Topological Berry Phase

• Topological Berry phase is a quantized geometric phase acquired during adiabatic evolution, serving as an order parameter for symmetry-protected topological transitions.
• Computational methods discretize symmetry-respecting loops in parameter space to robustly evaluate Berry phases, even in finite strongly correlated systems.
• Experimental techniques like many-body polarization, STM spectroscopy, and cold atom interferometry validate the theoretical predictions of quantized Berry phases in quantum materials.

A topological Berry phase is a geometric phase acquired by a quantum state under adiabatic evolution around a closed trajectory in a multidimensional parameter space, with the crucial property that its quantization is protected by discrete symmetries or topological invariants. This quantized phase distinguishes between distinct symmetry-protected topological (SPT) phases and encodes robust invariants that cannot change without a closing of the energy gap. In strongly correlated, interacting, or crystalline systems, topological Berry phases generalize to quantized values such as $2\pi/N$ (for $Z_N$-protected phases) and serve as order parameters for topological transitions, including those in higher-order SPT (HOSPT) phases, topological crystalline insulators, and interacting spin chains. The precise construction and quantization mechanisms, as well as physical consequences and computational methods, depend on the system's symmetry group and parameter manifold.

1. Formal Definition and Physical Foundations

Consider a quantum Hamiltonian $H(\lambda)$ depending on adiabatically controllable parameters $\lambda$ (e.g., local gauge twists, conserved quantities, or external fields). The instantaneous ground state $|\Psi(\lambda)\rangle$ forms a smooth (possibly multivalued) section over the parameter space. The Berry connection is defined as

$$\mathcal{A}(\lambda) = -i \langle \Psi(\lambda) | \nabla_\lambda \Psi(\lambda) \rangle,$$

and the Berry phase acquired along a closed loop $C$ is

$$\gamma = \oint_C \mathcal{A}(\lambda) \cdot d\lambda.$$

Topological quantization emerges when $\gamma$ is protected by symmetry or topology, e.g., particle-hole symmetry, discrete lattice symmetry, or the mapping class of the parameter space (Kariyado et al., 2017, Araki et al., 2019, Sun et al., 19 Dec 2024).

In SPT phases with $Z_N$ symmetry, appropriate choices of local twist operators and symmetry-respecting cycles yield

$$\gamma_N = \frac{2\pi m}{N},\quad m \in \mathbb{Z},$$

so the Berry phase becomes a $Z_N$ topological invariant, insensitive to local perturbations that respect the symmetry (Kariyado et al., 2017).

2. Construction of $Z_N$ (and $Z_Q$) Quantized Berry Phases

For systems with internal $SU(N)$ or $Z_N$ symmetry, one introduces $N$ orthogonal twist operators $\{\hat A_i\}$ satisfying $\sum_{i=1}^N \hat A_i = 1$. The general gauge twist is

$$U(\boldsymbol{\phi}) = \exp\Bigl[i \sum_{i=1}^N \hat A_i \phi_i\Bigr],$$

with $(N-1)$ independent parameters due to the constraint $\sum \phi_i$. Parameterizing these as coordinates of a synthetic Brillouin zone (sBZ), one defines a Berry connection vector $\mathcal{A}(\boldsymbol\phi)$ and computes the phase along a closed, symmetry-invariant loop $C$: $$\gamma_N = \oint_C \mathrm{Tr}\,\mathcal{A}(\boldsymbol\phi)\cdot d\boldsymbol\phi \mod 2\pi.$$ If $C$ is invariant under a cyclic $Z_N$ rotation, symmetry enforces exact quantization in units of $2\pi/N$ (Kariyado et al., 2017, Araki et al., 2019).

Table: $Z_N$ Berry phases in explicit SPT models

Model	Symmetry	sBZ Dimension	Quantized Values
SU(3) AKLT chain	$Z_3$	2	$0,\,2\pi/3,\,4\pi/3$
SU(4) biquadratic spin chain	$Z_4$	3	$0,\,\pi/2,\,\pi,\,3\pi/2$

The construction extends to HOSPTs, e.g., the extended BBH model hosts a $Z_4$ Berry phase characterizing corner modes, quantized by $C_4$ symmetry and the specific local twist protocol (Araki et al., 2019).

3. Physical Interpretation and Topological Phase Transitions

A quantized Berry phase is nontrivial only if the system remains gapped along the integration path $C$. Topological phase transitions appear when the gap closes at singular points (Dirac nodes, nodal lines), at which the Berry phase jumps to a different quantized value. In the SU(3) and SU(4) AKLT models, phase transitions occur at the points where the sBZ Hamiltonian develops gapless features (Dirac cones for $N=3$, nodal lines for $N=4$), resulting in discrete jumps in $\gamma_N$ (Kariyado et al., 2017).

Similarly, in HOSPTs and crystalline systems, gap closings at high-symmetry points or along crystal-imposed loops correspond to changes in the quantized Berry phase and mark topological transitions, e.g., between trivial, weak, and strong phases in topological crystalline insulators (Alexandradinata et al., 2014).

4. Concrete Applications: SPT Chains and Higher-Order Topological Phases

Explicit evaluation of $Z_N$ Berry phases in finite-size SPT models reveals the utility:

• SU(3) AKLT Chain: The phase $\gamma_3$ interpolates between $0$ and $2\pi/3$ as a tuning parameter passes through the topological transition, with perfect quantization except at the gapless critical point.
• SU(4) Chain: The phase $\gamma_4$ jumps from $0$ to $\pi/2$ at the transition, with nodal lines appearing in the sBZ at criticality (Kariyado et al., 2017).
• Extended BBH Model (HOSPT): The $Z_4$ Berry phase determines the presence and degeneracy of corner states, establishing a bulk-corner correspondence; the quantization survives up to the interaction-induced gap closing (Araki et al., 2019).

In these cases, the Berry phase is directly computable from the many-body ground state via numerical exact diagonalization, with discretization errors controlled to high precision.

5. Computational Methodology for Berry Phase Evaluation

For finite-size systems, the standard approach is to discretize the chosen path $C$ in the sBZ into $M$ points $\{\boldsymbol\phi_j\}$ and compute ground states $|\Psi_j\rangle$ at each point: $$\gamma_N \approx -\mathrm{Im}\,\ln\Bigl[\prod_{j=0}^{M-1} \langle\Psi_j|\Psi_{j+1}\rangle\Bigr].$$ The system size required for convergence is modest since the Berry phase probes the topological character of the many-body entanglement within the twisted region, not the entire chain (Kariyado et al., 2017).

Symmetry-respecting integration cycles are essential: the path $C$ must be closed and invariant under the protecting symmetry to guarantee quantization. For HOSPTs and crystalline SPTs, one constructs specialized loops (bent Wilson loops, multicorner paths) determined by point group and lattice symmetries (Alexandradinata et al., 2014, Araki et al., 2019).

6. Extension to Other Contexts: Interacting Systems, Topological Order, and Floquet Systems

The $Z_N$ topological Berry phase paradigm carries over to interacting strongly correlated systems, where it serves as an interacting topological invariant robust against local perturbations as long as the symmetry and bulk gap persist (Kariyado et al., 2017, Araki et al., 2019). In higher-order and crystalline SPTs, Wilson loop eigenvalues and non-Abelian parallel transport generalize the Abelian Berry phase, classifying phases beyond conventional insulators (Alexandradinata et al., 2014).

In time-periodically driven (Floquet) systems, Berry curvature engineering with light offers prospective ultrafast control of topological sectors and observables such as high-order harmonics, as the topological Berry phase imprints on the emission spectrum of electron-hole pairs under strong-field excitation (Bai et al., 9 Apr 2024).

7. Physical Significance and Experimental Implementation

The topological Berry phase is a directly observable quantity in various experimental protocols:

• Many-body polarization measurement: The Berry phase constructed by adiabatically inserting a $2\pi$ flux (gauge twist) on a bond yields quantized values directly linked to bulk topological invariants.
• STM spectroscopy: Quantization manifests as level splitting or edge/corner zero modes whose existence is dictated by the Berry phase (Zhang et al., 2022, Araki et al., 2019).
• Spin chains and cold atoms: Local twist and Ramsey interferometry protocols permit direct detection of cluster or corner Berry phases in engineered quantum simulators (Alexandradinata et al., 2014).
• Spectroscopy of condensed matter systems: The $Z_N$ jump in spectral quantities, topological phase transition markers, and surface state protection (e.g., $\pi$-phase in topological insulator surfaces) are experimentally accessible through ARPES, transport, and quantum oscillation methods (Taskin et al., 2011, Imura et al., 2012).

The universal quantization, topological robustness, and symmetry protection of the Berry phase establish it as a foundational tool for classifying and identifying SPT and HOSPT phases, as well as for diagnosing critical points and bulk-boundary correspondence in quantum materials (Kariyado et al., 2017, Araki et al., 2019, Sun et al., 19 Dec 2024).