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\chapter{The transverse-field Ising chain}

\noindent\textit{Bachelor thesis, E.~Example, University of Somewhere.
Chapter~2 of the draft of 26 September 2026.}
\bigskip

\section{Introduction}

The Ising model in a transverse field is the simplest quantum many-body
system that has a quantum phase transition. It consists of a chain of
spin-$\tfrac12$ particles which is coupled by a ferromagnetic exchange
interaction $J$ and which are placed in a magnetic field $h$ that points
perpendicular to the easy axis. As is well known, the competition between
the two terms leads to a transition between an ordered and a disordered
phase at zero temperature. In thermodynamic limit the transition happens
at a critical value of the field, and the behaviour of the chain close to
this point is universal.

It is worth noting that it is the case that the model can be solved
exactly. This was first done by Pfeuty in 1976 using a mapping to free
fermions, which had been introduced earlier by Jordan and Wigner
\cite{Pfeuty1970,JordanWigner1928}. Because of the exact solution, the
transverse-field Ising chain is the standard example against which
numerical methods are tested \cite{Sachdev2011}. In this chapter we
recieve the exact solution and compare it with numerics for finite
chains.

The chapter is organised as follows. Section~\ref{sec:model} introduces
the model and the notation. In Section~\ref{sec:jw} the Jordan--Wigner
transformation is applied and the dispersion of the fermionic excitations
is derived. Section~\ref{sec:phase} discusses the phase diagram, which is
shown in Figure~\ref{fig:phasediag}, and Section~\ref{sec:numerics}
compares the exact ground-state energy with exact diagonalisation of
short chains.

\section{Model and notation}
\label{sec:model}

We consider a chain of $L$ sites with periodic boundary conditions. On
every site $j$ there is a spin-$\tfrac12$ degree of freedom, described by
the Pauli matrices $\sigma^x_j$, $\sigma^y_j$ and $\sigma^z_j$. The
Hamiltonian reads
\begin{equation}
  H = -J \sum_{j=1}^{L} \sigma^z_j \sigma^z_{j+1}
      - h \sum_{j=1}^{L} \sigma^x_j ,
  \label{eq:hamiltonian}
\end{equation}
with $\sigma^z_{L+1} \equiv \sigma^z_1$. The first term favours states in
which neighbouring spins is aligned along the $z$ axis; the second term
favours states in which every spin points along $x$. Throughout this
chapter we take $J > 0$ and $h \geq 0$.

Two limits are easy to understand. For $h = 0$ the Hamiltonian is
diagonal in the $Z$ basis, and the two ground states are the fully
polarised states $\lvert \uparrow\uparrow\cdots\uparrow \rangle$ and
$\lvert \downarrow\downarrow\cdots\downarrow \rangle$. This is the
ferromagnetic phase, in which the $\mathbb{Z}_2$ symmetry generated by
$P = \prod_j \sigma^x_j$ is spontaneously broken. For $h \to \infty$ the
ground state is the unique product state in which every spin is polarised
along $z$, and the symmetry is unbroken. This is the paramagnetic phase.
Between the two limits there has to be a phase transition, and it is
the aim of this chapter to locate it and to analyse its properties.

We use the order parameter
\begin{equation}
  m_z = \frac{1}{N} \sum_{j=1}^{L} \langle \sigma^z_j \rangle ,
\end{equation}
which is nonzero in the ferromagnetic phase and vanishes in the
paramagnetic phase. In the paramagnetic phase this quantity is not an
unique measure of the disorder, so we will additionally use the energy
gap $\Delta$ between the ground state and the first excited state.

\section{Jordan--Wigner transformation}
\label{sec:jw}

The key idea of the exact solution is to represent the spins by
spinless fermions. We define
\begin{equation}
  c_j = \Bigl( \prod_{l<j} \sigma^x_l \Bigr)\, \sigma^-_j , \qquad
  \sigma^\pm_j = \tfrac12 \bigl( \sigma^z_j \pm i \sigma^y_j \bigr),
  \label{eq:jw}
\end{equation}
where the string of $\sigma^x$ operators ensures that the $c_j$ satisfy
the canonical anticommutation relations $\{ c_j, c_l^\dagger \} =
\delta_{jl}$. Inserting Equation~\eqref{eq:jw} into
Equation~\eqref{eq:hamiltonian} gives a quadratic form in the fermions,
\begin{equation}
  H = -J \sum_j \bigl( c_j^\dagger - c_j \bigr)
               \bigl( c_{j+1}^\dagger + c_{j+1} \bigr)
      - h \sum_j \bigl( 1 - 2 c_j^\dagger c_j \bigr) ,
  \label{eq:quadratic}
\end{equation}
up to a boundary term that depends on the fermion parity and that we
ignore for sufficiently large chains. Since the Hamiltonian is quadratic
it can be diagonalised by a Fourier transformation followed by a
Bogoliubov transformation. The result is
\begin{equation}
  H = \sum_k \varepsilon_k \Bigl( \gamma_k^\dagger \gamma_k - \tfrac12 \Bigr),
  \qquad
  \varepsilon_k = 2 \sqrt{J^2 + h^2 - 2 J h \cos k} ,
  \label{eq:dispersion}
\end{equation}
where the $\gamma_k$ are fermionic quasiparticle operators and $k$ runs
over the allowed momenta in the first Brillouin zone. The dispersion in
Equation~\eqref{eq:jw} is shown in the left panel of
Figure~\ref{fig:phasediag} for three values of the field.

The energy gap is the minimum of the dispersion, which is reached at
$k = 0$:
\begin{equation}
  \Delta = \varepsilon_0 = 2\,\lvert J - h \rvert .
  \label{eq:gap}
\end{equation}
Hence the gap closes exactly at $h = J$, and this is the location of the
quantum phase transition. Note that the gap $\Delta = 4\lvert J - h\rvert$
is linear in the distance from the critical point on both sides. The
ground-state energy follows from Equation~\eqref{eq:dispersion} as
\begin{equation}
  E_0 = -\frac12 \sum_k \varepsilon_k
  \;\xrightarrow{L \to \infty}\;
  -\frac{L}{2\pi} \int_{-\pi}^{\pi} \sqrt{J^2 + h^2 - 2Jh\cos k}\, \mathrm{d}k .
  \label{eq:e0}
\end{equation}
At the critical point $h = \Jc$ the integral can be done in closed form
and gives $E_0 / L = -4J/\pi$.

\begin{figure}[t]
  \centering
  \begin{tikzpicture}[scale=1.0]
    % left panel: dispersion
    \begin{scope}
      \draw[->] (-0.2,0) -- (3.6,0) node[right] {$k$};
      \draw[->] (0,-0.2) -- (0,3.2) node[above] {$\varepsilon_k / J$};
      \draw (0,0.05) -- (0,-0.05) node[below] {$-\pi$};
      \draw (1.6,0.05) -- (1.6,-0.05) node[below] {$0$};
      \draw (3.2,0.05) -- (3.2,-0.05) node[below] {$\pi$};
      \draw[thick,blue!70!black] plot[domain=0:3.2,samples=60]
        (\x,{0.7*sqrt(1+0.25-1*cos(deg(pi*(\x-1.6)/1.6)))});
      \draw[thick,red!70!black] plot[domain=0:3.2,samples=60]
        (\x,{0.7*sqrt(1+1-2*cos(deg(pi*(\x-1.6)/1.6)))});
      \draw[thick,green!50!black] plot[domain=0:3.2,samples=60]
        (\x,{0.7*sqrt(1+2.25-3*cos(deg(pi*(\x-1.6)/1.6)))});
      \node[blue!70!black,anchor=west] at (2.2,2.9) {$h = 0.5J$};
      \node[red!70!black,anchor=west] at (2.2,2.5) {$h = J$};
      \node[green!50!black,anchor=west] at (2.2,2.1) {$h = 1.5J$};
    \end{scope}
    % right panel: gap
    \begin{scope}[xshift=6cm]
      \draw[->] (-0.2,0) -- (3.6,0) node[right] {$h / J$};
      \draw[->] (0,-0.2) -- (0,3.2) node[above] {$\Delta / J$};
      \draw (1.5,0.05) -- (1.5,-0.05) node[below] {$1$};
      \draw (3.0,0.05) -- (3.0,-0.05) node[below] {$2$};
      \draw[thick] (0,2.4) -- (1.5,0) -- (3.3,2.88);
      \node[anchor=south east] at (1.4,1.4) {ordered};
      \node[anchor=south west] at (1.7,1.4) {disordered};
      \fill (1.5,0) circle (1.5pt);
    \end{scope}
  \end{tikzpicture}
  \caption{Left: the quasiparticle dispersion $\varepsilon_k$ of
    Equation~\eqref{eq:dispersion} for three values of the transverse
    field. Right: the gap $\Delta = 2\lvert J - h \rvert$ as a function of
    the field; it closes at the critical point $h_c = J$.}
  \label{fig:phase}
\end{figure}

\section{The phase diagram}
\label{sec:phase}

The two phases identified in Section~\ref{sec:model} are separated by a
single critical point at $h_c = J$, see Fig.~\ref{fig:phase}. In the
ordered phase, $h < h_c$, the order parameter $m_z$ is finite and the
gap is $\Delta = 2(J-h)$. In the disordered phase, $h > h_c$, the order
parameter vanishes and the gap is $\Delta = 2(h-J)$. Exactly at the
critical point the gap closes and the correlation length diverges as
$\xi \sim \lvert h - h_c \rvert^{-1/2}$, that is, with the critical
exponent $\nu = 1/2$ of the two-dimensional classical Ising model.

This mapping between the quantum chain and the classical two-dimensional
model is not a coincidence. A bunch of arguments show that the
transfer matrix of the classical model in the limit of an anisotropic
lattice is the exponential of the quantum Hamiltonian, so that the
universal properties of both are identical. This is the reason why the
transverse-field Ising chain was shown to have been found to be the
prototype of a quantum critical point.

The transversal field also has a simple interpretation in terms of the
fermions: it is a chemical potential. In the disordered phase the
fermionic band is empty, and the excitations are single fermions that
cost at least $\Delta$. In the ordered phase the picture is similar,
but the elementary excitations are domain walls between the two
ferromagnetic ground states. It is remarkable that these two very
different pictures are both described by the same dispersion
\eqref{eq:dispersion}.

\section{Numerical check}
\label{sec:numerics}

To analyze the accuracy of exact diagonalisation for short chains, we
compare the ground-state energy per site of chains with up to $L = 20$
sites with the exact result of Equation~\eqref{eq:e0}. The Hamiltonian
\eqref{eq:hamiltonian} is represented as a sparse matrix of dimension
$2^L$ and its lowest eigenvalue is obtained with the Lanczos algorithm.
Table~\ref{tab:energies} lists the results at the critical point.

\begin{table}[t]
  \centering
  \caption{Ground-state energy per site at $h = \Jc$ for periodic
    chains of length $L$ from exact diagonalisation, compared with the
    thermodynamic limit $-4/\pi$ of Equation~\eqref{eq:e0}. The values
    are given in units of $J$.}
  \label{tab:energies}
  \begin{tabular}{rll}
    \toprule
    $L$ & $E_0 / L$ & deviation \\
    \midrule
    8   & $-1.2872$ & $1.1 \times 10^{-2}$ \\
    12  & $-1.2795$ & $4.9 \times 10^{-3}$ \\
    16  & $-1.2769$ & $2.9 \times 10^{-3}$ \\
    20  & $-1.2757$ & $1.9 \times 10^{-3}$ \\
    $\infty$ & $-1.2732$ & --- \\
    \bottomrule
  \end{tabular}
\end{table}

The finite-size corrections decrease as $1/L^2$, as expected for a
critical point with dynamical exponent $z = 1$, where the leading
correction is set by the central charge of the conformal field theory.
Fitting the four data points to $E_0/L = e_\infty + a/L^2$ gives
$e_\infty = -1.2733$, in agreement with the exact value to four digits.
This confirms that the implementation is correct and that it can be
used for the perturbed models of the following chapters, for which no
exact solution exists.

\section{Summary}

The transverse-field Ising chain can be mapped to free fermions with the
dispersion of Equation~\eqref{eq:dispersion}. The gap closes at
$h_c = J$, which separates a ferromagnetic phase from a paramagnetic
phase. Its critical exponents are those of the two-dimensional
classical Ising model. Exact diagonalisation reproduces the exact
ground-state energy with an error below $10^{-2}$ already for
$L = 8$ sites, and the extrapolation in $1/L^2$ agrees with the exact
value to four digits.

\begin{thebibliography}{9}
\bibitem{Pfeuty1970}
  P.~Pfeuty, \emph{The one-dimensional Ising model with a transverse
  field}, Ann.\ Phys.\ \textbf{57}, 79 (1970).
\bibitem{JordanWigner1928}
  P.~Jordan and E.~Wigner, \emph{\"Uber das Paulische
  \"Aquivalenzverbot}, Z.\ Phys.\ \textbf{47}, 631 (1928).
\bibitem{Sachdev2011}
  S.~Sachdev, \emph{Quantum Phase Transitions}, 2nd ed.\ (Cambridge
  University Press, Cambridge, 2011).
\end{thebibliography}

\end{document}