# Studying the holographic Fermi surface in the scalar induced anisotropic
background

Sayan Chakrabarti, Debaprasad Maity, and Wadbor Wahlang  Department of
Physics, Indian Institute of Technology, Guwahati
Assam 781039, India

###### Abstract

Holographic properties of a finite density fermion system have been shown to
exhibit many interesting behaviours which can be observed in future. In this
paper, we study low energy fermion properties in the framework of the
holographic Mott-Insulator system. We study the nature of the Fermi surface
and its evolution by tuning two types of dipole couplings in the bulk. We
further introduce translational symmetry breaking complex scalar field, which
is assumed to couple with the holographic fermions. The symmetry breaking
background induced by the scalar field is known as Q-lattice. We calculate the
fermion spectral function, which captures the low energy behaviour of the
system. By tuning the dipole parameters and the non-normalizable component of
the scalar field, we observe interesting phenomena such as spectral weight
transfer, Fermi surface smearing, which has already been reported in various
real condensed matter experiments.

###### keywords:

AdS/CFT correspondence, AdS/CMT, Fermi arcs

##  1 Introduction

Holography has been shown to play a very interesting role in various branches
of physics, specifically at strong coupling. The holographic principle [1, 2]
has been successfully applied as a framework to understand various phenomena:
from non-linear hydrodynamics [3], Fermi liquid behavior [4, 5], transport
phenomena [6], to high temperature superconductors [7, 8, 9] to name a few and
it also helps in the understanding the condensed matter systems with the help
of classical gravity theories. One of the earlier important applications of
the holographic method is in the case of holographic superconductors, and the
results were remarkably similar to those obtained by conventional methods.
Since then, it was widely used as a tool to study condensed matter systems.
Several works have been done related to Fermi surfaces in order to study the
emergence of Fermi and non-Fermi liquid behaviours in the fermionic system
[10, 11, 12, 13, 14], also this method was used to study the high $T_{c}$
superconductor, Mott transition and the pseudogap phase [15, 16, 17] and there
of course exists many more applications in the field of nuclear physics. Apart
from many such known properties of real condensed matter systems, the
holographic study, particularly on the strongly coupled fermionic system at
finite density has been shown to predict many exotic properties and phases
which may be possible to observe in the laboratory in future. Therefore,
exploring various seemingly exotic phases of holographic fermionic system,
even without being able to be observed in the laboratory in the near future,
is being considered as an active research area. Following the same line, in
this paper, we study such holographic systems with some specific types of bulk
controlling parameters that have been introduced before in the context of
holographic Mott-Insulator physics. Our study reveals many interesting
features of the Fermi surface, such as spectral weight transfer, Fermi surface
smearing, to name a few, which we found to have some apparent similarities
with different experimental observations on the finite density fermion system.
We expect that this search for exotic holographic phases and their
experimental realisation will be helpful in understanding the nature of
duality between gravitational fields and their associated dual operators.

Along the above line of discussion, it may be mentioned here that
understanding some of the results from condensed matter experiments, such as
those of Angle-resolved photoemission spectroscopy (ARPES) [18, 19] is a hard
task for a theoretical physicist. ARPES experiment reveals the properties of
the electronic band structure, quasi-particle excitation, the nature of the
Fermi surface (FS) of many topological materials and high $T_{c}$
superconductors. The discovery of superconductivity in the LaBaCuO ceramics at
$30~{}K$ has opened a new horizon in the field of high-$T_{c}$
superconductivity [20].

An unconventional phase of this system which is puzzling physicists over the
decades is the pseudo-gap in the fermionic spectral function, which appears as
a

truncated Fermi surface termed as a Fermi Arc [21, 22, 23, 24, 25]. The
appearance of Fermi arc has also been observed in the ARPES experiments in
different real condensed matter system such as recently discovered topological
Dirac and Weyl semi-metals [25, 26, 27, 28, 29, 30, 31, 32] and a recent
review can be found in [33]. One of the common properties of all the systems
mentioned is that the Fermi arc appears in ($2+1$) dimensions either because
of the underlying planar structure of the superconductor or as surface states
of the bulk material for Dirac and Weyl semi-metals. This phenomenon possesses
a mystery, mainly because of the unconventional electronic properties, which
can not be explained by usual Landau’s theory. Furthermore, the systems are
generically shown to be strongly coupled, and this is where the holographic
method plays the role.

In the last few years, lots of research were performed to understand the
underlying mechanism behind these properties of the Fermi surface from the
perspective of condensed matter physics [34, 35, 36, 37], and the same has
been done using the holographic method [38, 39, 40, 41, 42]. In some recent
pioneering works [15, 43, 44], investigations were done to understand the
evolution of Fermi arc-like structures, taking RN-AdS black hole and other
holographic lattices as their gravity background.

In this paper, we will study a class of holographic models, where the
appearance of the Fermi arc is a generic feature. We will essentially
generalize our previous construction [45], where bulk dipole coupling a bulk
scalar field that can break the translational symmetry in the boundary. Our
goal is to look at the evolution of the Fermi surface and their band
structure, with particular emphasis on their observational significances.

Explicit translational symmetry breaking solutions of a holographically dual
[46, 47, 48, 49, 50] theory has been studied either by considering spatially
periodic chemical potential at the boundary or especially periodic scalar
field in the AdS background.

The specific framework we will consider here is Q-lattice [50] where
holographic lattice background is constructed by exploiting the solution of a
complex scalar field in the bulk. Fermionic spectral function has been studied
on the Q-lattice background [51] together with a dipole type of coupling in
the fermion sector. It is to be noted that the secondary Fermi surfaces were
noted long back in [52]. In that work, beginning with a dilatonic black hole
derived from a truncation of type IIB supergravity, fermionic Green’s function
dual to massless fermions in the bulk were studied. It was shown that the for
not too small bulk fermion charges, there are Fermi surfaces. The Fermi
momenta in this case are equally spaced, and there are a finite number of
them, proportional to the charge of the bulk fermion. Similar studies using
the anisotropic holographic backgrounds have been done in [53, 54, 55, 56].
Our goal would be to look into the nature of the holographic Fermi surface
evolution due to bulk coupling parameters which will correspond to different
dual controlling operators.

We organise the paper as follows: In the next section, we briefly review the
Q-lattice background solution, followed by section 3, where we write down the
fermion’s action and the equations of motion with our definition of the
spectral function $\rho\,(\omega,\,\vec{k})$. In section 4, we present the
results for different scenarios for massless fermion and discuss each of them.
For completeness, in section 5 we discuss the scenarios where by taking non-
zero fermion mass. Finally, in the last section, we conclude with the
interpretation of our results along with a brief discussion on future
directions.

##  2 The background geometry

As has already been mentioned in the introduction, Fermi arc behaviour has
been generically observed in two spatial dimension, we consider our
holographic system to be in ($2+1$) dimensions. Hence, we shall take a
($3+1$)-dimensional Einstein-Scalar-Maxwell system as a background set up,

$\displaystyle\mathcal{S}=\frac{1}{\kappa^{2}}\int
d^{3+1}x\sqrt{-g}\left[\mathcal{R}+\frac{6}{L^{2}}-\frac{1}{4}F^{2}-|\partial\phi|^{2}-m_{\phi}^{2}|\phi|^{2}\right]$
(1)

where $\mathcal{R}$ is a Ricci scalar, $L$ being the AdS radius which we will
set to be unity later and, Maxwell’s field strength $F=dA$, where $d$ is the
exterior derivative acting on the vector potential one form $A$. Finally,
$\kappa^{2}=16\pi G$ is the effective reduced gravitational constant which we
set to be unity too. As mentioned we consider a complex scalar field $\phi$
which will source the break down of translational invariance of the boundary
field theory. From the above action (1) we have the following equations of
motions

$\displaystyle R_{\mu\nu}=$ $\displaystyle
g_{\mu\nu}\left(-3+\frac{m_{\phi}^{2}}{2}|\phi|^{2}\right)+\partial_{(\mu}\phi\partial_{\nu)}\phi^{*}+\frac{1}{4}\left(2F_{\mu\nu}^{2}-\frac{1}{2}g_{\mu\nu}F^{2}\right)$
$\displaystyle\nabla_{\mu}F^{\mu\nu}=0\;,\;\;\left(\nabla^{2}-m_{\phi}^{2}\right)\phi=0.$
(2)

We take the ansatz for the metric and the complex scalar field to be of the
following form

$\displaystyle
ds^{2}=-g_{tt}(z)\;dt^{2}+g_{zz}(z)\;dz^{2}+g_{xx}(z)\;dx^{2}+g_{yy}(z)\;dy^{2}.$
(3) $\displaystyle\text{where},$ $\displaystyle
g_{tt}(z)=\frac{U(z)}{z^{2}};\;g_{zz}(z)=\frac{U(z)^{-1}}{z^{2}};\;g_{xx}(z)=\frac{V_{1}(z)}{z^{2}};$
$\displaystyle g_{yy}(z)=\frac{V_{2}(z)}{z^{2}},\;\;A=(1-z)\;a(z)dt,$
$\displaystyle\;U(z)\,=\,(1-z)\;u(z),\,\,\phi=e^{i\,k_{1}\,x+i\,k_{2}\,y}\;\chi(z)\,.$
(4)

With the above choice of ansatz for the scalar field, the breaking of
translational symmetry by the same scalar field is both in $x$ and $y$
directions. Here, $u,V_{1},V_{2},a,\chi$ are all functions of radial
coordinate $z$ and $k_{1},\,k_{2}$ are constants interpreted as wave numbers
of the lattice. From the above equation we have four second order coupled ODEs
for $V_{1},\,V_{2},\,a,\,\chi$ and one first order for $u$. For a general mass
$m_{\phi}^{2}$, the complex scalar field near the AdS boundary
($z\longrightarrow 0$) behaves as

$\displaystyle\chi(z)\,=\,z^{\alpha_{-}}\,\chi^{(1)}\,+\,z^{\alpha_{+}}\,\chi^{(2)}\,+...$
(5)

where, $\alpha_{\pm}=3/2\pm\sqrt{9/4+m_{\phi}^{2}}$ . The leading term
$\chi^{(1)}$ is associated with the source of the dual scalar operator in the
boundary theory, whose dimension is $\Delta=3-\alpha_{-}=\alpha_{+}$. To this
end we study in detail the case when, scalar field mass $m_{\phi}^{2}=0$,
which corresponds to $\alpha_{-}=0$ and the corresponding marginal dual scalar
operator in the (2+1)- dimensional boundary theory. For building a numerical
black hole solution, we solve these equations using the regularity conditions
near the horizon $z=1$ and, at the AdS boundary or UV, we assume the following
series expansion,

$\displaystyle u=1+O(z)\,\,,\,\,V_{1}=1+O(z)\,\,,\,\,V_{2}=1+O(z)\;,$
$\displaystyle a=\mu+O(z)$ (6)

To solve the non-linear equations first, we linearize the ODEs by discretizing
simultaneously both the equations of motions and the boundary conditions
following [57, 58] and then use the iterative Newton-Raphson’s (NR) method
[59, 60] in Mathematica. Moreover for fixed $m^{2}_{\phi}$, we found that the
solutions are specified by 3-dimensionless parameters namely,
$T/\mu\,,\,k/\mu$ and $\chi^{(1)}/\mu^{\alpha_{-}}$. We notice that for very
low temparature (small $T/\mu$), NR method does not converge as it is
sensitive to the initial guess values or seeds. We can overcome this by using
the solutions at higher temparature as a seeds to achieve another solution at
lower temparature. We illustrate the background solutions in Figure 1. We
consider this as our numerical background in our subsequent analysis for the
fermion spectral function. Most of our analysis will be for both fermion and
scalar mass to be zero. In this sense our ($3+1$) dimensional gravitational
bulk contains Weyl fermion. Our goal would be to understand the fermion
spectral function dual to those Weyl fermion. This could resemble the class of
Weyl semi-metals in the real condensed matter systems as pointed out in the
introduction.

Figure 1: Q-lattice profile with $m^{2}_{\phi}=-2$, for the parameters
$T/\mu\text{=1}$, $\chi^{(1)}/\mu\text{=0.5},k_{1}/\mu=1\text{ and
}k_{2}/\mu=0$

##  3 Fermions: Action and the spectral function.

Now that we have the solutions of the background, we can write down the action
for fermions with four non-trivial interaction terms given by

$\mathcal{S}_{fer}=\int
d^{4}x\sqrt{-g}i\bar{\psi}\bigg{(}\not{D}-m_{\psi}-i\,\wp_{1}\not{F}-i\,\wp_{2}\,|\phi|^{2}\not{F}\bigg{)}\psi\;.$
(7)

with $\phi$ and $m_{\psi}$ being the complex scalar field from the background
and fermion mass respectively. We have considered two coupling parameters: one
is simple dipole coupling parametrised by $\wp_{1}$ and another is similar to
dipole type coupling with the scalar field parametrised by $\wp_{2}$. At this
point, it is to be mentioned that these two types of couplings in the
holographic context has already been studied in understanding the Mott-
Insulator [16] transition or appearance of Fermi Arc [15]. One can interpret
the Fermi arc in the present context as the emergence of Mottness
anisotropically due to Lorentz violating couplings. In our previous paper
[45], we have further generalised and explored the influence of Lorentz
violating dipole coupling on the fermionic spectral function adding a scalar
field as a source term. We have constructed the solution associated with the
transition from circular to arc like Fermi surface controlled by the scalar
condensation in the bulk. In this paper, we look into such system with two
sets of couplings and also introduced one dimensional lattice in the boundary
theory. For the present case, we will mostly concentrate on the simultaneous
effect of those coupling parameters on the fermionic spectral function.
Various symbols in (7) are given by:

$\displaystyle\not{D}=e^{\mu}_{c}\Gamma^{c}\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\Gamma_{ab}-iqA_{\mu}\right)$
$\displaystyle\not{F}=\frac{1}{2}\Gamma^{ab}e^{\mu}_{a}e^{\nu}_{b}F_{\mu\nu}\;.$

where, $e^{\mu}_{a},\omega_{\mu}^{ab}$ are the vielbeins and spin connection,
$q$ is the charge of the bulk fermions. The Dirac equation from the above
action is given by:

$\displaystyle\left(\not{D}-m_{\psi}-i\,\wp_{1}\not{F}-i\,\wp_{2}\,|\phi|^{2}\not{F}\right)\psi=0.$
(8)

In order to further simplify the Dirac equation, we choose the following gamma
matrices

$\displaystyle\Gamma^{\underline{z}}=\left(\begin{array}[]{cc}-\sigma^{3}&0\\\
0&-\sigma^{3}\end{array}\right),\;\;\Gamma^{\underline{t}}=\left(\begin{array}[]{cc}i\sigma^{1}&0\\\
0&i\sigma^{1}\end{array}\right),$ (13)
$\displaystyle\Gamma^{\underline{x}}=\left(\begin{array}[]{cc}-\sigma^{2}&0\\\
0&\sigma^{2}\end{array}\right)\;\;,\;\;\Gamma^{\underline{y}}=\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right)$ (18)

Now, we can expand the Dirac equation as

$\displaystyle\bigg{(}e^{z}_{\underline{z}}\,$
$\displaystyle\Gamma^{\underline{z}}\,\partial_{z}+S.C+e^{t}_{\underline{t}}\,(\Gamma^{\underline{t}}\,\partial_{t}-iqA_{t})+e^{x}_{\underline{x}}\Gamma^{\underline{x}}\,\partial_{x}+e^{y}_{\underline{y}}\Gamma^{\underline{y}}\,\partial_{y}$
$\displaystyle-
m_{\psi}-i\,\wp_{1}\not{F}-i\,\wp_{2}\,|\phi|^{2}\not{F}\bigg{)}\;\psi=0$ (19)

where $S.C$ is a contribution from the spin connection that can be cancelled
by rescaling the field $\psi=(g_{tt}g_{xx}g_{yy})^{-\frac{1}{4}}e^{-i\omega
t+ik_{x}x+ik_{y}y}\xi\,(z,\vec{k})$, where
$\vec{k}\equiv(-\omega,k_{x},k_{y})$. Then we have

$\displaystyle\bigg{(}$
$\displaystyle\frac{1}{\sqrt{g_{zz}(z)}}\;\Gamma^{\underline{z}}\;\partial_{z}+\frac{1}{\sqrt{-g_{tt}(z)}}\;\left(\Gamma^{\underline{t}}\;(-i\omega)-iqA_{t}\right)+$
$\displaystyle\frac{1}{\sqrt{g_{xx}(z)}}\;\Gamma^{\underline{x}}\;(ik_{x})+\frac{1}{\sqrt{g_{yy}(z)}}\;\Gamma^{\underline{y}}\;(ik_{y})-m_{\psi}\;+$
$\displaystyle\frac{\partial_{z}A_{t}}{\sqrt{-g_{zz}(z)g_{tt}(z)}}\left(-i\,\wp_{1}\;\Gamma^{\underline{z}\;\underline{t}}-i\,\wp_{2}\;|\phi|^{2}\;\Gamma^{\underline{z}\;\underline{t}}\right)\bigg{)}\,\xi(z,\textbf{k})=0$
(20)

Using the basis (13) and by writing the spinors $\xi=(\xi_{1},\xi_{2})^{T}$,
on further splitting as $\xi_{j}=(\beta_{j},\alpha_{j})^{T}$, where $j=1,2$,
we have the following radial equations

$\displaystyle\left(\frac{1}{\sqrt{g_{zz}}}\partial_{z}\pm
m_{\psi}\right)\left(\begin{matrix}\beta_{1}\cr\alpha_{1}\end{matrix}\right)\mp\frac{(\omega+qA_{t})}{\sqrt{-g_{tt}(z)}}\left(\begin{matrix}\alpha_{1}\cr\beta_{1}\end{matrix}\right)+\frac{k_{x}}{\sqrt{g_{xx}}}\left(\begin{matrix}\alpha_{1}\cr\beta_{1}\end{matrix}\right)$
$\displaystyle-\frac{k_{y}}{\sqrt{g_{xx}}}\left(\begin{matrix}\alpha_{2}\cr\beta_{2}\end{matrix}\right)+\frac{\partial_{z}A_{t}}{\sqrt{-g_{zz}g_{tt}}}\;(\wp_{1}+\wp_{2}\;|\phi|^{2})\left(\begin{matrix}\alpha_{1}\cr\beta_{1}\end{matrix}\right)=0$
(21) $\displaystyle\left(\frac{1}{\sqrt{g_{zz}}}\partial_{z}\pm
m_{\psi}\right)\left(\begin{matrix}\beta_{2}\cr\alpha_{2}\end{matrix}\right)\mp\frac{(\omega+qA_{t})}{\sqrt{-g_{tt}(z)}}\left(\begin{matrix}\alpha_{2}\cr\beta_{2}\end{matrix}\right)-\frac{k_{x}}{\sqrt{g_{xx}}}\left(\begin{matrix}\alpha_{2}\cr\beta_{2}\end{matrix}\right)$
$\displaystyle-\frac{k_{y}}{\sqrt{g_{xx}}}\left(\begin{matrix}\alpha_{1}\cr\beta_{1}\end{matrix}\right)+\frac{\partial_{z}A_{t}}{\sqrt{-g_{zz}g_{tt}}}\;(\wp_{1}+\wp_{2}\;|\phi|^{2})\left(\begin{matrix}\alpha_{2}\cr\beta_{2}\end{matrix}\right)=0$
(22)

From these equations, we can expand near the horizon and one can find that the
leading terms from the equations of motions are given by

$\displaystyle\partial_{z}\left(\begin{matrix}\beta_{j}(z,\vec{k})\cr\alpha_{j}(z,\vec{k})\end{matrix}\right)\mp\frac{\omega}{4\pi
T}\frac{1}{1-z}\left(\begin{matrix}\alpha_{j}(z,\vec{k})\cr\beta_{j}(z,\vec{k})\end{matrix}\right)=0.$
(23)

From this equation, we determine the in-falling boundary condition at the
horizon for extracting the retarded Green’s function at the AdS boundary, and
the independent ingoing boundary condition should be imposed at the horizon,
i.e.,

$\left(\begin{matrix}\beta_{j}(z,\vec{k})\cr\alpha_{j}(z,\vec{k})\end{matrix}\right)\sim
c_{j}(1-z)^{-\frac{i\omega}{4\pi T}}.$ (24)

Furthermore, the asymptotic behaviour of the Dirac equations near the AdS
boundary $(z\rightarrow 0)$ has the following form

$\left(\begin{matrix}\beta_{j}\cr\alpha_{j}\end{matrix}\right){\approx}\;A_{j}z^{m_{\psi}}\left(\begin{matrix}1\cr
0\end{matrix}\right)+D_{j}z^{-m_{\psi}}\left(\begin{matrix}0\cr
1\end{matrix}\right).$ (25)

We can numerically read off the coefficients $A$ and $D$ to obtain the Green’s
functions. For the couplings considered here, we have a mixing of various
spinorial components. We followed the usual prescription used in [12, 13, 15]
to extract the Green’s function by using two sets of linearly independent
boundary conditions that are given by

$\displaystyle\left(\begin{array}[]{cc}\beta^{I}_{1}&\beta^{II}_{1}\\\
\beta^{I}_{2}&\beta^{II}_{2}\end{array}\right)=\left(\begin{array}[]{cc}s_{11}&s_{12}\\\
s_{21}&s_{22}\end{array}\right)\left(\begin{array}[]{cc}\alpha^{I}_{1}&\alpha^{II}_{1}\\\
\alpha^{I}_{2}&\alpha^{II}_{2}\end{array}\right),$ (32)

Now the retarded Green’s function is defined as

$\displaystyle
G^{R}(\omega,k_{x},k_{y})=-i\left(\begin{array}[]{cc}s_{11}&s_{12}\\\
s_{21}&s_{22}\end{array}\right)\cdot\gamma^{t}$ (35)

with gamma matrices in our basis as $\gamma^{t}=i\sigma_{1}$. The observable
quantity of interest, is the spectral function $\rho\,(\omega,k_{x},k_{y})$
which is given by

$\displaystyle\rho\,(\omega\,,\vec{k})=\text{Im}\left[\text{Tr}\,G^{R}(\omega\,,\vec{k})\right].$
(36)

where $\vec{k}\,\equiv\,(k_{x},k_{y})$ and $G^{R}(\omega\,,\vec{k})$ is the
retarded Green’s function. The properties of $\rho\,(\omega\,,\vec{k})$ is
what we will study in our subsequent analysis.

##  4 Results and Discussion

As already emphasized thorough out the earlier sections, our goal would be to
explore the behaviour of the Fermi surface considering various coupling
$\wp_{1}\,,\wp_{2}$ in a systematic manner.

We wish to discuss the properties of the Fermi surface close to zero
temperature. Before we explore different possibilities, as a consistency
check, we reproduce the result in Figure 3 obtained in [51, 53] assuming all
the coupling parameters set to zero.

Figure 2: Spectral function $\rho\,(k_{x},k_{y})$ with all $\wp$’s set to
zero. Panel $\bf{(a)-(d)}$ is for
$\frac{\chi^{(1)}}{\mu^{\alpha_{-}}}=0.2,\,1.0,\,2.0,\,3.0$ respectively. Here
$m_{\psi}=0\;,q=1$, background parameters $T/\mu=0.009$, $k_{1}/\mu=0.8.$ and
$k_{2}/\mu=0.$ Figure 3: Density plot of the spectral function
$\rho\,(k_{x},k_{y})$ with all couplings set to zero for the parameters
$m_{\psi}=0$ and $q=1$. In panel (a) to (d) the background parameter
$\chi^{(1)}=2.0,\,k_{1}/\mu=0.8,\,k_{2}/\mu=0,\,m_{\phi}^{2}=0$ and
$T/\mu=0.09,0.04,0.02,\text{and}\,0.009$ respectively. Clearly as we lowered
the temperature the Fermi surface becomes very sharp.  Figure 4: 2D slice
along $k_{y}=0$ (left) and $k_{x}=0$ (right) of Figure 3 to compare the peaks
for $T/\mu=0.02$ (blue) and $0.009$ (orange) respectively with other parameter
values same as in Figure 3.  Figure 5: Panel $\bf{(a)}$: The fermionic
spectral density in the $k_{x}-k_{y}$ space for $\wp_{1}=1$ near $\omega=0$.
Coupling $\wp_{2}$ set to zero. Panel $\bf{(b)\&(c)}$: The energy-momentum
distribution along $k_{x}$ and $k_{y}$ respectively. Here, the fermion mass
$m_{\psi}=0$ and charge $q=1$ with background parameters
$\chi^{(1)}=2.0,\,k_{1}/\mu=0.8,\;k_{2}/\mu=0,\,m_{\phi}^{2}=0$ and
$T/\mu=0.02\,.$ The thin white line is at $\omega=0$ which is the Fermi level.

###  4.1 Symmetry broken along $k_{1}$ only

Firstly, let us set our background parameter $k_{2}=0$ and consider only
$k_{1}$. The influence of the background scalar on the spectral function can
be seen from Figure 2, $\chi^{(1)}$ represents the source of the dual boundary
operator of the bulk scalar field. As we increase the strength of the boundary
source term, the amplitude of the fermionic spectral function reduces along
$k_{x}$. Also in the limit of $\chi^{(1)}\to 0$, the spectrum reduces to the
one observed in case of RN-AdS blackhole with a circular Fermi surface and no
gap (see Figure 2$\;{(a)}$). This suppression can be attributed to reducing a
quasi-particle lifetime near the $k_{x}^{F}\simeq\pm 2$ point. From Figure 3
($(b)-(d)$), at low temperature one notices that, without any coupling terms,
the spectral function gives rise to a stretched Fermi surface with its
amplitude suppression along $k_{x}-$direction $(k_{y}=0)$, arising due to the
broken translational symmetry along $x-$direction established by the
background scalar field. Whereas, along $k_{y}$ direction, the spectral
function indicates the existence of the Fermi surface with a large density of
states. Further, as one lowers the temperature, the peak of the Greens
function increases, which essentially enhances the quasi-particle lifetime. On
the other hand, turning on the pure dipole coupling $\wp_{1}=1$, the Fermi
surface is completely destructed along $k_{y}=0$ line giving rise to dipolar
Fermi arcs Figure 5(a). This destruction can be characterized by the opening
of the energy gap that can be clearly observed from panel $(b)$ of Figure 5.
With this consistency check of present calculations with the previous works,
we shall now examine the evolution of the spectral function or, in other
words, the nature of the Fermi surface due to change of different controlling
parameters such as coupling $(\wp_{1},\wp_{2})$, temperature $(T/\mu)$, and
source strength $\chi^{(1)}$ of the scalar field at the boundary.

Controlling both the dipole couplings $(\wp_{1},\wp_{2})$: No special features
appeared in the spectral function upon increasing the value of $\wp_{1}$, with
$\wp_{2}=0$, except the known two Fermi arcs becoming more flattened (see
[51]) and prominent. Similar behaviour is observed for the non-zero but
positive value of $\wp_{2}$, which we chose not to discuss any further here.
However, interesting behaviour of the spectral function emerges if the dipole
coupling $\wp_{2}$ assumes a negative value, which is further controlled by
the anisotropic scalar field profile in the background. The background
anisotropic scalar field plays a crucial role in controlling the shape and the
amplitude of the spectral function. As discussed, we will consider a complex
scalar field in the bulk that will have a non-normalizable source at the
boundary, which can be used as a tunable parameter. For this, we choose the
mass of the scalar field to be $m_{\phi}=0$, later on, we will comment on non-
zero masses which are outside the $\mbox{AdS}_{2}$ BF bound. Nevertheless, as
we independently increase the magnitude of the dipole coupling parameter
$\wp_{2}$, a pair of Fermi surfaces emerges as shown in Figure 6. The
appearance of multiple Fermi surfaces in holographic model was initially
reported in [52] for spherically symmetric background. Also recently in [55,
56], more complicated background geometry were introduced while breaking
translational invariance and Fermi arcs like features were observed triggered
by the effective mass term and the lattice effects. The emergence of this pair
of Fermi surfaces that are anisotropic is the interesting feature of our
results as they have been observed in the ARPES experiments in different real
condensed matter systems such as Topological insulators (TI), recently
discovered Dirac and Weyl semi-metals [25, 26]. For superconductivity, it is
the pseudogap region in the phase diagram where the Fermi arc appears with a
controlled doping concentration in a 2- dimensional layer of the
superconducting sample, and it is connected to the suppression in the Spectral
function. For the case of Dirac and Weyl semi-metals, Fermi arcs appear as
surface states which converge at the two distinct Dirac points in the three-
dimensional bulk. We also showed the presence of these gappedless surface
states in the energy-band dispersion in Figure 7 with only coupling $\wp_{2}$
and in Figure 10 with the two couplings parameters $\wp_{1}$ and $\wp_{2}$.
Further, from Figure 10, one can notice the existence of flat band near the
Fermi surface. Interestingly such a highly flat band has been observed in the
real condensed matter system, such as in the bilayer graphene [62]. The double
Fermi arcs for our holographic system can be thought of as holographic surface
states of a bulk material located in ($2+1$) dimensional boundary of the AdS
space [61].

Further observation in Figure 6 $(e)$ at $T/\mu=0.009$,though in the cuprates,
the presence of a secondary Fermi surfaces are not yet known but here it shows
that when these two surfaces crossed each other to form a $d-$wave like
structure Fermi surface. These phenomena have an interesting resemblance with
the usual high-temperature superconductor as discussed in a recent solvable
toy model using conventional method [63]. However, the shrinking of the Fermi
surface in the radial momentum direction can be tied with the reduction of the
available density of states for the system. This shrinking of the Fermi
surface does happen in the pseudogap region for the high-temperature
superconducting system as one decreases the temperature in the appropriate
range of doping concentration.

The same behaviour of the Fermi surface can be observed by keeping fixed
$\wp_{2}=-0.2$, and controlling source at the boundary of the scalar field
(see Figure 8). Therefore, the source identified by the non-normalizable part
of the bulk scalar field $\chi^{(1)}$ can be thought of as a doping parameter
in a certain boundary field theory. In the Figure 7, we plotted the evolution
of energy band diagram in both $k_{x}$ and $k_{y}$ direction. It clearly
captures the opening and closing of the energy gap depending on $\wp_{2}$
values specifically along $k_{y}$ direction.

Figure 6: Spectral density $\rho\,(k_{x},k_{y})$ for $T/\mu=0.001\,$(top) and
$T/\mu=0.009\,$(bottom) with $\chi^{(1)}/\mu^{\alpha_{-}}=2.0$ and
$k_{1}/\mu=0.8$. $\wp_{1}$ is set to zero and $\wp_{2}$ is non-zero and
$m_{\psi}=0\;,q=1$. Both the top and bottom panel from left to right
$\wp_{2}\,=\,[-0.1,-0.2,-0.3]$. Figure 7: Spectral density
$\rho\,(\omega,\vec{k})$ for $T/\mu=0.001$ with
$\chi^{(1)}/\mu^{\alpha_{-}}=2.0$, $k_{1}/\mu=0.8$ and $k_{2}/\mu=0$. Here,
parameter $\wp_{1}=0$ and $\wp_{2}$ is non-zero and $m_{\psi}=0\;,q=1$. In the
top and bottom panel from left to right $\wp_{2}\,=\,[-0.1,-0.2,-0.3]$. Figure
8: Plot of $\rho\,(k_{x},k_{y})$ with $\wp_{2}=-0.2$ and $\wp_{1}=0$. Panel
$(a)-(d)$ is for $\frac{\chi^{(1)}}{\mu^{\alpha_{-}}}=1.0,\,1.5,\,2.0,\,2.5$
respectively. Here, $m_{\psi}=0\;,q=1$, $k_{1}/\mu=0.8$, $k_{2}/\mu=0$, and
$T/\mu=0.009$. Figure 9: Spectral function $\rho\,(k_{x},k_{y})$ with
$\wp_{1}$, $\wp_{2}$ set to non-zero and $m_{\psi}=0\;,q=1$, $T/\mu=0.009$ ,
$\frac{\chi^{(1)}}{\mu^{\alpha_{-}}}=\,2.0$ , $m_{\phi}^{2}=0$ ,
$k_{1}/\mu=0.8$ and $k_{2}/\mu=0.0$. Panel (a)-(c) are for [
$(\wp_{1}\,,\wp_{2})\,]\,=\,[\,(0.2,-0.2),(0.2,-0.3),(0.2,-0.4)]$ and panel
$(d)-(f)$ are for [
$(\wp_{1}\,,\wp_{2})\,]\,=\,[\,(0.5,-0.3),(0.5,-0.4),(0.5,-0.5)]$
respectively. Figure 10: Energy-momentum dispersion with $\wp_{1}$, $\wp_{2}$
set to non-zero and $m_{\psi}=0,\;q=1$, $T/\mu=0.009$, $m_{\phi}^{2}=0$,
$k_{1}/\mu=0.8$ and $k_{2}/\mu=0$. Top and bottom panel are along $\omega-
k_{x}$ and $\omega-k_{y}$ respectively. From left to right the parameters [
$(\wp_{1}\,,\wp_{2})\,]\,=\,[\,(0.5,-0.2),(0.5,-0.3),(0.5,-0.4)]$ .

So far, we have discussed about controlling the dipole coupling independently.
Interesting behaviour of the Fermi surface also emerges (see Figure 9), once
we turn on both the coupling parameters. We assume two sample values of
$\wp_{1}=0.2,\,0.5$ , and vary $\wp_{2}$ across the range $(-0.2,-0.4)$
keeping the temperature fixed at $T/\mu=0.009$, and background translational
symmetry breaking parameter $k_{1}/\mu=0.8$ same. What is observed is that for
a given $\wp_{1}$, at a lower value of $\wp_{2}$, the outer dipolar Fermi arcs
is sharp with an additional closed Fermi surface inside the arc. Interestingly
with the increasing $\wp_{2}$, the outer Fermi arc is gradually disappearing
along with the prominent appearance of the inner closed Fermi surface through
the transfer of spectral weight from the outer to the inner one. Such transfer
of spectral weight has been observed in the real condensed matter system [64]
such as La1-xSrxMnO3. ARPES results on those materials shows the changes in
the electronic structure across the metal-insulator transition, which is
attributed to the gradual disappearance of the Fermi surface near the Mott-
Insulator transition. The reason has been given due to transfer of spectral
weight from the coherent band near the Fermi level to the lower Hubbard band.
Such transfer of spectral weight can also be seen in our holographic energy
band diagram shown in Figure 10. Furthermore, at lower $\wp_{2}$, $\omega=0$
Fermi surface seems to be smeared over a large momentum range. This
observation has striking similarities with the phenomena of Fermi surface
smearing observed experimentally in disordered condensed matter system [65].
The role of bulk dipole parameters can be thought of as the strength of the
disorderness in the boundary. As we increase $\wp_{2}$, the available smeared
FS states specifically near $k=0$ move toward the higher momentum state giving
rise to specific Fermi momentum. The emergence of such multiple Fermi surfaces
can also be understood from a different perspective that the boundary strongly
coupled system contains different species of emergent fermionic degrees of
freedom corresponding to their intrinsic properties, which must be coupled
with the dual operators associated with bulk dipole couplings.

###  4.2 Symmetry broken in both $k_{1}$ and $k_{2}$ spatial direction

So far we have look into the scenario where translational symmetry is broken
only in the $x-$direction. Now let us check what happens when symmetry is
broken in both $x$ and $y$ spatial directions. As expected, when
$k_{2}/k_{1}>1$, the gap along $k_{y}$ now opens up as shown in the first
panel of Figure 11. Similar interesting features were observed as shown in
Figure 11 and Figure 12. On zooming in on the inner Fermi surface Figure
11(e-f), the spectral weight is suppressed in the opposite direction in
comparison to that of the outer surface for smaller $\wp_{2}$ values. However,
for $k_{1}=k_{2}$ the Fermi surface is circular in shape as in the case of RN-
AdS blackhole background.

Figure 11: Spectral density with background parameters $T/\mu=0.009$,
$k_{1}/\mu=0.2$, $k_{2}/\mu=0.8$, $m_{\phi}^{2}=0$ and, $\chi^{(1)}=2$. Here,
the fermion mass $m_{\psi}=0\;,\text{and charge}\;q=1$. Left to right we vary
$\wp_{2}=0,-0.3,-0.5,\;\text{and}\;-1.2$ while fixing $\wp_{1}=0$. Right most
panel showing a zoom in version of the inner Fermi surface. Figure 12:
Spectral density with background parameters $T/\mu=0.009$, $k_{1}/\mu=0.8$,
$k_{2}/\mu=0.2$, $m_{\phi}^{2}=0$ and, $\chi^{(1)}=2$. Here, the fermion mass
$m_{\psi}=0,\;\text{ and charge}\;q=1$. In the panel $(a)-(d)$ we vary
$\wp_{2}=-0.3,-0.5,-0.7,-1.2$ and fixing $\wp_{1}=0.5$. Panel $(e)$: Zoom of
the inner FS in $(d)$ and $(f)$ is a zoom version for
$(\wp_{1},\wp_{2})=(0.5,-1.2)\;\text{and}\,(0.5,-1.4)$ respectively.

Furthermore, one can explore the interplay between the relative signs of
$\wp_{1}$ and $\wp_{2}$ such that the two interaction terms can cancel each
other either at the boundary or near the horizon. For instance, if we consider
a solution with the background parameters $\chi^{(1)}=2$, $T/\mu=0.009$,
$k_{1}/\mu=0.8$, $k_{2}/\mu=0.2$ and $m_{\phi}^{2}=0$, we can have the
effective coupling $(\wp_{1}+\wp_{2}|\phi|^{2})$ to be zero, negative or
positive for different $\wp_{1}$ and $\wp_{2}$ values. We have explored such a
scenario in Figure 13 where the plots show the cases when
$(\wp_{1}+\wp_{2}|\phi|^{2})\lessgtr 0$ going through the point
$(\wp_{1}+\wp_{2}|\phi|^{2})=0$. For fixed $\wp_{2}=-0.1$ and varying
$\wp_{1}$ such that the effective coupling becomes large positive value of we
saw a gap spectrum similar to Figure 5(a) and for large negative values, we
observed the presence of secondary FS with the outer surface showing almost of
negligible density.

Figure 13: Spectral density with background parameters $T/\mu=0.009$,
$k_{1}/\mu=0.8$, $k_{2}/\mu=0.2$, $m_{\phi}^{2}=0$ and, $\chi^{(1)}=2$. Here,
the fermion mass $m_{\psi}=0,\;\text{ and charge}\;q=1$. Top panel is for
combination of $\wp_{1}$ and $\wp_{2}$ in the IR while bottom panel is for the
UV. From left to right, the effective coupling is negative, zero and positive.

##  5 Effect of mass on the fermionic spectral function

For completeness, we also checked the case where fermion mass is non zero with
$m_{\psi}=1/4$ within the mass window $m_{\psi}<1/2$, where we have the
freedom to choose either the coefficient $A_{j}$ or $D_{j}$ as the sources for
fermionic operators. We did not find any significant differences and
interesting effects on the spectral function other than what we observed for
massless fermions. Nonetheless we present some of the results in Figure 14.

On the other hand, when the scalar mass is within the BF bound $-9/4\leq
m_{\phi}^{2}<0$, for mass $m_{\phi}^{2}$ close to $-9/4$, for example,
$m_{\phi}^{2}=-2$, the contribution from our coupling to the boundary
fermionic spectral function was highly suppressed, however, when $m_{\phi}$
approaches zero, the effect is much more significant as we have seen from the
results for $m_{\phi}^{2}=0$. We checked for $m_{\phi}^{2}=-11/16$, which
correspond to $\alpha_{-}=1/4$ and $\alpha_{+}=11/4$, and we obtained similar
results.

Figure 14: Spectral density with background parameters $T/\mu=0.009$,
$k_{1}/\mu=0.8$, $k_{2}/\mu=0$, $m_{\phi}^{2}=0$ and, $\chi^{(1)}=2$. Here,
the fermion mass $m_{\psi}=1/4\;,\text{and charge}\;q=1$. In the top panel we
vary $\wp_{2}=-0.1,-0.2,-0.3$ (left to right) and fixing $\wp_{1}=0.5$. In the
bottom $\wp_{1}=0$ and $\wp_{2}=-0.1,-0.2,-0.3$ from left to right.

##  6 Conclusion

We have considered the anisotropic lattice background, namely the Q-lattice,
and explored the fermionic spectral function in detail by considering two
specific fermion-gauge field couplings. In addition to the well-studied dipole
coupling, we have also introduced the coupling containing the background
scalar field $|\phi|^{2}$, which gives the interesting features.

In this paper, we adopt a similar generalisation considered in our previous
constructions by considering both types of coupling into the fermionic
spectral function. We further introduce the scalar field interaction with the
fermion and introduce the discrete spatial translation symmetry in two
dimensions on the boundary field theory. Apart from producing the anisotropic
background, the non-normalizable mode of the scalar field plays the role of
controlling the dipole coupling parameter in the bulk. We observed some
interesting features on the spectral function due to
$(\wp_{1}+\wp_{2}|\phi|^{2})$ coupling. For the first case, the background is
anisotropic without any Lorentz violation, and the spectral function passes
through some interesting phases (see Fig.10), where phenomena like spectral
transfer, Fermi surface smearing happen when simultaneously turning on both
the couplings $(\wp_{1},\wp_{2})$.

Acknowledgements We are very much grateful to our colleague Dr U. N. Maiti,
for illuminating discussions on the experimental aspects of our present work.

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# Generative error correction for code-switching speech recognition using
large language models

###### Abstract

Code-switching (CS) speech refers to the phenomenon of mixing two or more
languages within the same sentence. Despite the recent advances in automatic
speech recognition (ASR), CS-ASR is still a challenging task ought to the
grammatical structure complexity of the phenomenon and the data scarcity of
specific training corpus. In this work, we propose to leverage large language
models (LLMs) and lists of hypotheses generated by an ASR to address the CS
problem. Specifically, we first employ multiple well-trained ASR models for
N-best hypotheses generation, with the aim of increasing the diverse and
informative elements in the set of hypotheses. Next, we utilize the LLMs to
learn the hypotheses-to-transcription (H2T) mapping by adding a trainable low-
rank adapter. Such a generative error correction (GER) method directly
predicts the accurate transcription according to its expert linguistic
knowledge and N-best hypotheses, resulting in a paradigm shift from the
traditional language model rescoring or error correction techniques.
Experimental evidence demonstrates that GER significantly enhances CS-ASR
accuracy, in terms of reduced mixed error rate (MER). Furthermore, LLMs show
remarkable data efficiency for H2T learning, providing a potential solution to
the data scarcity problem of CS-ASR in low-resource languages.

Index Terms—  Code-switching speech recognition, large language model,
generative error correction.

Fig. 1: The main structure of the GER solution for Mandarin-English CS-ASR.
$M_{i}$ and $E_{i}$ denote Mandarin and English tokens respectively. Correct
tokens are highlighted in green, while wrong tokens appear in red.

##  1 Introduction

The code-switching (CS) concept refers to the practice of alternating between
more than one languages within a conversation or even within a single spoken
utterance [1]. This linguistic phenomenon is common in multilingual
communities, and individuals who are fluent in multiple languages [2]. CS
poses a great challenge to automatic speech recognition (ASR) systems, which
are typically trained on specific language resource [3, 4, 5, 6, 7]. When CS
occurs, ASR systems need to accurately transcribe the spoken item with
multiple alternatives, since words from different languages may sound similar
or have overlapping pronunciations [8]. Considering the grammatical structure
complexity, existing solutions focus on the technique of incorporating
linguistic knowledge in the ASR engine [9, 10, 11], also known as internal LM
estimation (ILME). More recently, [12] has shown that an external LM can lead
to a performance gain through rescoring the hypotheses generated by beam
search. However, LM’s potential for the CS task is still under-exploited. In
fact, (i) rescoring-based methods select only one hypothesis as their output,
which inevitably overlooks correct elements from other discarded hypotheses
[13], and (ii) a less powerful language model may struggle to handle complex
ambiguities and homophony when trained with limited in-domain data.

In this paper, we introduce a generative error correction (GER) approach that
utilizes large LMs (LLMs) to directly predict the correct transcription for
the CS-ASR task [14]. Different from traditional rescoring or correction
techniques, GER can effectively exploit token-level information in the N-best
hypotheses generated by an ASR system. Furthermore, LLMs have acquired vast
knowledge of cross-lingual combination during the pre-training stage, so that
the need for ad-hoc CS training data to build a robust ASR system is
effectively reduced. Specifically, GER consists of two key steps:

  * •

Multiple existing ASR models are used to transcribe spoken utterances into an
informative hypotheses list, thereby capturing more elements that potentially
contribute to a more accurate transcription.

  * •

A pre-trained foundation LLM with low-rank adapter[15] is employed to
efficiently learn to predict transcription based on the hypotheses list, with
limited in-domain hypotheses-transcription pairs.

We conduct experiments on the English-Mandarin ASRU dataset[16]. The
experimental results show that GER can significantly reduce the mixed error
rate (MER) for each independent ASR system, and further enhance the
performance after integrating their hypotheses lists. Furthermore, GER shows
remarkable data efficiency when learning hypotheses-to-transcription (H2T)
mapping. It requires less than 10% in-domain data to build a better CS-ASR
system than a Conformer-based baseline system.

##  2 Generative error correction

In the following, we first define GER in Section 2.1, and we then illustrate
the generation of N-best hypotheses list in Section 2.2. Finally, we introduce
the low-rank adaptation (LoRA) method to learn the hypotheses-to-transcription
(H2T) mapping.

###  2.1 Problem Setting

As shown in Fig. 1, given input speech $X$ with CS, the $i$-th ASR model,
$ASR_{i}$, transcribes $X$ to $n_{i}$ hypotheses list
$\\{Y_{1}^{i},Y_{2}^{i},\cdots,Y^{i}_{n_{i}}\\}$, where each $Y$ can be viewed
as an independent textual representation of $X$. After integrating candidates
from different $ASR_{i}$, we can obtain an N-best list $\mathcal{Y}_{N}^{I}$
consisting of $N=\sum_{i=1}^{I}n_{i}$ hypotheses transcribed from $I$ ASR
models. In GER, the goal is to learn a hypotheses-to-transcription mapping
$\mathcal{M}_{H2T}$ that predicts the transcription $Y$ based on
$\mathcal{Y}_{N}^{I}$ list, which can be expressed as:

$y=\mathcal{M}_{H2T}(\mathcal{Y}_{N}^{I})$ (1)

Given ground-truth transcription $Y^{*}$, we can optimize the large language
model to learn $\mathcal{M}_{H2T}$ in an auto-regressive manner. The cross-
entropy loss $\mathcal{L}_{H2T}$ can be denoted as:

$\mathcal{L}_{H2T}=\sum_{t=1}^{T}-\log\mathcal{P}_{\theta}(y_{t}^{*}|y_{t-1}^{*},\cdots,y_{1}^{*},\mathcal{Y}_{N}^{I})$
(2)

where $y_{t}$ is the $t$-th token of ground-truth transcription, and $\theta$
denotes the tunable module of LLM (i.e., LoRA in Fig. 2).

###  2.2 Hypotheses Generation

Mandarin-English is used as an example to illustrate the generation of an
informative hypotheses list. To transcribe speech into textual candidates, we
build a CS-ASR model ${ASR_{1}}$ from scratch and utilize two other existing
models, i.e., ${ASR_{2}}$ (CS-ASR) and ${ASR_{3}}$ (monolingual). The
underlying reason is that when applying ASR tasks in practical scenarios, we
would simultaneously consider training a specific model with in-domain data,
as well as seeking existing ASR models that can directly perform inference. It
is noted that the hypothesis quality of ${ASR_{1}}$ relies on the in-domain
data amount, while ${ASR_{2}}$ and ${ASR_{3}}$ are stable as they are well-
trained. Beam search is employed in the decoding stage by all ASR models used
to generate diverse hypotheses. The details for $ASR_{i}$ are as follows:

$\boldsymbol{ASR_{1}}$-Conformer (48.3M) is a mainstream end-to-end ASR model
with a remarkable performance [17]. It is typically viewed as the primary
solution to build a domain-specific ASR system. In practice, a Conformer
consists of a 12-layer encoder and 6-layer decoder, and we utilize in-domain
CS data to train it from scratch leveraging the ESPNet toolkit
111https://github.com/espnet/espnet/blob/master/egs2.

$\boldsymbol{ASR_{2}}$-Whisper-Large (1.5B) is a web-scale speech model that
is pre-trained on 680,000 hours of multilingual and multitask supervised data
[18]. We view it as a general ASR model since it has achieved remarkable ASR
performance in various speech domains. To perform zero-shot CS speech
recognition, we utilize the prompting technique in [19] to inspire Whisper as
a hypotheses generator.

$\boldsymbol{ASR_{3}}$-Transformer (31.0M) is a monolingual ASR model that is
pre-trained on Mandarin corpus AISHELL-1 [20]. This model consists of a
12-layer Transformer encoder and a 6-layer Transformer decoder that
incorporates the CTC probabilities into decoding [21]. We select such a
monolingual ASR model to provide sufficiently potential Chinese tokens, as the
target dataset is biased towards Mandarin. More importantly, the H2T learning
based on ${ASR_{3}}$-Transformer is equivalent to _transliteration_ [22],
where LLMs attempt to predict CS transcription according to Mandarin-only
hypotheses list, e.g., “他在等奥佛” $\rightarrow$“他在等offer”.

We then combine the hypotheses generated from $ASR_{1}\sim ASR_{3}$ to obtain
a new N-best list that includes more useful elements for predicting the
correct transcription. To validate our claim, we analyze the information on
each $ASR_{i}$ as well as the ensemble hypotheses in our experiments, and we
also conduct H2T learning on both single and ensemble N-best lists to evaluate
the performance gain by LLMs.

Fig. 2: Low-rank adaptation. Only $\mathcal{A}$ and $\mathcal{B}$ are
trainable.

###  2.3 H2T Learning with LoRA

To learn the H2T mapping, we integrate a trainable low-rank decomposition
matrix [15] into LLMs’ existing Transformer layers (as shown in Fig. 2),
enabling the model to adapt to new data while keeping the original LLMs fixed
to retain the previous knowledge. Specifically, LoRA performs a
reparameterization of each model layer expressed as a matrix multiplication by
injecting low-rank decomposition matrices. As a result, the representations
generated by the LLM are not distorted due to task-specific (H2T) tuning,
while the adapter module acquires the capability to predict the true
transcription from the N-best hypotheses [14].

We use an instruction-following prompt template to formulate the N-best list
as LLMs input, which is specified as: _task description_ \+ _the best
hypothesis_ \+ _other hypotheses_. Then LLMs perform auto-regressive decoding
according to the input and history sequence, with the cross-entropy training
criteria defined in Eq.(2).

##  3 Experiments

###  3.1 Dataset

We conduct our experiments on the ASRU dataset [16], which is a conversational
Mandarin-English CS corpus collected by smartphones in quiet rooms. The
transcripts of data cover many common fields, including entertainment, travel,
daily life, and social interaction. The speakers are from 30 provinces in
China, and 70 % of them were under 30 years old, with no significant
difference in the number of males and females. Each utterance contains 8.6
Chinese characters and 1.6 English words on average.

The original ASRU dataset consists of 200 hours of CS training set
$Train_{CS}$ and 20 hours of evaluation set $Dev_{CS}$. There is no domain
shift between $Train_{CS}$ and $Dev_{CS}$, so they are both defined as _in-
domain data_. In this paper, to show the efficiency of the proposed GER
method, we re-split the $Dev_{CS}$ to simulate the low-resource setting, where
15k utterances (19 hours) are randomly selected as low-resource training set
and other unseen 1152 samples (1.5 hours) are viewed as a new test set to
evacuate the performance.

###  3.2 LLM-LoRA Configuration

We use Chinese-Llama2-7b from HuggingFace
222https://huggingface.co/ziqingyang/chinese-llama-2-7b as the foundation
model to learn H2T mapping, which has been expanded and optimized with Chinese
vocabulary beyond the original Llama-2 333https://ai.meta.com/llama/.
Consequently, the subsequent training on Chinese corpus improves the
fundamental semantic understanding of both Chinese and English, which is
suitable for CS H2T learning. For LoRA training, $r$ is set to 4 and added in
each Q, K, and V layers in the Transformer blocks. Notably, the trainable
parameters are only 4.19M, accounting for only 0.06% of the Llama parameters.
The learning rate is set to $2\times e^{-4}$ to learn H2T mapping for 10
epochs, and the batch size is set to 128.

###  3.3 Evaluation Indicators

We utilize the mix error rate (MER) as the metric to evaluate the performance
of each ASR system. To gain more insights into our results, we also define two
oracles (i) $o_{nb}$, and (ii) $o_{cp}$ for a given N-best list. $o_{nb}$
denotes the WER of the best hypothesis in the N-best list. We measure $o_{nb}$
by calculating MER of all hypotheses one by one. $o_{cp}$ denotes the
theoretical minimum WER that recomposes all tokens in the N-best list. We
measure $\boldsymbol{o}_{cp}$ by counting the rate of missing tokens in N-best
list. Additionally, we use “1-_Best_ ” to denote the WER of the best
hypotheses in the list.

ID | ASR Model | Data Dur. (h) | Language | 1-_Best_ | GER | _Oracle_
---|---|---|---|---|---|---
In-D. | Out-D. | $o_{nb}$ | $o_{cp}$
1 | $ASR_{1}$-Conformer | 19 | - | M&E | 37.9 | - | 33.3 | 29.9
2 | 200 | - | 12.9 | $10.6_{{\color[rgb]{0,.5,.5}-17.8\%}}$ | 8.9 | 7.6
3 |  $ASR_{2}$-Whisper | 19 | 680k | Multi. | 11.0 | $8.6_{{\color[rgb]{0,.5,.5}-20.9\%}}$ | 9.5 | 7.8
4 |  $ASR_{3}$-Transformer | 19 | 150 | M | 41.1 | ${30.8}_{{\color[rgb]{0,.5,.5}-25.1\%}}$ | 38.1 | 22.8
5 | Ensemble | 19 | - | M&E | 11.0 | $\textbf{8.3}_{{\color[rgb]{0,.5,.5}-24.5\%}}$ | 8.6 | 6.5
Table 1: Main MER (%) results on ASRU test set. “In-D.” and “Out-D.”
respectively denote in-domain and out-domain training data, and “M” and “E”
stands for Mandarin and English. “1-_Best_ ”, “$o_{nb}$” and “$o_{cp}$” are
all defined in Section 3.3.

##  4 Results and Analysis

###  4.1 Main Results of GER

Table 1 shows the main MER results. For each $ASR_{i}$, we select the top-5
hypotheses to compose the N-best list for both oracle analysis and H2T
learning. Considering the MER result of each model, the ensemble N-best list
includes 8 hypotheses from ID 1,3,4 with a proportion of 1:5:1. Additionally,
it is worth noting that only ID-2 use full training set $Train_{CS}$, all
other systems except utilize 19 hours of in-domain training data for the H2T
learning, which accounts for less than 10% of other typical methods trained on
ASRU.

From Table 1, we observe that: 1) The ASR model with ID-1 shows that 19 hours
of training data is not enough to train the $ASR_{1}$ from scratch, therefore,
it cannot provide high-quality hypotheses for GER learning. 2) With 200 hours
of training data (ID-2), the Conformer baseline achieves 12.9 MER performance
while the GER can further reduce the relative MER by 17.8%. 3) Whisper can
achieve better performance than Conformer due to its large-scale pre-trained
data. Nonetheless, GER reaps significant performance gain (11.0% $\rightarrow$
8.6) using 19 hours of in-domain data. 4) $ASR_{3}$ achieves poor MER
performance as it is a monolingual ASR model, and GER can perform
transliteration with reducing MER by 25.1%, which predicts Mandarin-English
transcription according to Mandarin-only hypotheses. 5) When integrating the
hypotheses into an ensemble list, $o_{np}$ and $o_{cp}$ both drop, which
indicates a higher upper bound for GER. Accordingly, the GER obtains better
results than the single model that surpasses the 1-_best_ baseline by 24.5%
MER performance. In general, GER can enhance MER for both stand-alone and
ensemble ASR models. Furthermore, GER with Whisper and 19 hours of training
data allows us to build a better CS-ASR system than a Conformer baseline
trained on 200 hours.

1-_Best_ | Data Amount | GER | Oracle
---|---|---|---
#Pairs | Dur. | $o_{nb}$ | $o_{cp}$
11.0 | 1k | 1.3 h | $10.1_{{\color[rgb]{0,.5,.5}-8.2\%}}$ | 9.4 | 7.0
5k | 6.3 h | $9.3_{{\color[rgb]{0,.5,.5}-15.5\%}}$
10k | 12.6 h | $8.9_{{\color[rgb]{0,.5,.5}-19.1\%}}$
15k | 19.0 h | $8.6_{{\color[rgb]{0,.5,.5}-20.9\%}}$
Table 2: MER (%) results of GER method with different amounts of in-domain
training data. “#Pairs” denotes the number of hypotheses-transcription pairs,
and “Dur.” represents the corresponding speech duration.

###  4.2 Data Efficiency

We now examine the data efficiency of GER by gradually reducing the amount of
in-domain training data. As shown in the second column of Table 2, we generate
hypotheses-transcription pairs using $ASR_{2}$ and $ASR_{3}$ with a proportion
of 5:1. We do not use $ASR_{1}$ since it cannot provide reasonable hypotheses
with such a small amount of training data.

From Table 2, we observe that the relative MER can be reduced by 8.2% by using
only 1K in-domain pairs (1.3 hours) for H2T learning. This outcome
demonstrates that a multi-lingual LLM can significantly reduce the data
requirement of in-domain data for CS-ASR training. We believe it provides a
potential solution for those languages with limited CS training corpus.

###  4.3 Case Study

We conduct the case study in Table 3 to visualize the error correction
process. All three ASR models provide defective hypotheses with different
errors. Nevertheless, as all correct tokens are included in N-best list, GER
with LLM accurately predicts the true transcription according to semantics and
grammar of Mandarin and English: 1) “persistent data” is the most likely
object to be invented in this context, and 2) the subject of the invention
should be “他” rather than “它” according to Chinese grammar.

System | Utterance | MER
---|---|---
$ASR_{1}$ | persistent date这个东西当然不是他发明的 | 7.1
$ASR_{2}$ |  porsistent data这个东西当然不是它发明的 | 14.3
$ASR_{3}$ |  颇虽私人的队的这个东西当然不是他发明的 | 50.0
GER |  persistent data这个东西当然不是他发明的 | 0
GT | persistent data这个东西当然不是他发明的 | -

Table 3: Case study of GER. The utterance is drawn from the ASRU test set,
where error tokens are in red.

##  5 Conclusion

In this paper, we present a new paradigm called generative error correction
(GER) for CS-ASR. This GER approach first employs well-trained ASR models to
generate diverse hypotheses, and then learns an H2T mapping using LLMs with
LoRA adapter. The experiments show that GER can effectively enhance the CS-ASR
in terms of MER. Meanwhile, it shows remarkable data efficiency that provides
a potential solution to the data scarcity problem of CS-ASR in low-resource
language.

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