# Notes on steady state current through discrete-level quantum systems

Longwen Zhou<EMAIL_ADDRESS>
###### Abstract

In these notes, we take a naive approach to calculate electrical current
through a noninteracting quantum system with discrete energy levels. We do not
assume any prior knowledge on second quantization, scattering matrix and
nonequilibrium Green’s functions (NEGF). Instead, we will try to build our
solutions to the problem step by step from single-particle Schrodinger
equation and equilibrium Green’s functions. In the first section, we give the
definitions of retarded Green’s function, spectral function and density of
states for time-independent quantum systems. In the second section, we
introduce the left-center-right (LCR) system, which may be viewed as a minimum
model for the study of quantum transport. In the third section, we work out
the Green’s function and spectral function of the central part in the LCR
system. Important concepts like self-energy and level-width function will also
be introduced. In the fourth section, we will derive the Landauer formula for
steady state current using a scattering approach. The transmission coefficient
at a given energy will be derived. In the fifth section, we will apply
Landauer formula to the case in which the left and right leads are both semi-
infinite tight-binding chains, coupling to the central system only at its
boundaries. In this case, further simplifications can be made under wide-band
limit. In the sixth section, we take the zero temperature, zero bias limit to
obtain the linear conductance from Landauer formula. In the seventh section,
we apply the linear conductance formula under wide-band limit to two typical
models of the central system: a single-level quantum dot and a double-level
quantum dot. In both cases, we will calculate the spectral function and
electrical conductance analytically. The results illustrate how quantum
resonance and coherence can affect electrical transport in noninteracting
systems. In the eighth section, we use the Landauer formula to compute linear
conductance of a tight-binding chain. A simple example of metal-insulator
transition is discussed using the Aubry-André-Harper model. In the ninth
section, Landauer formula is applied to study edge state transport along the
boundary of a two-dimensional lattice. Using the Hofstadter model as an
example, we show the quantization of linear conductance in the spectral gap,
which indicates the topological nontrivial properties of the system. In the
last section, we will give a summary and discuss possible future extensions.

The first seven sections of these notes follow closely Chapter 3 of Ref.[1].
Other materials guiding the preparation of these notes are Chapter 13 of
Ref.[2], Chapter 3 of Ref.[3], the monograph [4] and the review paper [5].

###### Contents

  1. 1 Retarded Green’s function, spectral function and density of states
  2. 2 LCR system
  3. 3 Green’s function and spectral function of an LCR system
  4. 4 Landauer formula and transmission coefficient
  5. 5 Semi-infinite tight-binding leads and wide-band limit
  6. 6 Linear conductance at zero temperature
  7. 7 Steady state current in a quantum dot: resonance and coherence
    1. 7.1 Single-level quantum dot: spectral function and transmission coefficient
    2. 7.2 Double-level quantum dot: spectral function and transmission coefficient
  8. 8 Current across a tight-binding chain: metal-insulator transitions
  9. 9 Current along the edge of a two-dimensional lattice: topologically quantized transport
  10. 10 Summary and plan for future extensions

##  1 Retarded Green’s function, spectral function and density of states

The retarded Green’s function (or propagator) of a time-independent quantum
system described by Hamiltonian $H$ is defined as

$G(t,t^{\prime})=-\frac{i}{\hbar}\theta(t-t^{\prime})e^{-\frac{i}{\hbar}H(t-t^{\prime})},$
(1)

where $\hbar$ is the Planck constant, $t$ and $t^{\prime}$ are time variables.
The step function $\theta(t-t^{\prime})$ is defined as

$\theta(t-t^{\prime})=\begin{cases}1&t\geq t^{\prime}\\\
0&t<t^{\prime}\end{cases}.$ (2)

The retarded Green’s function is the solution of the equation of motion:

$\left(i\hbar\frac{\partial}{\partial
t}-H\right)G(t,t^{\prime})=\delta(t-t^{\prime})$ (3)

with boundary condition

$i\hbar\lim_{\eta\rightarrow 0^{+}}G(t^{\prime}+\eta,t^{\prime})=1,$ (4)

The Fourier transform of $G(t,t^{\prime})$ from time to energy domain is given
by

$G(E)=\int_{-\infty}^{\infty}dEe^{\frac{i}{\hbar}(E+i0^{+})(t-t^{\prime})}G(t,t^{\prime}),$
(5)

where $0^{+}$ represents an infinitesimal positive number. The Fourier
expansion of $\delta(t-t^{\prime})$ reads

$\delta(t-t^{\prime})=\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}e^{-\frac{i}{\hbar}E(t-t^{\prime})}dE.$
(6)

Combining Eqs. (3), (5) and (6), we obtain the retarded Green’s function in
energy domain as

$G(E)=\left(E+i0^{+}-H\right)^{-1}.$ (7)

Note that $G(E)$ here has the dimension of $E^{-1}$. Using $G(E)$, we can
introduce the spectral operator as

${\bf A}(E)=i\left[G(E)-G^{\dagger}(E)\right]=2\pi\delta(E-H),$ (8)

where we have used the Plemelj formula $\frac{1}{x\pm i0^{+}}={\cal
P}\frac{1}{x}\mp i\pi\delta(x)$, with ${\cal P}$ standing for the Cauchy
principle value. The spectral function $A$ and density of states $\varrho$ at
a given energy $E$ are obtained from the spectral operator as:

$\displaystyle A(E)$ $\displaystyle={\rm Tr}[{\bf A}(E)],$ (9)
$\displaystyle\varrho(E)$ $\displaystyle=\frac{1}{2\pi}A(E).$ (10)

Note that the trace is taken in the Hilbert space of Hamiltonian $H$. From now
on we will work in energy domain only.

##  2 LCR system

An LCR system may be regarded as the minimum model for the study of quantum
transport. It is usually adopted in the description of electrical current
through a noninteracting quantum dot or molecule junction. In matrix form, the
Hamiltonian of an LCR system can be expressed as

$H=\begin{pmatrix}H_{LL}&H_{LC}&0\\\ H_{CL}&H_{CC}&H_{CR}\\\
0&H_{RC}&H_{RR}\end{pmatrix},$ (11)

where $H_{LL}$ and $H_{RR}$ are Hamiltonians of left $(L)$ and right $(R)$
leads, respectively. $H_{CC}$ is the Hamiltonian of the central $(C)$ region,
whose transport property is of our interest. $H_{LC}=H_{CL}^{\dagger}$
describes the coupling between the central region and the left lead, and
$H_{RC}=H_{CR}^{\dagger}$ describes the coupling between the central region
and the right lead. There is no direct coupling between left and right leads.
The eigenvalue equation of Hamiltonian $H$ is given by

$\begin{pmatrix}H_{LL}&H_{LC}&0\\\ H_{CL}&H_{CC}&H_{CR}\\\
0&H_{RC}&H_{RR}\end{pmatrix}\begin{pmatrix}\Psi_{L}\\\ \Psi_{C}\\\
\Psi_{R}\end{pmatrix}=E\begin{pmatrix}\Psi_{L}\\\ \Psi_{C}\\\
\Psi_{R}\end{pmatrix}$ (12)

where the wave function components $\Psi_{L},\Psi_{C}$ and $\Psi_{R}$ are
written in a basis well-localized in each of the three regions $L,C$ and $R$.

##  3 Green’s function and spectral function of an LCR system

The retarded Green’s function of an LCR system can also be written in matrix
form as

$G=\begin{pmatrix}G_{LL}&G_{LC}&G_{LR}\\\ G_{CL}&G_{CC}&G_{CR}\\\
G_{RL}&G_{RC}&G_{RR}\end{pmatrix}.$ (13)

It satisfies Eq. (7) with $H$ given by Eq. (11):

$\begin{pmatrix}E-H_{LL}\pm i0^{+}&-H_{LC}&0\\\ -H_{CL}&E-H_{CC}\pm
i0^{+}&-H_{CR}\\\ 0&-H_{RC}&E-H_{RR}\pm
i0^{+}\end{pmatrix}\begin{pmatrix}G_{LL}&G_{LC}&G_{LR}\\\
G_{CL}&G_{CC}&G_{CR}\\\ G_{RL}&G_{RC}&G_{RR}\end{pmatrix}=1.$ (14)

The Green’s function of the central region $G_{CC}$ is then determined by the
following set of equations:

$\displaystyle G_{LC}=$ $\displaystyle G_{LL}^{0}H_{LC}G_{CC}$ (15)
$\displaystyle(E-H_{CC}\pm i0^{+})G_{CC}=$ $\displaystyle
H_{CL}G_{LC}+H_{CR}G_{RC}$ (16) $\displaystyle G_{RC}=$ $\displaystyle
G_{RR}^{0}H_{RC}G_{CC},$ (17)

where we have introduced Green’s functions of isolated left and right leads as

$G_{\alpha\alpha}^{0}=(E-H_{\alpha\alpha}\pm i0^{+})^{-1}\qquad\alpha=L,R.$
(18)

Solving these equations gives us

$G_{CC}=\frac{1}{E-H_{CC}-\Sigma_{L}-\Sigma_{R}+i0^{+}},$ (19)

where the retarded self-energy functions $\Sigma_{L,R}$ of the two leads are:

$\Sigma_{\alpha}\equiv H_{C\alpha}G_{\alpha\alpha}^{0}H_{\alpha
C}\qquad\alpha=L,R.$ (20)

The self-energy functions incorporate all effects of the lead on the central
system. One may combine $H_{CC}$ and the self-energies to obtain an effective
Hamiltonian for the “dressed” central region:

$H_{CC}^{{\rm eff}}\equiv H_{CC}+\Sigma_{L}+\Sigma_{R}.$ (21)

However, this Hamiltonian is non-Hermitian and its spectrum is in general not
real. Therefore it does not describe a closed quantum system. The open system
nature of $H_{CC}^{{\rm eff}}$ reflects the fact that all degrees of freedom
of the lead have been integrated out in order to obtain the self-energies. For
$H_{CC}^{{\rm eff}}$, the real part of its spectrum reflects the energy level
shift caused by coupling to the leads, and the imaginary part of its spectrum
determines the lifetime of the “dressed” energy level. As can be inspected
from Eq. (20), the lifetime of a “dressed” energy level is in general
proportional to the inverse square of the coupling strength between the lead
and the central region. To characterize the level-broadening caused by system-
lead coupling, we introduce the level width function as ($\alpha=L,R$):

$\Gamma_{\alpha}\equiv
i\left(\Sigma_{\alpha}-\Sigma_{\alpha}^{\dagger}\right)=H_{C\alpha}i\left(G_{\alpha\alpha}^{0}-G_{\alpha\alpha}^{0\dagger}\right)H_{\alpha
C}=2\pi H_{C\alpha}\delta(E-H_{\alpha\alpha})H_{\alpha C}.$ (22)

We see that the $\Gamma_{\alpha}$ is proportional to the density of states of
lead $\alpha$ and the square of the coupling strength between the lead
$\alpha$ and the central system.

Finally, the spectral function and density of states of the central region are
defined following Eq. (8) to (10) as:

$\displaystyle{\bf A}_{C}(E)$
$\displaystyle=i\left[G_{CC}(E)-G_{CC}^{\dagger}(E)\right],$ (23)
$\displaystyle A_{C}(E)$ $\displaystyle={\rm Tr}[{\bf A}_{C}(E)],$ (24)
$\displaystyle\varrho_{C}(E)$ $\displaystyle=\frac{1}{2\pi}A_{C}(E).$ (25)

With all these preparations, we will derive Landauer formula for steady state
transport using a scattering approach in the next section.

##  4 Landauer formula and transmission coefficient

We will take a scattering point of view to describe the charge transport in
our system. In all the derivations below, we assume there is no many-body
interactions. We start with the stationary Schrodinger equation of the LCR
system:

$\begin{pmatrix}H_{LL}&H_{LC}&0\\\ H_{CL}&H_{CC}&H_{CR}\\\
0&H_{RC}&H_{RR}\end{pmatrix}\begin{pmatrix}\Psi_{L}\\\ \Psi_{C}\\\
\Psi_{R}\end{pmatrix}=E\begin{pmatrix}\Psi_{L}\\\ \Psi_{C}\\\
\Psi_{R}\end{pmatrix}.$ (26)

Consider an incoming wave $\Psi_{L}^{0}$ from the left the lead to the central
region, which is an eigenstate of $H_{LL}$. This wave may be partially
transmitted into the central region, and partially reflected back to the left
lead, yielding a reflection wave $\Psi_{L}^{1}$. Therefore for such an
incoming wave, we can write the wave function in the left lead as

$\Psi_{L}=\Psi_{L}^{0}+\Psi_{L}^{1}.$ (27)

Plugging Eq. (27) into Eq. (26) gives us

$\displaystyle(E-H_{LL})(\Psi_{L}^{0}+\Psi_{L}^{1})$
$\displaystyle=H_{LC}\Psi_{C},$ (28) $\displaystyle(E-H_{CC})\Psi_{C}$
$\displaystyle=H_{CL}(\Psi_{L}^{0}+\Psi_{L}^{1})+H_{CR}\Psi_{R},$ (29)
$\displaystyle(E-H_{RR})\Psi_{R}$ $\displaystyle=H_{RC}\Psi_{C}.$ (30)

Using the eigenvalue equation $(E-H_{LL})\Psi_{L}^{0}=0$, definitions of lead
Green’s functions Eq. (18) and self-energies Eq. (20), we obtain the following
expressions for different wave function components in the left lead, central
region and right lead:

$\displaystyle\Psi_{L}$
$\displaystyle=(1+G_{LL}^{0}H_{LC}G_{CC}H_{CL})\Psi_{L}^{0},$ (31)
$\displaystyle\Psi_{C}$ $\displaystyle=G_{CC}H_{CL}\Psi_{L}^{0},$ (32)
$\displaystyle\Psi_{R}$
$\displaystyle=G_{RR}^{0}H_{RC}G_{CC}H_{CL}\Psi_{L}^{0}.$ (33)

When a steady state is established, the probability of the state in the
central region $|\Psi_{C}|^{2}$ should not change with time. This is the case
if the current approaching the central region equals the current leaving it.
The definition of steady state current can then be extracted from probability
conservation law as follows [6]:

$\displaystyle 0=\frac{d\Psi_{C}^{\dagger}\Psi_{C}}{dt}=$
$\displaystyle\Psi_{C}^{\dagger}\frac{d\Psi_{C}}{dt}+\frac{d\Psi_{C}^{\dagger}}{dt}\Psi_{C}$
$\displaystyle=$
$\displaystyle\Psi_{C}^{\dagger}\frac{1}{i\hbar}(H_{CL}\Psi_{L}+H_{CC}\Psi_{C}+H_{CR}\Psi_{R})$
$\displaystyle-$
$\displaystyle\frac{1}{i\hbar}(\Psi_{L}^{\dagger}H_{LC}+\Psi_{C}^{\dagger}H_{CC}+\Psi_{R}^{\dagger}H_{RC})\Psi_{C}$
$\displaystyle=$
$\displaystyle\frac{1}{i\hbar}(\Psi_{C}^{\dagger}H_{CL}\Psi_{L}-\Psi_{L}^{\dagger}H_{LC}\Psi_{C})$
(34) $\displaystyle+$
$\displaystyle\frac{1}{i\hbar}(\Psi_{C}^{\dagger}H_{CR}\Psi_{R}-\Psi_{R}^{\dagger}H_{RC}\Psi_{C}).$
(35)

We can now interpret the (local) probability current from lead $\alpha$ to the
central region as

$J_{\alpha}\equiv\frac{1}{i\hbar}(\Psi_{C}^{\dagger}H_{C\alpha}\Psi_{\alpha}-\Psi_{\alpha}^{\dagger}H_{\alpha
C}\Psi_{C})\qquad\alpha=L,R.$ (36)

The conservation law is then given by

$\sum_{\alpha=L,R}J_{\alpha}=0\Leftrightarrow J_{L}=-J_{R}.$ (37)

Using Eq. (36) and Eqs. (31) to (33), we can compute the contribution of the
incoming wave $\Psi_{L}^{0}$ to the current from the left lead to the central
region as:

$\displaystyle J_{L}=$
$\displaystyle\frac{i}{\hbar}(\Psi_{C}^{\dagger}H_{CR}\Psi_{R}-\Psi_{R}^{\dagger}H_{RC}\Psi_{C})$
$\displaystyle=$
$\displaystyle\frac{i}{\hbar}\Psi_{L}^{0\dagger}H_{LC}G_{CC}^{\dagger}(H_{CR}G_{RR}^{0}H_{RC}-H_{CR}G_{RR}^{0\dagger}H_{RC})G_{CC}H_{CL}\Psi_{L}^{0}$
$\displaystyle=$
$\displaystyle\frac{i}{\hbar}\Psi_{L}^{0\dagger}H_{LC}G_{CC}^{\dagger}(\Sigma_{R}-\Sigma_{R}^{\dagger})G_{CC}H_{CL}\Psi_{L}^{0}$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar}\Psi_{L}^{0\dagger}H_{LC}G_{CC}^{\dagger}\Gamma_{R}G_{CC}H_{CL}\Psi_{L}^{0},$
(38)

where we have used our definitions of self-energy and level width function
given by Eq. (20) and Eq. (22) in the last section. To proceed, we assume that
all possible incoming states are independent and originated from the same
Fermi function of the left lead $f_{L}(\epsilon_{kL})$, where $\epsilon_{kL}$
is the energy of incoming state $\Psi_{kL}^{0}$ with quantum number $k$. Also
we assume that there is no scattering among different incoming channels. Under
these conditions, the total probability current from the left lead to the
central region is given by

$\displaystyle I_{L}=$
$\displaystyle\sum_{k}\frac{1}{\hbar}\Psi_{kL}^{0\dagger}H_{LC}G_{CC}^{\dagger}\Gamma_{R}G_{CC}H_{CL}\Psi_{kL}^{0}f_{L}(\epsilon_{kL})$
(39) $\displaystyle=$
$\displaystyle\frac{1}{\hbar}\sum_{k}\sum_{q}\Psi_{kL}^{0\dagger}H_{LC}\Psi_{qC}^{0}\Psi_{qC}^{0\dagger}G_{CC}^{\dagger}\Gamma_{R}G_{CC}H_{CL}\Psi_{kL}^{0}f_{L}(\epsilon_{kL})$
$\displaystyle=$
$\displaystyle\frac{1}{h}\sum_{q}\Psi_{qC}^{0\dagger}G_{CC}^{\dagger}\Gamma_{R}G_{CC}H_{CL}\left[2\pi\sum_{k}\Psi_{kL}^{0}\Psi_{kL}^{0\dagger}f_{L}(\epsilon_{kL})\right]H_{LC}\Psi_{qC}^{0}$
$\displaystyle=$
$\displaystyle\frac{1}{h}\int_{-\infty}^{\infty}dE\sum_{q}\Psi_{qC}^{0\dagger}G_{CC}^{\dagger}\Gamma_{R}G_{CC}H_{CL}\left[2\pi\sum_{k}\delta(E-\epsilon_{kL})\Psi_{kL}^{0}\Psi_{kL}^{0\dagger}\right]H_{LC}\Psi_{qC}^{0}f_{L}(E),$

where we have inserted the completeness relation
$\sum_{q}\Psi_{qC}^{0}\Psi_{qC}^{0\dagger}=1_{C}$, with $q$ being the quantum
number of the central region. Recalling Eq. (22) for the level-width function,
we notice that

$H_{CL}\left[2\pi\sum_{k}\delta(E-\epsilon_{kL})\Psi_{kL}^{0}\Psi_{kL}^{0\dagger}\right]H_{LC}=\Gamma_{L}(E).$
(40)

Combining this observation with Eq. (39) yields

$\displaystyle I_{L}=$
$\displaystyle\frac{1}{h}\int_{-\infty}^{\infty}dE\sum_{q}\Psi_{qC}^{0\dagger}G_{CC}^{\dagger}\Gamma_{R}G_{CC}^{r}\Gamma_{L}\Psi_{qC}^{0}f_{L}(E)$
$\displaystyle=$ $\displaystyle\frac{1}{h}\int_{-\infty}^{\infty}dE{\rm
Tr}\left(G_{CC}^{\dagger}\Gamma_{R}G_{CC}\Gamma_{L}\right)f_{L}(E),$ (41)

where the trace is over localized basis of the central region. Finally, taking
into account the contributions from both left and right leads, the total
steady state probability current $I=I_{L}+I_{R}$ flowing into the central
region is given by:

$I=\frac{1}{h}\int_{-\infty}^{\infty}dET(E)\left[f_{L}(E)-f_{R}(E)\right],$
(42)

where the transmission coefficient $T(E)$ is defined as

$T(E)\equiv{\rm
Tr}\left[G_{CC}^{\dagger}(E)\Gamma_{R}(E)G_{CC}(E)\Gamma_{L}(E)\right].$ (43)

Eq. (42) is the Landauer formula for steady state transport in noninteracting
quantum systems. The current is determined by the transmission coefficient
times the difference of electron distributions in left and right leads at a
given energy $E$, and integrating over all possible energies of incoming
state. In this picture, we can roughly say that current is transmission. Other
popular approaches to the derivation of Landauer formula including the
scattering matrix formalism and NEGF. Interested readers can consult Refs.[2,
3] for further details.

In the next section, we will try to obtain a more explicit expression for
$T(E)$ by specifying the Hamiltonians of lead and coupling between leads and
central region.

##  5 Semi-infinite tight-binding leads and wide-band limit

Our derivations up to now are formal. In the following, we will specify the
Hamiltonians of lead and system-lead coupling in order to arrive at a more
explicit expression for the transmission coefficient. The Hamiltonians for the
lead and their coupling to the central region may be written quite generally
as (for $\alpha=L,R$):

$\displaystyle H_{\alpha\alpha}$
$\displaystyle=\sum_{m,n}t_{\alpha,mn}|m_{\alpha}\rangle\langle n_{\alpha}|,$
(44) $\displaystyle H_{C\alpha}$ $\displaystyle=\sum_{\mu,n}t_{C\alpha,\mu
n}|\mu\rangle\langle n_{\alpha}|=H_{\alpha C}^{\dagger}.$ (45)

where $\\{|m_{\alpha}\rangle\\}$ is a localized basis of lead $\alpha$.
$\\{|\mu\rangle\\}$ is a localized basis of the central region, which will be
assumed to have a finite dimension $N$.

Since in most cases, leads are just sources of electrons, we may simply choose
them to be semi-infinite tight-binding chains. Moreover, we allow each basis
$|\mu\rangle$ of the central region to be coupled separately to one tight-
binding chain, and require that all the tight-binding chains coupled to the
central region at the same side $\alpha$ are equivalent. The Hamiltonians for
lead may then be written as

$\displaystyle H_{LL}$
$\displaystyle=\sum_{\mu=0}^{N-1}\left[t_{L}\sum_{m=-\infty}^{-2}(|m,\mu\rangle\langle
m+1,\mu|+{\rm h.c.})+w_{L}\sum_{m=-\infty}^{-1}|m,\mu\rangle\langle
m,\mu|\right],$ (46) $\displaystyle H_{RR}$
$\displaystyle=\sum_{\mu=0}^{N-1}\left[t_{R}\sum_{m=1}^{\infty}(|m,\mu\rangle\langle
m+1,\mu|+{\rm h.c.})+w_{R}\sum_{m=1}^{\infty}|m,\mu\rangle\langle
m,\mu|\right],$ (47)

where $t_{L}\geq 0$ ($t_{R}\geq 0$) is the nearest neighbor hopping amplitude,
and $w_{L}$ ($w_{R}$) is the onsite potential of the left (right) tight-
binding chain. The lattice constant has been set to $1$, and each of the left
(right) chain extends from site $-\infty$ ($+1$) to site $-1$ ($+\infty$). The
tensor product basis $\\{|m,\mu\rangle\\}$ is complete, and satisfies the
orthonormal condition

$\langle m,\mu|n,\nu\rangle=\delta_{mn}\delta_{\mu\nu}.$ (48)

Next, we note that for each $\mu$, the semi-infinite tight-binding chain is a
tridiagonal Toeplitz matrices in our basis. Therefore we obtain the dispersion
relation $\epsilon_{k_{\alpha}\mu}$ and eigenfunctions
$\\{|k_{\alpha},\mu\rangle\\}$ of the $\mu$’s tight-binding chain as [7]:

$\displaystyle\epsilon_{k_{\alpha}\mu}$
$\displaystyle=2t_{\alpha}\cos(k_{\alpha})+w_{\alpha},$ (49)
$\displaystyle\langle m,\mu|k_{\alpha},\nu\rangle$
$\displaystyle=\sin(k_{\alpha}m)\delta_{\mu\nu},$ (50)

where $\alpha=L,R$ and $\mu=0,...,N-1$.

To evaluate the transmission coefficient $T(E)$, we need to calculate self-
energies and level width functions. Both of them require specific knowledge of
the couplings between leads and central system. Spatially, the central system
is located between sites $-1$ and $+1$. So we model the couplings between lead
and the central system as

$\displaystyle H_{CL}$ $\displaystyle=\sum_{\mu=0}^{N-1}t_{\mu
L}|\mu\rangle\langle-1,\mu|=H_{LC}^{\dagger},$ (51) $\displaystyle H_{CR}$
$\displaystyle=\sum_{\mu=0}^{N-1}t_{\mu
R}|\mu\rangle\langle+1,\mu|=H_{RC}^{\dagger},$ (52)

where $t_{\mu\alpha}$ is the coupling strength between the $\mu$’s basis of
the central region and lead $\alpha$. Note that this coupling is local in
space.

We are now ready to obtain a more explicit expression for transmission
coefficient $T(E)$ given by Eq. (43). From Eq. (22) and Eqs. (48) to (52), the
level-width function of lead $\alpha$ is given by

$\displaystyle\Gamma_{\alpha}=$ $\displaystyle 2\pi
H_{C\alpha}\delta(E-H_{\alpha\alpha})H_{\alpha C}$ $\displaystyle=$
$\displaystyle
2\pi\sum_{k_{\alpha},\mu}\delta(E-\epsilon_{k_{\alpha}\mu})H_{C\alpha}|k_{\alpha},\mu\rangle\langle
k_{\alpha},\mu|H_{\alpha C}$ $\displaystyle=$ $\displaystyle
2\pi\sum_{k_{\alpha},\mu}|t_{\mu\alpha}|^{2}\sin^{2}(k_{\alpha})\delta(E-\epsilon_{k_{\alpha}\mu})|\mu\rangle\langle\mu|.$
(53)

In the case that the lead $\alpha$ has a continuous spectrum, we can transform
the sum over quantum number $k_{\alpha}$ to an integral:

$\displaystyle\Gamma_{\alpha}=$
$\displaystyle\sum_{\mu}|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{-\pi}^{\pi}\sin^{2}(k_{\alpha})\delta(E-\epsilon_{k_{\alpha}\mu})dk_{\alpha}$
$\displaystyle=$
$\displaystyle\sum_{\mu}2|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{0}^{\pi}\sin^{2}(k_{\alpha})\delta(E-\epsilon_{k_{\alpha}\mu})dk_{\alpha}$
$\displaystyle=$
$\displaystyle\sum_{\mu}2|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{0}^{\pi}\sin^{2}(k_{\alpha})\delta(E-\epsilon_{k_{\alpha}\mu})d\arccos\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)$
$\displaystyle=$
$\displaystyle\sum_{\mu}\frac{|t_{\mu\alpha}|^{2}}{t_{\alpha}}|\mu\rangle\langle\mu|\int_{w_{\alpha}-2t_{\alpha}}^{w_{\alpha}+2t_{\alpha}}\frac{\sin^{2}\left[\arccos\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)\right]}{\sqrt{1-\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)^{2}}}\delta(E-\epsilon_{k_{\alpha}\mu})d\epsilon_{k_{\alpha}\mu}$
$\displaystyle=$
$\displaystyle\sum_{\mu}\frac{|t_{\mu\alpha}|^{2}}{t_{\alpha}}\frac{\sin^{2}\left[\arccos\left(\frac{E-w_{\alpha}}{2t_{\alpha}}\right)\right]}{\sqrt{1-\left(\frac{E-w_{\alpha}}{2t_{\alpha}}\right)^{2}}}|\mu\rangle\langle\mu|.$
(54)

We note that due to our choices of the center-lead coupling, $\Gamma_{\alpha}$
is diagonal in the basis of the central region $\\{|\mu\rangle\\}$. If the
band width $2t_{\alpha}$ of the lead $\alpha$ is much wider then
$E-w_{\alpha}$, we can take the so-called wide-band limit, in which we send
$\frac{E-w_{\alpha}}{2t_{\alpha}}\rightarrow 0$. In this limit, we have
$\sin^{2}\left[\arccos\left(\frac{E-w_{\alpha}}{2t_{\alpha}}\right)\right]\rightarrow
1$ and the level-width function $\Gamma_{\alpha}$ becomes independent of
energy $E$. We will denote the level width function in wide-band limit as

$\Gamma_{\alpha}^{w}=\sum_{\mu=0}^{N-1}\frac{|t_{\mu\alpha}|^{2}}{t_{\alpha}}|\mu\rangle\langle\mu|\qquad\alpha=L,R.$
(55)

In the same limit, the self-energies can also be worked out explicitly. From
Eq. (20) and Eqs. (48) to (52), we get

$\displaystyle\Sigma_{\alpha}=$
$\displaystyle\sum_{k_{\alpha},\mu}\frac{|t_{\mu\alpha}|^{2}\sin^{2}(k_{\alpha})}{E-\epsilon_{k_{\alpha}\mu}+i0^{+}}|\mu\rangle\langle\mu|$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{\mu}|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{-\pi}^{\pi}\frac{\sin^{2}(k_{\alpha})}{E-\epsilon_{k_{\alpha}\mu}+i0^{+}}dk_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\sum_{\mu}|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{0}^{\pi}\frac{\sin^{2}(k_{\alpha})}{E-\epsilon_{k_{\alpha}\mu}+i0^{+}}dk_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\sum_{\mu}|t_{\mu\alpha}|^{2}|\mu\rangle\langle\mu|\int_{0}^{\pi}\frac{\sin^{2}(k_{\alpha})}{E-\epsilon_{k_{\alpha}\mu}+i0^{+}}d\arccos\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)$
$\displaystyle=$
$\displaystyle\sum_{\mu}\frac{|t_{\mu\alpha}|^{2}}{2t_{\alpha}}|\mu\rangle\langle\mu|\frac{1}{\pi}\int_{w_{\alpha}-2t_{\alpha}}^{w_{\alpha}+2t_{\alpha}}\frac{\sin^{2}\left[\arccos\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)\right]}{\sqrt{1-\left(\frac{\epsilon_{k_{\alpha}\mu}-w_{\alpha}}{2t_{\alpha}}\right)^{2}}}\frac{1}{E-\epsilon_{k_{\alpha}\mu}+i0^{+}}d\epsilon_{k_{\alpha}\mu}$
$\displaystyle=$
$\displaystyle\frac{|t_{\mu\alpha}|^{2}}{2t_{\alpha}}\left[x+\sqrt{\frac{x+i0^{+}+1}{x+i0^{+}-1}}-(x+i0^{+})\sqrt{\frac{x+i0^{+}+1}{x+i0^{+}-1}}\right],$
(56)

where $x=\frac{E-w_{\alpha}}{2t_{\alpha}}$. In Fig. 1, we plot the real part
(red solid line) and imaginary part (blue dashed line) of
$2t_{\alpha}\Sigma_{\alpha}/|t_{\mu\alpha}|^{2}$ versus
$\frac{E-w_{\alpha}}{2t_{\alpha}}$. We observe that in the wide-band limit
$\frac{E-w_{\alpha}}{2t_{\alpha}}\rightarrow 0$, the imaginary part of
$2t_{\alpha}\Sigma_{\alpha}/|t_{\mu\alpha}|^{2}$ approaches one and its real
part vanishes. Therefore, we can safely ignore the level shift of the central
region caused by the real part of self-energy, retaining only the imaginary
part of self-energy. Therefore in the wide-band limit, the self-energy of lead
$\alpha$ ($=L,R$) becomes:

$\Sigma_{\alpha}^{w}=-\frac{i}{2}\sum_{\mu}\frac{|t_{\mu\alpha}|^{2}}{t_{\alpha}}|\mu\rangle\langle\mu|=-\frac{i}{2}\Gamma_{\alpha}^{w}.$
(57)

It is also energy independent and proportional to the level width function.

To summarize, for semi-infinite tight-binding leads coupled locally to the
central system, we find under wide-band limit the following expressions for
level width functions, Green’s functions, spectral functions and density of
states:

$\displaystyle\Gamma_{L}^{w}=$ $\displaystyle\sum_{\mu=0}^{N-1}\frac{|t_{\mu
L}|^{2}}{t_{L}}|\mu\rangle\langle\mu|,$ $\displaystyle\Gamma_{R}^{w}=$
$\displaystyle\sum_{\mu=0}^{N-1}\frac{|t_{\mu
R}|^{2}}{t_{R}}|\mu\rangle\langle\mu|,$ $\displaystyle G_{CC}^{w}(E)=$
$\displaystyle\frac{1}{E-H_{CC}+\frac{i}{2}\left(\Gamma_{L}^{w}+\Gamma_{R}^{w}\right)},$
$\displaystyle G_{CC}^{w\dagger}(E)=$
$\displaystyle\frac{1}{E-H_{CC}-\frac{i}{2}\left(\Gamma_{L}^{w}+\Gamma_{R}^{w}\right)},$
$\displaystyle A_{C}^{w}(E)=$ $\displaystyle 2\pi\varrho_{C}^{w}(E)=i{\rm
Tr}[G_{CC}^{w}(E)-G_{CC}^{w\dagger}(E)].$ (58)

These functions are enough for us to determine the transmission coefficient
$T(E)$ and current $I$ in wide-band limit, provided that the Hamiltonian of
the central region $H_{CC}$ is specified. Before doing that, we will introduce
a further simplification by taking the zero temperature, zero bias limit of
Eq. (42). In this limit, we obtain in the following section the linear
conductance of the central system as a response to perturbed electron
distributions of the lead caused by a small bias voltage.

##  6 Linear conductance at zero temperature

Let’s consider the case in which a small bias voltage $V$ is applied to the
left lead, making its chemical potential $\mu_{L}$ slightly differ from the
chemical potential of the right lead:

$-eV=\mu_{L}-\mu_{R},$ (59)

where $-e$ is the charge of electron. When $V$ is small enough, the difference
of Fermi functions of the two leads is given by

$f_{L}(E)-f_{R}(E)=f_{R}(E+eV)-f_{R}(E)\approx eV\frac{\partial
f_{R}(E)}{\partial E}.$ (60)

At zero temperature, only states below Fermi energy $E_{F}$ are filled. So the
Fermi function $f_{R}(E)=\theta(E_{F}-E)$ and its derivative $\frac{\partial
f_{R}(E)}{\partial E}=-\delta(E_{F}-E)$. So at zero temperature in low bias
regime, the current due to Landauer formula (42) is given by:

$I_{0}=-\frac{eV}{h}T(E_{F}),$ (61)

which is proportional to the bias voltage (thus a linear response) and the
transmission coefficient $T$ evaluated at the Fermi energy $E_{F}$. The
corresponding electrical conductance is then given by

${\cal C}_{0}=\frac{d(-eI_{0})}{dV}=\frac{e^{2}}{h}T(E_{F}),$ (62)

which is just the transmission coefficient multiplied by the conductance
quanta $\frac{e^{2}}{h}$. Prefect transmission $T(E_{F})=1$ at the Fermi
energy results in a quantized conductance $\frac{e^{2}}{h}$. Note here that
the spin degrees of freedom of incoming electrons have been ignored. For
spinful incoming electrons, the results for the current and conductance will
be twice of ${\cal C}_{0}$. Eq. (62) may be understood as a microscopic
version of Ohm’s law. The physical meaning of conductance ${\cal C}_{0}$ can
be interpreted as a linear response of the central system to a perturbation of
the lead’s chemical potential. Therefore it is a linear conductance. If the
perturbation $-eV$ makes $\mu_{L}$ slightly higher then $\mu_{R}$, a quantized
${\cal C}_{0}$ will count the number of available transport channels from left
to right lead. On the contrary, if the perturbation makes $\mu_{L}$ slightly
lower then $\mu_{R}$, a quantized ${\cal C}_{0}$ will be equal to the number
of available transport channels from right to left lead. If all transport
channels at $E_{F}$ are chiral and spatially separated, a quantized response
${\cal C}_{0}$ under a small bias voltage will only count the number of
transport channels with correct chirality _locally_. Therefore, in the Chern
insulator’s edge state transport calculation, the conductance ${\cal C}_{0}$
is equal to gap chirality instead of number of edge modes on the Fermi
surface.

The conductance ${\cal C}_{0}$ can also be derived directly from Kubo formula
of linear response theory. Interested readers can consult Chapter 2 of Ref.[3]
or Chapter 7 of Ref.[8].

In the following section, we will calculate the spectral function and linear
conductance of two explicit models in the wide-band limit. These model
calculations highlight the effects of resonance and coherence on electrical
transport in noninteracting quantum systems.

##  7 Steady state current in a quantum dot: resonance and coherence

In this section, we will apply the Landauer formula for linear conductance to
two explicitly examples: a single-level quantum dot and a double-level quantum
dot. In both examples, the spectral function and transmission coefficient of
the quantum dot can be obtained analytically in the wide-band limit. These
models serve as the starting point for study of coherent transport in discrete
level quantum systems. Some key concepts in the resonant transport can be
easily demonstrated in the following calculations.

###  7.1 Single-level quantum dot: spectral function and transmission
coefficient

The Hamiltonian of a single-level quantum dot is given by

$H_{CC}=\epsilon_{0}|0\rangle\langle 0|,$ (63)

where $\epsilon_{0}$ is the energy of level $|0\rangle$. A sketch of the model
is shown in Fig. 2. In the wide-band limit, we find from Eq. (58) the
following expressions for level-width functions and Green’s functions:

$\displaystyle\Gamma_{L}^{w}=$ $\displaystyle\gamma_{0L}|0\rangle\langle 0|,$
(64) $\displaystyle\Gamma_{R}^{w}=$ $\displaystyle\gamma_{0R}|0\rangle\langle
0|,$ (65) $\displaystyle G_{CC}^{w}(E)=$
$\displaystyle\frac{1}{E-\epsilon_{0}+\frac{i}{2}\gamma_{0}}|0\rangle\langle
0|,$ (66) $\displaystyle G_{CC}^{w\dagger}(E)=$
$\displaystyle\frac{1}{E-\epsilon_{0}-\frac{i}{2}\gamma_{0}}|0\rangle\langle
0|,$ (67)

where we have introduced simplified notations
$\gamma_{0\alpha}\equiv\frac{|t_{0\alpha}|^{2}}{t_{\alpha}}$ for $\alpha=L,R$
and $\gamma_{0}\equiv\gamma_{0L}+\gamma_{0R}$. The spectral function and
transmission coefficient are then given by

$\displaystyle A_{C}^{w}(E)=$ $\displaystyle i{\rm
Tr}[G_{CC}^{w}(E)-G_{CC}^{w\dagger}(E)]=\frac{\gamma_{0}}{(E-\epsilon_{0})^{2}+\frac{1}{4}\gamma_{0}^{2}}=2\pi\varrho_{C}^{w}(E),$
(68) $\displaystyle T^{w}(E)=$ $\displaystyle{\rm
Tr}\left[G_{CC}^{w\dagger}(E)\Gamma_{R}^{w}(E)G_{CC}^{w}(E)\Gamma_{L}^{w}(E)\right]=\frac{\gamma_{0L}\gamma_{0R}}{(E-\epsilon_{0})^{2}+\frac{1}{4}\gamma_{0}^{2}}.$
(69)

We see that both of them are of Lorentzian shape. In zero temperature, low
bias regime, the density of states and linear conductance are determined by
states at the Fermi energy:

$\displaystyle\varrho_{C}^{w}(E_{F})=$
$\displaystyle\frac{1}{2\pi}\frac{\gamma_{0}}{(E_{F}-\epsilon_{0})^{2}+\frac{1}{4}\gamma_{0}^{2}},$
(70) $\displaystyle{\cal C}_{0}=$
$\displaystyle\frac{e^{2}}{h}\frac{\gamma_{0L}\gamma_{0R}}{(E_{F}-\epsilon_{0})^{2}+\frac{1}{4}\gamma_{0}^{2}}=\frac{e^{2}}{\hbar}\frac{\gamma_{0L}\gamma_{0R}}{\gamma_{0L}+\gamma_{0R}}\varrho_{C}^{w}(E_{F}).$
(71)

The maximum of linear conductance is reached when the energy of an incoming
electron matches the energy of the quantum dot $\epsilon_{0}$. In this case,
we obtain the resonant transmission coefficient:

$T^{w}(E)=\frac{4\gamma_{0L}\gamma_{0R}}{(\gamma_{0L}+\gamma_{0R})^{2}},$ (72)

which is always equal to $1$ if the dot-lead coupling is symmetric
($\gamma_{0L}=\gamma_{0R}$), no matter how strong (or weak) of the coupling
is.

###  7.2 Double-level quantum dot: spectral function and transmission
coefficient

Let’s now apply the Landauer formula (42) to the electrical transport through
a two-level quantum dot. The Hamiltonian of the quantum dot is given by

$H_{CC}=\epsilon_{0}|0\rangle\langle 0|+\epsilon_{1}|1\rangle\langle
1|+t_{c}(|0\rangle\langle 1|+|1\rangle\langle
0|)=\begin{bmatrix}\epsilon_{0}&t_{c}\\\ t_{c}&\epsilon_{1}\end{bmatrix}.$
(73)

where $\epsilon_{0,1}$ are energies of decoupled levels $|0\rangle,|1\rangle$
and $t_{c}$ is the coupling strength between the two levels. In the wide-band
limit, we find from Eq. (58) the following expressions for level-width
function and Green’s function:

$\Gamma_{L}^{w}=\begin{bmatrix}\frac{|t_{0L}|^{2}}{t_{L}}&0\\\
0&\frac{|t_{1L}|^{2}}{t_{L}}\end{bmatrix}\qquad\Gamma_{R}^{w}=\begin{bmatrix}\frac{|t_{0R}|^{2}}{t_{R}}&0\\\
0&\frac{|t_{1R}|^{2}}{t_{R}}\end{bmatrix},$ (74) $\displaystyle
G_{CC}^{w}(E)=$
$\displaystyle\frac{1}{(E-\epsilon_{0}+i\gamma_{0})(E-\epsilon_{1}+i\gamma_{1})-t_{c}^{2}}\begin{bmatrix}E-\epsilon_{1}+i\gamma_{1}&t_{c}\\\
t_{c}&E-\epsilon_{0}+i\gamma_{0}\end{bmatrix},$ (75) $\displaystyle
G_{CC}^{w\dagger}(E)=$
$\displaystyle\frac{1}{(E-\epsilon_{0}-i\gamma_{0})(E-\epsilon_{1}-i\gamma_{1})-t_{c}^{2}}\begin{bmatrix}E-\epsilon_{1}-i\gamma_{1}&t_{c}\\\
t_{c}&E-\epsilon_{0}-i\gamma_{0}\end{bmatrix},$ (76)

where we have introduced the simplified notations $\gamma_{\mu}=\frac{|t_{\mu
L}|^{2}}{2t_{L}}+\frac{|t_{\mu R}|^{2}}{2t_{R}}$ for $\mu=0,1$. The spectral
function and transmission coefficient are then given by

$A_{C}^{w}(E)=-2{\rm Im}{\rm Tr}[G_{CC}^{w}(E)]=-2{\rm
Im}\left[\frac{(E-\epsilon_{0}+i\gamma_{0})+(E-\epsilon_{1}+i\gamma_{1})}{(E-\epsilon_{0}+i\gamma_{0})(E-\epsilon_{1}+i\gamma_{1})-t_{c}^{2}}\right],$
(77) $\displaystyle T^{w}(E)=$
$\displaystyle\frac{1}{|(E-\epsilon_{0}+i\gamma_{0})(E-\epsilon_{1}+i\gamma_{1})-t_{c}^{2}|^{2}}$
$\displaystyle\times$ $\displaystyle{\rm
Tr}\left(\begin{bmatrix}E-\epsilon_{1}-i\gamma_{1}&t_{c}\\\
t_{c}&E-\epsilon_{0}-i\gamma_{0}\end{bmatrix}\begin{bmatrix}\frac{|t_{0R}|^{2}}{t_{R}}&0\\\
0&\frac{|t_{1R}|^{2}}{t_{R}}\end{bmatrix}\begin{bmatrix}E-\epsilon_{1}+i\gamma_{1}&t_{c}\\\
t_{c}&E-\epsilon_{0}+i\gamma_{0}\end{bmatrix}\begin{bmatrix}\frac{|t_{0L}|^{2}}{t_{L}}&0\\\
0&\frac{|t_{1L}|^{2}}{t_{L}}\end{bmatrix}\right).$ (78)

To simplify these expressions, we focus on a minimum extension of single-level
quantum dot, i.e., only one level of the two-level quantum dot is coupled to
the lead. Our LCR system then forms a “T-junction” as shown in Fig. 3. In this
case, we may set $t_{1L}=t_{1R}=0$, which also results in $\gamma_{1}=0$.
Furthermore, we consider a symmetric dot-lead coupling by setting
$\frac{|t_{0L}|^{2}}{t_{L}}=\frac{|t_{0R}|^{2}}{t_{R}}=\gamma_{0}$. With these
simplifications, we obtain the following expressions for the spectral function
$A_{C}^{w}(E)$ and transmission coefficient $T^{w}(E)$ of the quantum dot:

$\displaystyle A_{C}^{w}(E)=$ $\displaystyle
2\gamma_{0}\frac{1+\frac{t_{c}^{2}}{(E-\epsilon_{1})^{2}}}{\left(E-\epsilon_{0}-\frac{t_{c}^{2}}{E-\epsilon_{1}}\right)^{2}+\gamma_{0}^{2}},$
(79) $\displaystyle T^{w}(E)=$
$\displaystyle\frac{\gamma_{0}^{2}}{\left(E-\epsilon_{0}-\frac{t_{c}^{2}}{E-\epsilon_{1}}\right)^{2}+\gamma_{0}^{2}}.$
(80)

We see that none of these functions are of Lorentzian shape. Moreover, the
transmission $T^{w}(E)=0$ if $E=\epsilon_{1}$. This is an anti-resonance
effect. Roughly speaking, an incoming electron has two possible paths to go
through the “T-junction” quantum dot. We denote them by our localized basis as
(i) $|-1,0\rangle\rightarrow|0\rangle\rightarrow|1,0\rangle$ and (ii)
$|-1,0\rangle\rightarrow|0\rangle\rightarrow|1\rangle\rightarrow|0\rangle\rightarrow|1,0\rangle$.
Along path (i), the incident electron will reach level $\epsilon_{0}$ and then
directly go through the quantum dot. Along path (ii), it will first hop from
level $\epsilon_{0}$ to $\epsilon_{1}$ through the coupling $t_{c}$, and then
hop back to the level $\epsilon_{0}$. If the energy $E$ of the incoming
electron matches $\epsilon_{1}$, waves separately following the two paths in
the quantum dot will have a $\pi$ phase difference when they meet with each
other at the level $\epsilon_{0}$, resulting in a destructive interference.
Therefore, under the condition $E=\epsilon_{1}$, we have a coherence-induced
destruction of tunneling in this simple example. In Fig. 4, the spectral and
transmission functions for three different values of the coupling strength
$t_{c}$ between two levels of the quantum dot are shown. In all cases, we
observe clear collapses of the transmission coefficient at $E=\epsilon_{1}$.

In low bias regime at zero temperature, the density of states and linear
conductance of this model are also determined by states at the Fermi energy:

$\displaystyle\varrho_{C}^{w}(E_{F})=$
$\displaystyle\frac{\gamma_{0}}{\pi}\frac{1+\frac{t_{c}^{2}}{(E_{F}-\epsilon_{1})^{2}}}{\left(E_{F}-\epsilon_{0}-\frac{t_{c}^{2}}{E_{F}-\epsilon_{1}}\right)^{2}+\gamma_{0}^{2}},$
(81) $\displaystyle{\cal C}_{0}=$
$\displaystyle\frac{e^{2}}{h}\frac{\gamma_{0}^{2}}{\left(E_{F}-\epsilon_{0}-\frac{t_{c}^{2}}{E_{F}-\epsilon_{1}}\right)^{2}+\gamma_{0}^{2}}=\frac{e^{2}}{\hbar}\frac{1}{2}\frac{\gamma_{0}}{1+\frac{t_{c}^{2}}{(E_{F}-\epsilon_{1})^{2}}}\varrho_{C}^{w}(E_{F}).$
(82)

With respect to $E_{F}$, both of them have two peaks in quite different
shapes, originating from nontrivial effects of quantum coherence in the
system.

##  8 Current across a tight-binding chain: metal-insulator transitions

The formulas we summarized in the collection of Eq. (58) are also appropriate
for the study of linear transport in a tight-binding chain. In this case, we
make the following choices for level-width function (in wide-band limit):

$\Gamma_{L}^{w}=\frac{|t_{CL}|^{2}}{t_{L}}|0\rangle\langle
0|,\qquad\Gamma_{R}^{w}=\frac{|t_{CR}|^{2}}{t_{R}}|N-1\rangle\langle N-1|,$
(83)

that is, the left (right) lead is only coupled to the first (last) site of the
central system. The Hamiltonian $H_{CC}$ describing the system of our interest
will be chosen to have the following form:

$H_{CC}=\sum_{\mu=0}^{N-1}\epsilon_{\mu}|\mu\rangle\langle\mu|+t_{C}\sum_{\mu=0}^{N-2}\left(|\mu\rangle\langle\mu+1|+{\rm
h.c.}\right),$ (84)

which represents a typical tight-binding chain with onsite potential
$\\{\epsilon_{\mu}|\mu=0,...,N-1\\}$ and nearest neighbor hopping $t_{C}$
(Other situations may include long-range, complex and site-dependent hopping
amplitudes. But it should be straightforward to generalize the treatment here
to those more complicated cases). The configuration of this LRC system is
illustrated in Fig. 5. The retarded Green’s function, spectral function and
transmission coefficient of the tight-binding chain are the computed by the
following formulas:

$\displaystyle G_{CC}^{w}(E)=$
$\displaystyle\frac{1}{E-H_{CC}+\frac{i}{2}\left(\gamma_{L}|0\rangle\langle
0|+\gamma_{R}|N-1\rangle\langle N-1|\right)},$ (85) $\displaystyle
A_{C}^{w}(E)=$ $\displaystyle i{\rm Tr}[G_{CC}^{w}(E)-G_{CC}^{w\dagger}(E)],$
(86) $\displaystyle T^{w}(E)=$ $\displaystyle\gamma_{L}\gamma_{R}|\langle
N-1|G_{CC}^{w}(E)|0\rangle|^{2}.$ (87)

where $\gamma_{\alpha}=\frac{|t_{C\alpha}|^{2}}{t_{\alpha}}$ for $\alpha=L,R$.
The last equality is obtained from Eq. (43). As a useful observation, the
transmission property of $H_{CC}$ at a given energy $E$ is simply determined
by a single corner matrix element $\langle N-1|G_{CC}^{w}(E)|0\rangle$ of the
retarded Green’s function.

To give an explicit demonstration, we consider the onsite potential of
$H_{CC}$ to have the following expression:

$\epsilon_{\mu}=\epsilon\cos\left(2\pi\frac{p}{q}\mu-
k_{y}\right)\qquad\mu=0,...,N-1\qquad p,q\in\mathbb{Z}.$ (88)

Here $k_{y}\in[0,2\pi)$ is a phase shift of the onsite potential, which will
be given a concrete physical meaning in our next example. With this choice,
the Hamiltonian of our central system is explicitly given by:

$H_{{\rm AAH}}=\epsilon\sum_{\mu=0}^{N-1}\cos\left(2\pi\frac{p}{q}\mu-
k_{y}\right)|\mu\rangle\langle\mu|+t_{C}\sum_{\mu=0}^{N-2}\left(|\mu\rangle\langle\mu+1|+{\rm
h.c.}\right).$ (89)

It is usually called Aubry-André-Harper (AAH) model, which has been thoroughly
explored in the study of metal-insulator transitions. Its two-dimensional
parent model, often called the Hofstadter model, is also a popular prototype
in the study quantum Hall effects and topological insulators.

As discussed in previous sections, at zero temperature, the linear transport
properties of $H_{{\rm AAH}}$ is determined by its spectral function
$A_{C}^{w}(E)$ and transmission coefficient $T^{w}(E)$ evaluated at the Fermi
energy $E_{F}$, giving us:

$\displaystyle A_{C}^{w}(E_{F})=$ $\displaystyle i{\rm
Tr}[G_{CC}^{w}(E_{F})-G_{CC}^{w\dagger}(E_{F})],$ (90) $\displaystyle
T^{w}(E_{F})=$ $\displaystyle\gamma_{L}\gamma_{R}|\langle
N-1|G_{CC}^{w}(E_{F})|0\rangle|^{2}.$ (91)

In Fig. 6, we show the numerical results of these two functions with respect
to $E_{F}$ and $k_{y}$ for two typical cases. There are third points to
mention. First, in the plots for spectral function $A_{C}^{w}(E_{F})$, the
states traversing the gaps are chiral edge modes. These modes are localized at
the edges of the AAH chain and therefore cannot contribute to the transport,
as reflected in the lower panels for the transmission coefficient
$T^{w}(E_{F})$. Second, there is a metal-insulator phase transition at
$\epsilon=2t_{C}$ for an irrational $\alpha$. When $\epsilon<2t_{C}$
($\epsilon>2t_{C}$), the bulk states (bright regions) separated by gaps in
$A_{C}^{w}(E_{F})$ are conducting (insulating) with non-vanishing (vanishing)
transmission coefficients $T^{w}(E_{F})$. Third, resolutions of edge mode in
spectral function are better for weaker couplings. Since the couplings are
introduced at the boundary of the AAH chain, it is expected that a strong
system-lead coupling will make the edge states less observable. On the
contrary, to get better transmission properties, we need to choose couplings
$\gamma_{L,R}$ closer to the energy range of the central system.

##  9 Current along the edge of a two-dimensional lattice: topologically
quantized transport

As a final example of these notes, we will apply the Landauer formula to
compute the conductance of a two-dimensional lattice. We choose the Hofstadter
model (the parent of AAH model) as the central system of our interest. It
describes noninteracting electrons hopping on a two-dimensional square lattice
in a perpendicular magnetic field. In real space, the Hamiltonian is given by:

$H_{CC}=t_{x}\sum_{x=0}^{N_{x}-2}\sum_{y=0}^{N_{y}-1}\left(|x\rangle\langle
x+1|\otimes|y\rangle\langle y|+{\rm
h.c.}\right)+t_{y}\sum_{x=0}^{N_{x}-1}\sum_{y=0}^{N_{y}-2}\left(e^{-i2\pi\frac{p}{q}x}|x\rangle\langle
x|\otimes|y\rangle\langle y+1|+{\rm h.c.}\right),$ (92)

where $t_{x}$ ($t_{y}$) is the nearest neighbor hopping amplitude along $x$
($y$) direction of the lattice. The system contains $(N_{x}+1)\times(N_{y}+1)$
lattice sites. This model can be reduced to the AAH model if we take periodic
boundary conditions along $y$-direction, and interpreting the parameter
$k_{y}$ in the AAH model as the quasimomentum along $y$-direction.

To evaluate the current, we now couple the central system described by
Hamiltonain (92) to leads. On the left hand side of $H_{CC}$, each lattice
site $(0,y)$ for $y=0,1,...,N_{y}-1$ is coupled to a semi-infinite tight-
binding chain extends from $(-\infty,y)$ to $(-1,y)$. On the right hand side
of $H_{CC}$, each lattice site $(N_{x}-1,y)$ for $y=0,1,...,N_{y}-1$ is also
coupled to a semi-infinite tight-binding chain extends from $(N_{x},y)$ to
$(+\infty,y)$. The configuration of this LRC system is illustrated in Fig. 7.
In the wide-band limit, the left and right level-width functions are given by:

$\displaystyle\Gamma_{L}^{w}=$ $\displaystyle|0\rangle\langle
0|\otimes\sum_{y=0}^{N_{y}-1}\gamma_{yL}|y\rangle\langle y|,$ (93)
$\displaystyle\Gamma_{R}^{w}=$ $\displaystyle|N_{x}-1\rangle\langle
N_{x}-1|\otimes\sum_{y=0}^{N_{y}-1}\gamma_{yR}|y\rangle\langle y|,$ (94)

where we took $\gamma_{y\alpha}=\frac{|t_{y\alpha}|^{2}}{t_{\alpha}}$ for
$\alpha=L,R$. The Hamiltonian $H_{CC}$, together with level-width functions
$\Gamma_{L,R}^{w}$, determines the retarded Green’s function, spectral
function and transmission coefficient of the central region:

$G_{CC}^{w}(E)=\frac{1}{E-H_{CC}+\frac{i}{2}\sum_{y=0}^{N_{y}-1}\left(\gamma_{yL}|0\rangle\langle
0|+\gamma_{yR}|N_{x}-1\rangle\langle N_{x}-1|\right)\otimes|y\rangle\langle
y|},$ (95) $\displaystyle A_{C}^{w}(E)=$ $\displaystyle-2{\rm Im}\\{{\rm
Tr}[G_{CC}^{w}(E)]\\},$ (96) $\displaystyle T^{w}(E)=$ $\displaystyle{\rm
Tr}\left[G_{CC}^{w\dagger}(E)\Gamma_{R}^{w}(E)G_{CC}^{w}(E)\Gamma_{L}^{w}(E)\right]$
$\displaystyle=$
$\displaystyle\sum_{y,y^{\prime}=0}^{N_{y}-1}\gamma_{yL}\gamma_{y^{\prime}R}|\langle
N_{x}-1,y^{\prime}|G_{CC}^{w}(E)|0,y\rangle|^{2}.$ (97)

The transmission coefficient $T^{w}(E)$ has a transparent interpretation as
summation over scattering amplitudes from left ($x=0$) to right ($x=N_{x}-1$)
edges for all possible incoming and outgoing sites.

As a demonstration of these formulas, we consider the transmission coefficient
of Hofstadter model at zero temperature in zero bias limit. For $p/q=1/3$ and
$p/q=1/5$, numerical results for $T^{w}(E_{F})$ vs. $E_{F}$ are shown in Fig.
8. In both cases, we observed quantized transmission coefficients (and
therefore conductances) in the spectral gap of the system. As illustrated in
the plots of spectral function. current-carrying states in the gap are chiral
edge modes localized at the boundary of the two-dimensional lattice. The
quantization of edge state conductance in a spectral gap of the Hofstadter
model is topological, equaling to the summation of bulk band Chern numbers
below the considered gap. This bulk-edge correspondence principle holds in
general for noninteracting fermionic Chern insulators. Interested readers can
consult Ref.[9] for more details.

##  10 Summary and plan for future extensions

In these notes, we give a brief introduction to matrix Green’s function,
focusing on its applications to steady state transport in discrete level
quantum systems without many-body interactions. The Landauer formula is
derived from a naive scattering viewpoint, and then applied to compute linear
conductance of a single-level quantum dot, a double-level quantum dot, a
tight-binding chain and a two-dimensional tight-binding lattice at zero
temperature under wide-band limit. Even though a lot of approximations and
simplifications have been made, the results still demonstrate the important
roles of _resonance, coherence, disorder and topology_ in quantum transport.

Further extensions of these notes may include the following topics:

  * •

Steady state transport in periodically driven quantum systems: a combination
of Green’s function and Floquet formalism

  * •

Initial states with inter-channel coherence

  * •

Effects of finite temperature and many-body interaction

  * •

Quantum pumping

### Acknowledgments

L. Zhou thanks Prof. Jian-Sheng Wang for his pedagogical lectures on NEGF, and
helpful discussions with Prof. Jiangbin Gong and Mr. Han Hoe Yap, which
motivated the preparation of these notes.

## References

  * [1] D. A. Ryndyk, _Theory of Quantum Transport at Nanoscale_ (Springer, 2016).
  * [2] R. A. Jishi, _Feynman Diagram Techniques in Condensed Matter Physics_ (Cambridge University Press, 2014).
  * [3] M. D. Ventra, _Electrical Transport in Nanoscale Systems_ (Cambridge University Press, 2008).
  * [4] J. C. Cuevas and E. Scheer, _Molecular Electronics: An Introduction to Theory and Experiment_ (World Scientific, 2010).
  * [5] J.-S. Wang, B. K. Agarwalla, H. Li and J. Thingna, Nonequilibrium Green’s function method for quantum thermal transport, Front. Phys. 9, 673-697 (2014).
  * [6] M. Paulsson, Non Equilibrium Green’s Functions for Dummies: Introduction to the One Particle NEGF equations, arXiv:cond-mat/0210519 (2006).
  * [7] S. Noschese, L. Pasquini, and L. Reichel, Tridiagonal Toeplitz Matrices: Properties and Novel Applications, Numerical Linear Algebra with Applications 20, 302-326 (2013).
  * [8] H. Bruus and K. Flensberg, _Many-Body Quantum Theory in Condensed Matter Physics_ (Oxford University Press, 2004).
  * [9] Y. Hatsugai, Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function, Phys. Rev. B 48, 11851 (1993).

Figure 1: Real part (red solid line) and imaginary part (blue dashed line) of
the lead $\alpha$’s self-energy $\Sigma_{\alpha}$, plotted with respect to the
difference between the energy of incoming electron $E$ and the band center
$w_{\alpha}$ of lead $\alpha$.  Figure 2: A single-level quantum dot coupled
to two semi-infinite tight-binding leads. Figure 3: A double-level quantum dot
coupled to semi-infinite tight-binding leads. The leads only couple to one of
the two levels, forming a T-shape junction. Figure 4: Spectral function $A(E)$
and transmission coefficient $T(E)$ of the double-level quantum dot coupled to
semi-infinite tight-binding leads in wide-band limit. System parameters are
$\epsilon_{0}=-2$, $\epsilon_{1}=2$ and $\gamma_{0}=0.5$. Only level
$\epsilon_{0}$ is coupled to the leads as shown in Fig. 3. The results for
three different coupling strengths between the two levels of the quantum dot
are shown. Figure 5: A tight-binding chain coupled to semi-infinite tight-
binding leads at its two boundary sites. Figure 6: Spectral function
$A_{C}^{w}(E_{F})$ and transmission coefficient $T^{w}(E_{F})$ of the AAH
model at zero temperature in zero bias, wide-band limit. Results are shown for
two different values of onsite potential $\epsilon=1.5$ and $2.5$. For
spectral function, we take $\gamma_{L}=\gamma_{R}=0.1$. For transmission
coefficient, we take $\gamma_{L}=\gamma_{R}=2.5$. The other system parameters
are $t_{C}=1$, $\frac{p}{q}=\frac{\sqrt{5}-1}{2}$, and $N=59$.  Figure 7: A
two-dimensional lattice (Hofstadter model) coupled to semi-infinite tight-
binding leads at its left and right boundary sites. Figure 8: Transmission
coefficients of the Hofstadter model at different $E_{F}$ for $p/q=1/3$ and
$p/q=1/5$. The other system parameters are chosen as $N_{x}=N_{y}=44$,
$t_{x}=t_{y}=1$ and $\gamma_{yL}=\gamma_{yR}=1$. The upper panels show
spectral functions calculated for the two cases with periodic boundary
conditions along $y$-direction. Bright lines traversing the spectral gaps
represent chiral edge modes. The lower panels show numerically evaluated
transmission coefficients vs. Fermi energy $E_{F}$ (in gray dots). Dashed
lines are guide to the eye to point out the quantization of transmission
coefficient (and conductance) in the spectral gap.


0 Research Representation and Interaction Darius Coelho, Harshit Barot, Naitik Rathod, and Klaus Mueller are with Stony Brook University. E-mail: {dcoelho | hbarot | nrathod |<EMAIL_ADDRESS>
# Can LLMs Generate Visualizations with Dataless Prompts?

Darius Coelho  Harshit Barot  Naitik Rathod  Klaus Mueller

###### Abstract

Recent advancements in large language models have revolutionized information
access, as these models harness data available on the web to address complex
queries, becoming the preferred information source for many users. In certain
cases, queries are about publicly available data, which can be effectively
answered with data visualizations. In this paper, we investigate the ability
of large language models to provide accurate data and relevant visualizations
in response to such queries. Specifically, we investigate the ability of GPT-3
and GPT-4 to generate visualizations with dataless prompts, where no data
accompanies the query. We evaluate the results of the models by comparing them
to visualization cheat sheets created by visualization experts.

###### keywords:

Large Language Models, Data Visualization, Captions, Prompt Engineering

Chart Cheat Sheet: A visualization cheat sheet to decide which chart is best
for a specific type of data and task. All of the charts were created by
prompting GPT4 with real-world queries. Notably, and this one of the major
findings of our paper, all returned charts matched what is commonly
recommended for the query’s task and scenario.

Introduction

Rapid advances in AI, specifically computer vision and natural language, have
led to the development of large language models (LLMs) such as GPT that can
generate contextually relevant and high-quality responses in the form of
images or text to natural language queries. LLM models are trained on large
amounts of text and image data sourced from a significant portion of the
internet which gives them seemingly all-encompassing knowledge. As a result,
LLMs have emerged as the de facto source of information for many users on the
web.

(a)

(b)

(c)

(d)

Figure 1: Visualizations showing the U.S. debt over the last two decades. (a)
is the ground truth visualization retrieved from Statista while the remaining
visualizations are generated with the prompt "Generate a chart showing the
national debt of the U.S. over the last 2 decades" with (b) DALL-E, (c)
GPT-3.5, and (d) GPT-4.

People often have queries about publicly available data on the web, even
though they do not possess the data themselves. For example, they might ask
"What is the US debt over the last two decades?" As LLMs are trained on data
available on the web, we hypothesize that they have captured publicly
available data and can respond to these queries accurately. Additionally,
given that visual representations can significantly enhance comprehension [2],
we believe that responses accompanied by data visualizations are ideally
suited to such queries.

We expect that when queried for a visual of the data, an LLM would provide us
with an appropriate visual of the data as these models have also captured
visualization rules [5]. For example, if we prompted an LLM "Generate a chart
showing the US debt over the last two decades", we would expect to receive a
line chart showing the US debt at each year over the past 20 years.

In this work, we investigate the ability of generative AI models to respond
appropriately to dataless queries. We define dataless queries as text-based
queries that can be answered with publicly available data but do not contain
the data themselves. We expect the model to provide the data, or an
approximation of the data, as part of its response. Our initial investigation
tested the ability of DALL-E 2, GPT-3.5, and GPT-4, to respond to dataless
queries. We found that GPT-4 appeared to be more adept at answering such
queries while responses from DALL-E 2 and GPT-3.5 were inadequate. This
motivated us to conduct a more in-depth investigation of GPT-4s abilities. We
generated 15 prompts based on popular data visuals on the web, aided by a
crowd-sourced survey to gather queries of popular interest. We then queried
GPT-4 using these prompts and evaluated the responses by comparing them to
visualization cheat sheets created by visualization experts and human-
generated charts on the web.

In the following. Section 2 provides relevant background. Section 3 details
the study we conducted. Section 4 presents a discussion and concludes our
paper.

##  1 Background

Generating data-visualizations based on user generated natural language (NL)
queries has been a longstanding research topic. Early approaches used
traditional natural language processing (NLP) techniques such as rule-based
and probabilistic grammar-based approaches to decompose NL queries into a set
of machine instructions. Cox et al. [3] created one of the first NL interfaces
for visualization using a grammar-based approach to convert NL questions into
database queries and visualize the results. Eviza [12] employed a
probabilistic grammar-based approach and a finite state machine to allow users
to interact with a given visualization using NL commands. NL4DV [10] is a
python library that translates NL queries into visualizations. It combines NLP
approaches to break down an NL input into the machine instructions required to
query a dataset and generate a chart.

More recently researchers have explored deep-learning based approaches to
convert NL queries to visualizations. Lui et al [6] proposed ADVISor, a deep-
learning based system that uses multiple models to process and transform data
and then generate annotated visualizations. Luo et al [8] devised a novel
approach using a transformer-based model with several novel visualization-
aware optimizations to take in a NL query along with an optional chart (e.g.,
a pie chart or a scatter plot) that serves as a template and generates a
visualization. RGVisNet [13] is a hybrid system that first looks up a large
data visualization query codebase for a query most relevant to the user’s data
visualization query. The retrieved query is then refined by a GNN-based deep-
learning model to create a visualization specification.

Recent advances in generative AI, especially LLMs such as OpenAI’s GPT
[1][11], have prompted researchers to investigate their application to the
field of data visualization. Maddigan and Susnjak [9] showed how ChatGPT and
GPT-3 can be leveraged to build an end-to-end solution for converting NL
queries to visualizations. Their system, Chat2Vis, takes in a NL query and a
dataset from the user and behind the scenes engineers prompts to precisely
query GPT-3. The queries are constructed such that GPT-3 responds with a
python script to visualize the data provided by the user. Dibia et al. [4]
proposed a four-stage approach with LIDA that combines LLMs with image
generation to turn datasets and analysis goals into charts and infographics.
Finally Li et al [5] evaluated GPT-3.5 for vega-lite specification generation
using multiple prompting strategies.

All of these previous approaches required a dataset as input, along with the
query to produce a visualization. Conversely, we have studied to what extent
LLMs can deliver a visualization with just the query, or prompt, alone.
Furthermore, we studied whether the visualizations so produced follow commonly
available visualization expert guidelines. In other words, have LLMS learnt
both data and visualization knowledge?

Figure 2: The visualization cheatsheet created by Patrik Lundblad at Qlik to
assist visualization designers pick appropriate charts for their data

##  2 Study

Our study aims to investigate the ability of generative AI to produce
visualizations when prompted with queries without data. This is much like the
way a user queries an image search engine for data visualizations e.g.
"Generate a plot of the most spoken languages". In this section, we discuss
how we chose to construct dataless prompts, how multiple models responded to a
small set of dataless prompts, and provide a detailed investigation into how
GPT-4 performs on dataless prompts.

###  2.1 Dataless Prompts

We define dataless prompts as text-based queries provided to LLMs that can be
answered with publicly available data but do not contain the data themselves.
An example of a dataless prompt is "What was the US GDP over the past 20
years." For this prompt, LLMs often provide a text-based response explaining
the GDP trends with values for some of the 20 years. However, providing this
prompt to a traditional search engine (e.g. Google or Yahoo) will provide the
user with links to articles or images showing the values. In our work we are
interested in the images, specifically the data visualizations. To have an LLM
produce a visualization, we need to explicitly tell it to provide a chart, for
example, "Generate a chart showing the US GDP over the past 20 years." Given
such a prompt, LLMs will respond with an image of a chart or code to generate
a chart. Thus for all dataless prompts in our study, we explicitly ask the
models to provide a chart.

###  2.2 Initial Investigation

We began with an initial investigation to test if generative AI modes could
produce appropriate responses to dataless prompts. We tested GPT-3.5, GPT-4
and DALL-E. We chose to add DALL-E to the investigation as using it is very
similar to how a user would use an image search engine. We queried all three
models with three prompts and visually inspected the results. Each model’s
response to the query "Generate a chart showing the national debt of the U.S.
over last 2 decades" along with a reference ground truth image sourced from
Google images is shown in figure 1 . Note that the GPT-3.5 responded with code
which we ran on our local machine to generate the chart in figure 1c.

In our tests, all three models were able to generate coherent responses.
GPT-3.5 did not have the capability to respond with an image however responses
contained mock data values or pointers to publicly available data and a python
code to generate a chart. GPT-4 could generate charts but it stated that it
used mock data. However, in both cases, the overall trend of the data was
similar to the original data. DALL-E on the other hand could only generate a
chart (shown in figure 1b) for one prompt while it generated graphics related
to the prompt topic for the others. In the case that it did generate the
chart, the overall trend did match that of the original data. Based on the
responses of each model, we concluded that GPT-4 was the only model that could
respond with an appropriate image or chart. GPT-3 was unable to provide a
visual result while DALL-E could not provide appropriate visuals.

###  2.3 Dataless Prompts with GPT-4

Having identified GPT-4 as a promising model to respond to dataless prompts,
we decided to test it on a wider range of prompts.

####  2.3.1 Procedure

To test GPT-4’s ability to respond to dataless prompts we first generated 15
prompts. We generated 7 prompts by asking Google Images for various chart
types. We then selected 7 random charts and constructed a prompt based on the
data used to generate it. For example, we selected a pie chart showing
Amazon’s 2022 revenue breakdown, and subsequently created the prompt "Create a
chart to display the revenue breakdown of Amazon in 2022."

To add more variety, we crowd-sourced the remaining 8 prompts. To crowd-source
the prompts we sent a Google form to graduate students in our university’s
computer science department asking "If you could ask ChatGPT to visualize
public data, what would you ask it to visualize for you? List as many queries
as you’d like". Seven students responded with a total of 22 responses. Next,
we tested if each response returned a relevant chart on Google images. We
discarded irrelevant responses such as "Make unique recipes using GPT". This
process resulted in us collecting 8 prompts with relevant charts on Google
images.

Once we finalized our set of 15 prompts we fed each prompt to GPT-4 and
compared the result to the Google image we collected. The result returned by
GPT-4 and the Google image result for each of the 15 prompts are shown in the
supplementary material. For each response, we evaluated if (a) GPT-4 returned
a chart with the data we requested, (b) it used an appropriate chart type, and
(c) the data matched the data in the chart retrieved with Google Images.

(a)

(b)

Figure 3: Visualizations retrieved from (a) Google images and (b) GPT-4
showing the U.S. cities with the highest average rent. When compared to the
visualization from Google images, we see that GPT-4 was able to generate a
similar ordering of cities however it provided different values for mean
rents, perhaps it retrieved data from a different year.

####  2.3.2 Results

Overall GPT-4 performed very well, producing coherent responses to all
prompts. For every prompt, it provided data requested in the prompt. We then
evaluated whether each returned chart followed commonly accepted visualization
design rules, given the specific data and query scenario. For this we used the
widely accepted visualization cheat sheet developed by Patrik Lundblad at Qlik
[7] shown in figure 2. We found that every chart generated was appropriate!
This can be gleaned from figure Can LLMs Generate Visualizations with Dataless
Prompts? which shows the placement of each GPT-4 generated chart into the
appropriate slot in Lundblad’s cheat sheet.

GPT-4 faltered, however, when it came to reproducing the exact same data shown
in the charts retrieved from Google Images. When generating line charts GPT-4
could procudce charts with the same general trend, however the values were not
identical and the trends were only correct at a larger scale. For example, in
figure 1d, GPT-4 introduced a dip in the U.S. debt between the years 2013 to
2016 however the original data shown in figure 1a shows that the debt was
monotonically increasing. When it came to generating rankings with bar charts
(e.g., top UEFA football clubs, most expensive cities to rent in etc) GPT-4
appeared to be more accurate. It was able to generate the same overall ranking
of items but the values represented by bars were inconsistent. See for example
figure 3, where a chart from Google images (a) reported the mean rent in New
York to be $3260 while GPT-4 (b) reported it as $4100.

##  3 Discussion and Conclusions

Our paper makes two contributions. First, we show that GPT-4 can return useful
visualizations from dataless queries – queries that are not accompanied by
associated data files, as has been common practice so far. It so absolves
mainstream users from having to look for data files first before generating
visualizations with GPT-4. While the visualizations so produced are not
perfect in every detail, they nevertheless indicate that GPT-4 has obtained a
sufficient amount of data knowledge plus the skill to produce a visual
rendition of it.

Second, we find that GPT-4 has been an excellent ’student’ of the art and
science of data visualization, having learned an impressive amount of
visualization knowledge from its ’teachers’, the many visualizations found on
the web. We put GPT-4 to the test by asking it to fill in a commonly used
visualization cheat sheet [7] where it reached a perfect score (compare figure
Can LLMs Generate Visualizations with Dataless Prompts? with the cheat sheet
published in [7]).

During our experimentation, we noticed that GPT-4 seemed to insist on using
specific visualizations for specific data queries, namely those shown in the
cheat sheet, even when the prompt was moderately altered to create confusion.
We value this as an indication that GPT-4’s visualization knowledge is fairly
well expressed. solid, and ingrained.

Further, we also found that while GPT3.5 was not as reliable in creating the
right type of visualization, it was still good at identifying the sources of
data. To that end, a current line of our work is to identify effective
mechanisms for LLMs to pull CSV files on its own from the respective online
repositories, followed by running the code that generates the appropriate
visualizations to provide users with results. Finally, we also believe that
there is high promise in Dall-E to eventually be able to produce high quality
infographics. We see this as another line of possible important future work.

## References

  * [1] T. Brown, B. Mann, et al.  Language models are few-shot learners.  In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, eds., Advances in Neural Information Processing Systems, vol. 33, pp. 1877–1901. Curran Associates, Inc., 2020.
  * [2] W. Cleveland and R. McGill.  Graphical perception: Theory, experimentation, and application to the development of graphical methods.  Journal American Statistical Association, 79(387):531–554, 1984.
  * [3] K. Cox, R. Grinter, S. Hibino, L. Jagadeesan, and D. Mantilla.  A multi-modal natural language interface to an information visualization environment.  Intern. Journal of Speech Technology, 4(3):297–314, 2001.
  * [4] V. Dibia.  LIDA: A tool for automatic generation of grammar-agnostic visualizations and infographics using large language models.  In D. Bollegala, R. Huang, and A. Ritter, eds., Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 3: System Demonstrations), pp. 113–126. Toronto, Canada, July 2023.
  * [5] G. Li, X. Wang, G. Aodeng, S. Zheng, Y. Zhang, C. Ou, S. Wang, and C. H. Liu.  Visualization generation with large language models: An evaluation, 2024. doi: 10 . 48550/arXiv . 2401 . 11255
  * [6] C. Liu, Y. Han, R. Jiang, and X. Yuan.  Advisor: Automatic visualization answer for natural-language question on tabular data.  In 2021 IEEE 14th Pacific Visualization Symposium (PacificVis), pp. 11–20. IEEE, 2021.
  * [7] P. Lundblad.  Qlik: How to choose the right chart for your data.  https://community.qlik.com/t5/Official-Support-Articles/How-to-Choose-the-Right-Chart/ta-p/1717241.  Accessed: 2024-04-29.
  * [8] Y. Luo, N. Tang, G. Li, J. Tang, C. Chai, and X. Qin.  Natural language to visualization by neural machine translation.  IEEE Transactions on Visualization and Computer Graphics, 28(1):217–226, 2022. doi: 10 . 1109/TVCG . 2021 . 3114848
  * [9] P. Maddigan and T. Susnjak.  Chat2vis: Generating data visualizations via natural language using chatgpt, codex and gpt-3 large language models.  IEEE Access, 11:45181–45193, 2023. doi: 10 . 1109/ACCESS . 2023 . 3274199
  * [10] A. Narechania, A. Srinivasan, and J. Stasko.  Nl4dv: A toolkit for generating analytic specifications for data visualization from natural language queries.  IEEE Transactions on Visualization and Computer Graphics, 27(2):369–379, 2021. doi: 10 . 1109/TVCG . 2020 . 3030378
  * [11] OpenAI, J. Achiam, et al.  Gpt-4 technical report, 2024. doi: 10 . 48550/arXiv . 2303 . 08774
  * [12] V. Setlur, S. E. Battersby, M. Tory, R. Gossweiler, and A. X. Chang.  Eviza: A natural language interface for visual analysis.  In Proc. UIST, 13 pages, p. 365–377, 2016.
  * [13] Y. Song, X. Zhao, R. C.-W. Wong, and D. Jiang.  Rgvisnet: A hybrid retrieval-generation neural framework towards automatic data visualization generation.  In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, p. 1646–1655, 2022.