The vertex code and its two-dimensional image

Hamiltonian and transfer matrix

Take a circle with L vertices and a qubit at each vertex. For p=0 the unaugmented complex has no A stabilizers. With only a transverse source,

\[H_z=-\frac K2\sum_j(1+Z_jZ_{j+1}),\qquad H_x=-g\sum_jX_j, \qquad H=H_z+H_x.\]

The source-free code space is spanned by the two aligned Z product states. The trace in this package includes the whole Hilbert space. Let \(D=e^{-\delta H_z}\), \(U=e^{-\delta H_x}\) and Q=DU, with delta=1/(TM). Inserting Z-basis states between factors gives

\[Z_M=C\sum_\sigma\exp\left[ \kappa_x\sum_{j,t}\sigma_{j,t}\sigma_{j+1,t} +\kappa_\tau\sum_{j,t}\sigma_{j,t}\sigma_{j,t+1}\right],\]

\[\kappa_x=\frac{\delta K}{2},\qquad \kappa_\tau=-\frac12\log\tanh(\delta g),\qquad C=e^{\beta KL/2} [\cosh(\delta g)\sinh(\delta g)]^{LM/2}.\]

Both directions are periodic for the ordinary trace. This is an exact finite-M representation of Q, including the normalization C. VertexModel uses L,M>=3 to avoid coincident-bond conventions.

The isotropic one-parameter family

At a chosen delta, set

\[K=\frac{2\kappa}{\delta},\qquad g=-\frac{1}{2\delta}\log\tanh\kappa.\]

Then kappa_x=kappa_tau=kappa and the normalized measure is the ordinary zero-field square-lattice Ising measure. Its exact bulk critical coupling is \(\kappa_c=\tfrac12\log(1+\sqrt2)\); see References. For a square spacetime torus L=M=n with delta=1, this path uses T=1/n. Changing n along that path changes the quantum temperature. It is not a Trotter refinement at fixed quantum H and T. A fixed-H refinement instead holds K,g,T fixed and increases M.

Wilson points and moments

Every collection of points is a Wilson 0-cycle. A same-slice quantum product \(\prod_j Z_j^{\nu_j}\) maps to the corresponding spin product. For diagonal products D_nu_t on different discrete time slices, the insertion is

\[\frac{\operatorname{Tr}(D_{\nu_0}Q D_{\nu_1}Q\cdots D_{\nu_{M-1}}Q)} {\operatorname{Tr}Q^M}.\]

This fixes the Euclidean multi-time meaning of a general VertexObservable point pattern. It is not a real-time correlation function.

For average=True the configuration value is the average of all translations of that product. A translated joint pattern and a product of two translated averages are different observables:

\[\frac1{LM}\sum_x A_xB_{x+r}\ne \left(\frac1{LM}\sum_x A_x\right) \left(\frac1{LM}\sum_y B_y\right)\]

in general. For a fixed-separation connected correlation, declare the joint point pattern explicitly and subtract the product of the two ensemble means after pooling. Repeated point insertions cancel modulo two.

The spacetime magnetization is \(m=(LM)^{-1}\sum_x\sigma_x\). Its powers sum products across all discrete times and positions. The Binder functional \(U_4=1-\langle m^4\rangle/(3\langle m^2\rangle^2)\) uses pooled moments. A finite torus has a rounded crossover; comparisons between sizes can provide finite-size critical indicators, with statistical, grid-interpolation and finite-size effects assessed separately.

Boundary twists: mathematical context

The current vertex executor samples the periodic-periodic model and Wilson point products. The following twist identities define useful terminology but do not advertise a twist API.

Let Z_ab be the spin partition sum with a reversed spatial seam if a=1 and a reversed temporal seam if b=1. The closed global-X operator \(\mathcal T=\prod_jX_j\) imposes temporal antiperiodicity:

\[\frac{\operatorname{Tr}(\mathcal T Q^M)}{\operatorname{Tr}Q^M} =\frac{Z_{01}}{Z_{00}}.\]

A repeated magnetic Euclidean insertion \(U_m=\exp[-\delta K Z_{L-1}Z_0]\) reverses the spatial bond on every slice. Thus \(\operatorname{Tr}[(U_m Q)^M]=C Z_{10}\). Both insertions give the corresponding combined seam choice at zero longitudinal source. U_m is a transfer-weight insertion, not a single same-time Pauli operator.

An expectation in a twisted background has its own denominator Z_ab. The four backgrounds are not equally weighted physical sectors of the ordinary periodic trace. For reweighting, reverse the desired bond signs explicitly. Reversing a seam twice composes modulo two; squaring its original reweighting factor is not the identity operation on the action. Practical twist-ratio sampling and its variance control require additional methods.