⚛️ quantum physics

Excitation spectra and rank tomography of linear matrix product tangent spaces

This paper introduces a tangent-space method and rank tomography for matrix product states to analyze excitation spectra and expressivity in non-uniform quantum many-body systems, demonstrating its effectiveness in reproducing low-lying excitations and capturing finite-size precursors of the Mott-insulator to superfluid transition using the Bose-Hubbard model.

Original authors: Otto T. P. Schmidt, Iacopo Carusotto

Published 2026-07-07

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CC BY 4.0

Original authors: Otto T. P. Schmidt, Iacopo Carusotto

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

The Big Picture: Mapping the "Vibrations" of Quantum Systems

Imagine you have a complex machine made of many tiny, interconnected gears (a quantum system). Usually, scientists use a method called Matrix Product States (MPS) to find the machine's "resting position" (its ground state). It's like finding the most comfortable way to sit on a couch.

But what if you want to know how the machine vibrates when you tap it? What are its "notes" or excitation spectra?

This paper introduces a new way to listen to those vibrations. Instead of just sitting on the couch, the authors describe how to stand up and take a tiny step in every possible direction to see how the system reacts. They call this the Tangent Space Method.

1. The "Couch" and the "Step" (The Tangent Space)

Think of the quantum system's possible states as a giant, multi-dimensional landscape.

The Ground State: This is the lowest valley in the landscape where the system naturally settles.
The Tangent Space: Imagine standing in that valley. The "tangent space" is the flat, smooth floor right under your feet. If you take a tiny step in any direction on this floor, you are creating a "linear perturbation" (a small vibration).

The authors show how to mathematically define this floor for systems with open ends (like a chain of atoms, not a circle). They also explain how to remove "fake" steps. In quantum mechanics, some steps just look like you're rotating your view of the system (gauge freedom) rather than actually changing the physics. The authors' method filters these out, keeping only the steps that represent real physical changes.

2. The "X-Ray" of the Floor (Rank Tomography)

The paper's second major contribution is a way to take an "X-ray" of this floor to see how detailed it is. They call this Rank Tomography.

Imagine the floor isn't just a flat sheet; it's made of different colored tiles, where each color represents a specific number of particles (like 10 particles, 11 particles, etc.).

The Problem: Sometimes, the floor has "holes" or missing tiles in certain colors. If you try to walk (simulate a vibration) in a direction that requires a missing tile, your simulation will be inaccurate.
The Solution: The authors developed a way to count exactly how many tiles exist for each particle number. They call this the Particle-Resolved Schmidt Rank Distribution (PRSR).

The Analogy:
Think of the "Bond Dimension" (a standard number used in these calculations) as the size of a suitcase.

Old Way: You just know you have a "Medium" suitcase.
New Way (PRSR): You open the suitcase and see exactly how many socks, shirts, and pants are inside. You might find that even though you have a "Medium" suitcase, you have packed so many socks that you have no room left for shirts.
Why it matters: This explains why two systems with the same "suitcase size" can give different results. One might be great at simulating "sock" vibrations (adding particles) but terrible at "shirt" vibrations (removing particles), simply because of how the internal space is arranged.

3. The Test Drive: The Bose-Hubbard Model

To prove their method works, the authors tested it on a famous quantum model called the Bose-Hubbard model. This model describes how atoms (bosons) move and interact on a grid.

The Scenario: They simulated a chain of 10 atoms. They wanted to see what happens as they change the "hopping" energy (how easily atoms move).
The Transition: They were looking for the moment the system changes from a Mott Insulator (atoms stuck in place, like people in a crowded elevator) to a Superfluid (atoms flowing freely, like water).
The Result: Their method successfully predicted the "notes" (energy levels) of the system. It even caught the subtle "softening" of the vibrations right before the atoms started flowing freely. This is like hearing a car engine sputter just before it shifts gears.

They also showed that if they broke the symmetry of the system (making the atoms not care about their exact count), the method still worked, proving it's a robust tool.

4. The Trade-Off: Size vs. Accuracy

The paper highlights a classic trade-off in computer simulations:

Small Suitcase (Low Bond Dimension): It's fast to pack and unpack (low computational cost), but you can only fit a few items. You get a rough approximation of the vibrations.
Big Suitcase (High Bond Dimension): It takes longer to pack, but you can fit more details. You get a very precise map of the vibrations.

The authors' "Tomography" tool helps scientists decide: Do I need a bigger suitcase, or is my current one just packed inefficiently?

Summary

In short, this paper does two main things:

Refines the Map: It gives a precise, algebraic way to calculate how quantum systems vibrate (excitation spectra) without getting confused by mathematical illusions.
Adds an X-Ray: It introduces a way to look inside the calculation to see exactly where the method is strong and where it is weak (based on particle numbers).

This allows scientists to understand not just what the energy levels are, but why the calculation is accurate or inaccurate, helping them choose the right tools for studying quantum materials.

Technical Summary: Excitation spectra and rank tomography of linear matrix product tangent spaces

Problem Statement
Matrix Product States (MPS) provide an efficient variational description for the ground states of one-dimensional quantum many-body systems. While the MPS tangent-space method is established for reconstructing low-energy excitation spectra, its application to finite, non-uniform systems with open boundary conditions (OBC) requires a rigorous formulation that accounts for the underlying algebraic structure. Furthermore, the expressivity and accuracy of the tangent-space method are not fully understood in terms of how the internal structure of the ground state controls the available directions in the tangent space. Specifically, it is unclear how the bond dimension alone dictates the ability to capture excitations in specific particle-number sectors, particularly near phase transitions where ground-state properties change abruptly.

Methodology
The authors formulate the variational space of MPS as an algebraic variety defined by rank constraints on tensor flattenings. By focusing on the smooth full-rank stratum of this variety, they derive an explicit representation of the tangent space that separates physical directions from those generated by gauge freedom.

Algebraic Formulation: The MPS variety is defined as the Zariski closure of the image of a polynomial contraction map. The authors utilize the left-canonical parametrization to define the parameter space as a product of Stiefel manifolds and a projective space. The tangent space is decomposed into vertical (gauge) and horizontal (physical) components. The horizontal tangent space is characterized by Hermiticity conditions on specific matrix products of tensor variations.
Linearized Excitation Method: Building on the Time-Dependent Variational Principle (TDVP), the authors derive a linearized method for excitation spectra. By considering linear perturbations around a variational ground state, they project the effective Hamiltonian onto the horizontal tangent space. This leads to a generalized eigenvalue problem $Bx = \omega Ax$ , where $A$ is the overlap matrix and $B$ is the projected Hamiltonian. The method involves compressing the basis to remove linear dependencies, resulting in a reduced Hermitian eigenvalue problem.
Rank Tomography: To analyze the expressivity of the method, the authors introduce "tangent-space tomography." This involves:
- Parametric Deficiency: Defining the deficiency $\delta^M_{par}$ in each particle-number sector $M$ as the difference between the target dimension of the sector and the dimension of the projected tangent space.
- Particle-Resolved Schmidt Rank (PRSR): Refining the standard bond dimension by decomposing the ground-state tensor flattening into blocks corresponding to specific particle numbers on either side of a cut. This yields a distribution $r_\ell(m)$ that captures the internal structure of the MPS.
- Block Structure: Showing that while ground-state tensors obey a particle-number selection rule ( $m' = m + n$ ), tangent variations decompose into blocks connecting all non-zero sectors, thereby generating particle- and hole-type excitations.

Key Results
The method is benchmarked using the Bose–Hubbard model on an open chain of $N=10$ sites.

Spectral Reconstruction: The linear MPS tangent-space method accurately reproduces the low-lying excitation spectra obtained via exact diagonalization. This holds true for both particle-number conserving Hamiltonians ( $\eta=0$ ) and those with explicit $U(1)$ symmetry breaking ( $\eta \neq 0$ ).
Finite-Size Precursors: The method successfully captures the softening of the particle-type mode near the finite-size precursor of the Mott-insulator to superfluid transition.
Accuracy and Bond Dimensions: The reconstruction accuracy depends on the bond dimension $D$ . While small bond dimensions provide approximations at lower computational cost, larger $D$ increases the available directions in the tangent space, improving spectral reconstruction.
Rank Profiles and Discontinuities: The study reveals that the accuracy of the reconstructed spectrum is not solely determined by the coarse bond dimension but by the particle-resolved rank profile. Even when the bond dimension pattern remains constant, a shift in the ground state's particle-number configuration (e.g., crossing from $M_0=10$ to $M_0=11$ ) alters the PRSR distribution. This change modifies the tangent rank in specific sectors, leading to discrete jumps in approximation error. For instance, in sectors where the tangent rank matches the target dimension, the spectrum is recovered exactly; where it falls short (parametric deficiency), errors arise.

Significance and Claims
The paper claims to provide a rigorous algebro-geometric foundation for the MPS tangent-space method applicable to finite, non-uniform systems with OBC. Its primary contributions are:

Explicit Construction: Deriving the tangent space and the associated effective Hamiltonian projection for non-uniform systems, explicitly handling gauge redundancy.
Expressivity Analysis: Introducing rank tomography (PRSR and parametric deficiency) to quantify the "missing" independent directions in the tangent space for specific particle sectors. This explains why states with identical bond dimensions can exhibit different spectral accuracies.
Practical Utility: Demonstrating that the method is a viable tool for calculating low-energy excitation spectra, provided the relevant sector ranks are sufficiently large. The authors suggest that the observed limitations (parametric deficiency) stem from the first-order nature of the ansatz and propose that extending to second-order perturbations (MPS double tangent-space) could address these shortcomings.

The work does not claim to solve the general many-body problem beyond the scope of the MPS ansatz but rather clarifies the structural limitations and capabilities of linear tangent-space methods within that framework.