⚛️ quantum physics

Fast Hubbard-Thouless Pumping with Local Counterdiabatic Driving

This paper demonstrates that while standard variational counterdiabatic driving fails to maintain quantized Thouless pumping in interacting many-body systems due to the energy-dependent structure of the exact gauge potential, a refined local ansatz incorporating interaction costs and ground-state weighting successfully suppresses excitations and preserves quantization even at ultra-fast cycle times.

Original authors: Alexander Chansky

Published 2026-10-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Chansky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the rules of motion are different from what we experience in daily life. To move a quantum system from one state to another without disturbing it, scientists usually rely on a principle called adiabaticity. Imagine trying to turn a delicate glass sculpture without cracking it; you must move your hands so slowly that the glass has time to adjust to every tiny shift. In physics, this means changing the conditions of a system so gradually that it stays perfectly in its lowest energy state, or ground state, the entire time. This slow, careful approach is essential for tasks like "Thouless pumping," a process where particles are transported across a material in a precise, quantized amount, like a conveyor belt moving exactly one item per turn. However, in the real world, moving too slowly is a luxury. Experiments are plagued by noise, heat, and decoherence, which can destroy the fragile quantum state before the process is even finished. Scientists have long sought a way to speed up these processes without breaking the delicate quantum order, a challenge that has remained difficult to solve, especially when the particles involved interact with one another.

A new study by Alexander Chansky addresses this challenge by developing a method to accelerate quantum pumping while keeping the system perfectly stable, even when the particles are interacting. The research focuses on a specific model of a one-dimensional chain of particles, known as the Rice-Mele model, which is a standard testbed for understanding how electrons move in materials. The goal was to perform the pumping cycle much faster than the traditional slow limit allows. The researchers found that for particles that do not interact with each other, a known mathematical trick called counterdiabatic driving works well. This trick involves adding a special, auxiliary force to the system that cancels out the unwanted jitters caused by moving too fast. However, when the particles do interact—repelling each other like charged magnets—the standard version of this trick fails completely. The researchers discovered that the usual method for calculating the necessary extra force treats all particle arrangements as if they were equally likely, which is a poor approximation for a system that is actually sitting in its lowest energy state. This mismatch causes the fast pumping to fail, with the particles getting lost and the precise transport breaking down.

To fix this, the team developed a new, more sophisticated approach that takes the specific energy costs of particle interactions into account. Instead of using a single, average force for every movement, they designed a force that changes depending on the local environment of the moving particle. Specifically, they realized that the "price" a particle pays to hop to a neighboring spot depends on how many other particles are sitting nearby. By creating a force that adjusts its strength based on this local energy cost, they could construct a highly accurate, local set of instructions for the system. They tested this new method in computer simulations on rings of eight and twelve sites, which are small but complex enough to capture the essential physics of interacting particles. The results were striking: with their new method, the system could be driven twenty-five to fifty times faster than previous successful experiments, yet it still transported the particles with perfect precision. The number of particles moved per cycle remained exactly what the laws of quantum mechanics predicted, and the system stayed in its desired state with very high accuracy.

The success of this method hinges on a specific insight about how to measure the "error" in the system. The researchers found that the standard way of measuring error, which assumes the system is hot and chaotic, was misleading for their cold, ordered quantum system. By switching to a method that focuses only on the system's lowest energy state, they could tune the extra forces to be exactly what was needed. They constructed a solution with fourteen adjustable parameters, each corresponding to a different possible energy cost for a particle hop. This allowed them to capture the complex behavior of the interacting particles without needing to build a force that connects every particle to every other particle across the entire chain, which would be impossible to build in a real lab. The study shows that for interacting quantum systems, the choice of how to calculate the necessary corrections is just as important as the corrections themselves.

The implications of this work extend beyond just one model. The researchers demonstrated that their approach works consistently regardless of the size of the system they simulated, suggesting that the method could scale up to larger, more realistic materials. They also noted that the forces they designed are "local," meaning they only involve particles that are close to each other, making them potentially feasible to build in real experiments using ultracold atoms or other quantum simulators. While the current results are based on computer simulations, the author points out that the mathematical tools they used are well-suited for handling much larger systems. This work provides a clear roadmap for how to speed up quantum processes in the presence of interactions, a critical step toward making quantum technologies practical and robust against the noise of the real world. By understanding exactly how the energy landscape changes when particles interact, scientists can now design faster, more reliable ways to manipulate quantum matter, opening the door to new experiments in topological physics and quantum computing.

Technical Summary: Fast Hubbard-Thouless Pumping with Local Counterdiabatic Driving

Problem Statement
Counterdiabatic (CD) driving, or transitionless quantum driving, offers a framework to suppress non-adiabatic excitations in rapidly driven quantum systems by adding an auxiliary Hamiltonian. However, in many-body and topological systems, the exact CD term is typically non-local and involves multi-body interactions, rendering it experimentally inaccessible. While variational approaches (specifically the Sels-Polkovnikov framework) have successfully generated local nearest-neighbor (NN) approximations for non-interacting topological pumps, this paper investigates whether these methods remain effective when interactions are introduced. Specifically, the author examines the "Hubbard-Thouless" pump in the interacting Rice-Mele model, where onsite ( $U$ ) and nearest-neighbor ( $V$ ) repulsions compete to stabilize quantized pumping. The central challenge is that standard variational constructions fail in this interacting regime, leading to the breakdown of quantized transport even at moderately fast cycle times.

Methodology
The study employs exact diagonalization on periodic rings of $L=8$ and $L=12$ sites for a spinful Rice-Mele chain with Hubbard interactions ( $U=8t$ ) and nearest-neighbor repulsion ( $V=4t$ ). The system is driven through a Thouless pumping cycle with periods $T=2/t$ and $T=1/t$ , significantly faster than the adiabatic limit ( $T=50/t$ ).

The author evaluates three distinct approaches to constructing the CD term:

Standard Variational Approach: Applying the Sels-Polkovnikov variational principle using an infinite-temperature action ( $S_\infty$ ) with a simple NN ansatz. This minimizes the Hilbert-Schmidt norm of the generalized force $G = \dot{H} + i[A, H]$ averaged over all states.
Ground-State-Weighted Action: Modifying the variational principle to use a ground-state-weighted action ( $S_{gs}$ ), which penalizes errors specifically in the subspace of the instantaneous ground state rather than averaging over all configurations.
Interaction-Resolved Ansatz: Recognizing that the exact many-body gauge potential's amplitude depends on the interaction energy cost ( $\Delta E$ ) of a hop, the author constructs a new ansatz. This ansatz resolves NN currents by the specific interaction energy change they induce. The operator is defined as:
$A_{\Delta E} = \sum_{\epsilon} \left[ \alpha^{\text{intra}}_{\epsilon} \sum_{j,\sigma} J^{\text{intra}}_{j\sigma} P^{\text{intra}}_{j\sigma,\epsilon} + \alpha^{\text{inter}}_{\epsilon} \sum_{j,\sigma} J^{\text{inter}}_{j\sigma} P^{\text{inter}}_{j\sigma,\epsilon} \right]$
where $P_{j\sigma,\epsilon}$ projects onto configurations where a hop changes the interaction energy by $\epsilon$ . For the specific parameters used ( $U=2V$ ), this yields 14 independent coefficients determined by minimizing the ground-state action $S_{gs}$ .

Key Contributions and Results

Failure of Standard Variational Methods: For non-interacting fermions, the exact CD term is non-local but well-approximated by a few NN terms. However, in the interacting regime, the standard variational approach using $S_\infty$ fails. The infinite-temperature average penalizes the CD amplitudes uniformly without accounting for the specific energy costs of hops. Consequently, the resulting two-parameter NN term is too weak to prevent the pump from breaking down at $T=2/t$ ; the pumped charge drifts from the quantized value, and fidelity drops to near zero.
Necessity of Ground-State Weighting: Switching to the ground-state-weighted action ( $S_{gs}$ ) improves performance but remains insufficient when paired with a simple NN ansatz. The fidelity still degrades significantly after a few cycles, indicating that the operator content (simple NN hops) is inadequate to capture the structure of the exact gauge potential.
Success of the Interaction-Resolved Ansatz: The primary contribution is the demonstration that resolving currents by the interaction energy cost $\Delta E$ restores quantized pumping. The 14-coefficient ansatz, derived from $S_{gs}$ , successfully maintains the pumped charge at the ideal quantized value ( $Q=10$ after 5 cycles) for both $T=2/t$ and $T=1/t$ .
Scalability and Accuracy: The method exhibits size-independent accuracy. The fidelity per site remains high ( $F^{1/L} \approx 0.99$ ) for both $L=8$ and $L=12$ rings. Furthermore, the loss of fidelity per cycle is nearly independent of the driving speed, contrasting sharply with the uncorrected pump where errors accumulate rapidly with speed.
Comparison to Exact CD: While the exact CD term (restricted to the ground state) yields perfect fidelity ( $F=1$ ), it is strongly non-local. The proposed local ansatz captures 95–98% of the exact gauge potential's action on the ground state, providing a practical compromise.

Significance
The paper claims that for interacting topological systems, the choice of the variational norm is as critical as the choice of operators. The standard infinite-temperature action is insufficient because it averages over high-energy configurations irrelevant to the pump's dynamics. The work demonstrates that the structure of the exact many-body gauge potential—specifically its dependence on the interaction energy denominators—provides a blueprint for constructing effective local operators. By conditioning hopping terms on the surrounding occupation numbers (via the $\Delta E$ resolution), one can achieve fast, quantized pumping in interacting systems at speeds 25 to 50 times faster than previous protocols. This approach offers a pathway to realizing fast topological protocols in experimental settings involving ultracold atoms or solid-state systems where interactions are unavoidable.

Technical Summary: Fast Hubbard-Thouless Pumping with Local Counterdiabatic Driving

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