the left ring. Parameters are:
$\Omega_{L}=1,\;\Omega_{R}=0$ to $\Omega_{L}=0.01,\;\Omega_{R}=0.01$,
$t_{l}=0$ to $t_{l}=0.05$, $N_{p}=2$, $N_{s}=5$ in each ring, $U=10$.
Triangles indicate the times at which the TOF snapshots where taken.

Our work provides a specific platform to observe QPS in cold atoms that is a
well known open problem in the field. At the same time, our work results are
relevant to progress in the implementation of integrated atomtronic circuits
Amico and et. al (2021); Amico  _et al._ (2021). Specifically, our results
effectively enable atomtronic circuit based on coupled atomic rings: In a
sense analogue to the ‘Rapid Single Flux Quantum Logic’ conceived with SQUID’s
Likharev and Semenov (1991), complex structures where the information is
encoded in the phase slips inherent to the different rings could be
implemented.

Acknowledgements. We thank Gianluigi Catelani and Wayne Jordan Chetcuti for
discussion. A. P-O. acknowledges financial support from Secretaria
d’Universitats i Recerca del Departament d’Empresa i Coneixement de la
Generalitat de Catalunya cofunded by the European Union Regional Development
Fund within the ERDF Operational Program of Catalunya (project QuantumCat,
ref. 001-P-001644).

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##  Appendix A Single particle analytical results in Fourier space

For one particle and $\Omega_{L}=1$, the initial state is
$|\psi_{+}\rangle=\frac{1}{\sqrt{2}}(b_{L,1}^{\dagger}+b_{R,0}^{\dagger})|0\rangle$,
where
$b_{\alpha,l}^{\dagger}=\frac{1}{\sqrt{N_{s}}}\sum_{j}e^{i\frac{2\pi(j\,l)}{N_{s}}}a_{\alpha,j}^{\dagger}$
creates a particle with momentum $l$ in the ring $\alpha$. It has an average
particle number per ring $n_{\alpha}=\frac{1}{2}$ and variance
$\sigma_{\alpha}^{2}=\frac{1}{4}$. In this case, $H_{BH}$ in momentum space
consists of a diagonal term, where the eigenvectors are vortex states,
$b_{\alpha,i}^{\dagger}|0\rangle$, and a perturbative one, that transfers
momentum modes between left and right rings,

$\displaystyle H_{BH}=$
$\displaystyle-2\,\sum_{\alpha=L,R}\sum_{i=0}^{N_{s}-1}\cos\left(2\pi(i-\Omega_{\alpha})/N_{s}\right)\hat{b}_{\alpha,i}^{\dagger}\hat{b}_{\alpha,i}$
$\displaystyle-\frac{t_{l}}{N_{s}}\sum_{i,j=0}^{N_{s}-1}\left(\hat{b}_{L,i}^{\dagger}\hat{b}_{R,j}+h.c.\right).$
(5)

The energy gap is $\Delta
E=\langle\psi_{-}|H_{p}|\psi_{-}\rangle-\langle\psi_{+}|H_{p}|\psi_{+}\rangle=\frac{2t_{l}}{N_{s}}+\mathcal{O}(t_{l}^{2})$,
where $H_{p}$ is the second term of Eq. (5) and
$|\psi_{-}\rangle=\frac{1}{\sqrt{2}}(b_{L,1}^{\dagger}-b_{R,0}^{\dagger})|0\rangle$.
If $U=0$, each particle behaves independently, and current oscillations are
due to the simultaneous but opposite transfer of the ground and first excited
momentum modes. The particle distribution is binomial, with
$n_{\alpha}=\frac{N_{p}}{2}$ and variance
$\sigma_{\alpha}^{2}=\frac{N_{p}}{4}$. The initial (and maximum) current
scales with the number of particles, and evaluates to
$J_{max}=2\,N_{p}\sin\left(\frac{2\pi}{N_{s}}\right)$.

##  Appendix B Two-state level simplification

All the current oscillations found can be understood in terms of dynamics
within a system of two quasi-degenerate states. For a given set of fluxes,
$\Omega_{L}=\Omega_{R}=\Omega\gtrsim 0$, and a weak enough interring coupling,
$t_{l}\gtrsim 0$, pairs of quasi-degenerate states can effectively be treated
as uncoupled from the rest of eigenstates. Here we show this explicitly with
two specific examples, one for $U=0$, which reduces to the case of one
particle, and another for large interactions, $U\to\infty$.

In the case of one particle, terms coupling different momentum modes in Eq.
(5) enter at order $\mathcal{O}(t_{l})$ when diagonalizing, assuming single
ring energy levels $E_{i}=-2\,t\cos\left(2\pi(i-\Omega)/N_{s}\right)$ are
separate enough. Neglecting those terms, $H_{BH}$ can be written in terms of
two-mode independent Hamiltonians,

$\displaystyle H_{BH}^{(i)}=$ $\displaystyle
E_{i}\left(\hat{b}_{L,i}^{\dagger}\hat{b}_{L,i}+\hat{b}_{R,i}^{\dagger}\hat{b}_{R,i}\right)$
$\displaystyle-\frac{t_{l}}{N_{s}}\left(\hat{b}_{L,i}^{\dagger}\hat{b}_{R,i}+h.c.\right).$
(6)

Within this picture, the ground and first excited states are
$|\phi_{0}^{\pm}\rangle=\frac{1}{\sqrt{2}}|0,1\rangle\pm\frac{1}{\sqrt{2}}|1,0\rangle$,
where $|N_{L},N_{R}\rangle$ define states with $N_{L}$ static particles in the
left ring and $N_{R}$ in the right one. Equivalently, the next excited states
are
$|\phi_{1}^{\pm}\rangle=\frac{1}{\sqrt{2}}|0,1^{+}\rangle\pm\frac{1}{\sqrt{2}}|1^{+},0\rangle$.
Here the subindex $1$ in $\phi$ and superindex $+$ in $1$ indicate one unit of
angular momentum in the direction of $\Omega$. These pair of eigenstates are
energetically separate from $|\phi_{0}^{\pm}\rangle$, and from the next
excited states,
$|\phi_{-1}^{\pm}\rangle=\frac{1}{\sqrt{2}}|0,1^{-}\rangle\pm\frac{1}{\sqrt{2}}|1^{-},0\rangle$,
in which the rotation is opposite to the fluxes $\Omega$ (see Fig. 6 (a)).
These energy gaps are much larger than the energy separating
$|\phi_{i}^{-}\rangle$ and $|\phi_{i}^{+}\rangle$ states in each pair, which
is $\Delta E=\frac{2t_{l}}{N_{s}}$ in all cases. The evolution of the initial
state from our protocol,
$|\psi_{+}\rangle=\frac{1}{\sqrt{2}}|1^{+},0\rangle+\frac{1}{\sqrt{2}}|0,1\rangle$,
consists of two independent evolutions with the same frequency
$\omega=\frac{\Delta E}{2\pi}$, one for $|1^{+},0\rangle$, and another for
$|0,1\rangle$,

$\displaystyle|\psi(\tau)\rangle$
$\displaystyle=\frac{e^{-i\,E_{1}\,\tau}}{\sqrt{2}}\left[\cos\left(\omega\,\tau\right)|1^{+},0\rangle+i\sin\left(\omega\,\tau\right)|0,1^{+}\rangle\right]$
$\displaystyle+\frac{e^{-i\,E_{0}\,\tau}}{\sqrt{2}}\left[\cos\left(\omega\,\tau\right)|0,1\rangle+i\sin\left(\omega\,\tau\right)|1,0\rangle\right]$
$\displaystyle\doteq\cos\left(\omega\,\tau\right)|L_{1};R_{0}\rangle+\sin\left(\omega\,\tau\right)|L_{0};R_{1}\rangle.$
(7)

Figure 6: Energy levels as a function of the interring coupling $t_{l}$ for
(a) $U=0$, $\Omega_{1}=\Omega_{2}=0.05$, $N_{p}=1$ and (b) $U=100$,
$\Omega_{1}=\Omega_{2}=0.1$, $N_{p}=2$ . In (a), the corresponding eigenstates
are, from bottom to top, $|\phi_{0}^{+}\rangle$, $|\phi_{0}^{-}\rangle$,
$|\phi_{1}^{+}\rangle$, $|\phi_{1}^{-}\rangle$, $|\phi_{-1}^{+}\rangle$, and
$|\phi_{-1}^{-}\rangle$, as described in the main text. The first and second
energy splittings are separated by
$E_{1}-E_{0}=-2\cos\left(2\pi(1-\Omega)/N_{s}\right)+2\cos\left(2\pi(\Omega)/N_{s}\right)$,
while the gap between the second and third is
$\Delta\epsilon_{\Omega}=\frac{8\pi}{N_{s}}\sin\left(\frac{2\pi\,j}{N_{s}}\right)\Omega+\mathcal{O}(\Omega^{3})$.
In (b), the energy levels correspond to eigenstates $|\chi_{0}\rangle$,
$|\chi_{0}^{+}\rangle$, $|\chi_{0}^{-}\rangle$, $|\chi_{1}^{+}\rangle$, and
$|\chi_{1}^{-}\rangle$.

For large interactions and even $N_{p}$, eigenstates have the same number of
particles $N=\frac{N_{p}}{2}$ in each ring. The ground state is
$|\chi_{0}\rangle=|N,N\rangle$, energetically separate from the the first
excited states, among which are
$|\chi_{1}^{\pm}\rangle=\frac{1}{\sqrt{2}}|N,N^{+}\rangle\pm\frac{1}{\sqrt{2}}|N^{+},N\rangle$,
and
$|\chi_{-1}^{\pm}\rangle=\frac{1}{\sqrt{2}}|N,N^{-}\rangle\pm\frac{1}{\sqrt{2}}|N^{-},N\rangle$.
In this case the energy gap $\Delta E$ separating $|\chi_{i}^{+}\rangle$ and
$|\chi_{i}^{-}\rangle$ states is much smaller and proportional to $t_{l}^{2}$,
see Fig. 6 for the case with $N=1$ in each ring. Given an initial state
$|\psi(t=0)\rangle=|N^{+},N\rangle$, it evolves with a relatively low
frequency $\omega=\frac{\Delta E}{2\pi}$,

$\displaystyle|\psi(\tau)\rangle$
$\displaystyle=e^{-i\,E_{1}\,\tau}\left[\cos\left(\omega\,\tau\right)|N^{+},N\rangle+i\sin\left(\omega\,\tau\right)|N,N^{+}\rangle\right]$
$\displaystyle\doteq\cos\left(\omega\,\tau\right)|L_{1};R_{0}\rangle+\sin\left(\omega\,\tau\right)|L_{0};R_{1}\rangle.$
(8)

##  Appendix C Spectrum for large interactions

The spectrum for $U=0$ does not depend on the number of particles, $N_{p}$,
and the relevant energy gap and evolution after the quench can be computed
analytically with first order perturbation theory. In contrast, for large
interactions, the energy gaps do depend on $N_{p}$, and scale as $t_{l}^{2}$,
instead of $t_{l}$, which implies that second-order perturbation theory is
needed. In this section we analyze how the spectrum for large interactions
depends on $\Omega_{\alpha}$, $t_{l}$, $U$, and $N_{p}$.

The energy spectrum for two particles and $\Omega_{R}\gtrsim 0$ as a function
of $\Omega_{L}$ is plotted in Fig. 7. At $\Omega_{L}=\Omega_{R}\gtrsim 0$,
states are symmetric with respect to the exchange of left and right rings.
They correspond to, from bottom to top and in the same notation as in Appendix
B,

$\displaystyle|\chi_{0}\rangle=$ $\displaystyle|1,1\rangle,$
$\displaystyle|\chi_{0}^{+}\rangle=$
$\displaystyle\frac{|0,2\rangle+|2,0\rangle}{\sqrt{2}},$
$\displaystyle|\chi_{0}^{-}\rangle=$
$\displaystyle\frac{|0,2\rangle-|2,0\rangle}{\sqrt{2}},$
$\displaystyle|\chi_{1}^{+}\rangle=$
$\displaystyle\frac{|1,1^{+}\rangle+|1^{+},1\rangle}{\sqrt{2}},$
$\displaystyle|\chi_{1}^{-}\rangle=$
$\displaystyle\frac{|1,1^{+}\rangle-|1^{+},1\rangle}{\sqrt{2}},$
$\displaystyle|\chi_{-1}^{+}\rangle=$
$\displaystyle\frac{|1,1^{-}\rangle+|1^{-},1\rangle}{\sqrt{2}},$
$\displaystyle|\chi_{-1}^{-}\rangle=$
$\displaystyle\frac{|1,1^{-}\rangle-|1^{-},1\rangle}{\sqrt{2}}.$ (9)

As $\Omega_{L}$ increases, this symmetry is broken, and eigenvectors turn to,
approximately, $|1,1\rangle$, $|0,2\rangle$, $|2,0\rangle$, $|1^{+},1\rangle$,
$|1,1^{+}\rangle$, $|1,1^{-}\rangle$, $|1^{-},1\rangle$. The energies increase
quadratically with $\Omega_{L}$, except for the state in which the left ring
is empty, $|0,2\rangle$, in which the energy remains constant. At
$\Omega_{L}\lesssim 1$, we have the same states except for a general shift in
one unit of angular momentum in the right ring. As $\Omega_{L}$ decreases from
$\Omega_{L}=1$, states with particles on the left also increase their energy
quadratically. Due to the small but finite $t_{l}$, the energy levels have
avoided crossings (some are too small to see in the figure). In terms of the
relation between the ground state at $\Omega_{L}=1$, $|1^{+},1\rangle$, the
initial state in our protocol, with the states at $\Omega_{L}\gtrsim 0$, these
avoided crossings imply
$|1^{+},1\rangle=\frac{1}{\sqrt{2}}|\chi_{1}^{+}\rangle+\frac{1}{\sqrt{2}}|\chi_{1}^{-}\rangle$.

Figure 7: Spectrum of energies depending on $\Omega_{L}$ (left) for $U=100$,
$N_{p}=2$, $t_{l}=0.1$, and $\Omega_{R}=0$. A quench can be made such that the
ground state at $\Omega_{L}=1$ is a superposition of the third and fourth
excited states at $\Omega_{L}=0$.

The corresponding energy gap, $\Delta E=E_{4}-E_{3}$, is plotted as a function
of $U$ and $t_{l}$ in Fig. 8. At $U=0$, $\Delta E$ is relatively large and
shows no dependence on $t_{l}$. In this case, the energy gap between the third
and fourth excited states corresponds to $\Delta E_{\Omega}$ (See Fig. 6). As
interactions $U$ increase, the energy gap $\Delta E$ closes and its dependence
on $t_{l}$ converges to a fixed gradient.

Figure 8:  Energy gap of the two occupied states as a function of the link
strength $t_{i}$ and interactions $U$ after a quench starting from
$\Omega_{L}=1$, $\Omega_{R}=0$ to $\Omega_{L}=\Omega_{R}=\Omega_{j}=0.1$ for
$N_{p}=2$ particles and $N_{s}=5$. All curves collapse for large enough
interactions and the gap behaves equally as a function of $t_{l}$.

As for the dependence of $\Delta E$ on $N_{p}$, it exhibits two main features.
First, the energy gaps for $N_{p}$ even are much smaller than for $N_{p}$ odd,
the former being proportional to $t_{l}^{2}$ and the latter to $t_{l}$.
Second, the energy gaps are the same under the exchange of particles and
holes. For large interactions, the BH model maps to a spinless FH model, in
which the spectrum is invariant, except for a general shift, under the
exchange $N_{p}\leftrightarrow 2N_{s}-N_{p}$. This implies that, in the case
of rings with five sites, the spectra and energy gaps are the same for the
pairs of particle numbers $N_{p}=1$, 9; 2, 8; 3, 7; and 4, 6. Apart from the
symmetry under the exchange $N_{p}\leftrightarrow 2N_{s}-N_{p}$, we find the
energy gap is largest at smallest or largest occupations, and decreases down
to a minimum at half filling.

Particle-hole symmetry still holds when $N_{p}>2N_{s}$, as shown in Fig. 4 in
the main text. If there are more particles than sites, such that there is a
number $2N_{s}\times N_{b}$ of particles evenly distributed in the background,
and $N_{e}$ excess particles, $N_{p}=2N_{s}\times N_{b}+N_{e}$, the energy gap
as a function of $N_{e}$ has the same qualitative behavior. Therefore, ring
systems with more particles than sites have very similar behavior in terms of
phase slips. If $N_{e}=0$, as in the case of full filling, the ground state
has $N_{b}$ particles per site, and is energetically very separate from the
first excited state (the gap being proportional to $U$, which we assume
large). In this case, our protocol does not allow for phase slips.


discussed above and with $st$ no more than a polynomial in the problem size,
the acceptance probability increases to one exponentially fast in the case of
a yes instance, and it decays to zero exponentially fast in the case of a no
instance, as desired.

##  Appendix L Local Reward Function

In this appendix, we develop a local reward function, as an alternative to the
global reward function considered in the main text, i.e., the acceptance
probability in Theorem 2. The acceptance probability in Theorem 2 can be
thought of as a global reward function because it corresponds to the
probability of measuring zero in every register. As indicated in [44], it is
helpful to employ a local reward function in order to mitigate the barren
plateau problem [43], which plagues all variational quantum algorithms.

Let us define the local and global reward functions. To begin, let $Z_{i}$ be
the event of measuring zero in the $i$th register. We then set the local
reward function to be the probability of measuring zero in a register chosen
uniformly at random; that is, it is given by the following:

$L\equiv\frac{1}{n}\sum_{i}\Pr\\!\left(Z_{i}\right).$ (154)

The event of measuring all zeros is given by $\bigcap_{i}Z_{i}$, and the
probability that this event occurs is
$G\equiv\Pr\\!\left(\bigcap_{i}Z_{i}\right)$, which is what we used in the
main text as the global reward function.

Now, we are interested in determining inequalities that relate the global and
local reward functions, and the following analysis employs the same ideas used
in [82, Appendix C]. Using DeMorgan’s laws, we find that

$\Pr\\!\left(\bigcap_{i}Z_{i}\right)=\Pr\\!\left(\left(\bigcup_{i}Z_{i}^{c}\right)^{c}\right)=1-\Pr\\!\left(\bigcup_{i}Z_{i}^{c}\right).$
(155)

We can then use the union bound to conclude that

$\Pr\\!\left(\bigcap_{i}Z_{i}\right)=1-\Pr\\!\left(\bigcup_{i}(Z_{i})^{c}\right)\geq
1-\sum_{i}\Pr\\!\left((Z_{i})^{c}\right).$ (156)

Finally, consider that

$\displaystyle G$ $\displaystyle=\Pr\\!\left(\bigcap_{i}Z_{i}\right)$ (157)
$\displaystyle\geq 1-\sum_{i}\Pr\\!\left(Z_{i}^{c}\right)$ (158)
$\displaystyle=\sum_{i}\Pr\\!\left(Z_{i}\right)-(n-1)$ (159)
$\displaystyle=nL-(n-1)$ (160) $\displaystyle=n(L-1)+1.$ (161)

We can also derive an upper bound on the global reward function in terms of
the local reward function. Recall the following inequality, which holds for
every set $\\{A_{1},A_{2},\ldots,A_{n}\\}$ of events:

$\Pr\\!\left(\bigcup_{i}A_{i}\right)\geq\frac{1}{n}\sum_{i}\Pr\\!\left(A_{i}\right).$
(162)

Setting $A_{i}=Z_{i}^{c}$, we get

$\Pr\\!\left(\bigcup_{i}Z_{i}^{c}\right)\geq\frac{1}{n}\sum_{i}\Pr\\!\left(Z_{i}^{c}\right).$
(163)

Using DeMorgan’s laws, we obtain the desired upper bound as follows:

$\displaystyle\Pr\\!\left(\bigcap_{i}Z_{i}\right)$ $\displaystyle\leq
1-\frac{1}{n}\sum_{i}\left(1-\Pr\\!\left(Z_{i}\right)\right)$ (164)
$\displaystyle=\frac{1}{n}\sum_{i}\Pr\\!\left(Z_{i}\right).$ (165)

In summary, we have established the following bounds:

$n(L-1)+1\leq G\leq L,$ (166)

so that $G=1$ if and only if $L=1$. Since we always have $G\in[0,1]$, the
lower bound is only nontrivial if $L$ is sufficiently large, i.e., if $L\geq
1-\frac{1}{n}$.


# ACC of plc threshold

Sung Rak Choi  and  Sungwook Jang  Department of Mathematics, Yonsei
University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

In this paper, we define potential log canonical threshold and prove that the
set of those thresholds satisfies the ascending chain condition (ACC). We also
consider collections of sequences of Fano type varieties and we study their
basic properties including boundedness.

##  1\. Introduction

Recently, various thresholds in birational geometry have attracted much
attention as their interesting behaviours (e.g., ascending chain condition,
ACC) turned out to have important implications especially in the minimal model
program. In this paper, we are interested in the so called potential
thresholds for the pair $(X,\Delta)$ such that $-(K_{X}+\Delta)$ is
pseudoeffective. It is a generalization of the well-known log canonical
thresholds. The notion of potential pairs was first introduced and studied in
[CP]. It was defined as a means to bound the singularities of the outcome of
the $-(K_{X}+\Delta)$-minimal model program (MMP). We believe that the deeper
understanding of this notion will be useful in establishing the
$-(K_{X}+\Delta)$-MMP.

For a pair $(X,\Delta)$ with an effective divisor $\Delta$ such that
$-(K_{X}+\Delta)$ is pseudoeffective, consider a birational morphism $f:Y\to
X$ and the divisorial Zariski decomposition $-f^{*}(K_{X}+\Delta)=P+N$. If $F$
is a prime divisor on $Y$, $a(X,\Delta,F)$ denotes the log discrepancy of
$(X,\Delta)$ at $F$. We define the potential log discrepancy of $(X,\Delta)$
at $F$ as $\bar{a}(X,\Delta,F):=a(X,\Delta,F)-\operatorname{mult}_{F}N$. The
given pair $(X,\Delta)$ is said to be potentially klt (resp. potentially
$\epsilon$-lc for $\epsilon\geq 0$) if $\inf_{f,F}\bar{a}(X,\Delta,F)>0$
(resp. $\geq\epsilon$) holds. See Definition 2.3 and [CP] for more details.

By construction, the notions of pklt, $\epsilon$-plc capture the singularities
of the pair $(X,\Delta)$ as well as the positivity of $-(K_{X}+\Delta)$.
However, if $-(K_{X}+\Delta)$ is nef, then the notions of pklt, $\epsilon$-plc
coincide with the usual klt, $\epsilon$-lc. One of the most important
properties these notions is their invariant property under
$-(K_{X}+\Delta)$-MMP (Proposition 2.5).

Recall that the log canonical threshold $\operatorname{lct}(X,\Delta,D)$ of a
klt pair $(X,\Delta)$ with respect to a given divisor $D$ is defined as the
supremum of the real numbers $t$ for which $(X,\Delta+tD)$ is lc. The
potential lct $\operatorname{p-lct}(X,\Delta,D)$ of a pklt pair $(X,\Delta)$
with respect to $D$ is defined similarly and it is at most the effective
threshold $t_{\operatorname{eff}}:=\\{t|-(K_{X}+\Delta+tD)\text{ is
pseudoeffective}\\}$. See Definition 2.3 for the precise definition.

We prove that the set of $\operatorname{p-lct}(X,\Delta,D)$ in a fixed
dimension satisfies the similar properties of the set of
$\operatorname{lct}(X,\Delta,D)$.

######  Theorem 1.1.

Suppose that $I$ is a DCC set of nonnegative real numbers, $n$ a natural
number, and $\epsilon$ a positive real number. Then there is an ACC set
$\Sigma$ of real numbers such that the plc threshold of $(X,\Delta)$ with
respect to $D$ belongs to $\Sigma$ whenever $(X,\Delta)$, $D$ satisfy the
following conditions

  1. (i)

$(X,\Delta)$ is a $\epsilon$-plc pair of dimension $n$,

  2. (ii)

$-(K_{X}+\Delta)$ is big,

  3. (iii)

$D$ is an effective nef and big divisor, and

  4. (iv)

the coefficients of $\Delta$ and $D$ belong to $I$.

If $-(K_{X}+\Delta)$ is nef, then the notion of $\epsilon$-plc coincides with
the usual lc for any $\epsilon\geq 0$. Thus in this case, if we could let
$\epsilon=0$ in Theorem 1.1, then it coincides with the main result Theorem
1.1 of [HMX]. However, since Theorem 3.1 is a special case of Theorem 1.1 and
Theorem 3.1 does not hold for $\epsilon=0$, Theorem 1.1 does not hold for
$\epsilon=0$ in general either. See Example 3.2.

######  Theorem 1.2.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then the
set of projective varieties $X$ satisfying the following conditions:

  1. (i)

$X$ is a projective variety of dimension $d$,

  2. (ii)

there is a boundary divisor $\Delta$ such that $(X,\Delta)$ is a
$\epsilon$-plc and $-(K_{X}+\Delta)$ is big

is a bounded family.

This paper is organized as follows. In Section 2, we give the definition of
potential lct and gather various materials for the proof of our main result.
In Section 3, we prove Theorem 1.1. In Section 4, we study exhausting
sequences for the set of varieties of Fano type.

##  2\. Preliminaries

In this section, we collect basic definitions and results that are necessary
in the proof of the main results. Throughout the paper, we consider
$\mathbb{Q}$-factorial normal projective varieties defined over some
algebraically closed field of characteristic $0$. Any divisor will be an
$\mathbb{R}$-divisor unless otherwise stated.

###  2.1. Positivity

Let $\Gamma$ be a prime divisor on a variety $X$. If $D$ is big, then we
define the asymptotic divisorial valuation of $\Gamma$ along $D$ as

$\sigma_{\Gamma}(D):=\inf\\{\operatorname{mult}_{\Gamma}D^{\prime}\leavevmode\nobreak\
|\leavevmode\nobreak\ \Delta\in|D|_{\mathbb{R}}\\},$

where $|D|_{\mathbb{R}}=\\{D^{\prime}\geq 0\leavevmode\nobreak\
|\leavevmode\nobreak\ D^{\prime}\sim_{\mathbb{R}}D\\}$. If $D$ is only
pseudoeffective, then we define $\sigma_{\Gamma}(D):=\lim_{\varepsilon\to
0}\sigma_{\Gamma}(D+\varepsilon A)$ where $A$ is an ample divisor. By [Nak,
Proposition III.1.10], it is known that there are only finitely many prime
divisors $\Gamma$ such that $\sigma_{\Gamma}(D)>0$.

######  Definition 2.1 (divisorial Zariski decomposition).

For a pseudoeffective divisor $D$ on $X$, we define the _negative part_ of $D$
as

$N_{\sigma}(D):=\sum_{\Gamma}\sigma_{\Gamma}(D)\Gamma$

where the sum is taken over all prime divisors on $X$, and the _positive part_
of $D$ as $P_{\sigma}(D):=D-N_{\sigma}(D)$. We call the expression
$D=P_{\sigma}(D)+N_{\sigma}(D)$ the _divisorial Zariski decomposition_ of $D$.
If $P_{\sigma}(D)$ is nef (resp. semiample), then we call
$D=P_{\sigma}(D)+N_{\sigma}(D)$ the Zariski decomposition (resp. good Zariski
decomposition) of $D$.

It is known that the positive part $P$ of $D$ carries much of the positivity
of $D$.

###  2.2. Singularity

By definition, a pair $(X,\Delta)$ consists of a $\mathbb{Q}$-factorial normal
projective variety $X$ and a boundary $\mathbb{Q}$-divisor $\Delta$ on $X$,
i.e., the coefficients of $\Delta$ are rational numbers in $[0,1]$.

######  Definition 2.2 (klt, lc).

Let $(X,\Delta)$ be a pair and $f:Y\to X$ be a proper birational morphism with
$Y$ normal projective variety. A prime divisor $F$ on such $Y$ is called a
prime divisor over $X$. Write $K_{Y}+\Delta_{Y}=f^{\ast}(K_{X}+\Delta)$ for
some divisor $\Delta_{Y}$ on $Y$. Then the log discrepancy of $(X,\Delta)$ at
$F$ is defined as $a(X,\Delta,F):=1-\operatorname{mult}_{F}\Delta_{Y}$. We say
that $(X,\Delta)$ is klt if $\inf_{F}a(X,\Delta,F)>0$, where the inifimum is
taken over all prime divisors $F$ over $X$. A pair $(X,\Delta)$ is
$\epsilon$-lc if $\inf_{F}(X,\Delta,F)\geq\epsilon$ and we just say a pair
$(X,\Delta)$ is lc if it is 0-lc.

To determine whether a given pair $(X,\Delta)$ is klt or $\epsilon$-lc, it
suffices to compute the log discrepancies for the divisors on a fixed log
resolution of $(X,\Delta)$.

######  Definition 2.3.

Let $(X,\Delta)$ be a pair such that $-(K_{X}+\Delta)$ is pseudoeffective and
let $F$ be a prime divisor over $X$.

  1. (1)

The potential log discrepancy $\bar{a}(X,\Delta,F)$ of $(X,\Delta)$ at $F$ is
defined as

$\bar{a}(X,\Delta,F):=a(X,\Delta,F)-\operatorname{mult}_{F}N,$

where $F$ is a prime divisor on some model $Y$ with $f:Y\to X$ and $N$ is the
negative part of $-f^{\ast}(K_{X}+\Delta)$. Since $a(X,\Delta,F)$ and
$\operatorname{mult}_{F}$ only depend on the valuation
$\operatorname{ord}_{F}$, potential log discrepancy also only depend on the
valuation $\operatorname{ord}_{F}$.

  2. (2)

The pair $(X,\Delta)$ is said to be potentially klt if
$\inf_{F}\bar{a}(X,\Delta,F)>0$ where $\inf$ is taken over all the prime
divisors $F$ over $X$. For a real number $\epsilon\geq 0$, the pair
$(X,\Delta)$ is said to be $\epsilon$-plc if
$\inf_{F}\bar{a}(X,\Delta,F)\geq\epsilon$.

######  Remark 2.4.

Unlike the usual klt or $\epsilon$-lc, to determine whether a given pair
$(X,\Delta)$ with pseudoeffective $-(K_{X}+\Delta)$ is pklt or $\epsilon$-plc,
we need to consider all the prime divisors over $X$ not just on a fixed model
of $X$. However, if there exists a log resolution $f:Y\to X$ of $(X,\Delta)$
such that $-f^{*}(K_{X}+\Delta)$ admits the good Zariski decomposition, then
it is enough to consider the potential log discrepancies only for the divisors
$F$ on such fixed $Y$ in order to determine whether the pair $(X,\Delta)$ is
pklt or not (see Lemma 3.3).

The following birational invariant property is what motivated the definition
of potential pairs.

######  Proposition 2.5 (cf. [CP, Proposition 3.11]).

Let $(X,\Delta)$ be an $\epsilon$-plc pair. Suppose that there is a
$-(K_{X}+\Delta)$-negative map $\varphi:X\dashrightarrow X^{\prime}$. Then for
any prime divisor $F$ over $X$, we have

$\bar{a}(X,\Delta,F)=\bar{a}(X^{\prime},\Delta^{\prime},F),$

where $\Delta^{\prime}$ is the birational transform of $\Delta$ on
$X^{\prime}$.

If we assume that $\varphi$ is the $-(K_{X}+\Delta)$-MMP, we have
$\bar{a}(X^{\prime},\Delta^{\prime},F)=a(X^{\prime},\Delta^{\prime},F)$ since
$-(K_{X^{\prime}}+\Delta^{\prime})$ is nef. Thus the resulting model
$(X^{\prime},\Delta^{\prime})$ is an $\epsilon$-lc pair.

The following is a characterization of Fano type varieties in terms of
potential pairs.

######  Theorem 2.6 ([CP, Theorem 5.1]).

Let $X$ be $\mathbb{Q}$-factorial normal projective variety. Then $X$ is a
Fano type variety if and only if $-K_{X}$ is big and $(X,\Delta)$ is a pklt
pair for some divisor $\Delta$.

Let us recall the the definitions of (asymptotic) multiplier ideals [Laz].

######  Definition 2.7.

Let $(X,\Delta)$ be a pair, $D$ a $\mathbb{Q}$-Cartier divisor on $X$, and let
$\mathfrak{b}_{\bullet}$ be a graded ideal sheaf on $X$, i.e., coherent ideal
sheaves on $X$ such that
$\mathfrak{b}_{m}\cdot\mathfrak{b}_{n}\subseteq\mathfrak{b}_{m+n}$ for any
nonnegative integers $m,n$.

  1. (1)

Let $f:Y\to X$ be a log resolution of $(X,\Delta+D)$. Then the multiplier
ideal sheaf $\mathcal{J}(X,\Delta;D)$ is defined as

$\mathcal{J}(X,\Delta;D):=f_{\ast}\mathcal{O}_{Y}(-\lfloor\Delta_{Y}+f^{\ast}D\rfloor),$

where $\Delta_{Y}$ is a $\mathbb{Q}$-divisor such that
$K_{Y}+\Delta_{Y}=f^{\ast}(K_{X}+\Delta)$. Let $\mathfrak{a}$ be a coherent
ideal sheaf on $X$ and suppose that $f:Y\to X$ is a log resolution for both
$(X,\Delta)$ and $\mathfrak{a}$. Then, for a rational number $c$, the
multiplier ideal sheaf $\mathcal{J}(X,\Delta;\mathfrak{a}^{c})$ is defined as

$\mathcal{J}(X,\Delta;\mathfrak{a}^{c}):=f_{\ast}\mathcal{O}_{Y}(-\lfloor\Delta_{Y}+cF\rfloor),$

where $F$ is a divisor determined so that
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$.

  2. (2)

The set $\\{\mathcal{J}(X,\Delta;\mathfrak{b}_{m}^{1/m})\\}$ of multiplier
ideal sheaves has the maximal element under the inclusion relation. We denote
the maximal element by $\mathcal{J}(X,\Delta;\mathfrak{b}_{\bullet})$, and
call it asymptotic multiplier ideal sheaf. Let $L$ be a $\mathbb{Q}$-Cartier
divisor on $X$ and let $\mathfrak{b}_{m}$ be the base ideal of $|mL|$. In this
case, we simply denote the asymptotic multiplier ideal sheaf by

$\mathcal{J}(X,\Delta;\mathfrak{b}_{\bullet})=\mathcal{J}(X,\Delta;\lVert
L\rVert).$

######  Definition 2.8.

Let $(X,\Delta)$ be a pair such that $-(K_{X}+\Delta)$ is pseudoeffective. We
define the non pklt locus as

$\mathrm{pNklt}(X,\Delta)=\bigcup\\{c_{X}(F)\leavevmode\nobreak\
|\leavevmode\nobreak\ \text{$F$ is a prime divisor over $X$ such that
$\bar{a}(F,X,\Delta)\leq 0$}\\},$

where $c_{X}(F)$ denotes the center of $F$ on $X$.

Let $\mathfrak{a}$ be a coherent ideal sheaf on $X$. Denote
$\mathcal{Z}(\mathfrak{a})$ by the vanishing locus of $\mathfrak{a}$, i.e.,
underlying Zariski closed set determined by $\mathfrak{a}$. The following
proposition tells us that the locus $\operatorname{pNklt}(X,\Delta)$ is closed
when $-(K_{X}+\Delta)$ is big. The proof is essentially the same as the proof
for [CP, Proposition 4.4].

######  Proposition 2.9.

Let $(X,\Delta)$ be a pair such that $-(K_{X}+\Delta)$ is big. Then we have

$\mathrm{pNklt}(X,\Delta)=\mathcal{Z}(\mathcal{J}(X,\Delta;\lVert-(K_{X}+\Delta)\rVert)).$

###### Proof.

Let $\mathcal{I}$ be the ideal sheaf defining the closed reduced scheme
structure on the stable base locus $\mathbf{B}(-(K_{X}+\Delta))$ and let
$\pi:Y\to X$ be a log resolution of both $\mathcal{I}$ and $(X,\Delta)$, with
$\mathcal{I}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$, where $F$ is an
effective divisor with simple normal crossing support. Then for a sufficiently
large and divisible integer $m>0$, $\pi$ is also a log resolution of
$|-m(K_{X}+\Delta)|$. Let $\pi^{\ast}(-(K_{X}+\Delta))=P+N$ be the divisorial
Zariski decomposition. Write

$K_{Y}+\Delta_{Y}=\pi^{\ast}(K_{X}+\Delta)$

and

$\pi^{\ast}|-m(K_{X}+\Delta)|=|M_{m}|+F_{m},$

where $|M_{m}|$ is free and $\operatorname{Supp}F_{m}=\operatorname{Supp}F$.

Suppose that $\Gamma$ is a prime divisor over $X$ such that
$A_{X,\Delta}(\Gamma)-\sigma_{\Gamma}(-(K_{X}+\Delta))\leq 0$. Blowing up
further, we may assume that $\Gamma$ is a divisor on $Y$. Since $mN\leq
F_{m}$, we have

$\mathrm{pNklt}(X,\Delta)\subseteq\mathcal{Z}(\mathcal{J}(X,\Delta;\lVert-(K_{X}+\Delta)\rVert)).$

Let $\varepsilon>0$ be a positive real number such that
$\operatorname{mult}_{\Gamma}(\Delta_{Y}+N)<1-\varepsilon$ for all prime
divisor $\Gamma\subset Y$ satisfying
$\operatorname{mult}_{\Gamma}(\Delta_{Y}+N)<1$. By [Nak, Lemma III.1.4], there
exists an effective $\mathbb{Q}$-divisor $P_{0}$ such that
$P_{0}\sim_{\mathbb{Q}}P$ and $\operatorname{mult}_{\Gamma}P_{0}<\varepsilon$
for all
$\Gamma\subseteq\operatorname{Supp}\Delta_{Y}\cup\operatorname{Supp}F$. For
sufficiently divisible $m>0$, $mP_{0}+mN$ is Cartier. Thus we have

$mP_{0}+mN\geq F_{m}.$

Let $\Gamma\subset Y$ be a prime divisor such that
$\operatorname{mult}_{\Gamma}(\Delta_{Y}+\tfrac{1}{m}F_{m})\geq 1$. If
$\operatorname{mult}_{\Gamma}(\Delta_{Y}+N)<1$, then

$1\leq\operatorname{mult}_{\Gamma}(\Delta_{Y}+\tfrac{1}{m}F_{m})\leq\operatorname{mult}_{\Gamma}(\Delta_{Y}+N+P_{0})<1.$

This is a contradiction. Therefore we have
$\operatorname{mult}_{\Gamma}(\Delta_{Y}+N)\geq 1$ and

$\mathrm{pNklt}(X,\Delta)\supseteq\mathcal{Z}(\mathcal{J}(X,\Delta;\lVert-(K_{X}+\Delta)\rVert)).$

∎

######  Remark 2.10 (cf. [CP, Proposition 4.4]).

Let $L$ be a $\mathbb{Q}$-Cartier divisor on $X$. Then the asymptotic
multiplier ideal sheaf $\mathcal{J}(X,\Delta;\lVert L\rVert)$ can be computed
by a general member of $|L|_{\mathbb{Q}}$. That is,
$\mathcal{J}(X,\Delta;\lVert L\rVert)=\mathcal{J}(X,\Delta;D)$ for some
$D\in|L|_{\mathbb{Q}}$. Therefore the above proposition shows that there
exists an effective $\mathbb{Q}$-Cartier divisor $D$ such that
$D\sim_{\mathbb{Q}}-(K_{X}+\Delta)$ and

$\mathrm{pNklt}(X,\Delta)=\mathcal{Z}(\mathcal{J}(X,\Delta;D))=\mathrm{Nklt}(X,\Delta+D).$

If $(X,\Delta)$ is a pklt pair with big $\mathbb{Q}$-Cartier divisor
$-(K_{X}+\Delta)$, then the pair $(X,\Delta+D)$ is klt and
$K_{X}+\Delta+D\sim_{\mathbb{Q}}0$. Thus $X$ of Fano type. This gives a proof
of if direction of Theorem 2.6.

If $-(K_{X}+\Delta)$ is pseudoeffective and $D$ is a divisor on $X$, then we
define the pseudoeffective threshold as follows

$t_{\operatorname{eff}}(X,\Delta,D):=\sup\\{\lambda|-(K_{X}+\Delta+\lambda
D)\;\text{is pseudoeffective}\\}.$

Suppose that $(X,\Delta)$ is a log canonical pair and $D$ is an effective
$\mathbb{R}$-Cartier divisor on $X$. The log canonical threshold of $D$ with
respect to $(X,\Delta)$ is

$\mathrm{lct}(X,\Delta;D):=\sup\\{t\in\mathbb{R}\leavevmode\nobreak\
|\leavevmode\nobreak\ \text{$(X,\Delta+tD)$ is log canonical}\\}.$

Note that the notion of pklt, plc can be defined only for the pairs
$(X,\Delta+tD)$ with $0\leq t\leq t_{\operatorname{eff}}(X,\Delta,D)$.

######  Definition 2.11.

Let $(X,\Delta)$ be a pair such that $-(K_{X}+\Delta)$ is pseudoeffective.
Assume that $(X,\Delta)$ is potentially klt. For a divisor $D$ on $X$, suppose
that $t_{\operatorname{eff}}=t_{\operatorname{eff}}(X,\Delta,D)>0$. The _plc
threshold_ $\text{p-lct}(X,\Delta,D)$ of $(X,\Delta)$ with respect to $D$ is
defined as

$\text{p-lct}(X,\Delta,D):=\begin{cases}\sup\\{\lambda|(X,\Delta+\lambda
D)\text{ is pklt}\\}&\text{if $(X,\Delta+t_{\operatorname{eff}}D)$ is not
pklt}\\\ t_{\operatorname{eff}}&\text{otherwise.}\end{cases}$

If $t_{\operatorname{eff}}=0$, the we define $\text{p-lct}(X,\Delta,D)=0$.

Note that if $t_{\operatorname{eff}}>0$ and $-(K_{X}+\Delta+tD)$ is nef for
any $t\in[0,t_{\operatorname{eff}}]$, then
$\text{p-lct}(X,\Delta,D)=\text{lct}(X,\Delta,D)$. Furthermore, if
$(X,\Delta+tD)$ is potentially klt for any $t\in[0,t_{\operatorname{eff}}]$,
then $\text{p-lct}(X,\Delta,D)=t_{\operatorname{eff}}(X,\Delta,D)$. Thus in
some cases, the effective threshold $t_{\operatorname{eff}}(X,\Delta,D)$ and
the log canonical threshold $\text{lct}(X,\Delta,D)$ appear as special cases
of potential lct.

We now recall the ascending chain condition (or ACC for short) of lct.
Descending chain condition is also abbreviated as DCC. Our main result is a
generalization of this result.

######  Theorem 2.12 ([HMX]).

Suppose that $n$ is a positive integer, $I$ is a subset of nonnegative real
numbers satisfying DCC. Then there exists an ACC set $I^{\prime}$ such that
the log canonical threshold $\mathrm{lct}(X,\Delta;D)$ belongs to
$I^{\prime}$, whenever the pair $(X,\Delta)$ and the divisor $D$ satisfy the
followings:

  1. (i)

$(X,\Delta)$ is a log canonical pair with $X$ of dimension $n$,

  2. (ii)

the coefficients of $\Delta$ and $D$ belong to $I$.

###  2.3. Finite generation

Let $X$ be a normal projective variety. For an $\mathbb{R}$-divisor $D$ on
$X$, we consider the group of global sections of $D$:

$H^{0}(X,D)=\\{f\in\mathbb{C}(X)\leavevmode\nobreak\ |\leavevmode\nobreak\
\mathrm{div}f+D\geq 0\\}.$

Let $D_{1},\cdots,D_{r}$ be given fixed $\mathbb{Q}$-divisors on $X$. Then we
define the section ring associated to $D_{1},\cdots,D_{r}$ as

$R=R(X;D_{1},\dots,D_{r}):=\bigoplus_{(m_{1},\cdots,m_{r})\in\mathbb{N}^{r}}H^{0}(X,m_{1}D_{1}+\cdots+m_{r}D_{r})$

and the support of $R$ is defined as

$\operatorname{Supp}R=\left\\{\left.D\in\sum\mathbb{R}_{\geq
0}D_{i}\leavevmode\nobreak\ \right|\leavevmode\nobreak\
|D|_{\mathbb{R}}\neq\emptyset\text{ in
}\operatorname{Div}_{\mathbb{R}}(X)\right\\}.$

######  Theorem 2.13 ([KKL]).

Let $X$ be a normal projective variety and let $D_{1},\cdots,D_{r}$ be
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors on $X$. Assume that the section
ring $R=R(X;D_{1},\dots,D_{r})$ is finitely generated. Then the following
hold:

  1. (1)

$\operatorname{Supp}R$ is a rational polyhedral cone.

  2. (2)

Suppose that $\operatorname{Supp}R$ contains a big divisor. If
$D\in\sum\mathbb{R}_{\geq 0}D_{i}$ is pseudoeffective, then $D$ is in
$\operatorname{Supp}R$.

  3. (3)

There is a finite decomposition $\operatorname{Supp}R=\bigcup\mathcal{C}_{i}$
such that each $\mathcal{C}_{i}$ is a rational polyhedral cone,
$\sigma_{\Gamma}$ is linear on $\mathcal{C}_{i}$ for every prime divisor
$\Gamma$ over $X$, and the cones $\mathcal{C}_{i}$ form a fan.

  4. (4)

There is a positive integer $d$ and a resolution $f:Y\to X$ such that
$\operatorname{Mob}f^{\ast}(dD)$ is base point free for every
$D\in\operatorname{Supp}R\cap\operatorname{Div}(X)$, and
$\operatorname{Mob}f^{\ast}(kdD)=k\operatorname{Mob}f^{\ast}(dD)$ for every
positive integer $k$.

##  3\. Proofs of Theorems

We first prove the ACC property of pseudoeffective thresholds
$t_{\operatorname{eff}}(X,\Delta,D)$.

######  Theorem 3.1.

Let $I$ be a DCC set of nonnegative real numbers and $n$ a natural number.
Then there exists an ACC set $I^{\prime}$ such that the pseudoeffective
threshold $t_{\operatorname{eff}}(X,\Delta,D)$ belongs to $I^{\prime}$,
whenever the pair $(X,\Delta)$ and the divisor $D$ satisfy the following
conditions:

  1. (i)

$(X,\Delta)$ is an $\epsilon$-plc pair with $X$ of dimension $n$,

  2. (ii)

$-(K_{X}+\Delta)$ is big,

  3. (iii)

$D$ is a nef divisor, and

  4. (iv)

the coefficients of $\Delta$ and $D$ belong to $I$.

###### Proof.

Since $(X,\Delta)$ is potentially $\epsilon$-lc and $-(K_{X}+\Delta)$ is big,
$X$ is of Fano type by Theorem 2.6. Therefore the section ring
$R=R(X;-(K_{X}+\Delta),-(K_{X}+\Delta+t_{\operatorname{eff}}D))$ is finitely
generated. For a log resolution $f$ as in Theorem 2.13 (4), let
$-f^{\ast}(K_{X}+\Delta+tD)=P(t)+N(t)$ be the good Zariski decomposition. Then
$P(t)$ is semiample for all $0\leq t\leq
t_{\operatorname{eff}}=t_{\operatorname{eff}}(X,\Delta,D)$. Note that
$t_{\operatorname{eff}}>0$ because $-(K_{X}+\Delta)$ is big. Take a
sufficiently small $\delta>0$ and let $g:Y\to X^{\prime}$ be the semiample
fibration associated to $P(t_{\operatorname{eff}}-\delta)$. Then by [KKL,
Theorem 4.2], the induced birational map $\varphi:X\dashrightarrow X^{\prime}$
is the ample model of $-(K_{X}+\Delta+tD)$ for all
$t_{\operatorname{eff}}-\delta\leq t<t_{\operatorname{eff}}$, and at the same
time, it is also a semiample model of
$-(K_{X}+\Delta+t_{\operatorname{eff}}D)$. Let $h:X^{\prime}\to Z^{\prime}$ be
the semiample fibration associated to
$-(K_{X^{\prime}}+\Delta^{\prime}+t_{\operatorname{eff}}D^{\prime})=-\varphi_{\ast}(K_{X}+\Delta+t_{\operatorname{eff}}D)$
and let $F^{\prime}$ be a general fiber of $h$. Note that $\dim F^{\prime}>0$
because $-(K_{X^{\prime}}+\Delta^{\prime}+t_{\operatorname{eff}}D^{\prime})$
is not big.

Now we claim that $F^{\prime}$ belongs to some bounded family. Let
$K_{Y}+\Delta_{Y}=f^{\ast}(K_{X}+\Delta)$,
$K_{Y}+\Delta_{Y}+E_{Y}=g^{\ast}(K_{X^{\prime}}+\Delta^{\prime})$ and
$P+N=-f^{\ast}(K_{X}+\Delta)$ the good Zariski decomposition. Then we can
write $-P-N+E_{Y}=g^{\ast}(K_{X^{\prime}}+\Delta^{\prime})$. Since $D$ is nef,
$N\leq N(t_{\operatorname{eff}})$. By [KKL, Lemma 4.1],
$N(t_{\operatorname{eff}})$ is $g$-exceptional and so is $N$. Moreover, by
definition, $E_{Y}$ is $g$-exceptional. Thus, by negativity lemma, $E_{Y}\leq
N$.

By assumption (i), we know that $(X^{\prime},\Delta^{\prime})$ is
$\epsilon$-lc. Thus the pair $(F^{\prime},\Delta^{\prime}|_{F^{\prime}})$ is
also $\epsilon$-lc. Now consider the restriction

$-(K_{X^{\prime}}+\Delta^{\prime}+t_{\operatorname{eff}}D^{\prime})|_{F^{\prime}}=-(K_{F^{\prime}}+\Delta^{\prime}|_{F^{\prime}}+t_{\operatorname{eff}}D^{\prime}|_{F^{\prime}})\equiv
0.$

Since

$\delta
D^{\prime}|_{F^{\prime}}\equiv-(K_{F^{\prime}}+\Delta^{\prime}|_{F^{\prime}}+(t_{\operatorname{eff}}-\delta)D^{\prime}|_{F^{\prime}})=-(K_{X^{\prime}}+\Delta^{\prime}+(t_{\operatorname{eff}}-\delta)D^{\prime})|_{F^{\prime}}$

and the right-hand side is ample, $D^{\prime}|_{F^{\prime}}$ is ample. This
implies that $-(K_{F^{\prime}}+\Delta^{\prime}|_{F^{\prime}})$ is ample. By
[Bir2, Theorem 1.1], $F^{\prime}$ belongs to a bounded family $\mathcal{F}$.

Finally by arguing as in the proof of [HL, Theorem 1.3], we finish the proof
as follows. Note first that we can write for some very ample Cartier divisor
$M_{F^{\prime}}$ on $F^{\prime}$

$t_{\operatorname{eff}}=\frac{1}{d}(-K_{F^{\prime}}\cdot M_{F^{\prime}}^{\dim
F^{\prime}-1}-\Delta^{\prime}|_{F^{\prime}}\cdot M_{F^{\prime}}^{\dim
F^{\prime}-1})$

where $d=D^{\prime}|_{F^{\prime}}\cdot M_{F^{\prime}}^{\dim F^{\prime}-1}$. By
the boundedness of $F^{\prime}$, we may assume that $-K_{F^{\prime}}\cdot
M_{F^{\prime}}^{\dim F^{\prime}-1}\leq r$ for some fixed number $r>0$. It is
easy to see that $d=D^{\prime}|_{F^{\prime}}\cdot M_{F^{\prime}}^{\dim
F^{\prime}-1}$ belongs to a DCC set (e.g., $I^{\prime}:=\\{a\cdot m|a\in
I,m\in\mathbb{N}\\}$). Note also that $\Delta^{\prime}|_{F^{\prime}}\cdot
M_{F^{\prime}}^{\dim F^{\prime}-1}$ belongs to the DCC set $I^{\prime}$.
Therefore, it is immediate that $t_{\operatorname{eff}}$ belongs to some ACC
set. ∎

In Theorem 3.1, the $\epsilon$-plc condition of $(X,\Delta)$ with $\epsilon>0$
cannot be strengthened to plc condition (i.e., $\epsilon=0$).

######  Example 3.2.

Let $X_{n}=\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(n))$ be a
Hirzebruch surface, $M$ the negative section, and $F$ a
$\mathbb{P}^{1}$-fiber. Then
$-K_{X_{n}}=((1+\frac{2}{n})M+(n+2)F)+(1-\frac{2}{n})M$ is the Zariski
decomposition with the negative part $N=(1-\frac{2}{n})M$. Let
$P=-K_{X_{n}}-N$. Then $P^{2}>0$ which implies that $-K_{X_{n}}$ is big.
Furthermore, $X_{n}$ is a Mori dream space. Hence $P$ is semiample and
$N=\frac{1}{m}\operatorname{Fix}(-mK_{X_{n}})$ for sufficiently divisible
integer $m>0$. Therefore $\mathcal{J}(X_{n};\lVert-
K_{X_{n}}\rVert)=\mathcal{O}_{X_{n}}(-\lfloor N\rfloor)=\mathcal{O}_{X_{n}}$.
By Lemma 3.3, we know that $X_{n}$ is pklt. If we let $D_{n}=M+(n+3)F$, then
one can see that $t_{\operatorname{eff}}(X_{n},0,D_{n})=\frac{n+2}{n+3}$.
Clearly, $\left\\{\frac{n+2}{n+3}\left|n\in\mathbb{N}\right.\right\\}$ does
not satisfy ACC. This shows that the condition being $\epsilon$-plc in Theorem
3.1 is necessary. Note that $\bar{a}(X_{n},0;M)=\frac{2}{n}$.

The following lemma is used in the proof of Theorem 1.1.

######  Lemma 3.3.

Let $(X,\Delta)$ be a pair such that $-(K_{X}+\Delta)$ is effective and
$R(X;-(K_{X}+\Delta))$ is finitely generated. Then we have

$\mathrm{pNklt}(X,\Delta)=\mathcal{Z}(\mathcal{J}(X,\Delta;\lVert-(K_{X}+\Delta)\rVert)).$

###### Proof.

By assumption, there are a positive integer $d$ and a resolution $f:Y\to X$
such that $M_{d}:=\operatorname{Mob}(-df^{\ast}(K_{X}+\Delta))$ is globally
generated and $M_{kd}=kM_{d}$ for every positive integer $k$. Let
$-f^{\ast}(K_{X}+\Delta)=P+N$ be the divisorial Zariski decomposition. Then we
have $P=\frac{1}{d}M_{d}$ and $N=\frac{1}{d}F_{d}$, where
$F_{d}=\operatorname{Fix}(-df^{\ast}(K_{X}+\Delta))$. Since $F_{kd}=kF_{d}$
for every positive integer $k$, we have

(3.1)
$\mathcal{J}(X,\Delta;\lVert-(K_{X}+\Delta)\rVert)=f_{\ast}\mathcal{O}_{Y}(-\lfloor\Delta_{Y}+\tfrac{1}{d}F_{d}\rfloor)=f_{\ast}\mathcal{O}_{Y}(-\lfloor\Delta_{Y}+N\rfloor)$

∎

Now we prove our main result, Theorem 1.1.

###### Proof of Theorem 1.1.

By Theorem 3.1, it is enough to show that $\text{p-lct}(X,\Delta,D)$ belongs
to an ACC set under the following conditions: $(X,\Delta)$ is a pklt pair of
dimension $n$ and $D$ is an effective divisor such that
$(X,\Delta+t_{\operatorname{eff}}D)$ is not pklt. Let
$\lambda_{1}=\operatorname{p-lct}(X,\Delta;D)\leq t_{\operatorname{eff}}$.
Since $-(K_{X}+\Delta)$ is big and $(X,\Delta)$ is pklt, $X$ is of Fano type
by Theorem 2.6. Therefore $R(X,-(K_{X}+\Delta),-(K_{X}+\Delta+\lambda_{1}D))$
is finitely generated. Take a resolution $f:Y\to X$ as in Theorem 2.13 (4).
Let $-f^{\ast}(K_{X}+\Delta+\lambda D)=P(\lambda)+N(\lambda)$ be the good
Zariski decomposition for $0\leq\lambda\leq t_{\operatorname{eff}}$. There
exists a semiample fibration $g:Y\to X^{\prime}$ associated to all
$P(\lambda_{1}-\delta)$ with sufficiently small $\delta>0$. By [KKL, Theorem
4.2], the induced rational map $\varphi:X\dashrightarrow X^{\prime}$ is a
semiample model for $-(K_{X}+\Delta+\lambda D)$ for any
$\lambda\in[\lambda_{1}-\delta,\lambda_{1}]$. Let
$K_{X^{\prime}}+\Delta^{\prime}+\lambda
D^{\prime}=\varphi_{\ast}(K_{X}+\Delta+\lambda D)$. Then
$K_{Y}+\Delta_{Y}+\lambda
f^{\ast}D+N(\lambda)=g^{\ast}(K_{X^{\prime}}+\Delta^{\prime}+\lambda
D^{\prime})$. By Lemma 3.3, $(X,\Delta+\lambda D)$ is pklt if and only if the
coefficients of $\Delta_{Y}+\lambda f^{\ast}D+N(\lambda)$ are less than 1.
Therefore we obtain that
$\operatorname{p-lct}(X,\Delta;D)=\operatorname{lct}(X^{\prime},\Delta^{\prime};D^{\prime})$.
Hence by Theorem 2.12, there is an ACC set $\Sigma_{1}$ containing
$\lambda_{1}$. ∎

As explained in Introduction, Theorem 1.1 does not hold for $\epsilon=0$ in
general.

##  4\. Boundedness of $\epsilon$-plc pairs

It is known that the ACC of log canonical thresholds (Theorem 2.12) is implied
by the following Birkar’s result on the boundedness of $\epsilon$-lc Fano
varieties ([HMX]).

######  Theorem 4.1 ([Bir2, Theorem 1.1]).

Let $d$ be a natural number and $\epsilon$ a positive real number. Then the
projective varieties $X$ such that

  1. $\bullet$

$(X,\Delta)$ is $\epsilon$-lc of dimension $d$ for some boundary $\Delta$, and

  2. $\bullet$

$-(K_{X}+\Delta)$ is nef and big,

form a bounded family.

Let $d$ be a positive integer and $\epsilon$ a nonnegative real number. Let us
consider the following collections of Fano type varieties:

$\begin{array}[]{rl}\operatorname{FT}(d,\epsilon)&:=\left\\{X\left|\begin{array}[]{l}\text{$X$
is a $\mathbb{Q}$-factorial normal projective variety of dimension $d$,}\\\
\text{$(X,\Delta)$ is $\epsilon$-lc and $-(K_{X}+\Delta)$ is nef and big for
some boundary $\Delta$}\end{array}\right.\right\\},\\\ &\\\
\operatorname{pFT}(d,\epsilon)&:=\left\\{X\left|\begin{array}[]{l}\text{$X$ is
a $\mathbb{Q}$-factorial normal projective variety of dimension $d$,}\\\
\text{$(X,\Delta)$ is $\epsilon$-plc and $-(K_{X}+\Delta)$ is big for some
boundary $\Delta$}\end{array}\right.\right\\}.\end{array}$

We will write $\operatorname{FT}(d,0)=\operatorname{FT}(d)$ and
$\operatorname{pFT}(d,0)=\operatorname{pFT}(d)$.

By definition,
$\operatorname{FT}(d,\epsilon)\subseteq\operatorname{pFT}(d,\epsilon)$ for any
$\epsilon\geq 0$ and the following holds:

$\operatorname{FT}(d)=\bigcup_{\epsilon>0}\operatorname{FT}(d,\epsilon)=\bigcup_{\epsilon>0}\operatorname{pFT}(d,\epsilon).$

Theorem 4.1 proves that $\operatorname{FT}(d,\epsilon)$ is bounded for any
$\epsilon>0$. It is well known that the Fano type varieties are Mori dream
spaces by [BCHM]. Thus the varieties $X\in\operatorname{FT}(d,\epsilon)$ can
be understood as the images of some
$X^{\prime}\in\operatorname{pFT}(d,\epsilon)$ that are obtained by running the
Minimal model program on some $\epsilon$-lc pair $(X^{\prime},\Delta)$.

######  Theorem 4.2.

Let $d$ be a positive integer and $\epsilon$ a positive real number. Then
$\operatorname{pFT}(d,\epsilon)$ is a bounded family.

###### Proof.

Let $X\in\operatorname{pFT}(d,\epsilon)$. Then $X$ is of Fano type and we can
run $-K_{X}$-MMP. Let $\phi:X\dashrightarrow X^{\prime}$ be a $-K_{X}$-MMP
which terminates with nef $\mathbb{Q}$-divisor $-K_{X^{\prime}}$. Then
$X^{\prime}$ is $\epsilon$-lc and $-K_{X^{\prime}}$ is nef and big. Thus, by
[Bir2, Theorem 1.1], $X^{\prime}$ belongs to a bounded family. This implies
that there is a positive integer $n$ such that $-nK_{X^{\prime}}$ is Cartier.
Now by the effective base point free theorem, we can also assume that
$|-nK_{X^{\prime}}|$ is base point free. Let $H^{\prime}\in|-nK_{X^{\prime}}|$
be a general element and let $B^{\prime}=\frac{1}{n}H^{\prime}$. Then
$K_{X^{\prime}}+B^{\prime}$ is a klt $n$-complement of $K_{X^{\prime}}$ and it
gives a klt $n$-complement $K_{X}+B$ of $K_{X}$ (cf. [Bir1, 6.1.(3)]).
Finally, by [HX, Theorem 1.3], $X$ belongs to a bounded family. ∎

We also have the following result.

######  Proposition 4.3.

For any $\epsilon>0$, there exists $\delta>0$ such taht
$\operatorname{FT}(n,\epsilon)\subsetneq\operatorname{pFT}(n,\delta)$.

###### Proof.

We know that for a fixed $\epsilon>0$,
$\operatorname{FT}(n,\epsilon)\subseteq\operatorname{pFT}(n,\epsilon)\subseteq\operatorname{pFT}(n,\epsilon^{\prime})$
for any $\epsilon^{\prime}<\epsilon$. Suppose that
$\operatorname{FT}(n,\epsilon)=\operatorname{pFT}(n,\epsilon^{\prime})$ for
any $0<\epsilon^{\prime}<\epsilon$. Note that

$\operatorname{FT}(n)=\bigcup_{r>0}\operatorname{pFT}(n,r)=\bigcup_{\epsilon>r>0}\operatorname{pFT}(n,r)=\operatorname{FT}(n,\epsilon).$

However, it is clear that
$\operatorname{FT}(n)\neq\operatorname{FT}(n,\epsilon)$. ∎

Proposition 4.3 implies that $\operatorname{pFT}(d,\epsilon)$ is not
necessarily of the form $\operatorname{FT}(d,\epsilon)$. Thus Theorem 4.1
cannot be applied directly to obtain the boundedness of
$\operatorname{pFT}(d,\epsilon)$. The following proposition gives an
alternative proof for the boundedness of $\operatorname{pFT}(d,\epsilon)$.

######  Proposition 4.4.

Let $\epsilon$ be a positive real number and $d$ a positive integer. Then
$\operatorname{pFT}(d,\epsilon)\subseteq\operatorname{FT}(d,\epsilon^{\prime})$
for any $0<\epsilon^{\prime}<\epsilon$.

###### Proof.

Suppose that $X\in\operatorname{pFT}(n,\epsilon)$. Then there exists a
boundary divisor $\Delta$ on $X$ such that $-(K_{X}+\Delta)$ is big and
$(X,\Delta)$ is $\epsilon$-plc. Since $X$ is of Fano type, the section ring
$R(-(K_{X}+\Delta))$ is finitely generated. Let $f:Y\to X$ be a log resolution
of $(X,\Delta)$ and let $-f^{\ast}(K_{X}+\Delta)=P+N$ be the good Zariski
decomposition. Write $K_{Y}+\Delta_{Y}=f^{\ast}(K_{X}+\Delta)$. Take a general
element $P_{0}\in|P|_{\mathbb{Q}}$ and let $D:=f_{\ast}(P_{0}+N)$. Then
$K_{X}+\Delta+D\sim_{\mathbb{Q}}0$ and
$f^{\ast}(K_{X}+\Delta+D)=K_{Y}+\Delta_{Y}+N+P_{0}$. Since $-(K_{X}+\Delta)$
is big, there are ample $\mathbb{Q}$-divisor $A$ and an effective
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $\Delta^{\prime}$ such that
$-(K_{X}+\Delta)\sim_{\mathbb{Q}}A+\Delta^{\prime}$. Let $\epsilon^{\prime}$
be a positive real number. Consider the pair
$(X,\Delta+\epsilon^{\prime}\Delta^{\prime}+(1-\epsilon^{\prime})D)$. By
construction,

$f^{\ast}(K_{X}+\Delta+\epsilon^{\prime}\Delta^{\prime}+(1-\epsilon^{\prime})D)=K_{Y}+\Delta_{Y}+N+\epsilon^{\prime}(f^{\ast}\Delta^{\prime}-N)+P_{0}.$

If the coefficients of $f^{\ast}\Delta^{\prime}$ are bounded, then we can show
that the assertion is true by taking sufficiently small $\epsilon^{\prime}$.
Since $-(K_{X}+\Delta)$ is not nef, there exists an rational curve $C$ on $X$
such that $0<-C\cdot(K_{X}+\Delta)\leq 2n$. Let $C^{\prime}$ be a curve on $Y$
such that $f(C^{\prime})=C$. Then we have

$C^{\prime}\cdot f^{\ast}\Delta^{\prime}<C^{\prime}\cdot
f^{\ast}(A+\Delta^{\prime})\leq 2n.$

Therefore, for any $y\in Y$,

$\operatorname{mult}_{y}C^{\prime}\cdot\operatorname{mult}_{y}f^{\ast}\Delta^{\prime}<2n.$

Hence the coefficients of $f^{\ast}\Delta^{\prime}$ is bounded. ∎

Since $\operatorname{FT}(d,\epsilon^{\prime})$ is bounded by Theorem 4.1, so
is $\operatorname{pFT}(d,\epsilon)$.

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