Partition function of the Kitaev quantum double model

1. Problem Statement

The paper addresses the challenge of computing the exact finite-temperature partition function for the Kitaev Quantum Double (KQD) model on arbitrary planar graphs that form the skeleton of a closed orientable surface of arbitrary genus $g$.

While the ground-state properties of KQD models (and related topological phases like the Toric code) are well-understood, understanding their behavior at finite temperature is crucial for assessing the stability of topological quantum memories. Previous studies established that topological order is destroyed at any finite temperature in the thermodynamic limit for local commuting projector Hamiltonians. However, for finite-size systems, a crossover temperature $T^* \propto 1/\ln L$ exists below which topological characteristics are preserved. To analyze these thermal fluctuations rigorously, an exact expression for the partition function $Z$ is required, which necessitates knowing the spectral degeneracies of all energy levels.

A specific difficulty arises in non-Abelian groups: the relationship between lattice excitations (vertex and plaquette violations) and the anyonic excitations of the underlying Topological Quantum Field Theory (TQFT) is not one-to-one. Specifically, "fluxons" in the TQFT (simple objects of the Drinfeld center) do not map directly to elementary plaquette excitations on the lattice when the group $G$ is non-Abelian.

2. Methodology

The authors employ a hybrid approach combining lattice gauge theory and topological quantum field theory (TQFT) concepts, specifically the theory of Drinfeld centers and fusion categories.

• Identification of Excitations:

  • They distinguish between energy eigenspaces (lattice excitations defined by violations of vertex/plaquette constraints) and anyons (simple objects of the Drinfeld center $Z(\text{Vec}_G)$).
  • Vertex excitations correspond to non-trivial irreducible representations (irreps) of $G$ (Chargeons).
  • Plaquette excitations correspond to non-trivial conjugacy classes of $G$.
  • Crucially, they identify that for non-Abelian groups, a single lattice plaquette excitation corresponds to only one subtype of a fluxon anyon, whereas the fluxon itself may have multiple subtypes (internal multiplicities) arising from the fusion of vertex and plaquette excitations.

• Degeneracy Calculation via Fusion Trees:

  • The core of the derivation relies on counting the number of ways to fuse excitations to the vacuum.
  • They utilize the Moore-Seiberg-Banks formula to calculate the fusion channel degeneracy (topological degeneracy) for a system with $n$ vertex excitations, $m$ plaquette excitations, and genus $g$.
  • They incorporate internal multiplicities (non-topological degeneracies) specific to vertex excitations in non-Abelian groups. The internal multiplicity of an anyon $J$ is shown to equal its quantum dimension $d_J$.

• Derivation of the Partition Function:

  • By summing over all possible numbers of excitations weighted by their Boltzmann factors and the derived degeneracies, they construct the full partition function.
  • They provide two independent proofs for the degeneracy formula:
    1. Lattice Gauge Theory Perspective (Appendix C): Decomposing the Hilbert space into gauge orbits and using group representation theory (induced representations, stabilizer subgroups) to count states.
    2. Lattice Manipulation (Appendix D): Using "restructuring lemmas" to transform any graph into a canonical geometry (a "lollipop" or "theta" graph) where the spectrum can be solved exactly using representation theory of the quantum double $D(G)$.

3. Key Contributions

1. Exact Partition Function Formula:
   The paper derives a compact, closed-form expression for the partition function $Z$ of the KQD model for any finite group $G$ and any genus $g$:
   $$Z = \sum_{A \in Z(\text{Vec}_G)} S_{1,A}^{2-2g} (z_A^{\text{Ve}})^{N_v} (z_A^{\text{Pl}})^{N_p}$$
   Where:

   • $S_{1,A}$ is the element of the modular $S$-matrix connecting the vacuum to anyon $A$.
   • $N_v$ and $N_p$ are the number of vertices and plaquettes.
   • $z_A^{\text{Ve}}$ and $z_A^{\text{Pl}}$ are effective single-site partition functions depending on the group structure and temperature.

2. Resolution of the Anyon-Lattice Excitation Mismatch:
   The authors rigorously clarify the distinction between energy eigenspaces (lattice excitations) and anyons (TQFT sectors). They demonstrate that for non-Abelian groups, the set of plaquette excitations is a proper subset of the fluxon anyons. They introduce the concept of subtypes and show how the internal multiplicity of vertex excitations ($n_J = d_J$) compensates for this mismatch in the counting of states.

3. Generalization of Previous Results:
   While the partition function for the Toric code ($G=\mathbb{Z}_2$) and string-net models was known, this work generalizes the result to arbitrary finite groups (including non-Abelian ones like $S_3$) and arbitrary topologies.

4. Thermodynamic Limit Analysis:
   They analyze the behavior of the system as $N_v, N_p \to \infty$. They show that the partition function factorizes into independent vertex and plaquette terms (resembling decoupled Potts models) multiplied by a topological factor $|G|^{2g-2}$. This confirms that topological order is lost at any $T > 0$ in the thermodynamic limit, consistent with Hastings' theorem.

4. Key Results

• Spectral Degeneracy: The degeneracy of an energy level with $n$ vertex excitations and $m$ plaquette excitations on a genus $g$ surface is given by:
  $$D_G(g, n, m) = \sum_{A \in Z(\text{Vec}_G)} S_{1,A}^{2-2g} (q_A^{\text{Ve}} - 1)^n (q_A^{\text{Pl}} - 1)^m$$
  where $q_A^{\text{Ve}}$ and $q_A^{\text{Pl}}$ are integers related to the quantum dimensions and fusion rules.

• Finite-Size Scaling: For finite systems, the topological order is preserved below a crossover temperature $T^* \propto 1/\ln L$. The partition function captures the competition between thermal fluctuations and the system size.

• Specific Examples:

  • Abelian Groups ($\mathbb{Z}_N$): The result recovers the known partition function of $N$-state Potts models, with complete symmetry between vertex and plaquette contributions.
  • Non-Abelian Groups ($S_3$): The partition function does not factorize into independent vertex and plaquette terms due to the lack of symmetry between chargeons and fluxons (since $Pl \neq Fl$). The formula correctly accounts for the complex fusion rules and internal multiplicities of the $S_3$ quantum double.

• Thermodynamic Quantities: The authors derive expressions for average energy, specific heat (showing Schottky anomalies), and entropy. The entropy in the low-temperature limit approaches $\ln |G|^{2g-2}$, which is strictly less than the ground-state degeneracy entropy $\ln(\text{GSD})$ in the thermodynamic limit, highlighting the non-commutativity of the $T \to 0$ and $L \to \infty$ limits.

5. Significance

• Theoretical Foundation: This work provides a rigorous, exact solution for the thermodynamics of a broad class of topological lattice models. It bridges the gap between the microscopic lattice description (Hamiltonian eigenstates) and the macroscopic topological description (TQFT anyons).
• Quantum Memory Stability: By providing the exact partition function, the paper offers a precise tool to quantify the thermal stability of topological quantum memories. It confirms that while finite-size systems offer protection, the protection is fragile against thermal noise in the large-system limit.
• Methodological Advance: The use of fusion trees and the explicit handling of internal multiplicities for non-Abelian groups provides a new toolkit for analyzing other commuting projector models, including string-net models and potentially twisted quantum double models.
• Experimental Relevance: With recent experimental realizations of topological order in quantum processors and Rydberg atom arrays, understanding the finite-temperature partition function is essential for interpreting experimental data and designing error-correction protocols.

In summary, the paper successfully solves the long-standing problem of obtaining the exact finite-temperature partition function for the Kitaev Quantum Double model, revealing the intricate interplay between topological degeneracy, non-Abelian internal multiplicities, and thermal fluctuations.