A while ago I found this excellent pdf on the arxiv, which provides an excellent pedagogical introduction to the quantum Ising chain. Not only that, but it achieves that rarity in physics which is often derisively called ‘rigor mortis,’ i.e. honest, full, mathematically correct definitions of all of the operators, spaces and fields up for grabs. It would be a waste of time to simply copy the PDF again but it is useful to summarise a few key exact duality transforms here should anyone need to find them in a hurry.
Formalism
Bosons
A boson is an eigenstate of the bosonic number operator \(a^\dagger a\), which is composed of operators satisfying \([a_i,a_j^\dagger] = \delta_{ij}\). A single pair \(a_i, a^\dagger_i\) acts on the Hilbert space \(\mathcal{F}_B = \operatorname{Span}\{ |n\rangle | n \in \mathbb{Z}_{\ge 0} \}\), while the full Hilbert space of \(N\) bosons is \(\mathcal{F}_B^{\otimes N}\). This space is sometimes called Fock space, hence the symbol \(\mathcal{F}\). A given many body boson state can be represented as a product of operators acting on the vacuum \(|\empty \rangle = |0,0,...,0\rangle\)
\[|n_1,n_2,...,n_N\rangle = \prod_{j=0}^N \frac{(a^\dagger)^{n_j}}{\sqrt{n_j!}} |\empty\rangle\]Note that the order is irrelevant, since differnet boson ‘flavours’ commute.
Fermions
A fermion is an eigenstate of the fermion number operator, \(c^\dagger c\), which is composed of operators satisfying \(\{c_i, c_j^\dagger\} = \delta_{ij}\). A single pair \(c_i, c^\dagger_i\) acts on the Hilbert space \(\mathcal{F}_F = \operatorname{Span}\{ |0\rangle, |1\rangle \}\), while the full Hilbert space of \(N\) (distinguishable) fermions is \(\mathcal{F}_F^{\otimes N}\). Again, products of operators can be used to signify the ‘bitstrings’
\[|n_1,n_2,...,n_N\rangle = \prod_{j=0}^N (a^\dagger)^{n_j} |\empty\rangle\]Note that a choice of what order to do the product in corresponds to a sign convention. This sign does not appear in expectation values, so is irrelevant.
An important kind of Fermion is a Majorana Fermion (sometimes called a ‘real Fermion’)
Spins
A spin is a collection of three operators acting on the Hilbert space \(\mathbb{C}^{2s+1}\). The collection of these operators also represents a \(2s+1\)-dimensional representation of \(SU(2)\), aka \(\text{Spin(s)}\). The generators of the Lie group (and therefore any of their representations) satisfy the canonical spin algebra
\[[S^i, S^j] = 2i \epsilon^{ijk}S^k\]It’s usually more helpful to reformulate these spins as ladder operators, \(S^\pm = \frac{1}{2}\left[S^x \pm i S^y\right]\).
This allows us to mimic two important bosonic commutators:
\([S^z, S^\pm] = \pm S^\pm\) \([a^\dagger a, a^\#] = \pm a^\#\)
where \(\# = \dagger\) if ‘+’ is chosen.
Things fall down when we look at the last one though: \([S^+, S^-] = S^z\) \([a, a^\dagger] = 1\)
Jordan-Wigner transformation
Define the string operator \(K_j = \prod_{l=0}^{j-1} \sigma_j^z\). The Jordan-Wigner fermion is defined as \(c_j = K_j S^+ _ j, c _ j^\dagger = (c _ j)^\dagger\).
These operators obey the correct anticommutation relations and have the correct dimensionality to form a complete set of commuting observables.
The Fourier Transform
Given some collection of operators \(c_r\) (fermions, bosons, spins, or something else) over a \(d\) dimensonal torus (With \(L\) sites in all directions), it is conventional to define a maximally nonlocal unitary roatioan of the Hilbert space using
\[c_k = L^{-d/2} \sum_j e^{-i k \cdot r } c_r\]Definitions
The transverse-field Ising Hamiltonian with open boundary conditions is given by
\[H_{OBC} = -J \sum_{j=1}^{L-1} \sigma_j^- \sigma_{j+1}^+ + \kappa \sigma_j^- \sigma_{j+1}^- + h.c. - h \sum_{j=1}^L \sigma_j^z\] \[= -J \sum_{j=1}^{L-1} c_j^\dagger c_{j+1} + \kappa c_j^\dagger c_{j+1}^\dagger + h.c. + h \sum_{j=1}^L (2c_j^\dagger c_j -1 )\]With periodic boundary conditions, end up with
\[H_{PBC} = H_{OBC} -J (\sigma_L^- \sigma_{1}^+ + \kappa \sigma_L^- \sigma_{1}^- + h.c.)\] \[= H_{OBC} -J (-)^{\hat{N}+1}(c_L^\dagger c_{1}^+ + \kappa c_L^\dagger c_{1}^\dagger + h.c.)\]Video Recordings and Exercises
Seminar 1: Introduction to fields
Filling in the gaps
Show that for Jordan-Wigner operators \(c_j, c_j^\dagger\)
- \[\{c_i, c_j^\dagger\} = \delta_{ij}\]
- \[\{c_i, c_j\} = \{c_i^\dagger, c_j^\dagger\} = 0\]
- \(\sigma^-_j \sigma^\pm_j = c^\dagger_j c^\#_j\), where \(\# =\dagger\) if \(\pm = -\)
Self Consistency Show that the fermion representaiton of \(\sigma^z_j = 1-2c^\dagger_j c_j\) still obeys the canonical commutators.
Operator Absorption Show that for any fermion \(c\), \(c^\dagger c c^\dagger = c^\dagger\). Hence show that \(c^\dagger = - \exp(i\pi c^\dagger c) c^\dagger\).
Essence of quantum mechanics Let \(\mathcal{H}\) be a finite dimensional vector space. Prove that if one has a collection of \(n\) commuting operators \(A_n\), each of which have \(m_i\) distinct eigenvalues \(\lambda_i\), then
- \[\operatorname{dim}(\mathcal{H}) = \prod_{i=1}^n m_i\]
- The set \(\{ \ket{\lambda_1} \otimes ... \otimes \ket{\lambda_n} \}\), where \(\ket{\lambda_i}\) run over all eigenvectors of \(A_i\), is a basis for \(\mathcal{H}\).
- If \(H = \sum_{i=1}^n \epsilon_i A_i\), where all of the \(A_i\) are positive semidifinite and \(\epsilon_i \ge 0\), then the lowest energy state corresponds to a choosing the minimum eigenvalues of all the \(A_i\)’s.
- Take two diagonalisable linear operators \(A,B\) on (finite dimensional) \(\mathcal{H}\). Show that \(A,B\) can be simultaneously diagonalised (i.e. there exists a common eigenbasis) if and only if \(A,B\) commute.
Seminar 2: The Fourier Transform
Rotations are a SU(2) automorphism Show that for any rotation matrix \(R \in SO(3)\) and \(SU(2)\) gnerators \(\sigma^i\) (i.e. operators satisfying \([\sigma^i, \sigma^j] = 2i\epsilon^{ij}_k \sigma^k\)),
\[[R^i_a \sigma^a, R^j_b \sigma^b] = 2i \epsilon^{ij}_k R^k_c \sigma^c\]Fourier Transforms
- Show that \(c_k\) satisfies the canonical commutator \([c_k, c_{k'}^\dagger]_\pm = \delta_{kk'}\) if and only if \(\frac{1}{L^d} \sum_r e^{-i(k-k')r} = \delta_{kk'}\). Show that this is the case if \(k = k_0 + \frac{2\pi n}{L}\) for \(n\in \mathbb{Z}^d\).
- Verify that, in 1D, the boundary conditions \(c_{L+1} = \pm c_1\) fix the value of \(k_0\) to \(0\) in the case of PBC and \(\pi/L\) in the case of ABC.
Filling in the gaps
- Determine what \(\mathcal{K}_p\) looks like in the case of an odd-length chain.
- Verify the following identites:
Commutation Station Let \(\Psi_k = \left( c_{k1}\ c_{k2},\, ...\, c_{kN},\ c^\dagger_{-k1},\,...,\,c^\dagger_{-kN} \right)^T\), i.e. regarded as a column vector. Show that the \(c\)’s are fermions (+) / bosons (-) if and only if
\[[\Psi_k^\alpha, \Psi_{k'}^\beta]_\pm = \delta_{kk'} \begin{pmatrix} 1& & & & \\ &\ddots& & & \\ & & 1 & & \\ & & & \pm 1 & \\ & & & & \ddots\\ & & & & & \pm 1 \end{pmatrix}^{\alpha \beta}\]