On leave of absence from the ] Institute of Physics and Electronics, Hanoi,
Vietnam

# Exact and approximate ensemble treatments
of thermal pairing in a multilevel model

N. Quang Hung1 [<EMAIL_ADDRESS>N. Dinh Dang1,2<EMAIL_ADDRESS>1) Heavy-Ion
Nuclear Physics Laboratory, RIKEN Nishina Center for Accelerator-Based
Science, 2-1 Hirosawa, Wako City, 351-0198 Saitama, Japan
2) Institute for Nuclear Science and Technique, Hanoi, Vietnam

###### Abstract

A systematic comparison is conducted for pairing properties of finite systems
at nonzero temperature as predicted by the exact solutions of the pairing
problem embedded in three principal statistical ensembles, as well as the
unprojected (FTBCS1+SCQRPA) and Lipkin-Nogami projected (FTLN1+SCQRPA)
theories that include the quasiparticle number fluctuation and coupling to
pair vibrations within the self-consistent quasiparticle random-phase
approximation. The numerical calculations are performed for the pairing gap,
total energy, heat capacity, entropy, and microcanonical temperature within
the doubly-folded equidistant multilevel pairing model. The FTLN1+SCQRPA
predictions agree best with the exact grand-canonical results. In general, all
approaches clearly show that the superfluid-normal phase transition is
smoothed out in finite systems. A novel formula is suggested for extracting
the empirical pairing gap in reasonable agreement with the exact canonical
results.

Suggested keywords

###### pacs:

21.60.-n, 21.60.Jz, 24.60.-k, 24.10.Pa, 21.10.Ma

##  I INTRODUCTION

Pairing correlations are a fundamental feature responsible for the
superconducting (superfluid) properties in many-body systems ranging from very
large ones as neutron stars to tiny ones as atomic nuclei. In macroscopic
systems such as superconductors, pairing correlations are destroyed as the
temperature $T$ increases and completely vanish at a value $T_{\rm c}\simeq
0.57\Delta(0)$, which is the critical temperature of the phase transition from
the superfluid state to the normal one. Here $\Delta(0)$ is the value of the
pairing gap at $T=$ 0\. The recent years witness a renewed interest in pairing
correlations, which is supported by the exact solutions of the pairing problem
in practice, the studies of unstable nuclei, the BCS to Bose-Einstein
condensation crossover, etc.

The superfluid properties and superfluid-normal (SN) phase transition of
infinite systems are accurately described by the Bardeen-Cooper-Schrieffer
theory BCS , where the average within the grand canonical ensemble (GCE) is
used to obtain the occupation number in the form of Fermi-Dirac distribution
for noninteracting fermions. The GCE consists of identically prepared systems
in thermal equilibrium, each of which shares its energy and number of
particles with an external heat bath. As compared to the other two principal
thermodynamic ensembles, namely the canonical ensemble (CE) and the
microcanonical one (MCE), the GCE is, perhaps, the most popular in theoretical
studies of systems at finite temperature because it is very convenient in the
calculations of average thermodynamic quantities 111For instance, the double-
time Green’s functions $G(t,t^{\prime})$, which are defined by using the GCE
Green , depend only on the time difference $(t-t^{\prime})$, which greatly
simplify the derivations in many statistical physics applications.. The CE is
also in contact with the heat bath, but the particle number is the same for
all systems. The MCE is an ensemble of thermally isolated systems sharing the
same energy and particle number. In thermodynamics limit (i.e. when the
system’s particle number $N$ and volume $V$ approach infinity, but $N/V$ is
finite), fluctuations of energy and particle number are zero, therefore three
types of ensembles offer the same average values for thermodynamic quantities.
Thermodynamics limit works quite well in large systems as well where these
fluctuations are negligible. The discrepancies between the predictions by
three types of ensembles arise when thermodynamics is applied to small systems
such as atomic nuclei or nanometer-size clusters. These systems have a fixed
and not very large number of particles, their single-particle energy spectra
are discrete with the level spacing comparable to the pairing gap. Under this
circumstance, the justification of using the GCE for these systems becomes
questionable. A number of theoretical studies have also shown that, in these
tiny systems, thermal fluctuations become so large that they smooth out the
sharp SN transition Moretto ; Goodman ; Egido ; SPA ; Zele ; MBCS ; FTBCS1 ;
AFTBCS . As the result, the pairing gap never collapses but decreases
monotonously with increasing $T$. These predictions are in qualitative
agreement with the results obtained by averaging the pairing energy in the CE
built on the eigenvalues obtained by exactly solving the pairing problem
Richardson ; EP . As a matter of fact, even at $T=$ 0, the exact pairing
solution in nuclei shows a sizable pairing energy in the region where the BCS
solution collapses EP . In the literature so far, under the pretext that a
nucleus is a system with a fixed number of particle, the thermodynamic
averages of the exact solutions of the pairing Hamiltonian are usually carried
out within the CE, and the results are compared with those obtained within
different theoretical approximations at $T\neq$ 0\. The latter, such as the
BCS, Hartree-Fock-Bogoliubov (HFB) theories, etc., as a rule, are always
derived within the GCE, where both energy and particle number fluctuate. On
the other hand, the well-known argument that the nuclear temperature should be
extracted from the MCE of thermally isolated nuclei is also quite often
debated and studied in detail Zele .

These results suggest that a thorough comparison of the predictions offered by
the exact pairing solutions averaged within three principal thermodynamic
ensembles, and those given by the recent microscopic approaches, which include
fluctuations around the thermal pairing mean field in nuclei, might be timely
and useful. This question is not new, but the answers to it have been so far
only partial. Already in the sixties Kubo Kubo  drew attention to the
thermodynamic effects in very small metal particles. Later, Denton et al.
Denton  used Kubo’s assumption to study the difference between the predictions
offered by the GCE and CE for the heat capacity and spin susceptibility within
a spinless equidistant level model for electrons. Very recently, the
predictions for thermodynamic quantities such as total energy, heat capacity,
entropy, and microcanonical temperature within three principal ensembles were
studied and compared in Ref. Sumaryada  by using the exact solutions of an
equidistant multilevel model with constant pairing interaction parameter.
However, no results for the pairing gaps as functions of temperature are
reported.

In the present paper, we carry out a systematic comparison of predictions for
nuclear pairing properties obtained by averaging the exact solutions in three
principal thermodynamic ensembles as well as those offered by recent
microscopic approaches to thermal pairing. For the latter, we choose the
unprojected and particle-number projected versions of the FTBCS1+SCQRPA, which
we recently developed in Ref. FTBCS1 . This approach has been rigorously
derived based on the same variational procedure used for the derivation of the
standard BCS theory, taking into account the effects due to quasiparticle-
number fluctuation (QNF) and coupling to the self-consistent quasiparticle
random-phase approximation (SCQRPA) in addition to those of the QNF.

The paper is organized as follows. The pairing Hamiltonian, its
diagonalization, ensemble treatments of the exact pairing solutions, the main
results of the FTBCS1 and FTBCS1+SCQRPA theories, as well as their particle-
number projected versions, are summarized in Sec. II. The numerical results
obtained within the doubly-folded equidistant multilevel model Richardson  are
analyzed in Sec. III. The paper is summarized in the last section, where
conclusions are drawn.

##  II Exact solution of pairing Hamiltonian

###  II.1 Exact solution at zero temperature

We consider a system of $N$ particles with single-particle energies
$\epsilon_{j}$, which are generated by particle creation operators
$a_{jm}^{\dagger}$ on $j$th orbitals with shell degeneracies $2\Omega_{j}$
($\Omega_{j}=j+1/2$), and interacting via a monopole-pairing force with a
constant parameter $G$. This system is described by the well-known pairing
Hamiltonian

$H=\sum_{jm}\epsilon_{j}\hat{N}_{j}-G\sum_{jj^{\prime}}\hat{P}_{j}^{\dagger}\hat{P}_{j^{\prime}}~{},$
(1)

where the particle-number operator $\hat{N}_{j}$ and pairing operator
$\hat{P}_{j}$ are given as

$\hat{N}_{j}=\sum_{m}a_{jm}^{\dagger}a_{jm}~{},\hskip
14.22636pt\hat{P}_{j}^{\dagger}=\sum_{m>0}a_{jm}^{\dagger}a_{j\widetilde{m}}^{\dagger}~{},\hskip
14.22636pt\hat{P}_{j}=(\hat{P}_{j}^{\dagger})^{\dagger}~{},$ (2)

with the symbol $~{}~{}\widetilde{}~{}~{}$ denoting the time-reversal
operator, namely $a_{j\widetilde{m}}=(-)^{j-m}a_{j-m}$. For a two-component
system with $Z$ protons and $N$ neutrons, the sums in Eq. (1) run also over
all $j_{\tau}m_{\tau}$, $j^{\prime}_{\tau}m^{\prime}_{\tau}$, and $G_{\tau}$
with $\tau=(Z,N)$.

The pairing Hamiltonian (1) was solved exactly for the first time in the
sixties by Richardson Richardson . By noticing that the operators
$\hat{J}_{j}^{z}\equiv(\hat{N}_{j}-\Omega_{j})/2$,
$\hat{J}^{+}_{j}\equiv\hat{P}_{j}^{\dagger}$, and
$\hat{J}^{-}_{j}\equiv\hat{P}_{j}$ close an SU(2) algebra of angular momentum,
the authors of Ref. EP  have reduced the problem of solving the Hamiltonian
(1) to its exact diagonalization in the subsets of representations, each of
which is given by a set basis states
$|k\rangle\equiv|\\{s_{j}\\},\\{N_{j}\\}\rangle$ characterized by the partial
occupation number $N_{j}\equiv(J_{j}^{z}+\Omega_{j})/2$ and partial seniority
(the number of unpaired particles) $s_{j}\equiv\Omega_{j}-2J_{j}$ of the $j$th
single-particle orbital. Here $J_{j}(J_{j}+1)$ is the eigenvalue of the total
angular momentum operator
$(\hat{J}_{j})^{2}\equiv\hat{J}^{+}_{j}\hat{J}^{-}_{j}+\hat{J}^{z}_{j}(\hat{J}^{z}_{j}-1)$.
The partial occupation number $N_{j}$ and seniority $s_{j}$ are bound by the
constraints of angular momentum algebra $s_{j}\leq N_{j}\leq
2\Omega_{j}-s_{j}$ with $s_{j}\leq\Omega_{j}$. In the present paper, we use
this exact diagonalization method to find the eigenvalues ${\cal E}_{s}$ and
eigenstates (eigenvectors) $|s\rangle$ of Hamiltonian (1). Each $s$th
eigenstate $|s\rangle$ is fragmented over the basis states $|k\rangle$
according to $|s\rangle=\sum_{k}C_{k}^{(s)}|k\rangle$ with the total seniority
$s=\sum_{j}s_{j}$, and degenerated by

$d_{s}=\prod_{j}\bigg{[}\frac{(2\Omega_{j})!}{s_{j}!(2\Omega-
s_{j})!}-\frac{(2\Omega_{j})!}{(s_{j}-2)!(2\Omega-s_{j}+2)!}\bigg{]}~{},$ (3)

where $(C_{k}^{(s)})^{2}$ determine the weights of the eigenvector components.
The state-dependent exact occupation number $f_{j}^{(s)}$ on the $j$th single-
particle orbital is then calculated as the average value of partial occupation
numbers $N_{j}^{(k)}$ weighted over the basis states $|k\rangle$ as

$f_{j}^{(s)}=\frac{\sum_{k}N_{j}^{(k)}(C_{k}^{(s)})^{2}}{\sum_{k}(C_{k}^{(s)})^{2}}=\sum_{k}N_{j}^{(k)}(C_{k}^{(s)})^{2}~{},\hskip
5.69054pt{\rm with}\hskip 5.69054pt\sum_{k}(C_{k}^{(s)})^{2}=1~{}.$ (4)

###  II.2 Exact solution embedded in thermodynamic ensembles

The properties of the nucleus as a system of $N$ interacting fermions at
energy ${\cal E}$ can be extracted from its level density $\rho({\cal E},N)$
Bohr

$\rho({\cal E},N)=\sum_{s,n}\delta({\cal E}-{\cal
E}_{s}^{(n)})\delta(N-n)~{},$ (5)

where ${\cal E}_{s}^{(n)}$ are the energies of the quantum states $|s\rangle$
of the $n$-particle system. Applying to the pairing problem, these energies
are the eigenvalues, which are obtained by exactly diagonalizing the pairing
Hamiltonian (1), as has been discussed in the previous section.

####  II.2.1 Grand canonical and canonical ensembles

Since each system in a GCE is exchanging its energy and particle number with
the heat bath at a given temperature $T=1/\beta$, both of its energy and
particle number are allowed to fluctuate. Therefore, instead of the level
density (5), it is convenient to use the average value, which is obtained by
integrating (5) over the intervals in ${\cal E}$ and $N$. This Laplace
transform of the level density $\rho({\cal E},N)$ defines the grand partition
function $Z(\beta,\lambda)$ Bohr ,

${\cal Z}(\beta,\lambda)=\int\int_{0}^{\infty}\rho({\cal E},N)e^{-\beta({\cal
E}-\lambda N)}dNd{\cal E}=\sum_{n}e^{\beta\lambda n}Z(\beta,n)~{},$ (6)

where $Z(\beta,n)$ denotes the partition function of the CE at temperature $T$
and particle number fixed at $n$, namely

${Z}(\beta,n)=\sum_{s}p_{s}(\beta,n)~{},\hskip
14.22636ptp_{s}(\beta,n)=d_{s}^{(n)}e^{-\beta{\cal E}_{s}^{(n)}}~{}.$ (7)

Within the GCE, the chemical potential $\lambda$ should be chosen as a
function of $T$ so that the average particle number $\langle N\rangle$ of the
system always remains equal to $N$. The summations over $n$ and $s$ in Eqs.
(6) and (7) are obtained by using Eq. (5) taking into account the degeneracy
$d_{s}^{(n)}$ (3) of each $s$th state in the $n$-particle system, and carrying
out the double integration with $\delta$ functions.

The thermodynamic quantities such as total energy ${\cal E}$, heat capacity
$C$ within the GCE and CE are as usual given as

$\langle{\cal E}\rangle_{\alpha}=-\frac{{\partial{\ln}{{\bf
Z}(\beta)_{\alpha}}}}{{\partial\beta}};\hskip
14.22636ptC^{(\alpha)}=\frac{\partial\langle{\cal E}\rangle_{\alpha}}{\partial
T}~{}$ (8)

where $\alpha$=GC for the GCE and $\alpha$=C for the CE.

The thermodynamic entropy $S^{(\alpha)}_{\rm th}$ is calculated within the GCE
or CE based on the general definition of the change of entropy (by Clausius):

$dS=\beta d{\cal E}~{}.$ (9)

By using the differentials of ${\rm ln}{\bf Z}(\beta)_{\alpha}$, one obtains

$S^{(\rm GC)}_{\rm th}=\beta(\langle{\cal E}\rangle_{\rm GC}-\lambda N)+{\rm
ln}{\cal Z}(\beta,\lambda)~{},\hskip 14.22636ptS^{(\rm C)}_{\rm
th}=\beta\langle{\cal E}\rangle_{\rm C}+{\rm ln}Z(\beta,N)~{}.$ (10)

The occupation number $f_{j}$ on the $j$th single-particle orbital is obtained
as the ensemble average of the state-dependent occupation numbers
$f_{j}^{(s)}$, namely

$f_{j}^{(\rm GC)}=\frac{1}{{\cal
Z}(\beta,\lambda)}\sum_{s,n}f_{j}^{(s,n)}d_{s}^{n}e^{-\beta({\cal
E}_{s}^{(n)}-\lambda n)}~{},\hskip 14.22636ptf_{j}^{(\rm
C)}=\frac{1}{{Z}(\beta,N)}\sum_{s}f_{j}^{(s,N)}d_{s}^{N}e^{-\beta{\cal
E}_{s}^{N}}~{},$ (11)

for the GCE and CE, respectively.

####  II.2.2 Microcanonical ensemble

Different from the GCE and CE, there is no heat bath within the MCE. A MCE
consists of thermodynamically isolated systems, each of which may be in a
different microstate (microscopic quantum state) $|s\rangle$, but has the same
total energy ${\cal E}$ and particle number $N$. Since the energy and particle
number of the system are fixed, one should use the level density (5) to
calculate directly the entropy by Boltzmann’s definition, namely

$S^{(\rm MC)}({\cal E})={\ln}\Omega({\cal E})~{},$ (12)

where $\Omega({\cal E})=\rho({\cal E})\Delta{\cal E}$ is the statistical
weight, i.e. the number of eigenstates of Hamiltonian (1) within a fixed
energy interval $\Delta{\cal E}$. The condition of thermal equilibrium then
leads to the standard definition of temperature within the MCE as Bohr

$\beta=\frac{\partial S^{(\rm MC)}({\cal E})}{\partial{\cal
E}}=\frac{1}{\rho({\cal E})}\frac{\partial\rho({\cal E})}{\partial{\cal
E}}~{},$ (13)

using which one can build a “thermometer” for each value of the excitation
energy ${\cal E}$ of the system.

In practical calculations, to handle the numerical derivative at the right-
hand side of Eq. (13), one needs a continuous energy dependence of $\rho({\cal
E})$ in the form of a distribution in Eq. (5). This is realized by replacing
the Dirac-$\delta$ function $\delta(x)$ at the right-hand side of Eq. (5),
where $x\equiv{\cal E}-{\cal E}_{s}$, with a nascent $\delta$-function
$\delta_{\sigma}(x)$ ($\sigma>$ 0), i.e. a function that becomes the original
$\delta(x)$ in the limit $\sigma\rightarrow$ 0 222This replacement is
equivalent to the folding procedure for the average density, discussed in Sec.
2.9.3 of the textbook Ring , taken at zero degree ($M=$ 0) of the Laguerre
polynomial $L_{M}^{1/2}(x)$.. Among the popular nascent $\delta$ functions are
the Gaussian (or normal) distribution, Breit-Wigner distribution BW , and
Lorentz (or relativistic Breit-Wigner) distribution, which are given as

$\delta_{\sigma}(x)_{\rm G}=\frac{1}{\sigma\sqrt{2\pi}}{\rm
e}^{-\frac{x^{2}}{2\sigma^{2}}}~{},\hskip 8.53581pt\delta_{\sigma}(x)_{\rm
BW}=\frac{1}{\pi}\frac{\sigma}{x^{2}+\sigma^{2}}~{},$ $\delta_{\sigma}({\cal
E}-{\cal E}_{s})_{\rm L}=\frac{1}{\pi}\frac{\sigma{\cal E}^{2}}{({\cal
E}^{2}-{\cal E}_{s}^{2})^{2}+\sigma^{2}{\cal E}^{2}}~{},$ (14)

respectively. In these distributions, $\sigma$ is a parameter, which defines
the width of the peak centered at ${\cal E}={\cal E}_{s}$. The full widths at
the half maximum (FWHM) $\Gamma$ of these distributions are
$\Gamma=2\sigma\sqrt{2{\ln}2}\simeq 2.36\sigma$ for the Gaussian distribution,
and $2\sigma$ for the Breit-Wigner and Lorentz ones. The disadvantage of using
such smoothing is that the temperature extracted from Eq. (13), of course,
depends on the chosen distribution as well as the value of the parameter
$\sigma$. It is worth mentioning that changing $\sigma$ is not equivalent to
changing $\Delta E$ for the discrete $\rho({\cal E})$ used in calculating the
statistical weight $\Omega({\cal E})$ in Eq. (12), since the wings of any
distributions in Eq. (14) extend with increasing $\sigma$, whereas for the
discrete spectrum, no more levels might be found in the low ${\cal E}$ by
enlarging $\Delta{\cal E}$.

The generalized form of Boltzmann’s entropy (12) is the definition by von
Neumann

$S=-{\rm Tr}[\rho{\rm ln}\rho]~{},$ (15)

which is the quantum mechanical correspondent of the Shannon’s entropy in
classical theory. By expressing the level density $\rho({\cal E})$ in Eq. (5)
in terms of the local densities of states $F_{k}({\cal E})$ Zele ,

$\rho({\cal E})=\sum_{k}F_{k}({\cal E})~{},\hskip 14.22636ptF_{k}({\cal
E})=\sum_{s}[C_{k}^{(s)}]^{2}\delta({\cal E}-{\cal E}_{s})~{},$ (16)

the entropy becomes

$S^{(s)}=-\sum_{k}[C_{k}^{(s)}]^{2}\ln[C_{k}^{(s)}]^{2}~{}.$ (17)

By using Eqs. (13) and (17) one can extract a quantity $T_{s}$ as the
“microcanonical temperature of each eigenstate $s$”.

###  II.3 Determination of pairing gaps from total energies

####  II.3.1 Ensemble-averaged pairing gaps

Although the exact solution of Hamiltonian (1) does not produce a pairing gap
per se, which is a quantity determined within the mean field, it is useful to
define an ensemble-averaged pairing gap to be closely compared with the gaps
predicted by the approximations within and beyond the mean field. In the
present paper, we define this ensemble-averaged gap $\Delta_{\alpha}$ from the
pairing energy ${\cal E}_{\rm pair}^{(\alpha)}$ of the system as follows

$\Delta_{\alpha}=\sqrt{-G{\cal E}_{\rm pair}^{(\alpha)}}~{},\hskip
14.22636pt{\cal E}_{\rm pair}^{(\alpha)}=\langle{\cal
E}\rangle_{\alpha}-\langle{\cal E}\rangle_{\alpha}^{(0)}~{},\hskip
14.22636pt\langle{\cal E}\rangle_{\alpha}^{(0)}\equiv
2\sum_{j}\Omega_{j}\big{[}\epsilon_{j}-\frac{G}{2}f_{j}^{(\alpha)}\big{]}f_{j}^{(\alpha)}~{},$
(18)

within the GCE ($\alpha={\rm GC}$), CE ($\alpha={\rm C}$), and MCE
($\alpha={\rm MC}$), where for the latter we put ${\cal E}_{\rm
pair}^{(\alpha)}\equiv{\cal E}_{\rm pair}(s)$ with $\langle{\cal
E}\rangle_{\rm MC}\equiv{\cal E}_{s}^{(N)}$. The term $\langle{\cal
E}\rangle_{\alpha}^{(0)}$ denotes the contribution from the energy
$2\sum_{j}\Omega_{j}\epsilon_{j}f_{j}^{(\alpha)}$ of the single-particle
motion described by the first term at the right-hand side of Hamiltonian (1),
and the energy $-G\sum_{j}\Omega_{j}[f_{j}^{(\alpha)}]^{2}$ of uncorrelated
single-particle configurations caused by the pairing interaction in
Hamiltonian (1). Therefore, subtracting the term $\langle{\cal
E}\rangle_{\alpha}^{(0)}$ from the total energy $\langle{\cal
E}\rangle_{\alpha}$ yields the residual that corresponds to the energy due to
pure pairing correlations. By replacing $f_{j}^{(\alpha)}$ with $v_{j}^{2}$,
one recovers from Eq. (18) the expression ${\cal E}_{\rm pair}^{(\rm
BCS)}=-\Delta^{2}_{\rm BCS}/G$ of the BCS theory. Given several definitions of
the ensemble-averaged gap existing in the literature, it is worth mentioning
that the definition (18) is very similar to that given by Eq. (52) of Ref.
Delft , whereas, even within the CE, the gap $\Delta_{\rm C}$ is different
from the canonical gap defined in Refs. Ross ; Frau , since in the latter, the
term $\langle{\cal E}\rangle_{\rm C}^{(0)}$ is taken at $G=$ 0\. The pairing
energy ${\cal E}_{\rm pair}^{(\alpha)}$ in Eq. (18) is also different from the
simple average value
$-G\sum_{jj^{\prime}}\langle{\hat{P}^{\dagger}_{j}\hat{P}_{j^{\prime}}}\rangle_{\alpha}$
of the last term of Hamiltonian (1) as the latter still contains the
uncorrelated term $-G\sum_{j}\Omega_{j}[f_{j}^{(\alpha)}]^{2}$.

####  II.3.2 Empirical determination of pairing gap at finite temperature

The simplest way to empirically determine the pairing gap of a system with $N$
particles in the ground state (at $T=$ 0) is to use the so-called three-point
formula $\Delta^{(3)}(N)$, which is given by the odd-even mass difference
between the ground-state energies of the $N$-particle system and the
neighboring systems with $N\pm 1$ particles Bohr . A straightforward extension
of this formula to $T\neq$ 0 reads

$\Delta^{(3)}(\beta,N)\simeq\frac{(-1)^{N}}{2}[\langle{\cal
E}(N+1)\rangle_{\alpha}-2\langle{\cal E}(N)\rangle_{\alpha}+\langle{\cal
E}(N-1)\rangle_{\alpha}]~{},$ (19)

which was used, e.g., in Ref. Kaneko  to extract the thermal pairing gaps in
wolfram and molybdenum isotopes. The four-point formula represents the
arithmetic average of the three-point gaps over in the neighboring systems
with $N$ and $N-1$ particles, namely

$\Delta^{(4)}(\beta,N)=\frac{1}{2}[\Delta^{(3)}(N)+\Delta^{(3)}(N-1)]~{}.$
(20)

A drawback of the gaps $\Delta^{(3)}(\beta,N)$ and $\Delta^{(4)}(\beta,N)$,
defined in this way, is that they still contain the admixture with the
contribution from uncorrelated single-particle configurations. The later
increases with increasing $T$. Therefore, Eq. (19) and, consequently, Eq. (20)
do not hold at finite temperature. To remove this contribution so that the
experimentally extracted pairing gap is comparable with $\Delta_{\alpha}$ in
Eq. (18), we propose in the present paper an improved odd-even mass difference
formula at $T\neq$ 0 as follows. Using Eq. (18) to express the total energy
$\langle{\cal E}(N)\rangle_{\alpha}$ of the system in terms of
$\Delta_{\alpha}(N)$ and $\langle{\cal E}(N)\rangle_{\alpha}^{0}$, we obtain

$\langle{\cal E}(N)\rangle_{\alpha}=\langle{\cal
E}(N)\rangle_{\alpha}^{(0)}-\frac{\widetilde{\Delta}^{2}(\beta,N)}{G}~{}.$
(21)

where $\langle{\cal E}(N)\rangle_{\alpha}$ is the experimentally known total
energy of the system with $N$ particles at $T\neq$ 0, whereas
$\widetilde{\Delta}(\beta,N)$ is the pairing gap of this system to be
determined. Replacing $\langle{\cal E}(N)\rangle_{\alpha}$ in the definition
of the odd-even mass difference (19) with the right-hand side of Eq. (21), we
obtain a quadratic equation for the three-point
$\widetilde{\Delta}^{(3)}(\beta,N)$

$\widetilde{\Delta}^{(3)}(\beta,N)=(-1)^{N}\bigg{\\{}\frac{1}{2}[\langle{\cal
E}(N+1)\rangle_{\alpha}+\langle{\cal E}(N-1)\rangle_{\alpha}]-\langle{\cal
E}\rangle_{\alpha}^{0}+\frac{[\widetilde{\Delta}^{(3)}(\beta,N)]^{2}}{G}\bigg{\\}}~{}.$
(22)

The discriminant of this equation is equal to $G\sqrt{1-4{S^{\prime}}/{G}}$,
where

$S^{\prime}=\frac{1}{2}\big{[}\langle{\cal
E}(N+1)\rangle_{\alpha}+\langle{\cal
E}(N-1)\rangle_{\alpha}\big{]}-\langle{\cal E}(N)\rangle_{\alpha}^{(0)}~{}.$
(23)

Therefore the condition for Eq. (22) to have real solutions is $S^{\prime}\leq
G/4$. Including both cases with even and odd $N$, the positive solution of Eq.
(22) is always possible provided $S^{\prime}<$ 0, which reads

$\widetilde{\Delta}^{(3)}(\beta,N)=\frac{G}{2}\bigg{[}(-1)^{N}+\sqrt{1-4\frac{S^{\prime}}{G}}\bigg{]}~{}.$
(24)

The quantity $S^{\prime}$ differs from the conventional odd-even mass
difference shown in the square brackets of (19) by the contribution due to the
uncorrelated single-particle motion, i.e. the last sum containing $G$ in the
definition of $\langle{\cal E}\rangle_{\alpha}^{(0)}$ in Eq. (18). The latter
is zero only at $G=$ 0, which yields $\widetilde{\Delta}^{(3)}(\beta,N)=$ 0,
as can be seen from Eq. (24) as well. In this case both $S^{\prime}$ and the
expression in the square brackets of Eq. (19) vanish as they are just the
difference of the Hartree-Fock energies $\langle
E(N+1)\rangle^{(0)}_{\alpha}+\langle E(N-1)\rangle^{(0)}_{\alpha}-2\langle
E(N)\rangle^{(0)}_{\alpha}$, which is zero at $G=0$. Moreover, while the odd-
even mass difference of Eq. (19) can be positive or negative depending on
whether $N$ is even or odd, $S^{\prime}$ should be always negative as
discussed above. The gap $\widetilde{\Delta}^{(3)}(\beta,N)$ extracted from
Eq. (24) is, therefore, consistent with the result of the exact calculation at
zero temperature, where the pairing gap is zero only at $G$ = 0, and increases
with $G$ (See e.g. Fig. 1 - (a) of Ref. Sumaryada ). As compared to the simple
finite-temperature extension of the odd-even mass (19), the modified gap
$\widetilde{\Delta}^{(3)}(\beta,N)$ is closer to the ensemble-averaged gap
$\Delta_{\alpha}(N)$ (18) since it is free from the contribution of
uncorrelated single-particle configurations. In Eq. (24), the energies
$\langle{\cal E}(N+1)\rangle_{\alpha}$ and $\langle{\cal
E}(N-1)\rangle_{\alpha}$ can be extracted from experiments, whereas the
pairing interaction parameter $G$ can be obtained by fitting the experimental
values of $\Delta(T=0,N)$. The energy $\langle{\cal
E}(N)\rangle_{\alpha}^{(0)}$ remains the only model-dependent quantity being
determined in terms of the single-particle energies $\epsilon_{j}$ and single-
particle occupation numbers $f_{j}^{(\alpha)}$. As a matter of fact, since the
pairing gap $\Delta^{(3)}(\beta,N)$ of the $N$-particle system at the left-
hand side of the expression (19) is also present in the total energy
$\langle{\cal E}(N)\rangle_{\alpha}$ of the same system at the right-hand side
of (19), the former is simply extracted from the latter by using Eqs. (18) and
(21). As the result, the modified gap $\widetilde{\Delta}^{(3)}(\beta,N)$ of
the system with $N$ particles explicitly depends now on $\langle{\cal
E}(N)\rangle_{\alpha}^{(0)}$ of the same system rather than on its total
energy $\langle{\cal E}(N)\rangle$. The modified four-point gap
$\widetilde{\Delta}^{(4)}(\beta,N)$ is then obtained from the modified three-
point gaps $\widetilde{\Delta}^{(3)}(\beta,N)$ and
$\widetilde{\Delta}^{(3)}(\beta,N-1)$ by using the definition (20).

##  III FTBCS1+SCQRPA and FTLN1+SCQRPA

The FTBCS1+SCQRPA includes the effects due to quasiparticle-number fluctuation
and coupling to the SCQRPA vibrations, which are neglected within the standard
BCS theory. The derivation of the FTBCS1+SCQRPA has already been given and
discussed in detail in Ref FTBCS1 . Therefore we give below only the main
results, which are necessary to follow the numerical calculations in the
present paper.

The rigorous derivation of the FTBCS1+SCQRPA theory follows the standard
variational procedure, which is used to derived the BCS theory. By using the
Bogoliubov’s canonical transformation Bogo

${\alpha}_{jm}^{\dagger}=u_{j}a_{jm}^{\dagger}-v_{j}a_{j\widetilde{m}}~{},\hskip
14.22636pt{\alpha}_{j\widetilde{m}}=u_{j}a_{j\widetilde{m}}+v_{j}a_{jm}^{\dagger}~{},$
(25)

the pairing Hamiltonian (1) is expressed in terms of quasiparticle operators,
$\alpha_{jm}^{\dagger}$ and $\alpha_{jm}$, as $H_{\rm q.p.}$, whose explicit
form is given in many papers, e.g., Eqs. (3), (8) – (14) of Ref. FTBCS1 . The
$u_{j}$ and $v_{j}$ coefficients of the Bogoliubov’s transformation (25) are
determined by minimizing the GCE average value of the Hamiltonian ${\cal
H}=H_{\rm q.p.}-\lambda\hat{N}$. This leads to the equation

$\langle[{\cal H},{\cal A}_{j}^{\dagger}]\rangle_{\rm GC}=0~{},\hskip
14.22636pt{\rm where}\hskip 5.69054pt{\cal
A}_{j}^{\dagger}=\frac{1}{\sqrt{\Omega_{j}}}\sum_{j=1}^{\Omega_{j}}\alpha_{jm}^{\dagger}\alpha_{j\widetilde{m}}^{\dagger}~{}.$
(26)

The final result yields the equation for the level-dependent pairing gap
$\Delta_{j}$ for a system with even number $N$ of particles,

$\Delta_{j}=\Delta+\delta\Delta_{j}~{},\hskip
14.22636pt\Delta=G\sum_{j^{\prime}}\Omega_{j^{\prime}}(1-2n_{j^{\prime}})u_{j^{\prime}}v_{j^{\prime}}~{},\hskip
14.22636pt\delta\Delta_{j}=2G\frac{\delta{\cal
N}_{j}^{2}}{1-2n_{j}}u_{j}v_{j}~{},$ (27)

with the QNF defined as

$\delta{\cal N}_{j}^{2}\equiv n_{j}(1-n_{j})~{},$ (28)

and the equation for the average particle number $N$,

$N=2\sum_{j}\Omega_{j}[v_{j}^{2}(1-2n_{j})+n_{j}]~{},$ (29)

The $u_{j}$ and $v_{j}$ coefficients in Eqs. (27) and (29) are determined as

$u_{j}^{2}=\frac{1}{2}\bigg{[}1+\frac{\epsilon_{j}^{\prime}-Gv_{j}^{2}-\lambda}{E_{j}}\bigg{]}~{},\hskip
14.22636ptv_{j}^{2}=\frac{1}{2}\bigg{[}1-\frac{\epsilon_{j}^{\prime}-Gv_{j}^{2}-\lambda}{E_{j}}\bigg{]}~{},$
(30)

with the renormalized single-particle energies $\epsilon_{j}^{\prime}$,

$\epsilon_{j}^{\prime}=\epsilon_{j}+\frac{G}{\sqrt{\Omega_{j}}(1-2n_{j})}\sum_{j^{\prime}}\sqrt{\Omega_{j^{\prime}}}(u_{j^{\prime}}^{2}-v_{j}^{2})(\langle{\cal
A}^{\dagger}_{j}{\cal A}^{\dagger}_{j^{\prime}\neq j}\rangle+\langle{\cal
A}^{\dagger}_{j}{\cal A}_{j^{\prime}}\rangle)~{},$ (31)

and the quasiparticle energies

$E_{j}=\sqrt{(\epsilon_{j}^{\prime}-Gv_{j}^{2}-\lambda)^{2}+\Delta_{j}^{2}}~{}.$
(32)

For a system with an odd number of particles, the blocking effect caused by
the unpaired particle should be taken into account in Eqs. (27) and (29). In
the present paper, for simplicity, we do not consider systems with odd
particle numbers within the FTBCS1+SCQRPA and FTLN1+SCQRPA.

The pair correlators $\langle{\cal A}^{\dagger}_{j}{\cal
A}^{\dagger}_{j^{\prime}\neq j}\rangle$ and $\langle{\cal
A}^{\dagger}_{j}{\cal A}_{j^{\prime}}\rangle$ in Eq. (31) are determined by
numerically solving a set of coupled equations (47) and (48) of Ref. FTBCS1 ,
which contain the ${\cal X}_{j}^{\mu}$ and ${\cal Y}_{j}^{\mu}$ amplitudes of
the SCQRPA equations. The details of the derivation of the SCQRPA are given in
Ref. SCQRPA . The SCQRPA equations are solved self-consistently with the gap
and number equations (27) and (29). The quasiparticle occupation numbers
$n_{j}$ are then found by solving a set of equations that include coupling of
quasiparticle density operators $\alpha_{jm}^{\dagger}\alpha_{jm}$ to the
SCQRPA phonon operators. The result yields the integral equation (69) of Ref.
FTBCS1  for $n_{j}$. To compare with the level-independent gap such as the BCS
one, the level-weighted gap,

$\overline{\Delta}=\sum_{j}\Omega_{j}\Delta_{j}/\sum_{j}\Omega_{j}~{},$ (33)

is used instead of Eq. (27). The total energy $\langle{\cal E}\rangle_{\rm
FTBCS1+SCQRPA}$ is calculated by averaging the quasiparticle representation
$H_{\rm q.p.}$ of the Hamiltonian (1) within the GCE, i.e.

$\langle{\cal E}\rangle=\langle H_{\rm q.p.}\rangle_{\rm GC}~{},$ (34)

and the heat capacity $C$ is then found from Eq. (8). The thermodynamic
entropy is obtained by integrating Eq. (9)

$S_{\rm th}=\int_{0}^{T}\frac{1}{\tau}Cd\tau~{}.$ (35)

The BCS equations BCS ; Bogo  are recovered from Eqs. (27) and (29) by
neglecting the QNF $\delta{\cal N}_{j}^{2}$ [i.e. $\delta\Delta_{j}=0$ in Eq.
(27)] together with the pair correlators $\langle{\cal A}^{\dagger}_{j}{\cal
A}^{\dagger}_{j^{\prime}\neq j}\rangle$ and $\langle{\cal
A}^{\dagger}_{j}{\cal A}_{j^{\prime}}\rangle$ in Eq. (31), and assuming
$n_{j}$ to have the form of Fermi-Dirac distribution for noninteracting
quasiparticles, i.e. setting

$n_{j}=n_{j}^{\rm FD}\equiv\frac{1}{e^{\beta E_{j}}+1}~{}.$ (36)

The BCS is known to violate the particle number. This causes a certain quantal
particle-number fluctuation around the average value determined by Eq. (29)
even at $T=$ 0\. The FTBCS1+SCQRPA takes into account only the thermal effect
in terms of QNF $\delta{\cal N}_{j}^{2}$, but does not remove the quantal
fluctuation, which is a feature inherent in the BCS wave functions. To cure
this inconsistency, a proper particle-number projection (PNP) needs to be
carried out. The Lipkin-Nogami (LN) method LN  is an approximated PNP before
variation, which is widely used in nuclear study because of its simplicity.
This method has been implemented into the FTBCS1+SCQRPA, and the ensuing
approach is called the FTLN1+SCQRPA (See Sec. II.C.2. of Ref. FTBCS1 ).
However, the LN method can approximately eliminate only the quantal
fluctuations due to particle-number violation within the BCS theory. These
quantal fluctuations are different from the thermal particle-number
fluctuations, which always arise from the exchange of particles between the
systems in the GCE. The LN method is, therefore, not sufficient to remove the
particle-number fluctuations within the GCE. To avoid the thermal particle-
number fluctuations, the average in the CE should be used instead.
Unfortunately, the methods of equilibrium statistical physics applied to the
nuclear theories such as the Matsubara-Green’s function and/or double-time
Green’s function techniques, which are used to derived the BCS and QRPA
equations at finite temperature, are all based on the GCE. The complete
particle-number projection based on the applying particle-number projection
operator PNP  at finite temperature for these approaches still remains a
subject under study [See, e.g. Ref. PMBCS ].

In this paper, the numerical results obtained within both particle-number
unprojected (FTBCS1+SCQRPA) and projected (FTLN1+SCQRPA) approaches will be
compared with those given by averaging the exact pairing solution with the
principal thermodynamic ensembles.

##  IV NUMERICAL RESULTS

###  IV.1 Details of numerical calculations

The schematic model employed for numerical calculations consists of $N$
particles, which are distributed over $\Omega=N$ doubly-folded equidistant
levels (i.e. with the level degeneracy $2\Omega_{j}=2$). These levels, whose
energies are $\epsilon_{j}=\epsilon[j-(\Omega+1)/2]$ ($j=$ 1, …, $\Omega$),
interact via the pairing force with a constant parameter $G$. The model is
half-filled, namely, in the absence of the pairing interaction, all the lowest
$\Omega/2$ levels (with negative single-particle energies) are filled up with
$N$ particles, leaving $\Omega/2$ upper levels (with positive single-particle
energies) empty 333This model is also called Richardson’s model, picket-fence
model, ladder model, multilevel pairing model, etc. in the literature.. It is
worth mentioning that the extension of the exact solution to $T\neq$ 0 is not
possible at a large value of $\Omega=N$. For the present schematic model, the
number of eigenstates $n_{S}$, each of which is $2^{S}$-degenerated, increases
almost exponentially with $\Omega$

$n_{S}(\Omega)={\rm C}_{N_{\rm pair}}^{\Omega}+\sum_{S}{{\rm
C}_{S}^{\Omega}\times{\rm C}_{N_{\rm pair}-\frac{S}{2}}^{\Omega-S}}~{},$ (37)

where ${\rm C}_{n}^{m}={m!}/[n!(m-n)!]$ and N${}_{\rm pair}=N/2$ is the
numbers of pairs distributed over $\Omega$ single-particle levels. The sum in
Eq. (37) runs over all the values of total seniorities S = 0, 2,
$\ldots,\Omega$ . Therefore, at $\Omega=N=$ 16 there are 5196627 states, which
corresponds to the order $\sim$ 2.7$\times 10^{13}$ for the square matrix to
be diagonalized. This makes the finite-temperature extension of the exact
pairing solution practically impossible for $\Omega>$ 16 since all the
eigenvalues must be included in the partition function. Therefore, in the
present paper, we limit the calculations up to $\Omega=N=$ 14, for which there
are 73789 eigenstates. For the GCE average with respect to the system with $N$
particles and $\Omega=N$ levels, the sum over particle numbers $n$ runs from
$n_{min}=1$ to $n_{max}=2\Omega-1$ with the blocking effect caused by the odd
particle properly taken into account. The calculations are carried out by
using the level distance $\epsilon=$ 1 MeV and the pairing interaction
parameter $G=$ 0.9 MeV. With these parameters, the values of the pairing gap
obtained at $T=$ 0 are equal to around 3, 3.5, and 4.5 MeV for $N=$ 8, 10, and
12 in qualitative agreement with the empirical systematic for realistic nuclei
Bohr .

###  IV.2 Results within GCE, CE and BCS-based approaches

Figure 1: (Color online) Pairing gaps $\overline{\Delta}$, total energies
$\langle{\cal E}\rangle$, heat capacities $C$ and thermodynamic entropies,
obtained for $N=$ 8, 10, and 12 ($G=$ 0.9 MeV) within the FTBCS (dotted
lines), FTBCS1+SCQRPA (thin dashed lines), FTLN1+SCQRPA (thick dashed lines),
CE (dash-dotted lines), and GCE (solid lines) vs temperature $T$.

Shown in Fig. 1 are the pairing gaps, total energies, and heat capacity
obtained for $N=$ 8, 10, and 12 as functions of temperature $T$ within the
FTBCS, FTBCS1+SCQRPA, and FTLN1+SCQRPA, along with the corresponding results
obtained by embedding the exact solutions (eigenvalues) of the Hamiltonian (1)
in the CE and GCE. These latter results using the exact pairing solutions are
referred to as CE and GCE results hereafter. For the FTBCS1+SCQRPA and
FTLN1+SCQRPA the level-weighted gaps $\overline{\Delta}$ (33) are plotted in
Figs. 1 (a1) – 1 (a3).

It is seen from this figure that the GCE results are close to the CE ones for
the gaps, obtained from Eq. (18), as well as for the total energies, obtained
by using Eq. (8). Among three systems under consideration, the largest
discrepancies between the CE and GCE results are seen in the lightest system
($N=$ 8), for which the GCE gap is slightly lower than the CE one, and
consequently, the GCE total energy is slightly higher than that obtained
within the CE. As $N$ increases, the high-$T$ values of the GCE and CE gaps
become closer, so do the corresponding total energies. Different from the BCS
results (dotted lines), which show a collapse of the gap and a spike in the
temperature dependence of the heat capacity at $T=T_{\rm c}$, no singularity
occurs in the GCE and CE results. Both GCE and CE gaps decrease monotonously
with increasing $T$ and remain finite even at $T\gg$ 5 MeV. The FTBCS1+SCQRPA
and FTLN1+SCQRPA predictions for the pairing gap are found in qualitative
agreement with the GCE and CE results [Figs. 1 (a1) – 1 (a3)]. Because of a
different definition of the pairing gap in the exact solutions within the GCE
and/or CE, where actually no mean-field gap exists (See Sec. II C1), one
cannot expect a more quantitative agreement between the predictions by the
FTLN1+SCQRPA and the GCE (CE) results. In this respect, the modified gap (24)
yields a better quantitative agreement with the GCE (CE) results, as will be
seen later in Sec. IV.4. The two BCS-based approaches differ noticeably only
at $T\leq T_{\rm c}$, where the FTLN1+SCQRPA gap, due to PNP, practically
coincides with the GCE and CE results at $T<$ 0.5 – 1 MeV. At $T>T_{\rm c}$
the predictions by two approaches start to converge to the same value, which
decreases with increasing $T$, remaining smaller than the GCE and CE gaps.

For the same reason, at $T<T_{\rm c}$, where the total energy predicted by the
FTLN1+SCQRPA agrees very well with the GCE and CE results, the FTBCS1+SCQRPA
energy is significantly larger [Figs 1 (b1) – 1 (b3)]. At $T>T_{\rm c}$, one
finds a remarkable agreement between the energies predicted by the
FTBCS1+SCQRPA, FTLN1+SCQRPA, and that obtained within the GCE. This seems to
be a natural consequence, given the fact that the two BCS-based approaches are
derived by using the variational procedure within the GCE. The energies do not
suffer either from the difference in the definitions as the pairing gaps do,
as has been discussed above. For the heat capacities [Figs 1 (c1) – 1 (c3)],
the spike obtained at $T=T_{\rm c}$ within the FTBCS theory is completely
smeared out within the GCE, CE as well as the FTBCS1+SCQRPA and FTLN1+SCQRPA,
where only a broad bump is seen in a large temperature region between 0 and 3
MeV. At $T<$ 1 – 1.2 MeV, the difference between the GCE and CE energies leads
to a significant discrepancy between the GCE and CE values for the heat
capacity. As the FTBCS1+SCQRPA and FTLN1+SCQRPA heat capacities are close to
the GCE values, this explains the discrepancy reported in Figs. 4 (b) and 4
(c) of Ref. FTBCS1  between these results and the predictions obtained within
the CE by the quantum Monte-Carlo calculations, which use a model Hamiltonian
with same monopole pairing interaction. The thermodynamic entropies $S_{\rm
th}$ are shown as functions of $T$ in Figs. 1 (d1) – 1 (d3). Once again the
FTLN1+SCQRPA results for the thermodynamic entropy $S_{\rm th}$ agree very
well with $S_{\rm th}$ obtained within the GCE, whereas such good agreement is
seen for the FTBCS1-SCQRPA results only at $T<$ 1 MeV. The CE thermodynamic
entropy is significantly lower than the values obtained in all other
approaches under consideration, which are based on averaging within the GCE.
On the other hand, the BCS theory strongly overestimates the thermodynamic
entropy at $T>T_{\rm c}$.

###  IV.3 Results within MCE

Figure 2: (Color online) Temperature extracted from Eq. (13) within the MCE
(dots) vs excitation energy ${\cal E}^{*}$ in comparison with the CE results
(dash-dotted line) for $N=$ 8, 10, and 12. The results in the top panels, (a)
– (c), are obtained from the definition of Neumann for entropy Eq. (15),
whereas those in the middle panels, (d) – (f), and bottom ones, (g) – (i), are
calculated from Boltzmann’s entropy Eq. (12) with two different values of the
energy interval $\Delta E$ in the statistical weight $\Omega$.

The values of temperature within the MCE as extracted by using Eq. (13) are
plotted in Fig. 2 along with the CE results against the excitation energy
${\cal E}^{*}$. The CE definition of the latter is ${\cal E}^{*}_{\rm
C}\equiv\langle{\cal E}(T)\rangle_{\rm C}-\langle{\cal E}(T=0)\rangle_{\rm
C}$. For comparison, the results obtained by using the entropy (17) are also
presented in the top panels [Figs. 2 (a) – 2 (c)]. They show the values of the
eigenstate temperatures $T_{s}$, which scatter widely around the heat bath
temperature (i.e. the CE result) [Figs. 2 (a) – 2 (c)]. Many of these values
are even negative. Since the eigenstate temperatures $T_{s}$ are related with
the spread of the exact eigenfunctions over the “unperturbed” basis states $k$
with the weights $[C_{k}^{(s)}]^{2}$, they need not follow the trend of the
heat bath (or canonical) temperature, which depends just on the level density
[Eqs. (12) and (13)]. In fact, with increasing the energy interval $\Delta E$,
within which the levels are counted, the values of the MCE temperature
determined by Eq.(13) gradually converge to the CE values [Figs. 2 (a) – 2
(i)]. This means that, the thermal equilibrium within the MCE for the present
isolated pairing model can be reached only at large $N$ and dense spectrum
(small level spacing). Moreover, a full thermalization in a system with pure
pairing is a subject under question. In Ref. Zele ; Volya1 , the
thermalization of the system is characterized by the single-particle
temperature, which is obtained by fitting the occupation numbers of the
individual eigenstates to those given by the Fermi-Dirac distribution. The
numerical results Fig. 12 of Ref. Volya1  show that, the temperature extracted
from the density of states in 116Sn by using Eq. (13) agrees with the single-
particle temperature only when all the residual interactions are taken into
account. With the pure pairing interaction alone, the single-particle
temperature shows a low temperature of the whole system, whereas the MCE
temperature (13) is a hyperbola as a function of the excitation energy with a
singularity in the middle of the spectrum, where it turns negative [See also
Figs. 53 and 54 of Ref. Zele ].

Figure 3: (Color online) Temperatures [(a) – (c)], entropies [(d) – (f)], and
pairing gaps [(g) – (i)] within the MCE as functions of excitation energy
${\cal E}^{*}$ for $N=$ 8 ($G=$ 0.9 MeV) obtained by using the Gaussian,
Lorentz, and Breit-Wigner distributions from Eq. (14) for the level density at
different values of the parameter $\sigma$. The dash-dotted lines show the CE
values.

The values of MCE temperature, entropy and gap obtained by using the Gaussian,
Lorentz, and Breit-Wigner distributions from Eq. (14) are shown in Fig. 3 as
functions of excitation energy ${\cal E}^{*}$. While the fluctuating behavior
of the microcanonical temperature can be smoothed out by increasing the
parameter $\sigma$ in all three distributions, we found that only the Gaussian
distribution can simultaneously fit both the temperature and entropy [Figs. 3
(a) and 3 (d)]. The Lorentz distribution can fit only the MCE temperature to
the CE one, but fails to do so for the entropy [Figs. 3 (b) and 3 (e)],
whereas the Breit-Wigner distribution can fit the MCE temperature to the CE
value only at high excitation energies [Figs. 3 (c)]. A similar result is seen
for the pairing gaps as functions of ${\cal E}^{*}$, where the Gaussian fit
gives the best performance among the three distributions [Compare Figs. 3 (g),
3 (h), and 3 (i)]444The non-vanishing values of $T$ and $S$ at ${\cal E}^{*}=$
0 in Figs. 3 (a), 3 (c), 3 (d), 3 (f) are the artifacts due to the use of the
Gaussian and Breit-Wigner distribution functions (14) to smooth out the
discrete level density. These two distributions have non-zero values $\sim
1/\sigma$ at $x=$ 0, where the Lorentz distribution vanishes.. We conclude
that the Gaussian distribution should be chosen as the best one for smoothing
the level density $\rho({\cal E})$ in Eq. (5) to extract the MCE temperature.

###  IV.4 Pairing gaps extracted from odd-even mass differences

Figure 4: (Color online) Pairing gaps extracted from the odd-even mass
differences as functions of $T$ for $N=$ 10 (a,c) and $N=$ 9 (b,d) ($\Omega=$
10, $G=$ 0.9 MeV). The thin solid and thick solid lines denote the gaps
$\Delta^{(i)}(\beta,N)$ ($i=$ 3, 4) from Eq. (19), and the modified gaps
$\widetilde{\Delta}^{(i)}(\beta,N)$ from Eq. (24), respectively. The dash-
dotted lines are the canonical results $\Delta^{(i)}_{\rm C}$. The upper
panels (a) and (b) show the three-points gaps ($i=$ 3), whereas the the
corresponding four-points gaps ($i=$ 4) are shown in the lower panels (c) and
(d).

We extracted the pairing gaps $\Delta^{(i)}(\beta,N)$ ($i=$ 3 and 4) by using
the simple extension of the odd-even mass formula to $T\neq$ 0 in Eq. (19) as
well as the modified gaps $\widetilde{\Delta}^{(i)}(\beta,N)$ from Eq. (24),
and the canonical gaps $\Delta^{(i)}_{\rm C}$ from Eq. (18) for several
particle numbers up to $N=$ 12\. In these calculations, the blocking effect
caused by the unpaired particle in the systems with an odd particle number is
properly taken into account in constructing the basis states when
diagonalizing the pairing Hamiltonian. The results obtained for $N=$ 9 and 10
($\Omega=$ 10) are displayed in Fig. 4. First of all, the values for
$S^{\prime}$ are found to be always negative, and increases with $T>$ 1 MeV to
reach a value of around $-$2 MeV at $T=$ 5 MeV, and vanish at very high $T$.
By comparing Figs. 4 (a) and 4 (b) one can see a clear manifestation of the
parity effect Delft , which causes the reduction of the three-point gap the in
the system with an odd particle number ($N=$ 9) due to the blocking effect.
For lights systems as those considered here, this reduction is rather strong
(about 1 MeV at $T=$ 0). With increasing $T$, as the thermodynamics weakens
the effect of a single-unpaired particle, the parity effect starts to fade out
in such a way that the three-point canonical gap $\Delta^{(3)}_{\rm C}(N=9)$
slightly increases with $T$ up to $T\simeq$ 1 MeV, starting from which it is
even slightly larger than $\Delta^{(4)}_{\rm C}(N=9)$. This feature is found
in qualitative agreement with the results obtained in Fig. 25 of Ref. Delft
for ultrasmall metallic grains. It is also seen in Fig. 4 that the naive
extension of the odd-even mass formula to $T\neq$ 0, resulting in the gap
$\Delta^{(i)}(\beta,N)$ (thin solid lines), fails to match the temperature-
dependence of the canonical gap $\Delta^{(i)}_{\rm C}$ (dash-dotted lines).
The former even increases with $T$ at $T<$ 1 MeV, whereas it drastically drops
at $T>$ 1 – 1.5 MeV, resulting in a very depleted tail at $T>$ 2 MeV as
compared to the canonical gap $\Delta^{(i)}_{\rm C}$. Moreover, the three-
point gap $\Delta^{(3)}(\beta,N=9)$ even turns negative at $T>$ 2.4 MeV,
suggesting that such simple extension of the odd-even mass difference to
finite $T$ is invalid. At the same time, the modified gap
$\widetilde{\Delta}^{(i)}(\beta,N)$ (thick solid line) given by Eq. (24) is
found in much better agreement with the canonical one. At $T<$ 1.5 MeV, the
three point-gap $\widetilde{\Delta}^{(3)}(\beta,N)$ is almost the same as the
canonical gap. At higher $T$, it becomes larger (smaller) than the canonical
value for the system with an even (odd) $N$, however the systematic of the
results of our calculations up to $N=$ 12 show we that this discrepancy
decreases with increasing the particle number. The source of the discrepancy
resides in the assumption of the odd-even mass formula that the gap obtained
as the energy difference between the systems with $N+1$ and $N$ particles is
the same as that obtained from the energy difference between systems with $N$
and $N-1$ particles. This assumption does not hold for small $N$ (Cf. Ref.
cons ). The average in the four-point gap nearly eliminates this difference.
As a result, the modified four-point gaps $\widetilde{\Delta}^{(4)}(\beta,N)$
practically coincide with the canonical gaps. A natural consequence of the
average in the definition of the four-point gap (20) is that the gaps obtained
in the systems with $N$ and $N-1$ particles are now nearly the same. The
pairing gaps predicted by a number of alternative theories Moretto ; Goodman ;
Egido ; SPA ; MBCS , including those discussed in the present work, are in
closer agreement to the GCE and CE gaps rather than to the gap
$\Delta(\beta,N)$ from Eq. (19). Therefore, the comparison in Fig. 4 suggests
that formula (24) is a much better candidate for the experimental gap at
$T\neq$ 0, rather than the simple odd-even mass difference (19).

##  V Conclusions

In the present work, a systematic comparison is conducted for pairing
properties of finite systems at finite temperature as predicted by the exact
solutions of the pairing problem embedded in three principal statistical
ensembles, as well as by the recently developed FTBCS1 (FTLN1)+SCQRPA. The
analysis of numerical results obtained within the doubly-folded equidistant
multilevel model for the pairing gap, total energy, heat capacity, entropies,
and MCE temperature allows us to draw following conclusions.

1) The sharp SN phase transition is indeed smoothed out in exact calculations
within all three principal ensembles. The results obtained within GCE and CE
are very close to each other even for systems with small number of particles.
As for the MCE, although it can also be used to study the pairing properties
of isolated systems at high-excitation energies, there is a certain ambiguity
in the temperature extracted from the level density due to the discreteness of
a small-size system. This ambiguity, therefore, depends on the shape and
parameter of the distribution employed to smooth the discrete level density.
We found that, in this respect, the normal (Gaussian) distribution gives the
best fit for both of the temperature and entropy to the canonical values. The
wide fluctuations of MCE temperature obtained here also indicate that thermal
equilibrium within thermally isolated pure-pairing systems might not be
reached. On the other hand, it opens an interesting perspective of studying
the behavior of phase transitions in finite systems within microcanonical
thermodynamics Gross  by using the exact solutions of pairing problem.

2) The predictions by the FTBCS1+SCQRPA and FTLN1+SCQRPA are found in
reasonable agreement with the results obtained by using the exact solutions
embedded in the GCE and CE. The best agreement is see between the FTLN1+SCQRPA
and the GCE results. Once again, this is a robust confirmation that
quasiparticle-number fluctuation, included in these approximations, is indeed
the microscopic origin of the strong thermal fluctuations that smooth out the
sharp SN phase transition in finite systems.

3) We suggest a novel formula to extract the pairing gap at finite temperature
from the difference of total energies of even and odd systems where the
contribution of uncorrelated single-particle motion is subtracted. The new
formula predicts a pairing gap in much better agreement with the canonical gap
than the simple finite-temperature extension of the odd-even mass formula.

###### Acknowledgements.

Discussions with Peter Schuck (Orsay) are gratefully acknowledges. One of us
(N.Q.H.) also thanks S. Frauendorf (Notre Dame), and V. Zelevinsky (East
Lansing) for discussions and hospitality during his visit at the University of
Notre Dame and Cyclotron Institute of the Michigan State University, where a
part of this work was presented. NQH is a RIKEN Asian Program Associate. The
numerical calculations were carried out using the FORTRAN IMSL Library by
Visual Numerics on the RIKEN Super Combined Cluster (RSCC) system.

## References

  * (1) J. Bardeen, L. Cooper, and Schrieffer, Phys. Rev. 108, 1175 (1957).
  * (2) N.N. Bogoliubov and S.V. Tyablikov, Sov. Phys. Dokl. 4, 589 (1959) [Dokl. Akad. Nauk. SSSR 126, 53 (1959)]; D.N. Zubarev, Sov. Phys. Usp. 3, 320 (1960) [Usp. Fiz. Nauk 71, 71 (1960); R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II - Nonequilibrium Statistical Mechanics (Springer, Berlin-Heidelberg, 1985).
  * (3) L.G. Moretto, Phys. Lett. B 40, 1 (1972).
  * (4) A.L. Goodman, Phys. Rev. C 29, 1887 (1984).
  * (5) J.L. Egido, P. Ring, S. Iwasaki, and H.J. Mang, Phys. Lett. B 154, 1 (1985).
  * (6) R. Rossignoli, P. Ring and N.D. Dang, Phys. Lett. B 297, 9 (1992); N.D. Dang, P. Ring and R. Rossignoli, Phys. Rev. C 47, 606 (1993).
  * (7) V. Zelevinsky, B.A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276, 85 (1996).
  * (8) N. Dinh Dang and V. Zelevinsky, Phys. Rev. C 64, 064319 (2001); N.D. Dang and A. Arima, Phys. Rev. C 67, 014304 (2003); N.D. Dang and A. Arima, Phys. Rev. C 68, 014318 (2003); N.D. Dang, Nucl. Phys. A 784, 147 (2007).
  * (9) N.D. Dang and N.Q. Hung, Phys. Rev. C 77, 064315 (2008).
  * (10) N.Q. Hung and N.D. Dang, Phys. Rev. C 78, 064315 (2008).
  * (11) R.W. Richardson, Phys. Lett. 3, 277 (1963); Ibid. 14, 325 (1965); R.W. Richardson and N. Sherman, Nucl. Phys. 52, 221 (1964).
  * (12) A. Volya, B.A. Brown, and V. Zelevinsky, Phys. Lett. B 509 (2001) 37.
  * (13) R. Kubo, J. Phys. Soc. Japan 17, 975 (1962).
  * (14) R. Denton, B. Mühlschlegel, and D.J. Scalapino, Phys. Rev. B 8, 3589 (1973).
  * (15) T. Sumaryada and A. Volya, Phys. Rev. C 76, 024319 (2007).
  * (16) A. Bohr and B.R. Mottelson, Nuclear structure I (Bejamin, NY, 1969).
  * (17) P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, Heidelberg, 2004).
  * (18) J. von Delft and D.C. Ralph, Phys. Rep. 345, 61 (2001).
  * (19) R. Rossignoli, N. Canoza, and P. Ring, Ann. Phys. 275, 1 (1999).
  * (20) S. Frauendorf, N.K. Kuzmenko, V.M. Mikhajlov, and J.A. Sheikh, Phys. Rev. B 68, 024518 (2003).
  * (21) G. Breit, Phys. Rev. 58, 506 (1940); E.P. Wigner, Phys. Rev. 70, 15 (1946); M. Danos and W. Greiner, Phys. Rev. 134, B284 (1964).
  * (22) K. Kaneko and M. Hasegawa, Phys. Rev. C 72, 024307 (2005), K. Kaneko et al., Phys. Rev. C 74, 024325 (2006).
  * (23) N.N. Bogoliubov, JETP 34, 58 (1958).
  * (24) N.Q. Hung and N.D. Dang, Phys. Rev. C 76, 054302 (2007); 77, 029905(E) (2008).
  * (25) H.J. Lipkin, Ann. Phys. 9, 272 (1960); Y. Nogami, Phys. Rev. 134, B313 (1964); H.C. Pradhan, Y. Nogami, and J. Law, Nucl. Phys. A 201, 357 (1973).
  * (26) H. Olofsson, R. Bengtsson, P. Möller, Nucl. Phys. A 784, 104 (2007).
  * (27) N. Dinh Dang, Phys. Rev. C 76, 064320 (2007).
  * (28) A. Volya, V. Zelevinsky, and B. Alex Brown, Phys. Rev. C 65, 054312 (2002).
  * (29) N. Dinh Dang, Phys. Rev. C 74, 024318 (2006).
  * (30) D.H.E. Gross and J.F. Kenney, J. Chem. Phys. 112, 224111 (2005).


# Continuous Streaming Multi-Talker ASR with Dual-path Transducers

###### Abstract

Streaming recognition of multi-talker conversations has so far been evaluated
only for 2-speaker single-turn sessions. In this paper, we investigate it for
multi-turn meetings containing multiple speakers using the Streaming Unmixing
and Recognition Transducer (SURT) model, and show that naively extending the
single-turn model to this harder setting incurs a performance penalty. As a
solution, we propose the dual-path (DP) modeling strategy first used for time-
domain speech separation. We experiment with LSTM and Transformer based DP
models, and show that they improve word error rate (WER) performance while
yielding faster convergence. We also explore training strategies such as chunk
width randomization and curriculum learning for these models, and demonstrate
their importance through ablation studies. Finally, we evaluate our models on
the LibriCSS meeting data, where they perform competitively with offline
separation-based methods.

Index Terms—  Multi-talker ASR, long-form meeting transcription, dual-path
RNN, transducer

##  1 Introduction

With advancements in clean, single-speaker transcription [1, 2], researchers
are focusing on harder speech recognition settings with interference in the
form of noise, reverberation, or overlapping speakers [3, 4, 5]. Of these, the
latter is particularly irksome, while also being prevalent in meetings and
natural conversations [6, 7, 8]. The conventional approach to handle
overlapping speech is through a cascade of separation and recognition modules,
which may be sub-optimal since the modules are independently optimized. It
also requires considerable engineering efforts to maintain the pipeline.

An alternative solution to this problem is through the application of end-to-
end models that are trained with text-based supervision [9, 10, 11, 12, 13,
14, 15, 16]. Several model paradigms have been explored, including attention-
based encoder-decoders [17, 18, 19], and recurrent neural network transducers
(RNN-T) [20, 21], with most of the models functioning in an offline mode. For
the streaming case, Lu et al. recently proposed Streaming Unmixing and
Recognition Transducer (SURT) [22] which simultaneously transcribes
overlapping speech into two channels through “unmixing” and “transcription”
modules that are jointly optimized using an RNN-T based heuristic error
assignment training (HEAT) loss. We will review SURT in § 2.

SURT was originally evaluated on 2-speaker single-turn sessions and showed
promising results. However, real-life meetings often contain multiple speakers
and several turns of conversation, and it is not immediately clear whether the
“vanilla” SURT model would transfer well to these harder settings. In this
paper, we show that the train-test mismatch does, in fact, incur performance
costs that cannot be easily recovered using multi-turn training. We then
investigate dual-path (DP) LSTM and Transformer models, which have been
successfully applied in speech separation [23, 24], and show that they are
better at modeling the longer sequences that occur in multi-turn meetings. Our
contributions include training strategies for DP models, such as curriculum
learning and chunk width randomization, and a theoretical analysis of DP
Transformers. We evaluate our models on several tiers of simulated multi-
talker mixtures and on the LibriCSS meeting corpus [25], where it performs
competitively with offline separation-based approaches.

Fig. 1: An overview of the RNN-T based SURT model for the 2-speaker
overlapping case.

##  2 SURT: Review

Fig. 1 shows an overview of the RNN-T based SURT model. Given input features
$X$ obtained from an overlapping speech session, a mask-based unmixing module
extracts speaker-dependent representations $H_{1}$ and $H_{2}$ as

$\displaystyle H_{1}=M\ast\bar{X},\quad H_{2}=(\mathbbm{1}-M)\ast\bar{X},$ (1)
$\displaystyle
M=\sigma(\texttt{MaskEnc}(X))~{}\text{and}~{}\bar{X}=\texttt{MixEnc}(X).$

Here, $\sigma$ denotes the Sigmoid function, $\ast$ is the Hadamard product,
and $\mathbbm{1}$ has the same shape as $M$. Both the MaskEnc and MixEnc use a
4-layer 2D convolutional architecture. $H_{1}$ and $H_{2}$ are fed into the
RNN-T module, and produce hypotheses $\hat{Y}_{1}$ and $\hat{Y}_{2}$ under the
HEAT assumption (i.e., $Y_{1}$ starts before $Y_{2}$). The final loss is given
as

$\mathcal{L}_{\text{heat}}(X,Y_{1},Y_{2})=\mathcal{L}_{\text{rnnt}}(Y_{1},H_{1})+\mathcal{L}_{\text{rnnt}}(Y_{2},H_{2}),$
(2)

where $\mathcal{L}_{\text{rnnt}}$ is the standard RNN-T loss. Although Lu et
al. only evaluated HEAT on 2-speaker single-turn sessions, it is
straightforward to extend HEAT to long-form sessions.

###  2.1 HEAT vs. PIT

(a) Delay = 2.0 s

(b) Delay = 0.0 s

Fig. 2: Training dynamics for HEAT versus PIT based loss for different
utterance delays: (a) 2.0 s, and (b) 0.0 s.

The output label permutation problem in overlapped ASR is often addressed
using utterance-level permutation invariant training (PIT), which computes the
optimal permutation of reference and hypothesis sequences as the training
loss. For the SURT problem formulated above, it is given as

$\displaystyle\mathcal{L}_{\text{pit}}(X,Y_{1},Y_{2})=\min$
$\displaystyle\big{(}\mathcal{L}_{\text{rnnt}}(Y_{1},H_{1})+\mathcal{L}_{\text{rnnt}}(Y_{2},H_{2}),$
$\displaystyle\mathcal{L}_{\text{rnnt}}(Y_{1},H_{2})+\mathcal{L}_{\text{rnnt}}(Y_{2},H_{1})\big{)}.$
(3)

In real use cases, we usually have information (such as pitch or gender) which
can disambiguate the label sequences. For long-form meetings with partially
overlapped utterances, utterance start time can be a good heuristic for output
matching, and has been shown to outperform PIT [22]. To investigate this
phenomenon further, we set up simple experiments on 2-utterance mixtures
generated from LibriSpeech train-clean set. We prepared two kinds of mixtures
with utterance delays of 2.0 and 0.0 seconds, respectively, and trained the
vanilla SURT model using both HEAT and PIT losses (equations 2 and 2.1,
respectively). In Fig. 2, we show the training dynamics for both the
experiments and also plot the % correct output assignment. This quantity
represents how often the model assigns $Y_{1}$ to output channel 1.

As expected, for the case of mixtures with 2.0s delay, HEAT quickly learned
the output assignment order. In fact, even when training with PIT, the same
heuristic was learned (albeit slower), and both models started to converge
only after this point was reached (denoted by the vertical line in Fig. 2(a).
Thereafter, using PIT is wasteful, especially in our case of the expensive
RNN-T loss computation. In the absence of utterance delay (Fig. 2(b)), PIT
produced a random output assignment. Surprisingly, HEAT still learned the
correct assignment, but on decoding with the trained model, we found that it
learned a degenerate solution where both output channels produce the exact
same hypothesis.

Besides HEAT and PIT being empirically equivalent for non-zero delays, it is
infeasible to compute all permutations when training on longer sessions, since
the complexity grows as $\mathcal{O}(N!)$, with $N$ being the number of
utterances. Even if we use linear sum assignment (which has a complexity of
$\mathcal{O}(N^{3})$ [26]), it still requires computing the loss between all
$N^{2}$ pairs of references and hypotheses, which is computationally
prohibitive when using RNN-T since it implicitly sums over all possible
alignments. For these reasons, we use HEAT for all experiments in the
remainder of this paper.

Table 1: Summary of evaluation sets used in this work.

Name Description # spk. # utt. dev test Tier-1 2-speaker single-turn 2 2 1355
1310 Tier-2 2-speaker multi-turn 2 2-4 892 885 Tier-3 Multi-speaker multi-turn
2-4 2-12 462 450

##  3 Multi-turn evaluation

Our objective is to investigate SURT for long-form multi-talker sessions. For
this, we first prepared 3 tiers of evaluation data with increasing difficulty,
all of which are obtained by mixing LibriSpeech test set utterances. Tier-1
(T1) contains single-turn sessions similar to the original SURT evaluation
data [22]. Tier-2 (T2) extends this to sessions containing up to 4 utterances,
while retaining 2 speakers. Finally, Tier-3 (T3) contains multi-turn sessions
with up to 4 speakers. All the tiers were generated to contain an overlap
ratio between 0% and 40% per session. Table 1 summarizes the statistics for
these evaluation sets. For training, we prepared 2-speaker single and multi-
turn sessions which are comparable to the T1 and T2 evaluation sets,
respectively. It is infeasible to train on longer sessions (like T3) because
of memory constraints. Throughout this paper, we evaluated our models using
the word error rate (WER) metric computed between the best permutation of
reference utterances and the output channels111Since the utterances are
ordered by start time, this process elicits $2^{N}$ permutations, which makes
evaluation infeasible for $N$ greater than 12.. For our preliminary
experiments, we considered the vanilla SURT model which uses a 6-layer
1024-dim LSTM network in the RNN-T encoder (see yellow box in Fig. 1). The
prediction network is a 2-layer 1024-dim LSTM model. The model performance
when trained on single and multi-turn sessions are shown below:

Train \Eval Tier-1 Tier-2 Tier-3 Single-turn 11.1 17.6 24.9 Multi-turn 13.6
15.9 20.9

Can the vanilla SURT model trained on single-turn sessions generalize to
multi-turn evaluation? They cannot. WER degrades significantly as tier
complexity increases (row 1), presumably because the model had never seen any
instances with multiple utterances in an output channel.

Can we generalize to multi-turn sessions by simply training on matching data?
Although the WER on T2 and T3 improved, such training degraded T1 results (row
2), suggesting that it is challenging to make standard LSTM models generalize
to diverse session lengths. An alternate model may be required which is better
suited to this problem.

##  4 Streaming Dual-path Transducers

Single channel time-domain speech separation, which requires modeling
extremely long sequences, has benefited from the application of dual-path (DP)
networks such as DP-RNN [23] and DP-Transformers [24]. Here, we investigate
streaming versions of these models as encoders in the SURT RNN-T module.

###  4.1 Dual-path LSTM

Fig. 3: Overview of the streaming dual path LSTM model. The intra-chunk LSTM
is bidirectional, whereas the inter-chunk LSTM is unidirectional.

DP-LSTM consists of an intra and an inter LSTM per layer [23], as shown in
Fig. 3. Input sequences are segmented into (overlapping) chunks and first fed
into the bidirectional intra-LSTM, which processes each chunk independently.
The output is then passed into the inter-LSTM which is unidirectional and
performs strided processing over chunks. By choosing the chunk width to be
approximately square root of the sequence length $l$, we can ensure that both
the LSTMs get similar length inputs. Since the intra-LSTM is bidirectional, a
latency equal to the chunk width is introduced in this model.

###  4.2 Dual-path Transformer

On the surface, DP-Transformer is similar to the DP-LSTM model, with the LSTM
blocks replaced with self-attention blocks [24]. The intra-Transformer uses a
full attention matrix, while the inter-Transformer is constrained to use
causal attention. Consequently, the overall attention pattern for the sequence
can be visualized as shown in Fig. 4(a). Similar to the DP-LSTM, we can ensure
$\mathcal{O}(l)$ attention computation for each chunk by choosing
$\mathcal{O}(\sqrt{l})$ chunk sizes. Consequently, the DP-Transformer has a
complexity of $\mathcal{O}(l\sqrt{l})$, as opposed to the quadratic complexity
of the vanilla Transformer. Sparse transformers with similar attention
patterns have previously been proposed for long sequences (Fig.4).

(a) Dual-path

(b) Strided [27]

(c) Block [28]

(d) Axial [29]

Fig. 4: Comparison of self-attention patterns in different
$\mathcal{O}(l\sqrt{l})$ transformer architectures.

Despite this reduced complexity, the DP-Transformer (in non-streaming mode) is
a universal function approximator. To see this, consider the following
properties for the DP-Transformer. First, every token in the sequence attends
to itself. Second, the directed graph $\mathcal{D}$ corresponding to the
attention pattern contains a Hamiltonian path through all tokens. Third, any
token can directly/indirectly access all other tokens after exactly 2 layers
(in non-streaming mode). Therefore, the model satisfies all necessary
conditions for universal approximability of sparse transformers [30].

Table 2: WER results for SURT models with regular and dual-path encoders,
trained on multi-turn data with curriculum learning.

Encoder Size Tier-1 Tier-2 Tier-3 dev test dev test dev test LSTM 75.6 M 13.6
13.8 15.9 17.1 20.9 21.0 DP-LSTM 65.4 M 11.1 11.7 13.7 14.7 19.9 20.2 \+ CWR
11.1 11.4 13.0 14.1 19.6 19.6 DP-Transformer 42.9 M 11.5 12.4 13.7 15.1 19.1
20.4 \+ CWR 11.1 12.2 13.5 14.5 17.9 18.6

###  4.3 Chunk width randomization

Dual-path models trained with a fixed chunk width may not be suitable for
evaluation on diverse sequence lengths due to mismatch in train-test input
size for the inter block. We propose training with chunk width randomization
(CWR), wherein we vary the CW between a minimum and maximum value for each
mini-batch. CWR increases the train time diversity in sequence length for both
the intra and inter blocks and makes the model robust to such variations at
test time.

##  5 Experimental setup

The architecture of the vanilla SURT model is detailed in §3. For the DP-based
models, we replaced the RNN-T encoder with the corresponding dual path
modules: a 6-layer DP-LSTM with 512-dim intra and inter blocks, and a 12-layer
DP-Transformer consisting of 256-dim self-attention split into 8 heads with a
1024-dim feedforward layer. All models were trained to convergence using a
learning rate schedule consisting of 10k steps of warmup to 0.0003 and a
linear decay thereafter. We used AdamW as the optimizer and clipped gradients
to a norm of 5. The models were trained on 16 V100 GPUs using FP16 precision.
We used decoding beams of size 4 and 8 for the dev and test sets,
respectively, in all experiments. For the fixed chunk dual-path models, a
chunk width of 30 was used for training and decoding. For models with CWR, we
trained with chunk widths between 15 and 45 and decoded with chunks of width
35. These values were tuned on the dev set. For the LibriCSS experiments,
since the data contains far-field recordings [25], we additionally trained
using simulated noise and reverberations [31]. For these evaluations, we
decoded on the approx. 1-minute long segments provided with the corpus (cf.
note 1).

##  6 Results & discussion

Multi-turn evaluation. Table 2 shows the results for our DP models compared to
the LSTM-based SURT. Both the DP-LSTM and DP-Transformer provide WER
improvements across all evaluation tiers without degrading performance on
single-turn sessions (§3). CWR training helped both models, with substantial
gains for DP-Transformer on T3 evaluation. DP-Transformer are competitive with
DP-LSTM, and outperform them on T3, with only two-thirds the model size.

Fig. 5: Training curves for the dual-path models compared with the regular
LSTM model, plotted w.r.t. wall clock time (mins). Table 3: Effect of
curriculum learning on dev set. We compare models trained only on multi-turn
sessions versus those initialized from 20k steps of single-turn training.

Model Tier-1 Tier-2 Tier-3 None 20k None 20k None 20k LSTM 15.8 13.6
($\downarrow$14.1%) 17.8 15.9 ($\downarrow$10.7%) 21.8 20.9 ($\downarrow$4.1%)
DP-LSTM 11.4 11.1 ($\downarrow$2.6%) 13.6 13.0 ($\downarrow$4.4%) 18.6 19.6
($\uparrow$5.4%) DP-Transformer 12.5 11.1 ($\downarrow$11.2%) 13.7 13.5
($\downarrow$1.5%) 18.2 17.9 ($\downarrow$1.6%)

Training dynamics and curriculum learning. Fig. 5 shows the training curves
for the dual-path models compared with the LSTM-based SURT model. Both the DP-
LSTM and DP-Transformer converged faster and to a better minimum, despite
smaller model sizes. We also found it important to use a curriculum learning
strategy for all the models, where we train them first on single-turn
sessions, and then introduce multi-turn sessions after 20k training steps.
Table 3 compares performances on the dev set for models trained with and
without a curriculum. Since the multi-turn training data also contains single-
turn sessions, and the benefits are seen across all tiers, these improvements
cannot be attributed solely to variation in training data.

Accuracy vs. latency. Besides improving robustness to session lengths, CWR
also enables models to be deployed with lower latency at the cost of a small
WER degradation. Fig. 6 shows this trade-off between decoding latency
(determined by chunk width) and WER performance. WER tends to degrade slightly
for low latency decoding, and the best performance is obtained for chunk sizes
of 35.

(a) DP-LSTM

(b) DP-Transformer

Fig. 6: Accuracy vs. latency trade-off for dual-path models trained with chunk
width randomization (CWR), evaluated on dev set.

Evaluation on LibriCSS. Table 4 compares our models against offline modular
systems composed of continuous speech separation (CSS) and ASR modules. The
“conformer CSS + E2E ASR” model [32] contains 197M parameters (59M for CSS,
and 138M for ASR [33]), and uses language model fusion and rescoring. On high
overlap conditions, our models are competitive with these state-of-the-art
offline modular systems, with most errors arising from leakage in single-
speaker regions, i.e., when both channels transcribed an utterance even when
no overlap occurred. For the low overlap cases, omissions were another major
error source, where some utterances were completely missed by both channels.
DP-Transformer does not outperform DP-LSTM in this case possibly due to
overfitting to the simulated noise and reverberation, as evident from their
performance in the clean (digitally mixed clean audio) setting.

Table 4: WER results for single-channel LibriCSS. 0L and 0S denote 0% overlap
with long and short silences, respectively.

Model Overlap ratio in % 0L 0S 10 20 30 40 BLSTM CSS + Hybrid ASR [25] 16.3
17.6 20.9 26.1 32.6 36.1 Conformer CSS + E2E ASR [32] 6.1 6.9 9.1 12.5 16.7
19.3 SURT w/ DP-LSTM replayed 9.8 19.1 20.6 20.4 23.9 26.8 SURT w/ DP-LSTM
clean 6.6 21.6 21.7 20.6 25.4 28.4 SURT w/ DP-Transformer replayed 9.3 21.1
21.2 25.9 28.2 31.7 SURT w/ DP-Transformer clean 6.9 18.9 19.6 21.9 23.9 28.7

##  7 Conclusion

We investigated SURT for continuous streaming multi-talker ASR, and
demonstrated the effectiveness of dual-path LSTMs and Transformers for
generalization to diverse session lengths. Training strategies such as
curriculum learning and chunk width randomization provided WER improvements
and enabled the model to be deployed with different latencies. We also showed
that DP-Transformers are universal function approximators. Our models
demonstrated encouraging results across different overlap levels on the
LibriCSS dataset while being smaller, faster, and simpler than modular
systems. In future work, we will explore segmentation in the SURT output
channels, which will naturally enable speaker-attributed ASR.

## References

  * [1] D. Amodei et al.,  “Deep speech 2 : End-to-end speech recognition in english and mandarin,”  in Proceedings of The 33rd International Conference on Machine Learning, 2016.
  * [2] W. Xiong, J. Droppo, X. Huang, F. Seide, M. Seltzer, A. Stolcke, D. Yu, and G. Zweig,  “Toward human parity in conversational speech recognition,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2017.
  * [3] J. Barker, R. Marxer, E. Vincent, and S. Watanabe,  “The third ‘CHiME’ speech separation and recognition challenge: Dataset, task and baselines,”  in IEEE ASRU, 2015.
  * [4] K. Kinoshita, M. Delcroix, T. Yoshioka, T. Nakatani, A. Sehr, W. Kellermann, and R. Maas,  “The REVERB challenge: A common evaluation framework for dereverberation and recognition of reverberant speech,”  in IEEE WASPAA, 2013.
  * [5] S. Watanabe, M. Mandel, J. Barker, and E. Vincent,  “CHiME-6 challenge: Tackling multispeaker speech recognition for unsegmented recordings,”  ArXiv, vol. abs/2004.09249, 2020.
  * [6] J. Carletta et al.,  “The AMI meeting corpus: A pre-announcement,”  in MLMI, 2005.
  * [7] E. Shriberg, A. Stolcke, and D. Baron,  “Observations on overlap: findings and implications for automatic processing of multi-party conversation,”  in INTERSPEECH, 2001.
  * [8] T. Yoshioka, D. Dimitriadis, A. Stolcke, W. Hinthorn, Z. Chen, M. Zeng, and X. Huang,  “Meeting transcription using asynchronous distant microphones,”  in INTERSPEECH, 2019.
  * [9] A. Graves, S. Fernández, F. Gomez, and J. Schmidhuber,  “Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks,”  in ICML, 2006.
  * [10] A. Graves,  “Sequence transduction with recurrent neural networks,”  in ICML Representation Learning Workshop, 2012.
  * [11] L. Lu, X. Zhang, and S. Renals,  “On training the recurrent neural network encoder-decoder for large vocabulary end-to-end speech recognition,”  in IEEE ICASSP, 2016.
  * [12] J. Chorowski, D. Bahdanau, D. Serdyuk, K. Cho, and Y. Bengio,  “Attention-based models for speech recognition,”  in Advances in Neural Information Processing Systems, 2015.
  * [13] C. Chiu, T. Sainath, Y. Wu, R. Prabhavalkar, P. Nguyen, Z. Chen, A. Kannan, R. J. Weiss, K. Rao, K. Gonina, N. Jaitly, B. Li, J. Chorowski, and M. Bacchiani,  “State-of-the-art speech recognition with sequence-to-sequence models,”  in IEEE ICASSP, 2018.
  * [14] Y. He, T. N. Sainath, R. Prabhavalkar, I. McGraw, R. Alvarez, D. Zhao, D. Rybach, A. Kannan, Y. Wu, R. Pang, et al.,  “Streaming end-to-end speech recognition for mobile devices,”  in IEEE ICASSP, 2019.
  * [15] J. Li, R. Zhao, H. Hu, and Y. Gong,  “Improving RNN transducer modeling for end-to-end speech recognition,”  in IEEE ASRU, 2019.
  * [16] J. Li,  “Recent advances in end-to-end automatic speech recognition,”  arXiv preprint arXiv:2111.01690, 2021.
  * [17] S. Settle, J. Le Roux, T. Hori, S. Watanabe, and J. Hershey,  “End-to-end multi-speaker speech recognition,”  in IEEE ICASSP, 2018.
  * [18] X. Chang, Y. Qian, K. Yu, and S. Watanabe,  “End-to-end monaural multi-speaker ASR system without pretraining,”  in IEEE ICASSP, 2019.
  * [19] N. Kanda, Y. Gaur, X. Wang, Z. Meng, and T. Yoshioka,  “Serialized output training for end-to-end overlapped speech recognition,”  in INTERSPEECH, 2020.
  * [20] A. Tripathi, H. Lu, and H. Sak,  “End-to-end multi-talker overlapping speech recognition,”  in IEEE ICASSP, 2020.
  * [21] I. Sklyar, A. Piunova, and Y. Liu,  “Streaming multi-speaker ASR with RNN-T,”  in IEEE ICASSP, 2021.
  * [22] L. Lu, N. Kanda, J. Li, and Y. Gong,  “Streaming end-to-end multi-talker speech recognition,”  IEEE Signal Processing Letters, 2021.
  * [23] Y. Luo, Z. Chen, and T. Yoshioka,  “Dual-path RNN: Efficient long sequence modeling for time-domain single-channel speech separation,”  in IEEE ICASSP, 2020.
  * [24] J-J. Chen, Q. Mao, and D. Liu,  “Dual-path transformer network: Direct context-aware modeling for end-to-end monaural speech separation,”  in INTERSPEECH, 2020.
  * [25] Z. Chen, T. Yoshioka, L. Lu, T. Zhou, Z. Meng, Y. Luo, J. Wu, and J. Li,  “Continuous speech separation: Dataset and analysis,”  in IEEE ICASSP, 2020.
  * [26] H. Kuhn,  “The hungarian method for the assignment problem,”  Naval Research Logistics Quarterly, vol. 2, pp. 83–97, 1955.
  * [27] R. Child, S. Gray, A. Radford, and I. Sutskever,  “Generating long sequences with sparse transformers,”  ArXiv, vol. abs/1904.10509, 2019.
  * [28] J. Qiu, H. Ma, O. Levy, W-T. Yih, S. Wang, and J. Tang,  “Blockwise self-attention for long document understanding,”  in Findings of the Association for Computational Linguistics: EMNLP 2020, 2020.
  * [29] J. Ho, N. Kalchbrenner, D. Weissenborn, and T. Salimans,  “Axial attention in multidimensional transformers,”  ArXiv, 2019.
  * [30] C. Yun, Y-W. Chang, S. Bhojanapalli, A. S. Rawat, S. Reddi, and S. Kumar,  “O(n) connections are expressive enough: Universal approximability of sparse transformers,”  in Advances in Neural Information Processing Systems, 2020.
  * [31] N. Kanda, G. Ye, Y. Gaur, X. Wang, Z. Meng, Z. Chen, and T. Yoshioka,  “End-to-end speaker-attributed ASR with transformer,”  in INTERSPEECH, 2021.
  * [32] S. Chen, Y. Wu, Z. Chen, J. Li, C. Wang, S. Liu, and M. Zhou,  “Continuous speech separation with conformer,”  in IEEE ICASSP, 2021.
  * [33] C. Wang, Y. Wu, Y. Du, J. Li, S. Liu, L. Lu, S. Ren, G. Ye, S. Zhao, and M. Zhou,  “Semantic mask for transformer based end-to-end speech recognition,”  in INTERSPEECH, 2020.


33Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
34Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
39Institute for High Energy Physics (IHEP), Protvino, Russia
40ICCUB, Universitat de Barcelona, Barcelona, Spain
41Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
42European Organization for Nuclear Research (CERN), Geneva, Switzerland
43Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
44Physik-Institut, Universität Zürich, Zürich, Switzerland
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 27
69National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
70National University of Science and Technology ”MISIS”, Moscow, Russia,
associated to 34
71National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
72Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 40
73University of Michigan, Ann Arbor, United States, associated to 61
74Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
ySezione INFN di Trieste, Trieste, Italy
zEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
aaSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
abPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased