[challenge]: Exact diagonalization benchmark for interacting Thouless pumps

[chengchen1987]

Released by

Chen Cheng, Lanzhou University

Contact email

chengchen@lzu.edu.cn

Method

Exact Diagonalization

Challenge issue

value: |

Challenge: Exact diagonalization benchmark for interacting Thouless pumps

Summary

This challenge uses exact diagonalization (ED) to study an interacting Thouless pump, using the spinful Rice--Mele--Hubbard chain as the minimal model. The goal is to compare four quantities:

$$ C_{\rm MB},\qquad \Delta_{\min},\qquad Q_{\rm adiabatic},\qquad Q_{\rm real\text{-}time}(T). $$

Here (C_{\rm MB}) is the many-body Chern number, (\Delta_{\min}) is the minimum many-body gap during the pump cycle, (Q_{\rm adiabatic}) is the pumped charge inferred from polarization winding, and (Q_{\rm real\text{-}time}(T)) is the transported charge during a pump with finite period (T).

The central question is:

$$ \textit{When do interactions preserve, destroy, or generate quantized Thouless pumping?} $$

Physics motivation

Thouless pumping is a classic example of quantized transport in one dimension. In the noninteracting Rice--Mele model, the pumped charge is determined by single-particle band topology. With Hubbard interactions, however, the problem becomes genuinely many-body: topology must be diagnosed using many-body eigenstates, gaps, Berry phases, and real-time dynamics.

This topic is still active. Recent experiments and theory have studied Hubbard--Thouless pumps, interaction-induced breakdown of pumping, and interaction-enabled pumping in nonsliding lattices.

Relevant references include:

• D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
• A.-S. Walter et al., Nature Physics 19, 1471 (2023).
• K. Viebahn et al., Phys. Rev. X 14, 021049 (2024).

Model

The baseline model is the spinful Rice--Mele--Hubbard chain,

$$ H(\phi) = -\sum_{j,\sigma} \left[t+(-1)^j\delta(\phi)\right] \left(c^\dagger_{j\sigma}c_{j+1,\sigma}+{\rm h.c.}\right) + \Delta(\phi) \sum_{j,\sigma} (-1)^j n_{j\sigma} + U\sum_j n_{j\uparrow}n_{j\downarrow}. $$

A standard pump cycle is

$$ \delta(\phi)=\delta_0\cos\phi, \qquad \Delta(\phi)=\Delta_0\sin\phi, \qquad \phi:0\rightarrow 2\pi. $$

The calculation should be performed in fixed particle-number sectors, for example at half filling,

$$ N_\uparrow=N_\downarrow=L/2. $$

Suggested system sizes are (L=6,8) as mandatory targets, and (L=10) as an optional advanced target.

Why exact diagonalization?

ED is not the best method for large one-dimensional systems; DMRG/MPS is better for thermodynamic-limit ground states and long-time dynamics. However, ED is ideal here because it gives exact finite-size access to

$$ E_n(\theta,\phi),\qquad |\psi_n(\theta,\phi)\rangle,\qquad \langle \psi(\lambda)|\psi(\lambda+\Delta\lambda)\rangle, \qquad |\psi(t)\rangle. $$

These are precisely the ingredients needed to compute many-body gaps, many-body Chern numbers, polarization winding, and exact finite-time pumped charge. ED should therefore be viewed as the exact small-system benchmark, while DMRG/MPS is the natural large-system follow-up.

Key ED skills

The implementation should pay careful attention to:

• fermionic signs in the spinful occupation basis;
• fixed (N_\uparrow,N_\downarrow) sectors;
• twisted boundary conditions,
  (c_{L+1,\sigma}=e^{i\theta}c_{1,\sigma});
• gauge-invariant Berry curvature, for example using the Fukui--Hatsugai--Suzuki method;
• minimum-gap tracking over the full ((\theta,\phi)) parameter grid;
• polarization phase unwrapping;
• norm-conserving real-time evolution.

Required measurements

The solution should compute and compare:

1. The low-energy spectrum (E_n(\phi)) and the minimum many-body gap
   [
   \Delta_{\min}(U)=\min_{\theta,\phi}\left[E_1(\theta,\phi)-E_0(\theta,\phi)\right].
   ]

2. The many-body Chern number (C_{\rm MB}(U)) on the ((\theta,\phi)) torus.

3. The many-body polarization (P(\phi)) and the adiabatic pumped charge (Q_{\rm adiabatic}).

4. The finite-time pumped charge (Q_{\rm real\text{-}time}(U,T)).

The key comparison is

$$ C_{\rm MB} \quad \text{versus} \quad Q_{\rm real\text{-}time}(T). $$

This distinguishes true topological breakdown from finite-time nonadiabatic breakdown.

Possible extensions

Possible directions toward new physics include:

• interaction-induced pumping, where the (U=0) system is trivial but finite (U) produces pumping;
• adding nearest-neighbor repulsion (V) and mapping the (U)-(V) plane;
• optimizing pump paths to maximize (\Delta_{\min});
• studying spin-charge diagnostics during the pump;
• benchmarking the ED results against larger-system DMRG/MPS calculations.

A possible paper-level goal is:

$$ \textit{Exact finite-size topology versus real-time transport in interacting Thouless pumps.} $$