Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 343 / 4

The Transverse-field Ising modelcommutes with translations, reflections, and the global spin flipFor $\lambda \gg 1$, the unique product ground state has $Z_i=-1$ at every site; for $\lambda \ll -1$, it has $Z_i=+1$. Both are symmetric paramagnets. For $|\lambda |\ll 1$, the two thermodynamic ground states are ferromagnets $|+{⟩}_X^{\otimes L}$ and $|-{⟩}_X^{\otimes L}$, exchanged by $G$; on a finite ring their even and odd cat combinations have an exponentially small splitting.

The large-field phase has a unique symmetry-preserving ground state, whereas the small-field phase spontaneously breaks the ${\mathbb{Z}}_2$ symmetry. These distinct symmetry realizations cannot be connected while retaining both a nonzero thermodynamic gap and the symmetry, so a phase transition must intervene. Kramers--Wannier duality locates the self-dual transitions at $|\lambda |=1$.

Use the $X$ eigenbasis $|s_i{⟩}_X$, $s_i\in {\mathbb{Z}}_2$, and put a dual qubit on every edge with $t_i=s_i+s_{i+1}(\mathrm{mod}2)$. The Kramers--Wannier intertwiner isAn explicit matrix product operator tensor isContracting neighboring virtual indices ${\beta }_i={\alpha }_{i+1}$ around the ring is the graphical MPO: each tensor copies its input bit to the left virtual leg and outputs the XOR of its two virtual legs.

Directly from the domain-wall definition,Therefore $DH(\lambda )=\tilde{H}(\lambda )D$, whereAfter exchanging $X$ and $Z$ and rescaling, this is the same Ising family at reciprocal coupling, so corresponding symmetry sectors have the same spectrum and $H(\lambda )$ is dual to $|\lambda |H(1/\lambda )$ up to the elementary sign conventions. In particular the Ising spectrum is self-dual at $|\lambda |=1$.

The original symmetry flips every $X$-basis bit, $s_i\mapsto s_i+1$, without changing any domain wall. Hencewith the present normalization. On the dual side, periodic domain walls obeyso the original global symmetry becomes a constraint on the dual symmetry sector.

Periodic versus antiperiodic boundary conditions determine whether the product of dual domain walls is $+1$ or $-1$. Conversely, the original even and odd $G$ sectors correspond to choices of dual boundary twist. Keeping all sectors therefore requires summing over both symmetry charges and both boundary conditions; within each matched sector the intertwiner is invertible up to normalization.

If an operator $O$ is symmetric, $[O,G]=0$, it descends consistently to a dual operator satisfying $DO=\tilde{O}D$. If it is nonsymmetric, it changes the global charge and cannot be represented by a local operator within one fixed dual boundary sector; its dual either changes the twist, acquires a disorder string, or is annihilated by the projected intertwiner.

The two-dimensional Kramers--Wannier projected entangled pair operator puts an input bit $s_v$ at every vertex and an output bit $t_e$ at every edge. A COPY tensor at each vertex sends $s_v$ to every incident virtual leg, and an XOR tensor on $e=(uv)$ imposesContracting the vertex-edge virtual legs gives the requested PEPO. Algebraically it is

The output domain walls are not independent. Around every plaquette $p$, each vertex bit occurs twice, soProducts of these loop operators generate a ${\mathbb{Z}}_2$ one-form symmetry: its charged objects are line operators, and contractible closed loops act trivially on the PEPO image. This symmetry is unavoidable because edge variables obtained as discrete gradients have zero flux around every contractible loop.

On the square lattice the PEPO intertwines the local operators asIts image also obeys the automatic zero-flux constraints $B_p={\prod }_{e\in \partial p}{\tilde{Z}}_e=1$. Thus the transverse-field Ising Hamiltonian maps to a ${\mathbb{Z}}_2$ lattice-gauge Hamiltonian generated by edge fields and stars in the zero-flux sector.

At the commuting-projector fixed point, promoting the image constraints to energetic terms givesthe toric code Hamiltonian. Conversely, within a simply connected zero-flux sector one may solve ${\tilde{Z}}_{uv}=X_u{X}_v$ and recover vertex Ising variables. On a torus, the remaining noncontractible loop eigenvalues correspond to the different Ising boundary twists, which accounts for the toric code's topological ground-state sectors.