[challenge]: Computing the geometric phase of 2d-TFIM

Released by

Si-Yuan Chen

Contact email

chance.siyuan@gmail.com

Method

PEPS Based Algorithm

Challenge issue

Background

The two-dimensional transverse-field Ising model (2D TFIM) is a standard benchmark for equilibrium and nonequilibrium quantum many-body physics. Its ground-state properties and quantum phase transition can be accurately studied using sign-problem-free quantum Monte Carlo (QMC), infinite projected entangled-pair states (iPEPS), and, as a low-cost qualitative baseline, mean-field theory.

Generic real-time evolution in two spatial dimensions is substantially more difficult. Recent large-scale comparisons of MPS, tree tensor networks, two-dimensional tensor networks, and neural quantum states show that the available methods are complementary: no single method remains systematically accurate across all annealing and post-quench regimes [7].

We consider the square-lattice Hamiltonian
$H_0(\Omega,\Delta)=\Omega \sum_i X_i+J\sum_{\langle i,j\rangle}Z_iZ_j+\Delta\sum_i Z_i $

where $\Omega$ is the transverse field or Rabi drive, $J$ is the nearest-neighbor Ising interaction, and $\Delta$ is a longitudinal field or detuning.

The first target will be the uniform, unfrustrated square-lattice model. Spatially varying $J_{ij}$, long-range interactions, and inhomogeneous detunings are possible extensions, but they are outside the minimal initial scope.

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Research objective

The objective of this issue is to compute the ground-state adiabatic geometric phase, i.e. the Berry phase, of the 2D TFIM along closed loops in a controlled Hamiltonian parameter space.

For a nondegenerate instantaneous ground state $H(\boldsymbol\lambda)|\psi_0(\boldsymbol\lambda)\rangle = E_0(\boldsymbol\lambda)|\psi_0(\boldsymbol\lambda)\rangle$,
and a closed parameter-space loop $C$, the Berry phase is

$\gamma_B[C]=\oint_Ci\langle\psi_0(\boldsymbol\lambda)|\nabla_{\boldsymbol\lambda}\psi_0(\boldsymbol\lambda)\rangle\cdot d\boldsymbol\lambda.$

The corresponding Berry curvature is

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,\qquad A_\mu=i\langle\psi_0|\partial_\mu\psi_0\rangle .$

The main quantities to calculate are:

1. the Berry phase or geometric-phase density along selected closed loops;
2. the local Berry-curvature density over a two-parameter manifold;
3. its convergence with the iPEPS bond dimension $D$, CTMRG environment dimension $\chi$, and parameter discretization;
4. its behavior near the 2D Ising quantum critical region;
5. its finite-rate correction under an explicit slow evolution.

Because the total Berry phase is generally extensive, the thermodynamic-limit calculation should focus on either $\bar\gamma_B=\lim_{N\rightarrow\infty}\frac{\gamma_B}{N},$
or on a baseline-subtracted phase difference between two loops.

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Parameterizations to study

Two related parameterizations should be implemented.

A. Reference parameterization: the Kolodrubetz rotation

As the first benchmark, reproduce the global spin rotation introduced for the TFI model in Ref. [4]. In that construction, all spins are rotated about the transverse-field axis. The transverse-field term remains invariant, while the Ising interaction acquires a phase in its pair-creation and pair-annihilation components.

This parameterization has three major advantages:

• the Berry curvature has already been calculated for the 1D and 2D TFI models using QMC [4];
• the 1D model provides an exactly solvable benchmark;
• the associated curvature is nontrivial at zero longitudinal field and has a known critical-scaling interpretation.

This track should be treated as the primary validation target.

B. Experiment-native laser-phase parameterization

For Rydberg or globally driven spin implementations, introduce the phase $\phi$ of the transverse drive:

$H_{\mathrm{R}}(\Omega,\Delta,\phi)=\Omega\sum_i\left(\cos\phi,X_i+\sin\phi,Y_i\right)+J\sum_{\langle i,j\rangle}Z_iZ_j+\Delta\sum_i Z_i$.

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Why this phase is important

1. Quantum geometry and topology

The Berry connection and Berry curvature characterize the geometry of the ground-state manifold. Curvature integrals define Chern numbers and control several quantized and geometric response phenomena [1,5].

2. Detection of quantum critical behavior

Geometric phases and their derivatives can display singular or universal scaling near quantum phase transitions. This was established for one-dimensional spin chains [2,3] and was subsequently formulated in terms of the full quantum geometric tensor and adiabatic gauge potentials [5].

For the 2D TFI model, Ref. [4] found that the curvature density does not diverge in the same way as in one dimension, but can contain a weaker, nonanalytic critical contribution. A thermodynamic-limit iPEPS calculation may provide a complementary way to resolve this structure.

3. Connection to measurable nonadiabatic response

Berry curvature is not only a formal wave-function quantity. Under a slow ramp $\lambda_\mu(t)$, it appears as the leading velocity-dependent correction to the generalized force associated with another parameter $\lambda_\nu$ [4,5].

Schematically,

$\langle-\partial_{\lambda_\nu}H\rangle_v = \langle-\partial_{\lambda_\nu}H\rangle_0 + v_\mu F_{\nu\mu} +O(v_\mu^2).$

This gives a practical route for validating an equilibrium Berry-curvature calculation through finite-rate dynamics without directly extracting the global phase of a many-body wave function.

4. Relevance to Rydberg control and geometric gates

The phase of the Rabi drive is an experimentally programmable control parameter in Rydberg arrays [10]. Geometric and holonomic phases have also been proposed for controlled rotations and controlled-phase gates in interacting Rydberg atoms [8,9].

The present project is not itself a gate-design project, but computing the many-body geometric phase of a two-dimensional interacting array is a prerequisite for assessing whether such phases can be controlled, enhanced, or made robust at scale.

5. A benchmark for two-dimensional many-body methods

The same quantity can be approached through exact diagonalization, QMC, iPEPS ground states, gauge-invariant overlap formulas, and slow real-time evolution. Agreement among these methods would provide a useful benchmark for two-dimensional tensor-network algorithms.

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Previous work

Berry phase and criticality

• Berry introduced the geometric phase accumulated under cyclic adiabatic evolution [1].
• Carollo and Pachos showed that geometric phases can diagnose criticality in spin chains [2].
• Zhu studied scaling of the geometric phase near the XY-chain quantum critical point [3].
• Kolodrubetz developed a sign-problem-free QMC procedure for Berry curvature and applied it directly to the 1D and 2D TFI models [4].
• Kolodrubetz, Sels, Mehta, and Polkovnikov reviewed the relation between Berry curvature, the quantum geometric tensor, adiabatic gauge potentials, and nonadiabatic response [5].
• Fukui, Hatsugai, and Suzuki introduced a manifestly gauge-invariant discretization of Berry curvature and Chern numbers using link variables and plaquette Wilson loops [6].
  -Recent study on computational complexity of Berry phase estimation in topological phases of matter [11]

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References

1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984). DOI

2. A. C. M. Carollo and J. K. Pachos, “Geometric Phases and Criticality in Spin-Chain Systems,” Phys. Rev. Lett. 95, 157203 (2005). DOI

3. S.-L. Zhu, “Scaling of Geometric Phases Close to the Quantum Phase Transition in the XY Spin Chain,” Phys. Rev. Lett. 96, 077206 (2006). DOI

4. M. Kolodrubetz, “Measuring Berry curvature with quantum Monte Carlo,” Phys. Rev. B 89, 045107 (2014). DOI

5. M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, “Geometry and non-adiabatic response in quantum and classical systems,” Physics Reports 697, 1–87 (2017). DOI

6. T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances,” J. Phys. Soc. Jpn. 74, 1674–1677 (2005). DOI

7. J. Vovrosh et al., “Simulating dynamics of the two-dimensional transverse-field Ising model: A comparative study of large-scale classical numerics,” Phys. Rev. Research 8, 023311 (2026). DOI, arXiv

8. Z.-Y. Jin and J. Jing, “Geometric quantum gates via dark paths in Rydberg atoms,” Phys. Rev. A 109, 012619 (2024). DOI

9. L. S. Yagüe Bosch, T. Ehret, F. Petiziol, E. Arimondo, and S. Wimberger, “Shortcut-to-Adiabatic Controlled-Phase Gate in Rydberg Atoms,” Ann. Phys. (Berlin) 535, 2300275 (2023). DOI

10. S. Ebadi et al., “Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays,” Science 376, 1209–1215 (2022). DOI

11. Ryu Hayakawaet al., "Computational complexity of Berry phase estimation in topological phases of matter," arxiv

Start by locating the repository's existing PEPS or iPEPS implementation and its numerical validation entry points; no specific files or tests are identified in the issue. Reproduce the Kolodrubetz rotation benchmark first, then add Berry-phase or curvature calculations for the stated 2D TFIM parameterizations. Done means convergence is assessed across bond dimension, environment dimension, and discretization, with comparison near criticality and under slow evolution.