⚛️ quantum physics

Site Basis Excitation Ansatz for Matrix Product States

This paper introduces the Site Basis Excitation Ansatz (SBEA), a highly efficient method for computing elementary excitation spectra in one-dimensional quantum lattice systems using infinite matrix product states, which leverages a non-orthogonal basis constructed via a single-site diagonalization and avoids gauge constraints to achieve high accuracy in calculating dispersion relations and reconstructing Wannier excitations.

Original authors: Steven R. White

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Steven R. White

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

Imagine you are trying to understand the music of a very long, complex instrument, like a giant piano with infinite keys. You want to know what notes (excitations) it can play and how they sound at different speeds (momentum).

In the world of quantum physics, this "instrument" is a chain of atoms, and the "notes" are tiny ripples of energy called magnons (or spin waves). For decades, scientists have used a powerful tool called DMRG (Density Matrix Renormalization Group) to study these systems. It's like having a super-smart calculator that can figure out the ground state (the quietest, most stable state) of the instrument perfectly.

However, figuring out the other notes (the excited states) has been tricky. The old method, called the Excitation Ansatz (EA), was like trying to find a specific note by guessing a single key to press, then checking if it sounds right. If you wanted to know the sound at a different speed, you had to start the whole guessing game over again. It was accurate but slow and repetitive.

Steven White, a physicist at UC Irvine, has introduced a new, clever shortcut called the Site Basis Excitation Ansatz (SBEA). Here is how it works, using some everyday analogies:

1. The "Infinite Piano" Problem

First, you need to know how the piano sounds when it's completely silent (the ground state). Usually, simulating an "infinite" piano is hard.

The Old Way: You try to build the infinite piano directly, which can be mathematically messy and sometimes the computer gets confused and stops working.
White's Trick: He suggests building a very long but finite piano first (like 150 keys). Once you have the sound of the middle of that piano, you realize the middle keys look exactly the same no matter how long the piano is. So, you just take that middle section and pretend it repeats forever. It's a much simpler way to get the "infinite" sound without the headache.

2. The "One-Note" vs. The "Orchestra"

The old method (EA) tried to find the perfect "single note" (a specific tensor) to create a ripple at a specific speed. To do this, it had to solve a massive, complicated puzzle for every single speed you wanted to check.

The Analogy: Imagine you want to know how a drum sounds when hit at different speeds. The old way was to hit the drum, listen, adjust the drum skin, hit it again, and repeat this for every single speed.

White's New Method (SBEA) changes the strategy:
Instead of finding the perfect note for every speed, he finds a small library of "building block" notes (a basis set) that can be mixed together to create any speed.

The Analogy: Instead of tuning the drum for every speed, you find 7 or 8 specific "drum sounds" (like a thud, a tap, a boom) that, when mixed together in different proportions, can recreate any sound you want.
How he finds them: He uses a quick, one-time calculation (like a single "Lanczos" step) to find the most important "drum sounds" that naturally occur in the system.

3. The "Non-Orthogonal" Secret Sauce

Here is the most surprising part. In math, we usually like our building blocks to be "orthogonal," meaning they are completely independent of each other (like the X, Y, and Z axes on a graph).

The Old Rule: Previous methods forced these building blocks to be perfectly independent. White found that this actually made the math harder and the results worse.
The New Rule: He lets the building blocks overlap and mix (non-orthogonal).
The Analogy: Imagine trying to describe a color. The old way said, "You must use pure Red, pure Green, and pure Blue, and they can't touch." White says, "Let's use a Red that's slightly orange, a Green that's slightly yellow, and a Blue that's slightly purple. Even though they overlap, mixing them is actually easier and gets you the exact color you want faster."
By allowing this "messy" overlap, the math becomes incredibly simple. Once you have your library of 7 building blocks, finding the sound for any speed is just a tiny, fast calculation (like solving a small puzzle) rather than a massive one.

4. The "Wannier" Excitations (The Localized Ripple)

Finally, the paper shows how to create a "Wannier excitation."

The Analogy: In band theory (physics of solids), we have "Bloch waves" (ripples that travel everywhere) and "Wannier functions" (ripples that stay stuck in one spot).
White shows how to take his traveling ripples and combine them to make a "localized ripple" that stays in one spot on the chain, looking like a little packet of energy. This is useful because it's easier to think about two particles interacting if they are sitting in specific "parking spots" (Wannier functions) rather than floating everywhere.

Why Does This Matter?

Speed: It turns a slow, repetitive process into a quick, one-time setup followed by instant results.
Simplicity: It removes the need for complex, confusing mathematical rules (gauge conditions) that used to trip up computers.
Accuracy: It produces results that are almost perfect for simple systems (like the spin-1 chain) and is ready to tackle much harder problems, like 2D materials (sheets of atoms), which are currently too difficult for computers to solve.

In summary: Steven White figured out that instead of trying to solve a unique, hard puzzle for every single question, you can solve one small, smart puzzle to build a "toolbox." Once you have the toolbox, you can answer any question instantly by just mixing the tools together. It's faster, simpler, and surprisingly more accurate.

1. Problem Statement

Calculating the elementary excitation spectra of one-dimensional quantum lattice systems is fundamental for theoretical understanding and experimental comparison. While Density Matrix Renormalization Group (DMRG) methods are highly effective for ground states, obtaining full dispersion relations (energy vs. momentum) for excited states is challenging.

Limitations of Existing Methods:
- Standard DMRG: Limited to a few excited states; obtaining a full dispersion curve is difficult.
- Excitation Ansatz (EA): A Matrix Product State (MPS) based method where an excitation is a plane-wave superposition of a ground-state MPS with a single local tensor replaced. While efficient, the standard EA requires a separate, full diagonalization for every momentum $k$ , which is computationally redundant.
- VUMPS (Variational Uniform MPS): The standard method for obtaining infinite MPS ground states can suffer from convergence issues depending on initial matrices and involves complex diagrammatic formulations.
- Gauge Constraints: Previous implementations of EA often imposed a left-orthonormal gauge condition to ensure orthogonality between excitations on different sites. The author argues this severely hampers convergence and efficiency.

2. Methodology: Site Basis Excitation Ansatz (SBEA)

The paper introduces the Site Basis Excitation Ansatz (SBEA), a variation of the EA that decouples the basis construction from the momentum diagonalization.

A. Ground State Preparation (Simplified iMPS)

Instead of relying on VUMPS or recursive iDMRG, the author proposes a simple method to generate a uniform infinite MPS (iMPS) from a finite system DMRG calculation:

Perform finite-system DMRG on a long open chain ( $N \gg \xi$ , where $\xi$ is the correlation length).
Insert a new site in the center of the chain and optimize the new tensor using a single Lanczos step (similar to a single-site DMRG update).
Because the new tensor sits between two orthogonal half-chains, it naturally forms a uniform structure ( $\dots AAA \dots$ ).
Apply the Orus-Vidal procedure to convert this into the canonical form ( $\dots \lambda \Gamma \lambda \Gamma \dots$ ) or symmetric form ( $\dots AAA \dots$ ).

Advantage: This approach is simple, robust, and avoids the initialization sensitivity of VUMPS.

B. Constructing the Excitation Basis

Rather than optimizing a specific tensor $B(k)$ for each momentum, SBEA constructs a small, fixed basis $\{B_\alpha\}$ that spans the excitation space:

Single-Site Lanczos: A local effective Hamiltonian is constructed for a single site embedded in the infinite ground state.
Eigenstate Extraction: The lowest $N_\alpha$ eigenstates of this effective Hamiltonian are computed. These states serve as the basis tensors $B_\alpha$ .
Pre-truncation: To keep the basis size manageable, the environment of the excitation tensor is pre-truncated to a smaller bond dimension $\chi_{tr}$ before diagonalization. This retains the most relevant Schmidt vectors while discarding extended, low-probability states.
Gauge Choice (Crucial Innovation): The method does not impose a left-orthonormal gauge. The author demonstrates that imposing this gauge pushes eigenvalues to high energies and destroys the locality of the basis. By leaving the basis non-orthogonal, the method captures the low-energy spectrum with a very small number of basis states ( $N_\alpha$ ).

C. Non-Orthogonal Band Theory

Once the basis $\{B_\alpha\}$ is established:

Kernel Calculation: Compute the overlap kernel $O_{\alpha\alpha'}(d)$ and Hamiltonian kernel $H_{\alpha\alpha'}(d)$ in real space, where $d$ is the distance between sites. Due to the finite correlation length, these kernels decay exponentially, allowing for a finite cutoff $L_c$ .
Fourier Transform: Transform the kernels to momentum space: $\tilde{O}(k)$ and $\tilde{H}(k)$ .
Generalized Eigenvalue Problem: For any momentum $k$ , solve the small $N_\alpha \times N_\alpha$ generalized eigenvalue problem:
$\tilde{H}(k) \mathbf{c} = E(k) \tilde{O}(k) \mathbf{c}$
This step is computationally trivial (akin to non-orthogonal band theory) compared to the full diagonalization required by standard EA.

D. Wannier Excitations

The paper constructs Wannier excitations by linearly combining the momentum eigenstates. These are localized, orthonormal states that span the single-magnon space. They are shown to be well-localized (within a few correlation lengths) and can serve as a basis for renormalized multi-magnon effective Hamiltonians.

3. Key Contributions

SBEA Algorithm: A highly efficient method to compute dispersion relations by solving a tiny generalized eigenvalue problem for all momenta after a single basis construction step.
Gauge Freedom: The identification that not imposing a left-orthonormal gauge is critical. It allows for a compact basis ( $N_\alpha \approx 7$ ) and rapid convergence, whereas the gauge-constrained approach requires a prohibitively large basis.
Simplified iMPS Generation: A "black-box" approach to generating infinite ground states using finite DMRG and a single Lanczos insertion, bypassing VUMPS initialization issues.
Wannier Excitations: A formalism to construct localized, orthonormal excitation states from the non-orthogonal MPS basis, bridging the gap between MPS methods and tight-binding/Wannier function concepts in band theory.

4. Results

The method was tested on the $S=1$ Heisenberg antiferromagnetic chain (Haldane chain), a gapped system with a known gap $\Delta/J \approx 0.41$ .

Accuracy: With a pre-truncation bond dimension $\chi_{tr}=8$ $χ_{t r} = 8$ and a basis size $N_\alpha=7$ $N_{α} = 7$ , the method reproduces the one-magnon dispersion curve with high accuracy.
- At $k=\pi$ , the calculated Haldane gap is $\Delta/J = 0.4107$ , compared to the exact value of $0.410479$.
- The dispersion matches time-dependent DMRG (tDMRG) results perfectly up to the band maximum ( $E_{max} \approx 2.7J$ ).
Efficiency:
- Calculating the overlap and Hamiltonian kernels takes less than a minute.
- Solving for 500 different $k$ values takes only a few seconds on a standard laptop.
- The method is significantly faster than standard EA, which requires a full diagonalization for each $k$ .
Limitations: The method works best for single-particle excitations. In the region where the single-magnon line enters the two-magnon continuum ( $k < 0.72\pi$ ), the single-particle ansatz is less accurate, and the convergence with $N_\alpha$ slows down.

5. Significance

Accessibility: The SBEA method lowers the technical barrier for computing excitation spectra. It replaces complex diagrammatic derivations and convergence-prone algorithms with simple linear algebra and standard DMRG steps.
Scalability: The approach is naturally extendable to larger unit cells and potentially to two-dimensional systems (via PEPS or similar extensions), where full dispersion calculations are currently computationally prohibitive.
Theoretical Insight: The work clarifies the role of gauge choices in MPS excitation ansatzes, showing that non-orthogonal bases can be more powerful and efficient than orthogonal ones for capturing low-energy physics.
Future Applications: The construction of localized Wannier excitations provides a pathway to study multi-particle interactions and construct effective low-energy Hamiltonians for complex quantum materials.