---
title: Generalized Toric Code Hamiltonian
url: https://www.emergentmind.com/topics/generalized-toric-code-hamiltonian
type: topic
---

# Generalized Toric Code Hamiltonian

The generalized toric code Hamiltonian is a family of lattice Hamiltonians that extends Kitaev’s \(\mathbb Z_2\) toric code by modifying the gauge algebra, star terms, plaquette terms, symmetry content, perturbative origin, or emergent phase structure while retaining a toric-code-like organization of constraints, Wilson loops, or anyon sectors. In the literature surveyed here, this phrase encompasses commuting-projector quantum doubles, symmetry-deformed toric codes, Hamiltonians obtained from partial anyon condensation in the color code, modified toric codes with holes or flux attachment, and microscopic Hamiltonians whose low-energy or Floquet-effective descriptions reproduce toric-code physics [1011.1942] [2506.00114] [2508.19877].

## 1. Canonical structure and the baseline model

The standard toric code is defined on a square lattice with qubits on edges and Hamiltonian
\[
H_{\mathrm{TC}}=-J_e\sum_v A_v - J_m\sum_p B_p,
\qquad
A_v=\prod_{e\in +_v}\sigma_e^x,
\qquad
B_p=\prod_{e\in\partial p}\sigma_e^z.
\]
The vertex and plaquette stabilizers commute, and the ground state is the simultaneous \(+1\) eigenspace of all \(A_v\) and \(B_p\). On a closed surface of genus \(g\), the ground-state degeneracy is \(4^g\); on a torus it is four, labeled by non-contractible Wilson loops \(W_x\) and \(W_y\) [1608.04565] [2506.00114].

This canonical form supplies the reference point for essentially all generalizations. In one direction, the same topological order can be written in locally equivalent forms, such as Wen’s plaquette model,
\[
H_w=-\mathcal J\sum_{i,j} P_{i,j},
\qquad
P_{i,j}=X_{i,j}Z_{i,j+1}Z_{i+1,j}X_{i+1,j+1},
\]
which is mapped to the toric code by Hadamard gates on one checkerboard sublattice [2211.09724]. In another direction, the toric code is the \(G=\mathbb Z_2\) member of the broader quantum double family, where star and plaquette operators are replaced by group-theoretic projectors [1011.1942]. Generalization therefore does not refer to a single deformation, but to a controlled enlargement of the toric-code Hamiltonian paradigm.

## 2. Partial anyon condensation and modified toric-code phases

A particularly explicit route to a generalized toric code Hamiltonian arises from the 2D color code on a three-colorable hexagonal lattice. With qubits on vertices, the unperturbed color-code Hamiltonian is
\[
H_{\mathrm{CC}}=-\sum_p B_p^z-\sum_p B_p^x,
\qquad
B_p^x=\prod_{i\in p}X_i,
\qquad
B_p^z=\prod_{i\in p}Z_i.
\]
The perturbed model introduces color-resolved Ising interactions on colored edges,
\[
H=
-\sum_p B_p^x
-\sum_{c=r,g,b}(1-J_c)\sum_{p\in c} B_p^z
-\sum_{c=r,g,b} J_c \sum_{\langle i,j\rangle\in c} X_iX_j.
\]
The couplings \(J_c\in[0,1]\) act as physical control parameters for color-selective anyon condensation [2508.19877].

A key step is the Ising-basis transformation
\[
|\{\mu_p\}\rangle_{\mathrm{Ising}}
=
\prod_p [1+(-1)^{\mu_p}B_p^z]\,|+\rangle^{\otimes N},
\]
under which the Hamiltonian decomposes into three independent transverse-field Ising models on triangular lattices,
\[
H_{\mathrm{eff}}^{(c)}
=
- J_c \sum_{\langle i,j\rangle\in E_c}\bar X_i\bar X_j
-(1-J_c)\sum_{i\in c}\bar Z_i.
\]
This maps the color code with anisotropic Ising perturbations onto three decoupled color sectors and places the transition at the triangular-lattice TFIM critical value \(J_c^\ast\approx 0.17\) [2508.19877].

Condensing a single color sector produces an emergent toric code. For example, with \((J_r,J_g,J_b)=(1,0,0)\), a red-link basis transformation yields a toric-code Hamiltonian on the dual triangular lattice,
\[
H_{\mathrm{TC}}=-J_e\sum_v A_v - J_m\sum_t B_t,
\qquad
A_v=\prod_{e\ni v}\tilde X_e,
\qquad
B_t=\prod_{e\in\partial t}\tilde Z_e,
\]
where \(A_v\) is a six-body vertex operator and \(B_t\) is a three-body plaquette operator. In this sense, single-color anyon condensation realizes a phase transition from the \(\mathbb Z_2\times\mathbb Z_2\) color code to a \(\mathbb Z_2\) toric-code phase [2508.19877].

Condensing two color sectors produces a different generalized toric code. For \(J_g=J_b=1\) and \(J_r=0\), the effective Hamiltonian on green link qubits is
\[
\hat H
=
-\sum_{p\in G}\hat B_p^x
-\sum_{p\in R}\hat B_p^z
-\sum_{\langle i,j\rangle\in G}\hat X_i\hat X_j,
\]
which is a toric code on the green triangular lattice with missing plaquette terms on blue triangles and additional Ising couplings. The resulting ground state contains open \(X\)-strings ending on the blue triangles, so \(m\)-type fluxes are partially condensed only on a subset of plaquettes. The paper terms this regime a “partially topological phase,” and the emergent Hamiltonian is a modified toric code with holes and reduced anyon content [2508.19877].

The phase structure is diagnosed by color-resolved string order parameters
\[
S_c(\gamma_c)=\left\langle \prod_{(k,k')\in\gamma_c} X_kX_{k'} \right\rangle,
\]
which map to long-distance TFIM correlators \(\langle \bar X_i\bar X_j\rangle\to M_c^2\). The patterns \(S_r=S_g=S_b=0\), one nonzero \(S_c\), two nonzero \(S_c\), and all nonzero \(S_c\) distinguish the color-code phase, toric-code phases, partially topological phases, and trivial phase, respectively [2508.19877].

## 3. Symmetry-deformed toric codes

Another major class of generalized toric code Hamiltonians is obtained by deforming the Gauss-law star terms while keeping plaquette flux constraints. A general staggered form is
\[
H=-J_{s_1}\sum_{s\in s_1}A_s - J_{s_2}\sum_{s\in s_2}A_s - J_m\sum_p B_p,
\]
with model-dependent \(A_s\) and the standard \(B_p=\prod_{e\in\partial p}\sigma_e^z\). The “term-dropping” protocol expands the toric-code star operator into monomials in \(\sigma^\pm\) and selectively removes monomials to impose additional global or subsystem symmetries [2506.00114].

In the \(U(1)\)-symmetric toric code, the star term is the six-term charge-conserving projection of \(\prod_{l\in s}(\sigma_l^+ + \sigma_l^-)\), preserving total \(S^z\) on the star. In the \(XY\) toric code,
\[
A_s^{XY}=\

Source: https://www.emergentmind.com/topics/generalized-toric-code-hamiltonian