An algebraic approach to intertwined quantum phase transitions in the Zr
isotopes

N. Gavrielov1,2$\star$

1 Center for Theoretical Physics, Sloane Physics Laboratory, Yale University,
New Haven, Connecticut 06520-8120, USA

2 Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

*<EMAIL_ADDRESS>
34th International Colloquium on Group Theoretical Methods in Physics
Strasbourg, 18-22 July 2022 10.21468/SciPostPhysProc.?

## Abstract

The algebraic framework of the interacting boson model with configuration
mixing is employed to demonstrate the occurrence of intertwined quantum phase
transitions (IQPTs) in the 40Zr isotopes with neutron number 52–70. The
detailed quantum and classical analyses reveal a QPT of crossing normal and
intruder configurations superimposed on a QPT of the intruder configuration
from U(5) to SU(3) and a crossover from SU(3) to SO(6) dynamical symmetries.

###### Contents

  1. 1 Introduction
  2. 2 Theoretical framework
    1. 2.1 The interacting boson model with configuration mixing
    2. 2.2 Wave functions structure
    3. 2.3 Geometry
    4. 2.4 QPTs and order parameters
  3. 3 QPTs in the Zr isotopes
    1. 3.1 Model space
    2. 3.2 Hamiltonian and $E2$ transitions operator
  4. 4 Results
    1. 4.1 Evolution of energy levels
    2. 4.2 Evolution of configuration content
    3. 4.3 Evolution of symmetry content
    4. 4.4 Evolution of order parameters
    5. 4.5 Classical analysis
  5. 5 Conclusions and outlook

##  1 Introduction

Quantum phase transitions [1, 2, 3] are qualitative changes in the structure
of a physical system that occur as a function of one (or more) parameters that
appear in the quantum Hamiltonian describing the system. In nuclear physics
[4], we vary the number of nucleons and examine mainly two types of quantum
phase transitions (QPTs). The first describes shape phase transitions in a
single configuration, denoted as Type I. When interpolating between two
shapes, for example, the Hamiltonian can be written as a sum of two parts

$\hat{H}=(1-\xi)\hat{H}_{1}+\xi\hat{H}_{2}~{},$ (1)

with $\xi$ the control parameter. As we vary $\xi$ with nucleon number from 0
to 1, the equilibrium shape and symmetry of the Hamiltonian vary from those of
$\hat{H}_{1}$ to those of $\hat{H}_{2}$. QPTs of this type have been studied
extensively in the framework of the interacting boson model (IBM) [5, 6, 4,
7]. One example of such QPT is the 62Sm region with neutron number 84–94,
where the shape evolves from spherical to axially-deformed, with a critical
point at neutron number 90.

The second type of QPT occurs when the ground state configuration changes its
character, typically from normal to intruder type of states, denoted as Type
II QPT. In such cases, the Hamiltonian can be written in matrix form [8]. For
two configurations $A$ and $B$ we have

$\hat{H}=\left[\begin{array}[]{cc}\hat{H}_{A}(\xi_{A})&\hat{W}(\omega)\\\
\hat{W}(\omega)&\hat{H}_{B}(\xi_{B})\end{array}\right]~{},$ (2)

with $\xi_{i}$ ($i\\!=\\!A,B$), the control parameter of configuration ($i$),
and $\hat{W}$, the coupling between them with parameter $\omega$. QPTs of this
type are manifested empirically near (sub-) shell closure, e.g. in the light
Pb-Hg isotopes, with strong mixing between the configurations [9, 10].

Recently, we have introduced a new type of phase-transitions in even-even [11,
12] and odd-mass [13] nuclei called intertwined quantum phase transitions
(IQPTs). The latter refers to a scenario where as we vary the control
parameters ($\xi_{A},\xi_{B},\omega$) in Eq. 2, each of the Hamiltonians
$\hat{H}_{A}$ and $\hat{H}_{B}$ undergoes a separate and clearly distinguished
shape-phase transition (Type I), and the combined Hamiltonian simultaneously
experiences a crossing of configurations $A$ and $B$ (Type II).

##  2 Theoretical framework

A convenient framework to study the different types of QPTs together is the
extension of the IBM to include configuration mixing (IBM-CM) [14, 15, 16].

###  2.1 The interacting boson model with configuration mixing

The IBM for a single shell model configuration has been widely used to
describe low-lying quadrupole collective states in nuclei in terms of $N$
monopole ($s^{\dagger}$) and quadrupole ($d^{\dagger}$) bosons, representing
valence nucleon pairs. The model has U(6) as a spectrum generating algebra,
where the Hamiltonian is expanded in terms of its generators,
$\\{s^{\dagger}s,s^{\dagger}d_{\mu},d^{\dagger}_{\mu}s,d^{\dagger}_{\mu}d_{\mu^{\prime}}\\}$,
and consists of Hermitian, rotational-scalar interactions which conserve the
total number of $s$\- and $d$-bosons
$\hat{N}=\hat{n}_{s}+\hat{n}_{d}\\!=\allowbreak\\!s^{\dagger}s+\sum_{\mu}d^{\dagger}_{\mu}d_{\mu}~{}$.
The boson number is fixed by the microscopic interpretation of the IBM [17] to
be $N\\!=\\!N_{\pi}+N_{\nu}$, where $N_{\pi}$ ($N_{\nu}$) is the number of
proton (neutron) particle or hole pairs counted from the nearest closed shell.

The solvable limits of the model correspond to dynamical symmetries (DSs)
associated with chains of nested sub-algebras of U(6), terminating in the
invariant SO(3) algebra. In the IBM there are three DS limits

$\displaystyle\text{U(6)}\supset\begin{cases}&\text{U(5)}\supset\text{SO(5)}\supset\text{SO(3)},\\\
&\text{SU(3)}\supset\text{SO(3)},\\\
&\text{SO(6)}\supset\text{SO(5)}\supset\text{SO(3)}.\end{cases}$ (3)

In a DS, the Hamiltonian is written in terms of Casimir operators of the
algebras of a given chain. In such a case, the spectrum is completely solvable
and resembles known paradigms of collective motion: spherical vibrator [U(5)],
axially symmetric [SU(3)] and $\gamma$-soft deformed rotor [SO(6)]. In each
case, the energies and eigenstates are labeled by quantum numbers that are the
labels of irreducible representations (irreps) of the algebras in the chain.
The corresponding basis states for each of the chains (3) are

$\displaystyle\text{U(5)}:$
$\displaystyle\quad\ket{N,n_{d},\tau,n_{\Delta},L},$ (4a)
$\displaystyle\text{SU(3)}:$ $\displaystyle\quad\ket{N,(\lambda,\mu),K,L},$
(4b) $\displaystyle\text{SO(6)}:$
$\displaystyle\quad\ket{N,\sigma,\tau,n_{\Delta},L},$ (4c)

where $N,n_{d},(\lambda,\mu),\sigma,\tau,L$ label the irreps of U(6), U(5),
SU(3), SO(6), SO(5) and SO(3), respectively, and $n_{\Delta},K$ are
multiplicity labels.

An extension of the IBM to include intruder excitations is based on
associating the different shell-model spaces of 0p-0h, 2p-2h, 4p-4h, $\dots$
particle-hole excitations, with the corresponding boson spaces with
$N,\,N\\!+\\!2,\,N\\!+\\!4,\ldots$ bosons, which are subsequently mixed [15,
16]. For two configurations the resulting IBM-CM Hamiltonian can be
transcribed in a form equivalent to that of Eq. 2

$\hat{H}=\hat{H}_{A}^{(N)}+\hat{H}_{B}^{(N+2)}+\hat{W}^{(N,N+2)}~{}.$ (5)

Here, the notations $\hat{\cal O}^{(N)}\\!=\\!\hat{P}_{N}^{{\dagger}}\hat{\cal
O}\hat{P}_{N}$ and $\hat{\cal
O}^{(N,N^{\prime})}\\!=\\!\hat{P}_{N}^{{\dagger}}\hat{\cal
O}\hat{P}_{N^{\prime}}$, stand for an operator $\hat{\cal O}$, with
$\hat{P}_{N}$, a projection operator onto the $N$ boson space. The Hamiltonian
$\hat{H}_{A}^{(N)}$ represents the $N$ boson space (normal $A$ configuration)
and $\hat{H}_{B}^{(N+2)}$ represents the $N\\!+\\!2$ boson space (intruder $B$
configuration).

###  2.2 Wave functions structure

The eigenstates $\ket{\Psi;L}$ of the Hamiltonian (5) with angular momentum
$L$, are linear combinations of the wave functions, $\Psi_{A}$ and $\Psi_{B}$,
in the two spaces $[N]$ and $[N+2]$,

$\ket{\Psi;L}=a\ket{\Psi_{A};[N],L}+b\ket{\Psi_{B};[N\\!+\\!2],L}\,,\;$ (6)

with $a^{2}+b^{2}\\!=\\!1$. We note that each of the components in Eq. 6,
$\ket{\Psi_{A};[N],L}$ and $\ket{\Psi_{B};[N\\!+\\!2],L}$, can be expanded in
terms of the different DS limits with its corresponding boson number in the
following manner

$\ket{\Psi_{i};[N_{i}],L}=\sum_{\alpha}C^{(N_{i},L)}_{\alpha}\ket{N_{i},\alpha,L},$
(7)

where $N_{A}\\!=\\!N$ and $N_{B}\\!=\\!N+2$, and
$\alpha=\\{n_{d},\tau,n_{\Delta}\\},\\{(\lambda,\mu),K\\},\\{\sigma,\tau,n_{\Delta}\\}$
are the quantum numbers of the DS eigenstates. The coefficients
$C^{(N,L)}_{\alpha}$ give the weight of each component in the wave function.
Using them, we can calculate the wave function probability of having definite
quantum numbers of a given symmetry in the DS bases, Eq. 7, for its $A$ or $B$
parts

$\displaystyle\text{U(5)}:\quad$ $\displaystyle
P^{(N_{i},L)}_{n_{d}}=\sum_{\tau,n_{\Delta}}[C^{(N_{i},L)}_{n_{d},\tau,n_{\Delta}}]^{2},$
$\displaystyle\text{SO(6)}:\quad$ $\displaystyle
P^{(N_{i},L)}_{\sigma}=\sum_{\tau,n_{\Delta}}[C^{(N_{i},L)}_{\sigma,\tau,n_{\Delta}}]^{2},$
(8a) $\displaystyle\text{SU(3)}:\quad$ $\displaystyle
P^{(N_{i},L)}_{(\lambda,\mu)}=\sum_{K}[C^{(N_{i},L)}_{(\lambda,\mu),K}]^{2},$
$\displaystyle\text{SO(5)}:\quad$ $\displaystyle
P^{(N_{i},L)}_{\tau}=\sum_{n_{d},n_{\Delta}}[C^{(N_{i},L)}_{n_{d},\tau,n_{\Delta}}]^{2}.$
(8b)

Here the subscripts $i\\!=\\!A,B$ denote the different configurations, i.e.,
$N_{A}\\!=\\!N$ and $N_{B}\\!=\\!N+2$. Furthermore, for each eigenstate (6),
we can also examine its coefficients $a$ and $b$, which portray the
probability of the normal-intruder mixing. They are evaluated from the sum of
the squared coefficients of an IBM basis. For the U(5) basis, we have

$P^{(N_{A},L)}_{a}\equiv
a^{2}=\sum_{n_{d},\tau,n_{\Delta}}|C^{(N_{A},L)}_{n_{d},\tau,n_{\Delta}}|^{2};\qquad
P^{(N_{B},L)}_{b}\equiv
b^{2}=\sum_{n_{d},\tau,n_{\Delta}}|C^{(N_{B},L)}_{n_{d},\tau,n_{\Delta}}|^{2}.$
(9)

where the sum goes over all possible values of $(n_{d},\tau,n_{\Delta})$ in
the $(N_{i},L)$ space, $i=A,B$, and $a^{2}+b^{2}\\!=\\!1$.

###  2.3 Geometry

To obtain a geometric interpretation of the IBM is we take the expectation
value of the Hamiltonian between coherent (intrinsic) states [18, 5] to form
an energy surface

$\displaystyle E_{N}(\beta,\gamma)$
$\displaystyle=\bra{\beta,\gamma;N}\hat{H}\ket{\beta,\gamma;N}~{}.$ (10)

The $(\beta,\gamma)$ of Eq. 10 are quadrupole shape parameters whose values,
$(\beta_{\rm eq},\gamma_{\rm eq})$, at the global minimum of
$E_{N}(\beta,\gamma)$ define the equilibrium shape for a given Hamiltonian.
The values are $(\beta_{\rm eq}\\!=\\!0)$, $(\beta_{\rm
eq}\\!=\\!\sqrt{2},\gamma_{\rm eq}\\!=\\!0)$ and $(\beta_{\rm
eq}\\!=\\!1,\gamma_{\rm eq}\text{~{}arbitrary})$ for the U(5), SU(3) and SO(6)
DS limits, respectively. Furthermore, for these values the ground-band
intrinsic state, $\ket{\beta_{\rm eq},\gamma_{\rm eq};N}$, becomes a lowest
weight state in the irrep of the leading subalgebra of the DS chain, with
quantum numbers $(n_{d}\\!=\\!0)$, $(\lambda,\mu)\\!=\\!(2N,0)$ and
$(\sigma\\!=\\!N)$ for the U(5), SU(3) and SO(6) DS limits, respectively.

For the IBM-CM Hamiltonian, the energy surface takes a matrix form [19]

$\displaystyle
E(\beta,\gamma)=\left[\begin{array}[]{cc}E_{A}(\beta,\gamma;\xi_{A})&\Omega(\beta,\gamma;\omega)\\\
\Omega(\beta,\gamma;\omega)&E_{B}(\beta,\gamma;\xi_{B})\end{array}\right],$
(13)

where the entries are the matrix elements of the corresponding terms in the
Hamiltonian (2), between the intrinsic states of each of the configurations,
with the appropriate boson number. Diagonalization of this two-by-two matrix
produces the so-called eigen-potentials, $E_{\pm}(\beta,\gamma)$.

###  2.4 QPTs and order parameters

The energy surface depends also on the Hamiltonian parameters and serves as
the Landau potential whose topology determines the type of phase transition.
In QPTs involving a single configuration (Type I), the ground state shape
defines the phase of the system, which also identifies the corresponding DS as
the phase of the system. Such Type I QPTs can be studied using a Hamiltonian
as in Eq. 1, that interpolates between different DS limits (phases) by varying
its control parameters $\xi$. The order parameter is taken to be the
expectation value of the $d$-boson number operator, $\hat{n}_{d}$, in the
ground state, $\braket{\hat{n}_{d}}_{0^{+}_{1}}$, and measures the amount of
deformation in the ground state.

In QPTs involving multiple configurations (Type II), the dominant
configuration in the ground state defines the phase of the system. Such Type
II QPTs can be studied using a Hamiltonian as in Eq. 5, that interpolates
between the different configurations by varying its control parameters
$\xi_{A},\xi_{B},\omega$. The order parameters are taken to be the expectation
value of $\hat{n}_{d}$ in the ground state wave function,
$\ket{\Psi;L=0^{+}_{1}}$, and in its $\Psi_{A}$ and $\Psi_{B}$ components, Eq.
6, denoted by $\braket{\hat{n}_{d}}_{0^{+}_{1}}$, $\braket{\hat{n}_{d}}_{A}$
and $\braket{\hat{n}_{d}}_{B}$, respectively. The shape-evolution in each of
the configurations $A$ and $B$ is encapsulated in $\braket{\hat{n}_{d}}_{A}$
and $\braket{\hat{n}_{d}}_{B}$, respectively. Their sum weighted by the
probabilities of the $\Psi_{A}$ and $\Psi_{B}$ components
$\braket{\hat{n}_{d}}_{0^{+}_{1}}=a^{2}\braket{\hat{n}_{d}}_{A}+b^{2}\braket{\hat{n}_{d}}_{B}$,
portrays the evolution of the normal-intruder mixing.

##  3 QPTs in the Zr isotopes

Along the years, the $Z\approx 40,~{}A\approx 100$ region was suggested by
many works to have a ground state that is dominated by a normal spherical
configuration for neutron numbers 50–58 and by an intruder deformed
configuration for 60 onward. This dramatic change in structure is explained in
the shell model by the isoscalar proton-neutron interaction between non-
identical nucleons that occupy the spin-orbit partner orbitals $\pi 1g_{9/2}$
and $\nu 1g_{7/2}$ [20]. The crossing between configurations arises from the
promotion of protons across the Z=40 subsell gap. The interaction energy
results in a gain that compensates the loss in single-particle and pairing
energy and a mutual polarization effect is enabled. Therefore, the single-
particle orbitals at higher intruder configurations are lowered near the
ground state normal configuration, which effectively reverses their order.

###  3.1 Model space

Using the framework of the IBM-CM, we consider ${}_{40}^{90}$Zr as a core and
valence neutrons in the 50–82 major shell. The normal $A$ configuration
corresponds to having no active protons above $Z\\!=\\!40$ sub-shell gap, and
the intruder $B$ configuration corresponds to two-proton excitation from below
to above this gap, creating 2p-2h states. Therefore, the IBM-CM model space
employed in this study, consists of $[N]\oplus[N+2]$ boson spaces with total
boson number $N=1,2,\ldots 8$ for 92-106Zr and $\bar{N}=\bar{7},\bar{6}$ for
108,110Zr, respectively, where the bar over a number indicates that these are
hole bosons.

###  3.2 Hamiltonian and $E2$ transitions operator

In order to describe the spectrum of the Zr isotopes, we take a Hamiltonian
that has a form as in Eq. 5 with entries

$\displaystyle\hat{H}_{A}(\epsilon^{(A)}_{d},\kappa^{(A)},\chi)$
$\displaystyle=\epsilon^{(A)}_{d}\,\hat{n}_{d}+\kappa^{(A)}\,\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}~{},$
(14a) $\displaystyle\hat{H}_{B}(\epsilon^{(B)}_{d},\kappa^{(B)},\chi)$
$\displaystyle=\epsilon^{(B)}_{d}\,\hat{n}_{d}+\kappa^{(B)}\,\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}+\kappa^{\prime(B)}\hat{L}\cdot\hat{L}+\Delta_{p}~{},$
(14b)

where the quadrupole operator is given by
$\hat{Q}_{\chi}=d^{\dagger}s+s^{\dagger}\tilde{d}\\!+\\!\chi(d^{\dagger}\times\tilde{d})^{(2)}$,
and $\hat{L}=\sqrt{10}(d^{\dagger}\tilde{d})^{(1)}$ is the angular momentum
operator. Here $\tilde{d}_{m}=(-1)^{m}d_{-m}$ and standard notation of angular
momentum coupling is used. The off-set energy between configurations $A$ and
$B$ is $\Delta_{p}$, where the index $p$ denotes the fact that this is a
proton excitation. The mixing term in Eq. 5 between configurations $(A)$ and
$(B)$ has the form [14, 15, 16] $\hat{W}=\omega\,[\,(d^{\dagger}\times
d^{\dagger})^{(0)}+(s^{\dagger})^{2}\,]+{\rm H.c.}$, where H.c. stands for
Hermitian conjugate. The parameters are obtained from a fit, elaborated in the
appendix of Ref. [12].

The $E2$ operator for two configurations is written as
$\hat{T}(E2)\\!=\\!e^{(A)}\hat{Q}^{(N)}_{\chi}+e^{(B)}\hat{Q}^{(N+2)}_{\chi}$,
with
$\hat{Q}_{\chi}^{(N)}\\!=\\!\hat{P}_{N}^{{\dagger}}\hat{Q}_{\chi}\hat{P}_{N}$
and
$\hat{Q}_{\chi}^{(N+2)}\\!=\\!P_{N+2}^{{\dagger}}\hat{Q}_{\chi}\hat{P}_{N+2}$.
The boson effective charges $e^{(A)}$ and $e^{(B)}$ are determined from the
$2^{+}\\!\to\\!0^{+}$ transition within each configuration [12], and $\chi$ is
the same parameter as in the Hamiltonian (14).

For the energy surface matrix (13), we calculate the expectation values of the
Hamiltonians $\hat{H}_{A}$ (14a) and $\hat{H}_{B}$ (14b) in the intrinsic
state of Section 2.3 with $N$ and $N\\!+\\!2$ bosons respectively, and a non-
diagonal matrix element of the mixing term $\hat{W}$ between them. The
explicit expressions can be found in [12].

##  4 Results

In order to understand the change in structure of the Zr isotopes, it is
insightful to examine the evolution of different properties along the chain.

###  4.1 Evolution of energy levels

\begin{overpic}[height=101.17645pt]{fig_energies.pdf} \put(33.5,48.0){(a) {\bf
Exp}} \put(56.0,48.0){(b) {\bf Calc}} \end{overpic} Figure 1: Comparison
between (a) experimental and (b) calculated energy levels
$0_{1}^{+},2_{1}^{+},4_{1}^{+},0_{2}^{+},2_{2}^{+},4_{2}^{+}$. Empty (filled)
symbols indicate a state dominated by the normal $A$ configuration (intruder
$B$ configuration), with assignments based on Eq. 9. The
symbol[${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\CIRCLE}$,
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangle}$,
${\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\blacklozenge}$],
indicates the closest dynamical symmetry [U(5), SU(3), SO(6)] to the level
considered, based on Eq. 8. Note that the calculated values start at neutron
number 52, while the experimental values include the closed shell at 50.
References for the data can be found in [12].

In Fig. 1, we show a comparison between selected experimental and calculated
levels, along with assignments to configurations based on Eq. 9 and to the
closest DS based on Eq. 8, for each state. In the region between neutron
number 50 and 56, there appear to be two configurations, one spherical
(seniority-like), ($A$), and one weakly deformed, ($B$), as evidenced by the
ratio $R_{4/2}$, which is $R^{(A)}_{4/2}\cong 1.6$ and $R^{(B)}_{4/2}\cong
2.3$ at at 52–56. From neutron number 58, there is a pronounced drop in energy
for the configuration ($B$) states and at 60, the two configurations exchange
their role, indicating a Type II QPT. At this stage, the $B$ configuration
appears to undergo a U(5)-SU(3) Type I QPT, similarly to case of the Sm region
[14, 21, 22]. Beyond neutron number 60, the $B$ configuration is strongly
deformed, as evidenced by the small value of the excitation energy of the
state $2_{1}^{+}$, $E_{2_{1}^{+}}\\!=\\!139.3$ keV and by the ratio
$R^{(B)}_{4/2}\\!=\\!3.24$ in 104Zr. At still larger neutron number 66, the
ground state band becomes $\gamma$-unstable (or triaxial) as evidenced by the
close energy of the states $2_{2}^{+}$ and $4_{1}^{+}$,
$E_{2_{2}^{+}}\\!=\\!607.0$ keV, $E_{4_{1}^{+}}\\!=\\!476.5$ keV, in 106Zr,
and especially by the results $E_{4^{+}_{1}}\\!=\\!565$ keV and
$E_{2^{+}_{2}}\\!=\\!485$ keV for 110Zr of Ref. [23], a signature of the SO(6)
symmetry. In this region, the $B$ configuration undergoes a crossover from
SU(3) to SO(6).

###  4.2 Evolution of configuration content

Figure 2: Left panels: percentage of the wave functions within the intruder
B-configuration [the $b^{2}$ probability in Eq. (6)], for the ground
$0^{+}_{1}$ (bottom) and excited $2^{+}_{1}$ (top) states in 92-110Zr. Right
panels: evolution of symmetries for the lowest $0^{+}$ (bottom) and $2^{+}$
(top) state of configuration $B$ along the Zr chain. Shown are the
probabilities of selected components of U(5)
(${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\CIRCLE}$),
SU(3)
(${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangle}$),
SO(6)
(${\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\blacklozenge}$)
and SO(5) ( ), obtained from Eq. 8. For neutron numbers 52–58 (60–70),
$0^{+}_{B}$ corresponds to the experimental $0^{+}_{2}$ ($0^{+}_{1}$) state.
For neutron numbers 52–56 (58–70), $2^{+}_{B}$ corresponds to the experimental
$2^{+}_{2}$ ($2^{+}_{1}$) state.

We examine the configuration change for each isotope, by calculating the
evolution of the probability $b^{2}$, Eq. 9, of the $0^{+}_{1}$ and
$2^{+}_{1}$ states. The left panels of Fig. 2 shows the percentage of the wave
function within the $B$ configuration as a function of neutron number across
the Zr chain. The rapid change in structure of the $0^{+}_{1}$ state (bottom
left panel) from the normal $A$ configuration in 92-98Zr (small $b^{2}$
probability) to the intruder $B$ configuration in 100-110Zr (large $b^{2}$
probability) is clearly evident, signaling a Type II QPT. The configuration
change appears however sooner in the $2^{+}_{1}$ state (top left panel), which
changes to configuration $B$ already in 98Zr, in line with [24]. Outside a
narrow region near neutron number 60, where the crossing occurs, the two
configurations are weakly mixed and the states retain a high level of purity,
especially for neutron number larger than 60.

###  4.3 Evolution of symmetry content

We examine the changes in symmetry of the lowest $0^{+}$ and $2^{+}$ states
within the $B$ configuration, which undergoes a Type I QPT. In the right
bottom panel of Fig. 2 the red dots represent the percentage of the U(5)
$n_{d}\\!=\\!0$ component in the wave function, $P^{(N+2,L=0)}_{n_{d}=0}$ of
Eq. 8. It is large ($\approx 90\%$) for neutron number 52–58 and drops
drastically ($\approx 30\%$) at 60. The drop means that other
$n_{d}\\!\not=\\!0$ components are present in the wave function and therefore
this state becomes deformed. Above neutron number 60, the $n_{d}\\!=\\!0$
component drops almost to zero (and rises again a little at 70), indicating
the state is strongly deformed. To understand the type of DS associated with
the deformation above neutron number 60, we add in blue triangles the
percentage of the SU(3) $(\lambda,\mu)=(2N+4,0)$ component,
$P^{(N+2,L=0)}_{(\lambda,\mu)=(2N+4,0)}$ of Eq. 8 for 60–66. For neutron
number 60, it is moderately small ($\approx 35\%$), at neutron number 62 it
jumps ($\approx 85\%$) and becomes maximal at 64 ($\approx 92\%$). This serves
as a clear evidence for a U(5)-SU(3) Type I QPT. At neutron number 66 the
SU(3) $(\lambda,\mu)\\!=\\!(2N+4,0)$ component it is lowered, and one sees by
the green diamonds the percentage of the SO(6) $\sigma=N+2$ component,
$P^{(N+2,L=0)}_{\sigma=N+2}$ of Eq. 8. The latter becomes dominant for 66–70
($\approx 99\%$), suggesting a crossover from SU(3) to SO(6).

In order to further elaborate the Type I QPT within configuration $B$ from
U(5) to SU(3) and the subsequent crossover to SO(6), we examine also the
evolution of SO(5) symmetry. The gray histograms in the right panel of Fig. 2
depict the probability of the $\tau\\!=\\!0$ component of SO(5),
$P^{(N+2,L=0)}_{\tau=0}$ of Eq. 8, for $0^{+}_{B}$. For neutron numbers 52–56,
the $0^{+}_{B}$ state is composed mainly of a single
($n_{d}\\!=\\!0,\tau\\!=\\!0$) component, appropriate for a with state good
U(5) DS. For neutron number 58, the larger $\tau\\!=\\!0$ but smaller
$n_{d}\\!=\\!0$ probabilities imply the presence of additional components with
($n_{d}\\!\neq\\!0,\tau\\!=\\!0$). For neutron numbers 60–64, the
$\tau\\!=\\!0$ probability decreases, implying admixtures of components with
($n_{d}\neq 0,\tau\neq 0$), appropriate for a state with good SU(3) DS. For
neutron numbers 66–70, the $\tau\\!=\\!0$ probability increases towards its
maximum value at 70, appropriate for a crossover to SO(6) structure with good
SO(5) symmetry.

In the top right panel of Fig. 2 we observe a similar trend for the
$2^{+}_{B}$ state. For neutron numbers 52–58, it is dominated by a single
($n_{d}\\!=\\!1,\tau\\!=\\!1$) component. For neutron number 60,
$P^{(N+2,L=2^{+}_{B})}_{n_{d}=1}$ is smaller than
$P^{(N+2,L=2^{+}_{B})}_{\tau=1}$, indicating the onset of deformation. For
62–64, $P^{(N+2,L=2^{+}_{B})}_{n_{d}=1}$ is much smaller than
$P^{(N+2,L=2^{+}_{B})}_{\tau=1}$, implying admixtures of components with
($n_{d}\\!\neq\\!1,\tau\\!\neq\\!1$). For neutron numbers 66–70,
$P^{(N+2,L=2^{+}_{B})}_{n_{d}=1}$ remains small but
$P^{(N+2,L=2^{+}_{B})}_{\tau=1}$ increases towards its maximum value at 70.

###  4.4 Evolution of order parameters

\begin{overpic}[width=195.12767pt]{fig_nd.pdf} \put(90.0,40.0){(a)}
\end{overpic}\begin{overpic}[width=195.12767pt]{fig_be2.pdf}
\put(90.0,40.0){(b)} \end{overpic}

Figure 3: (a) Evolution of order parameters along the Zr chain, normalized
(see text). (b) $B(E2)$ values in W.u. for $2^{+}\rightarrow 0^{+}$
transitions in the Zr chain. The solid line (symbols
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\CIRCLE},~{}{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\blacksquare},~{}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangle},~{}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\blacklozenge}$)
denote calculated results (experimental results). Dotted lines denote
calculated $E2$ transitions within a configuration. The data for 94Zr, 96Zr,
100Zr, 102Zr and (104Zr, 106Zr) are taken from [25], [26], [27], [28], [29],
respectively. For 98Zr (neutron number 58), the experimental values are from
[30] ($\blacklozenge$), from [31] ($\blacktriangle$), and the upper and lower
limits (black bars) are from [27, 24].

The configuration and symmetry analysis of Sections 4.2 and 4.3 suggest a
situation of simultaneous occurrence of Type I and Type II QPTs. The order
parameters can give further insight to these QPTs. Fig. 3(a) shows the
evolution along the Zr chain of the order parameters
($\braket{\hat{n}_{d}}_{A},\,\braket{\hat{n}_{d}}_{B}$ in dotted and
$\braket{\hat{n}_{d}}_{0^{+}_{1}}$ in solid lines), normalized by the
respective boson numbers, $\braket{\hat{N}}_{A}\\!=\\!N$,
$\braket{\hat{N}}_{B}\\!=\\!N\\!+\\!2$,
$\braket{\hat{N}}_{0^{+}_{1}}\\!=\\!a^{2}N\\!+\\!b^{2}(N\\!+\\!2)$. The order
parameter $\braket{\hat{n}_{d}}_{0^{+}_{1}}$ is close to
$\braket{\hat{n}_{d}}_{A}$ for neutron number 52–58 and coincides with
$\braket{\hat{n}_{d}}_{B}$ at 60 and above. The clear jump and change in
configuration content from 58 to 60 indicates a Type II phase transition [8],
with weak mixing between the configurations. Configuration $A$ is spherical
for all neutron numbers, and configuration $B$ is weakly-deformed for neutron
number 52–58. From neutron number 58 to 60 we see a sudden increase in
$\braket{\hat{n}_{d}}_{B}$ that continues towards 64, indicating a U(5)-SU(3)
Type I phase transition. Then, we observe a decrease from neutron number 66
onward, due in part to the crossover from SU(3) to SO(6) and in part to the
shift from boson particles to boson holes after the middle of the major shell
50–82. These conclusions are stressed by an analysis of other observables
[12], in particular, the $B(E2)$ values. As shown in Fig. 3(b), the calculated
$B(E2)$’s agree with the experimental values and follow the same trends as the
respective order parameters.

###  4.5 Classical analysis

\begin{overpic}[width=78.04842pt]{fig_92ZrSurf.pdf} \put(-5.0,69.0){(a)}
\put(-5.0,50.0){ ${}^{92}$Zr} \put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_94ZrSurf.pdf}
\put(-5.0,69.0){(b)} \put(-5.0,50.0){ ${}^{94}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_96ZrSurf.pdf}
\put(-5.0,69.0){(c)} \put(-5.0,50.0){ ${}^{96}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_98ZrSurf.pdf}
\put(-5.0,69.0){(d)} \put(-5.0,50.0){ ${}^{98}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_100ZrSurf.pdf}
\put(-5.0,69.0){(e)} \put(-5.0,50.0){ ${}^{100}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)} \end{overpic}

\begin{overpic}[width=78.04842pt]{fig_102ZrSurf.pdf} \put(-5.0,69.0){(f)}
\put(-5.0,50.0){ ${}^{102}$Zr} \put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_104ZrSurf.pdf}
\put(-5.0,69.0){(g)} \put(-5.0,50.0){ ${}^{104}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_106ZrSurf.pdf}
\put(-5.0,69.0){(h)} \put(-5.0,50.0){${}^{106}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_108ZrSurf.pdf}
\put(-5.0,69.0){(i)} \put(-5.0,50.0){ ${}^{108}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)}
\end{overpic}\begin{overpic}[width=78.04842pt]{fig_110ZrSurf.pdf}
\put(-5.0,69.0){(j)} \put(-5.0,50.0){ ${}^{110}$Zr}
\put(60.0,70.0){\small$\gamma$(deg)} \end{overpic}

Figure 4:  Contour plots in the $(\beta,\gamma)$ plane of the lowest eigen-
potential surface, $E_{-}(\beta,\gamma)$, for the 92-110Zr isotopes.

In Fig. 4, we show the calculated lowest eigen-potential
$E_{-}(\beta,\gamma)$, which is the lowest eigenvalue of the matrix Eq. 13.
These classical potentials confirm the quantum results, as they show a
transition from spherical (92-98Zr), Figs. 4(a)-(d), to a double-minima
potential that is almost flat-bottomed at 100Zr, Fig. 4(e), to prolate axially
deformed (102-104Zr), Figs. 4(f)-(g), and finally to $\gamma$-unstable
(106-110Zr), Figs. 4(h)-(j).

##  5 Conclusions and outlook

The algebraic framework of the IBM-CM allows us to examine QPTs using both
quantum and classical analyses. We have employed this analysis to the Zr
isotopes with $A\\!=\\!92\text{--}110$, which exhibit a complex structure that
involves a shape-phase transition within the intruder configuration (Type I
QPT) and a configuration-change between normal and intruder (Type II QPT),
namely IQPTs. This was done by analyzing the energies, configuration and
symmetry content of the wave functions, order parameters and $E2$ transition
rates, and the energy surfaces. Further analysis of other observables
supporting this scenario is presented in [12]. Recently, we have also
exemplified the notion IQPTs in the odd-mass 41Nb isotopes [13] and it would
be interesting to examine the notion of IQPTs in other even-even and odd-mass
chains of isotopes in the $Z\approx 40,~{}A\approx 100$ region and other
physical systems.

## Acknowledgments

This work was done in collaboration with F. Iachello (Yale university) and A.
Leviatan (Hebrew university).

#### Funding information

This work was supported in part by the US-Israel Binational Science Foundation
Grant No. 2016032 and the Israel Academy of Sciences for a Postdoctoral
Fellowship Program in Nuclear Physics.

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# Rapid solution for searching similar audio items

Kastriot Kadriu  Faculty of Computer and Information Science
University of Ljubljana
Email<EMAIL_ADDRESS>
###### Abstract

A naive approach for finding similar audio items would be to compare each
entry from the feature vector of the test example with each feature vector of
the candidates in a k-nearest neighbors fashion. There are already two
problems with this approach: audio signals are represented by high dimensional
vectors and the number of candidates can be very large - think thousands. The
search process would have a high complexity. Our paper will treat this problem
through hashing methodologies more specifically the Locality Sensitive
Hashing.
This project will be in the spirit of classification and clustering problems.
The computer sound production principles will be used to determine which
features that describe an audio signal are the most useful. That will down-
sample the size of the feature vectors and speed up the process subsequently.

###### Keywords:

near neighbors search, lsh, audio signals, audio fingerprinting

##  I Introduction

Audio-visual data is represented by high dimensional vectors. Searching and
comparing distances in such large collections has a large complexity. The k-NN
problem represents a naive approach for collecting near (similar) items. For
low dimensions, k-d trees can be used to answer queries in logarithmic and
linear space. In high dimensions, the k-NN problem suffers from the curse of
dimensionality where either the running time or the space requirements grow
exponentially in dimension $d$.
Locality Sensitive hashing (LSH) technique consists of projecting each feature
vector onto a random distribution of values resulting into random buckets.
This random projecting, which is not so random, is repeated several times. The
theorem follows the principle that similar items end up in similar buckets.
These buckets, that can be represented by hash values, allow us for near-
constant time accession of results that represent the near items of a given
query example that is also hashed in a similar manner.

##  II Related work

A locality sensitive hashing model has been initally proposed in [1] where we
have a direct approach for reducing computation complexity caused by a high
dimensional data. Multiple versions of LSH have since been developed. ANN in
[2] is an efficient library for querying approximate near neighbors in
abritrarily high dimensions. It implements different data structures based on
kd-trees and box-decomposition trees and it employes a couple of different
strategies. MinHash [5] is often seen as an instance of locality sensitive
hashing. It is a technique for estimating the similarity of two sets in an
efficient way, mainly by using the Jaccard similarity coefficient. It is
highly used for detecting duplicates of documents, using the so-called
shingles.
SimHash [6] is a version of LSH used for similar purposes, for finding similar
items. It has lower memory requirements than MinHash, however its similarity
detection is limited in large distances.
One variant of LSH is based on bands from which we are also able to
fingerprint (find the uniqueness) of one item that we can later use in
identifying duplicate or similar other items.

##  III Locality-sensitive hashing

Locality-sensitive hashing is a set of techniques that consist of grouping
similar items through various metrics that eventually speed up the querying
process for near neighbors, or in some cases near-duplication detection, of
data.
The LSH model is a geo-spatial structure that contains the points (items) that
have been projected onto a random distribution.
The projected matrix is equal to:

$[Projected(P)]_{k\times n}=[Random(R)]_{k\times d}[Original(D)]_{d\times n}$

The random matrix is generated by using a Gaussian distribution.
After projection, we check whether our query point lies above or below the
hyper-planes defined by the random projection points, denoted by 1 and 0
respectively. After that, we deal with a matrix in which we perform some basic
bit-wise operations to generate the hash value, although the opportunities
here are not strict as long as the proper similarity between item is able to
be kept by the generated hashes.
After hashing items into buckets, it’s possible that two similar items are
close to each other but on the opposite sides of the border. To handle this
issue, the random projection (and hashing) is repeated for a selected number
of times. This ensures that similar items will fall into the same bucket at
least once.
The number of buckets that contain the points is $\approx 2^{s}$ where $s$ is
the hash size. A LSH model can be configured with other hashing functions that
are based on deterministic projections. However, random projections are
desirable because they lead us to a data-independent evaluation of the model.

Figure 1: LSH model architecture

##  IV Methodology

###  IV-A Dataset

The dataset used in this project is the Free Music Archive (FMA) dataset [3],
[4]. This extensive dataset contains metadata and pre-computed features for
106,574 tracks from 16,341 artists and 14,854 albums arranged in a
hierarchical taxonomy of 161 genres. The dataset offers features that describe
different properties of the audio. The ones that we will be dealing with are
the ones that are computed by Librosa. Librosa decomposes audios into features
that describe the following properties such as Chroma Energy Normalized
(CENS), Mel-frequency cepstral coefficients, Tonnetz, and Spectral Contracts.
For each of these properties, the following values are reported: kurtosis,
max, min, mean and standard deviation.

###  IV-B The model

A working model in the form of a script has been implemented on Python. The
model contains three classes: FastAudioSearch, LSH and Table. FastAudioSearch
encapsulates the integration and sets up the train and testing procedures.
Parameters such as hash size, number of tables and data directory can be
passed when running the script.
The training process consists of setting up the geo-spatial structure where
each item (row) is hashed into a bucket, as shown in Algorithm 1.

Algorithm 1 Algorithm for generating LSH given multi dimensional data

1:function build_lsh_tables($P$)

2: for table $t$ in tables $T$ do

3: generate random projections

4: for table $t$ in tables $T$ do

5: generate hashes using dot projection of $P$ and random projections of $t$

6: organize document in buckets defined by the respective hashes

Algorithm 2 Algorithm for querying similar items in a LSH model

function query_lsh_tables($D$, $k$)

2: for table $t$ in tables $T$ do

$H\leftarrow$ generate hashes using dot product of $D$ and random projections
of $t$

4: for test point $d$ in $D$ do

if $t$ contains a bucket hashed by $H_{d}$ then

6: return distances of $i$ with points in $t$ as near neighbor candidates

for test point $d$ in $D$ do

8: $sorted\\_dists\leftarrow$ sort distances of d

for each candidate point $c$ in $sorted\\_dists$ do

10: return $c$ only if it has the lowest distance with $d$ in any table

The testing process consists of generating the hashes for the query item and
getting the buckets with those hashes from the geo-spatial structure -
Algorithm 2. Table class contains the hashing procedure for our data whereas
LSH sets up the strategy for hashing and retrieving results from multiple
tables.

###  IV-C Complexity

In the brute force method of the k-NN problem training is considered a
constant operation since in this approach there’s not an actual training
process happening, whereas the querying of the results has a complexity of
$O(n\times d\times k)$. In the k-d tree method, building the tree represents
the training phase that has the complexity of $O(d\times n\times logn)$.
Usually, we don’t care much about the training times since the process isn’t
expected to be repeated much, but as we can see, in k-d trees the process is
dependent on the dimension size and for large dimension, we can see how it can
effect the time. The querying times, representing the prediction phase, with
this approach are reduced to $O(k\times logn)$. When the dimensionality is
big, those trees break down and we end up testing all the nodes, bringing the
complexity to $O(k\times logn)$.

In the case of LSH with random projections, considering a query, table lookup
by its hash value is done in constant time. The process of checking the
distances with the near neighbor candidates with the same hash value is
$O(d\times\frac{n}{2^{h_{s}}})$. The generation of the hash value is done by
taking the dot product of the vector of the query point with the random
projection and it has a complexity $O(d\times h_{s})$. This process is
repeated for $t$ number of tables. So the overall complexity for querying
approximate near neighbors for a point is considered to be $O(t\times((d\times
h_{s})+(d\times\frac{n}{2^{h_{s}}}))$.

##  V Experiments

Experiments have been done on a Macbook Pro late 2013 machine running on a
quad-core Intel i7 processor and 16gb of RAM. The aim of the experiments is to
report on several properties of the model. First, we show that hashing the
content on blocks for a selected number of tables is a relatively rapid
process. Next we show that the random projection is a safe process in which
most important data attributes can be saved. We also show that querying
similar items candidates for a test example is a robust process and finally we
show that those candidates are indeed similar with the queried example.
For testing the speed of buckets building, we time our operations.
Since the dataset doesn’t provide a collection of ground truth near items that
can be used for evaluating the accuracy of the model, we don’t use a
traditional train-test split. Instead, a sequence of random samples that de-
facto represent our testing data is picked from the original dataset. This
sequence remains the same for all experiments reported in this paper.

The average time for building buckets for a table with different hash sizes is
shown in Fig. 2. The process for building the table includes the dot product
with the randomly generated matrix, hashing each row (item) and organising
buckets in a dictionary data structure. Subsequently, the training time is
increased linearly proportional to the number of tables.

The random projection matrix is controlled by the hash size value that
determines the downsampled dimensions. However, if we downsample the original
matrix too much, we risk losing important data. For a randomly selected
sample, we test how accurate the hash size parameter selection is by testing
if the predicted similar items candidates do indeed hold the distance relation
with each other when we check their actual distance in the original data
matrix in the Euclidean space. This means, that we are looking for a
monotonically increasing sequence of distances between the query item and
candidates. This process is repeated for 100 re-iterations with a different
random sample each time. Then the values are averaged.

Figure 2: Average time for hashing items in buckets in a single table for
different hash sizes, (data size = 100k). Figure 3: Query processing time for
different number of tables, (data size = $100k$, averaged per $100$ queries).
Figure 4: Accuracy of the model evaluated for different hash size values.
Figure 5: Distance relation of our randomly selected samples, $h^{s}=250,t=1$.

A sensible hash size parameter is important to be selected because we want to
optimize accuracy (higher value) and speed (lower value).

##  VI Discussions

From the experiments seen in the previous section, we can observe several
properties.
First, a lower hash size leads to faster training times (Fig. 2). That is
because of the small number of buckets that are generated and subsequent less
operations needed to process the query. But on the other hand, a lower hash
size leads to longer querying times because a small number of buckets means a
large list of items in those buckets that is retrieved and needs to be sorted.
In addition, the data projected in very small dimensions loses the distance
relation between items thus affecting the accuracy in retrieving ordered
similar items (Fig. 4).
A higher hash size guarantees higher accuracy but with a cost in time. Fine
tuning the hash size value altogether with a sensible number of tables can
achieve good results for our purposes.
From Fig. 2, we can see also that the time for building tables is increased
linearly with the hash size but that should not hinder the process much
because the model is supposed to be built (trained) only once.
In Fig. 3, the query processing times drop exponentially and that happens
because the number of buckets is increased exponentially with the hash size
value.
The evaluated accuracy in Fig. 4 follows a linear growth, with the growth of
the hash size for reasons describe above. In the plot, the growth isn’t
exactly uniform and that can be justified with two reasons:

  * •

the results aren’t very deterministic - although the margin of error of
reported values isn’t large, the values fluctuate because of the random nature
of the model.

  * •

the improvement of accuracy throttles in high hash size values - the data can
be sparse and unrelated to each other and there’s only so much we can do to
improve the model.

Euclidean distance has been used as a metric. It has shown to be able of
proper accounting of distances. Alternatively, the hash values can be kept in
bits of 1 and 0 and then the Hamming distance metric can be used to determine
which buckets are closest to the current one.
Fig. 5 is another accuracy plot where the distance relations for each of our
testing samples are displayed. There we can see that the sequences are mainly
monotonically increased.

The model has some limitations. Our paper takes an agnostic approach to space
complexity. To improve querying times, a more efficient way to handle the
retrieved candidates post-bucket collection could’ve been implemented.
Different data structures could’ve been used. An analysis from an I/O
complexity standpoint could be useful for this case.
Another issue that isn’t addressed is that the width of the bucket which isn’t
fixed. The side effect of non fixed bucket widths is that sometimes random
projections being really random might encapsulate the data in massively
unbalanced buckets. In our experiments, we have found this could be the case
if we are testing with a low number of samples ( 100). On the other hand, a
large number of samples (100k as in our case) ensures that the data is
balanced enough that the average size of $2^{s}$ is achieved and the buckets
are balanced, and since the true purpose of LSH models is high dimensional
data, this issue can be disregarded.
Finally, the testing phase was impaired by a lack of ’ground truths’ similar
items that could’ve been used to directly compare the retrieved candidates.

##  VII Conclusions

In this paper, we presented to you an approach for constructing a model which
is able to query similar items in sub linear time. That is a tremendous
improvement from a brute-force approach, which we could regard as impossible
to be used in the experiment of this paper, as the dimensionality of the data
is way too high. This implementation was done using pre-computed audio
features. It can very easily be tweaked and use other features that describe
the audio such as information about genre, the nature of the song and the
singer. Our implementation lacks an opinion on the space complexity that an
approach like this could have. Although, from the experiments done, this
approach should be better even in that regard compared to other methods of
near neighbor search such as the brute-force or some sort of dimensional
trees. In the present times when we are dealing with big data, it’s very
important to consider approaches like this that have already become the norm
in solving similar items searching tasks.

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