Fermions on a Lattice

Abstract

All fundamental microscopic theories of nature contain fermions. Thus we briefly recall the basic properties of a Dirac field in Euclidean space. By using anticommuting Grassmann variables we formulate the path integral for Fermi fields. However, when one tries to define quark or lepton fields on a space-time lattice then one encounters the species-doubling problem: a naively discretized Dirac field describes more particles than expected. This also follows from the celebrated Nielsen–Ninomiya theorem which is proven in the chapter. We present several proposals to discretize fermion fields without or with less doublers, these include Wilson fermions, staggered fermions and Ginsparg–Wilson fermions. At the end we briefly discuss (supersymmetric) Yukawa models and gauge theories with fermions at zero and finite temperature and comment on the sign problem for equilibrium lattice systems with finite baryon number density. In the end-of-chapter problems we deal with the Pfaffian which shows up in (supersymmetric) lattice theories containing Majorana fermions.

Appendix: The SLAC Derivative

We introduce the SLAC derivative on a one-dimensional periodic lattice with equidistant sampling points

$$ x_k=x_0+k,\quad k=1,\dots,N . $$

(15.118)

The set of lattice functions x _k → ψ _k ∈ ℂ, equipped with the scalar product

$$ \left( {\phi ,\psi } \right)=\sum\limits_{{k=1}}^{N} {{{{\bar{\phi }}}_{k}}{{\psi }_{k}}} , $$

(15.119)

define a Hilbert space. If ψ is normalized to one we may interpret |ψ _k |² as the probability of finding the particle described by the wave function ψ at site x _k . Then the expectation value of the position operator is given by

$$ {{\left\langle {\hat{x}} \right\rangle }_{\psi }}=\left\langle {\bar{\psi }\left| {\hat{x}} \right|\psi } \right\rangle =\sum {{{x}_{k}}{{{\left| {{{\psi }_{k}}} \right|}}^{2}}} \equiv \sum\limits_{{k{k}^{\prime}}} {{{{\bar{\psi }}}_{k}}{{x}_{{k{k}^{\prime}}}}{{\psi }_{{{k}^{\prime}}}}} . $$

(15.120)

As expected, the position operator $\hat{x}$ is diagonal in real space such that its matrix elements vanish if k ≠ k′. To introduce the SLAC derivative we switch to momentum space with wave functions $\tilde{\psi}(p_{\ell})\equiv \tilde{\psi}_{\ell}$ given by

$$ \begin{array}{*{20}{c}} {{{{\tilde{\psi }}}_{\ell }}=\frac{1}{{\sqrt{N}}}\sum\limits_{{k=1}}^{N} {{{\text{e}}^{{-\text{i}\,{{p}_{\ell }}{{x}_{k}}}}}{{\psi }_{k}}} ,} & {\ell =1,\ldots ,N} \\ \end{array}. $$

(15.121)

The inverse Fourier transformation reads

$$ \begin{array}{*{20}{c}} {{{{\tilde{\psi }}}_{k}}=\frac{1}{{\sqrt{N}}}\sum\limits_{{\ell =1}}^{N} {{{\text{e}}^{{-\text{i}\,{{p}_{\ell }}{{x}_{k}}}}}{{{\tilde{\psi }}}_{\ell }}} ,} & {k=1,\ldots ,N} \\ \end{array}. $$

(15.122)

We choose the {p _ℓ } symmetric with respect to the origin,

$$ p_\ell=\frac{2\pi}{N} \biggl(\ell-\frac{N+1}{2} \biggr) , $$

(15.123)

and with this choice the number of sites must be odd to obtain periodic wave functions and it must be even to obtain antiperiodic wave functions.

Now we seek a lattice momentum operator $\hat{p}$ which is diagonal in momentum space and has eigenvalues p _ℓ . This means that below the cutoff it has exactly the same eigenvalues as the continuum operator on the interval. Similarly as in the continuum we interpret $\vert\tilde{\psi}_{\ell}\vert^{2}$ as probability for finding the eigenvalue p _ℓ of $\hat{p}$. Then the mean value of $f(\hat{p})$ is

$$ \begin{gathered} {{\left\langle {f\left( {\hat{p}} \right)} \right\rangle }_{\psi }}=\sum\limits_{\ell } {f\left( {{{p}_{\ell }}} \right){{{\left| {{{{\tilde{\psi }}}_{\ell }}} \right|}}^{2}}=\frac{1}{N}} \sum\limits_{\ell } {\sum\limits_{{k{k}^{\prime}}} {{{\text{e}}^{{\text{i}\,{{p}_{\ell }}\left( {{{x}_{k}}-{{x}_{{{k}^{\prime}}}}} \right)}}}} f\left( {{{p}_{\ell }}} \right){{{\bar{\psi }}}_{k}}{{\psi }_{{{k}^{\prime}}}}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {=\sum\limits_{{k{k}^{\prime}}} {{{{\bar{\psi }}}_{k}}f{{{\left( p \right)}}_{{k{k}^{\prime}}}}{{\psi }_{{{k}^{\prime}}}},} } & {f{{{\left( p \right)}}_{{k{k}^{\prime}}}}=\frac{1}{N}} \\ \end{array}\sum\limits_{\ell } {{{\text{e}}^{{\text{i}\,{{p}_{\ell }}\left( {{{x}_{k}}-{{x}_{{{k}^{\prime}}}}} \right)}}}f\left( {{{p}_{\ell }}} \right)} . \hfill \\ \end{gathered} $$

(15.124)

Of course the operator $f(\hat{p})$ is non-diagonal in position space and to find its matrix elements f(p) _kk′ we define the generating function

$$ Z\left( x \right)=\frac{1}{N}\sum\limits_{{\ell =1}}^{N} {{{\text{e}}^{{-\text{i}\,N{{p}_{\ell }}x}}}} =\frac{{\sin \left( {\pi Nx} \right)}}{{N\sin \left( {\pi x} \right)}}. $$

(15.125)

The matrix elements are obtained by differentiation,

$$ f(p)_{kk'}= f \biggl(\frac{1}{\mathrm{i}N}\frac{\mathrm{d}}{\mathrm {d}x} \biggr) Z(x) \bigg\vert_{x=t_{kk'}},\quad t_{kk'}=\frac{k-k'}{N} . $$

(15.126)

In particular we find

$$ p_{kk}=0,\quad p_{k\neq k'}=\frac{\pi}{\mathrm{i}N}(-)^{k-k'} \frac{1}{ \sin(\pi t_{kk'})} , $$

(15.127)

and these matrix elements define the SLAC derivative $\partial_{\mathrm{slac}}=\mathrm{i} \hat{p}$ in position space.