Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond

Technical Deep Dive: Matrix Product States for Modulated Symmetries

Overview

This technical deep dive summarizes the key results from the paper "Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond". The paper develops a generalized matrix product state (MPS) framework for characterizing one-dimensional systems with modulated symmetries, and uses this framework to:

1. Classify symmetry-protected topological (SPT) phases protected by modulated symmetries
2. Derive generalized Lieb-Schultz-Mattis (LSM)-type constraints for systems with modulated symmetries

The core technical contribution is the derivation of a "generalized push-through condition" that describes how modulated symmetry operations are represented in the virtual bond degrees of freedom of an injective MPS.

Methodology

The key steps are:

1. Derive a generalized "push-through" condition for how modulated symmetry operations act on the virtual legs of an injective, translationally-invariant MPS.
2. Use this generalized push-through condition to classify 1D SPT phases protected by modulated symmetries.
3. Apply the push-through condition to derive generalized LSM-type constraints for systems with modulated symmetries.

The paper considers several examples to illustrate the framework, including:

• Exponential symmetries
• Mixed charge-exponential symmetries
• Non-Abelian symmetries with an exponential structure

Results

SPT Classification with Modulated Symmetries

The paper derives the following key results for classifying SPT phases with modulated symmetries:

• For exponential symmetries with automorphism T(g) = g^b, the virtual bond cocycles ω(g,h) must satisfy ω(g,h)ω(h,g)^-1 = exp(2πik/(b^2-1)), where k ∈ ℤN. This classifies the SPT phases as ℤ{gcd(b^2-1,N)}.
• For mixed charge-exponential symmetries, the allowed virtual bond cocycles are restricted by the condition ω(ga, gb) ω(gb, ga)^-1 = exp(2πik(a-b)/(b-1)), where k ∈ ℤN. This yields SPT phases classified by ℤ{gcd(b-1,N)}.

LSM Theorems with Modulated Symmetries

The paper also derives generalized LSM-type constraints for systems with modulated symmetries:

• For two exponential symmetries with coprime exponents a and b, the system can only have a symmetric, gapped, and non-degenerate ground state if c | n, where c = gcd(ab-1, N) and n characterizes the on-site projective representation. Otherwise, an LSM constraint forbids such a ground state.
• For a non-Abelian (dihedral) symmetry group with an exponential component, the authors construct a model that cannot realize a unique gapped symmetric ground state, due to an LSM obstruction.

Interpretation

The generalized MPS framework developed in this work provides a unified approach for understanding the interplay between symmetries, topology, and constraints in 1D quantum systems. By deriving a push-through condition for modulated symmetries, the authors are able to systematically classify SPT phases and derive LSM-type theorems in this more general setting.

The examples showcased illustrate how this framework can capture a wide range of modulated symmetry structures, including exponential, dipole, and mixed charge-exponential symmetries. This extends the applicability of the MPS formalism beyond the previously studied case of global, unmodulated symmetries.

Limitations & Uncertainties

The paper focuses solely on 1D systems, leaving open the question of how to generalize these results to higher dimensions. Additionally, the framework is limited to injective, translationally-invariant MPSs, and does not address the classification of non-injective or symmetry-broken phases.

What Comes Next

The authors suggest several promising directions for future work, including:

• Extending the MPS framework to non-injective tensors, which could capture symmetry-breaking phases and other long-range-entangled states with modulated symmetries.
• Investigating how the virtual bond data manifests at boundaries of open chains with modulated symmetries.
• Generalizing the formalism to continuous modulated symmetries and more general non-Abelian symmetry groups.
• Developing fermionic and higher-dimensional generalizations of the modulated symmetry MPS framework.