The Uhlmann phase of Higher-Order Topological Insulators at Finite Temperature

Abstract: We have studied the finite-temperature topology of higher-order topological insulators (HOTIs) based on the Uhlmann phase, which is a phase angle of the Uhlmann overlap. As an example of HOTIs, the Hamiltonian of the Benalcazar-Bernevig-Hughes (BBH) model is constructed from Gamma matrices satisfying the Clifford algebra. This specific algebraic structure underpins the model's higher-order topological properties, including the quantization of the Uhlmann phase to $0$ or $π$. This quantization enables us to treat the abrupt jumps of the Uhlmann phase as an indication of the nontrivial topological phase of the BBH model at finite temperature. From the disappearance of these jumps, we determine the critical temperature at which the topological transition occurs. For a special choice of parameters, the Uhlmann overlap and the critical temperature can be computed analytically.

• The paper demonstrates that the Uhlmann phase quantizes to 0 or π, marking clear thermal topological transitions in HOTIs.
• It employs both analytical and numerical methods on the BBH model to establish the role of symmetry protection in preserving topological order at finite temperatures.
• The study provides precise critical temperature estimates and informs experimental measurement protocols for mixed-state topological phases.

Uhlmann Phase Analysis of Higher-Order Topological Insulators at Finite Temperature

Overview and Motivation

The paper "The Uhlmann phase of Higher-Order Topological Insulators at Finite Temperature" (2606.00479) investigates the topological properties of higher-order topological insulators (HOTIs) in thermal equilibrium, using the Uhlmann phase as a finite-temperature topological index. The focus is on the Benalcazar-Bernevig-Hughes (BBH) model, a prototypical HOTI showcasing quantized edge and corner states protected by reflection and chiral symmetries. Extending topological characterization from pure states via the Berry phase to mixed states via the Uhlmann connection, the authors provide a rigorous description of thermal topological transitions and derive analytic expressions for the critical temperature in special cases.

BBH Model and Topological Characterization

The BBH model, realized as a two-dimensional tight-binding Hamiltonian with alternating hopping, is constructed from Gamma matrices satisfying the Clifford algebra. This mathematical structure ensures reflection and chiral symmetries, prohibiting nontrivial mixing between distinct bands and enforcing quantization of topological invariants. At zero temperature, nontrivial topology manifests as quantized electric dipole moments and corner modes. Standard diagnostics exploit nested Wilson loops of the Berry connection to extract polarization quantization ($p_x = p_y = 1/2$ in the topological phase) and quadrupole moments.

At finite temperature, the primary challenge is generalizing topological invariants to mixed states. The density matrix, encompassing all eigenstates via Boltzmann weights, replaces the pure state wave function. Prior works have adapted the Berry phase to mixed states using the Uhlmann phase, a geometric phase acquired by parallel transporting amplitudes in the space of full-rank density matrices [Viyuela-1d, Diehl]. The Uhlmann phase is defined through the inner product of initial and parallel transported amplitudes, and its quantization is shown to be robust under the symmetry constraints of the BBH model.

Uhlmann Connection and Phase Quantization

The Uhlmann connection behaves as a non-Abelian gauge field on the amplitude space of density matrices, closely paralleling the Berry connection's role for pure states. For the BBH model, the Uhlmann Wilson loop—computed along discretized paths in momentum space—generates Uhlmann-Wannier centers analogous to Wannier centers in the Berry formalism. However, degeneracy in the Uhlmann-Wannier bands complicates nested Wilson loop construction for HOTIs at finite temperature, motivating the use of the Uhlmann phase as the primary index.

Numerical analysis reveals that the Uhlmann phase for the BBH model is strictly quantized to $0$ or $\pi$ across the Brillouin zone. Abrupt jumps between these values mark nontrivial topological regimes. The mathematical origin of quantization is traced to the block-diagonal structure of the BBH Hamiltonian, which is a linear combination of Clifford Gamma matrices. Detailed calculations demonstrate that only models built exclusively from these matrices (preserving reflection and chiral symmetry) yield quantization; admixtures violating this algebra abolish sharp jumps and quantization, producing continuous Uhlmann phase variation.

Analytically, the phase angle $\Phi^U = \arg[\cos r_1 + \cos r_2]$ strictly confines possible values, enforcing quantization under the Clifford algebraic structure. For the BBH model, this result is rigorously established.

Numerical Results: Topological Transitions and Critical Temperature

Numerical computations of the Uhlmann phase as a function of momentum and temperature demonstrate:

• At low $T$: Robust phase jumps between $0$ and $\pi$, indicating the persistence of topological order in thermal mixed states.
• At high $T$: Absence of jumps; the Uhlmann phase remains zero, signalling a trivial phase.

For each value of the parameter $m$, a unique critical temperature $T_c$ exists, above which the topological signature vanishes. The authors provide strong numerical results for $0$0 as a function of $0$1, with $0$2 near the zero-temperature phase boundary ($0$3) and a pronounced dip at $0$4 due to closing of the energy gap ($0$5).

For the special case $0$6, analytic computation is feasible: the Uhlmann overlap and phase transitions are derived explicitly, yielding a precise critical temperature $0$7. The analytic and numerical results are in rigorous agreement.

Implications, Theoretical and Practical

This work establishes the Uhlmann phase as a robust, quantized topological indicator for HOTIs under thermal fluctuations. The explicit connection between quantization and symmetry constraints advances the theoretical understanding of topological phase transitions in mixed states. Practically, the analytic expression for $0$8 in flat-band models provides a benchmark for experimental and numerical verification.

The findings have broader implications for quantum materials:

• Finite-$0$9 Topology: The extension of topological indices to density matrices enables prediction and classification of thermal topological transitions in realistic systems.
• Symmetry Protection: Quantization failure upon symmetry violation underscores the necessity of precise Hamiltonian engineering to preserve thermal topological phases.
• Measurement Protocols: The results inform strategies for quantifying topological order using mixed state geometric phases in experimental platforms, including cold atom systems, superconducting qubits, and optical lattices.

Future developments may include:

• Generalizing analytic treatments to wider classes of HOTIs and interacting systems.
• Exploring the Uhlmann connection as a universal diagnostic for thermal topological phenomena beyond reflection/chiral symmetry-protected models.
• Developing protocols for real-time measurement of the Uhlmann phase and topological transitions in noisy quantum hardware.

Conclusion

This paper rigorously demonstrates the quantization and diagnostic utility of the Uhlmann phase as a finite-temperature topological index for higher-order topological insulators, exemplified by the BBH model. The quantized $\pi$0/$\pi$1 phase structure—protected by Clifford algebraic symmetries—enables precise detection of thermal topological transitions and exact calculation of critical temperatures. The theoretical framework and analytic/numerical results deepen understanding of mixed-state topology, providing a foundation for further studies on thermal topological phenomena in quantum materials and engineered quantum systems.