🔬 mesoscale physics

Mixed Floquet Lattice model for gapless topology

This paper demonstrates that while a mixed Floquet lattice model with two incommensurate drives can realize momentum-resolved Weyl semimetal topology through power transfer, the total real-space response deviates from static Weyl physics and instead follows an effective Rice-Mele pumping structure, highlighting the unique challenges of translating gapless semimetallic phases into driven synthetic dimensions.

Original authors: Goutham Vinjamuri, Ashutosh Dubey, Ankur Das

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Goutham Vinjamuri, Ashutosh Dubey, Ankur Das

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Imagine you are trying to build a complex 3D city, but you only have a 1D street. How do you create the extra dimensions? In this paper, the authors use a clever trick called "synthetic dimensions." Instead of building physical roads, they use rhythmic shaking (driving forces) to create invisible "floors" in a mathematical city.

Here is the story of their discovery, broken down into simple concepts:

1. The Setup: A Shaking Street

Think of a single line of atoms (a 1D chain). Now, imagine you shake this line with two different rhythms at the same time. These rhythms are "incommensurate," meaning they never quite line up perfectly (like the ratio of a golden spiral).

The Real World: The physical line of atoms (1 dimension).
The Synthetic World: The two shaking rhythms act as two invisible, extra dimensions.
The Result: You have a "Mixed Floquet Lattice." It's a 3D world built from a 1D street and two shaking hands.

2. The Goal: Finding "Weyl Points"

In physics, there are special materials called Weyl semimetals. Imagine these as a 3D map where the energy landscape has "mountain peaks" and "valleys" that touch at specific points. These touching points are called Weyl nodes. They are like the "hubs" of a subway system where different lines cross.

The authors wanted to see if they could recreate these 3D "hubs" in their 1D shaking street. They wanted to know: If we shake a 1D line with two rhythms, can we create the same magical 3D topology?

3. The Discovery: It Depends on How You Look

The answer was surprising. The system behaves differently depending on whether you look at the whole street or just one specific spot on the street.

The "Momentum-Resolved" View (Zooming In)

Imagine you are a detective looking at just one specific house on the street (a specific momentum, $k_x$ ).

When you zoom in on this single house, the shaking creates a perfect 2D map.
On this map, you can clearly see the "Weyl hubs" (the topological features).
The Measurement: By measuring how much energy is transferred between the two shaking rhythms, the authors found a "count" (a number called the Chern number). This count perfectly matched the number of hubs for that specific house.
The Metaphor: It's like checking a single floor of a building and finding a perfect, working elevator shaft.

The "Total Real-Space" View (Zooming Out)

Now, imagine you step back and look at the entire street at once.

You might expect the whole street to act like a giant 3D Weyl semimetal.
The Surprise: It doesn't. The total energy transfer across the whole street does not show the complex 3D map of the Weyl hubs.
Instead, the whole street acts like a simple, 1D "pump" (specifically, a Rice-Mele pump). It's like a simple bucket brigade passing water from one end to the other.
The Metaphor: Even though every single floor has a perfect elevator, if you look at the whole building from the outside, it just looks like a simple water tower. The complex 3D structure gets "washed out" when you average everything together.

4. Why Does This Happen? (The Topological Obstruction)

The authors explain this with a concept called a "Topological Obstruction."

The Problem: The position of the Weyl hubs (the "hubs") can slide around continuously along the street.
The Rule: A "total" measurement (looking at the whole street) can only give you a whole number (like 0, 1, or 2). It cannot give you a sliding, continuous number.
The Conflict: You cannot map a sliding, continuous position (where the hubs are) onto a single, fixed whole number without losing information.
The Result: The "whole street" view is forced to forget the exact location of the hubs and only remembers a simple "on/off" switch (is the pump topological or not?). The detailed map of the hubs is only visible if you look at the street slice-by-slice.

5. The Big Takeaway

This paper proves that gapless systems (systems with touching points like Weyl semimetals) do not simply copy themselves into these synthetic shaking worlds.

Fully Gapped Systems (Insulators): If you have a system with no touching points (like a standard insulator), the whole system behaves nicely and copies the 3D topology perfectly.
Gapless Systems (Weyl Semimetals): If you have touching points, the "whole picture" view breaks down. The complex 3D topology only survives if you look at the system momentum-by-momentum (slice-by-slice).

Summary Analogy

Imagine a 3D hologram projected onto a 1D laser beam.

If you look at a single slice of the beam, you see the perfect, detailed 3D image (the Weyl nodes).
But if you try to look at the entire beam at once, the 3D image blurs into a simple, flat line. The complexity of the 3D shape cannot be captured by the total brightness of the whole beam; you have to scan through it to see the details.

The authors have shown that in these "shaking" synthetic worlds, you must scan through the system to find the topological secrets; you can't just look at the whole thing at once.

Technical Summary: Mixed Floquet Lattice Model for Gapless Topology

Problem Statement
While periodic driving (Floquet engineering) has successfully realized fully gapped topological phases, such as topological insulators and superconductors, in synthetic dimensions, the realization of gapless topological phases remains an open challenge. Specifically, it is unclear whether the characteristic physics of Weyl semimetals—characterized by band degeneracies (Weyl nodes) and gapless spectra—can be faithfully reproduced in Floquet synthetic dimensions. Previous work has largely focused on gapped systems where topological invariants (like Chern numbers) are well-defined over the entire Brillouin zone. This paper investigates whether a time-reversal-broken Weyl semimetal can be constructed in a mixed (1 real + 2 synthetic)-dimensional setting and whether its topological responses translate directly to the driven system.

Methodology
The authors construct a theoretical model of a one-dimensional chain of two-level systems subjected to two incommensurate driving frequencies, $\omega_1$ and $\omega_2$ (with ratio $\omega_1/\omega_2$ set to the golden ratio).

Synthetic Dimensions: The driving phases, $\theta_1 = \omega_1 t + \theta_{01}$ and $\theta_2 = \omega_2 t + \theta_{02}$ , are treated as synthetic momenta, effectively creating a 3D system where one dimension is real space ( $k_x$ ) and two are synthetic ( $n_1, n_2$ ).
Hamiltonian Construction: Starting from a static time-reversal-broken Weyl semimetal Hamiltonian, the crystal momenta $k_y$ and $k_z$ are replaced by the drive phases. This yields a time-dependent Hamiltonian $H(t) = \vec{\sigma} \cdot \vec{B}(t)$ .
Floquet Lattice Formalism: The system is mapped onto a tight-binding problem on a Floquet lattice. The Schrödinger equation is reformulated as a hopping problem where the driving frequencies act as an effective electric field $\vec{\omega} = (\omega_1, \omega_2)$ , creating a linear potential (Wannier-Stark ladder) in the synthetic directions.
Observables: The study analyzes two distinct responses:
1. Momentum-resolved response: Calculating the Chern number $C(k_x)$ for fixed real momentum $k_x$ by measuring the rate of energy (power) transfer between the two drives.
2. Real-space (total) response: Calculating the total power transfer across the entire system by Fourier transforming the Hamiltonian to real space and analyzing the dynamics of the full wavefunction.

Key Results

Momentum-Resolved Topology: For a fixed real momentum $k_x$ , the system behaves as a 2D gapped Floquet synthetic band. The rate of energy pumping between the drives is quantized and directly measures the $k_x$ -resolved Chern number, $C(k_x)$ .
- In the LCI (Low-Chern Insulator) phase, $C(k_x) = 1$ for all $k_x$ .
- In the W2 (Weyl-2) phase, $C(k_x)$ exhibits a step-like structure: it is 1 for $|k_x| < |k_0|$ (where $k_0$ is the Weyl node position) and 0 otherwise. This step function correctly identifies the separation of Weyl nodes.
- In the trivial phase, $C(k_x) = 0$ everywhere.
- These results align with the static Weyl semimetal phase diagram when viewed slice-by-slice.
Real-Space (Total) Response: In contrast to the momentum-resolved view, the total power transfer (integrated over all $k_x$ ) does not reproduce the static Weyl semimetal phase diagram.
- Instead, the total response follows the topology of an effective Rice–Mele pump.
- The system transitions between topological and trivial regimes based on whether the time-dependent trajectory of the intercell hopping parameter crosses a specific circle in the complex plane, rather than based on the presence of Weyl nodes.
- Consequently, the total power transfer fails to distinguish between different Weyl phases (e.g., W2 vs. W4) or to detect the continuous movement of Weyl nodes.
Topological Obstruction: The authors identify a fundamental topological obstruction preventing the total real-space pump from encoding the full Weyl topology.
- A total pump is a 1D gapped system characterized by a single integer invariant (winding number).
- The position of Weyl nodes ( $k_0$ ) varies continuously within a fixed Weyl phase.
- There is no injective map from a continuous parameter ( $k_0$ ) to a discrete integer group ( $\mathbb{Z}$ ). Therefore, a single integer invariant cannot encode the continuous location of Weyl nodes. The momentum-resolved response avoids this by providing a function $C(k_x)$ rather than a single integer.

Significance and Claims
The paper establishes that gapless semimetallic phases do not straightforwardly translate to Floquet synthetic dimensions in the same manner as gapped insulating phases.

Distinct Dynamical Structure: The driven system exhibits a "distinct dynamical phase structure" where the global response is governed by effective 1D pumping mechanisms (Rice–Mele) rather than the parent 3D Weyl topology.
Partial Realization: While the full Weyl topology is not captured in the total real-space response, the mixed Floquet lattice successfully captures the Weyl topology in a momentum-resolved sense. The $k_x$ -resolved power transfer serves as a robust marker for the Weyl nodes and their chirality.
Platform for Non-Equilibrium Topology: The work proposes mixed Floquet lattices as a viable platform for exploring non-equilibrium gapless topology, revealing that the dynamical response of driven gapless systems is qualitatively different from their static counterparts.

The authors suggest that these results are experimentally accessible using current platforms, such as driven cold-atom optical lattices, nitrogen-vacancy centers in diamond, or superconducting quantum circuits, where quantized energy pumping and momentum-resolved responses can be measured.