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

1. Problem Statement

The interplay between symmetries and topology in one-dimensional (1D) quantum systems is traditionally understood through Matrix Product States (MPS). Standard MPS formalism successfully classifies Symmetry-Protected Topological (SPT) phases and formulates Lieb-Schultz-Mattis (LSM) theorems for systems with global (unmodulated) symmetries.

However, a new frontier has emerged involving modulated symmetries, where the unitary action of a symmetry operator varies spatially (e.g., dipole conservation, exponential symmetries, or fractonic constraints). While theoretical progress has been made on these systems, a unified MPS framework to:

1. Classify SPT phases protected by arbitrary spatial modulations.
2. Derive generalized LSM constraints for such systems.
   has been lacking. Existing methods often rely on specific models or cohomology formalisms that do not directly translate to the MPS tensor network language.

2. Methodology

The authors generalize the standard MPS formalism to translationally invariant systems with discrete modulated symmetries.

• Generalized Symmetry Push-Through Condition:
  In standard global symmetry, a symmetry operator $U_g$ acting on physical degrees of freedom can be "pushed through" the MPS tensor $A$ to act as unitaries $v$ and $v^\dagger$ on the virtual bonds, typically satisfying $U_g \cdot A \propto v^\dagger A v$.
  For modulated symmetries, the physical action depends on the site index $j$ (e.g., $U_{g,j} = U_g^{f(j)}$). The authors derive a site-dependent push-through condition:
  $$U_{g,j} \cdot A \doteq v_{j-1}^\dagger(g) A v_j(g)$$
  where $v_j(g)$ are site-dependent unitaries acting on the virtual bonds.

• Translation Invariance Constraints:
  By enforcing translational invariance on the MPS, the authors derive a consistency condition linking the virtual unitaries at different sites via the automorphism $T(g)$ induced by translation on the symmetry group $G$:
  $$v_j(g) \doteq v_{j-1}(T(g))$$
  This leads to a constraint on the 2-cocycles $\omega$ characterizing the projective representations of the virtual bonds:
  $$[\omega(g, h)] = [\omega(T(g), T(h))]$$
  This equation dictates which projective classes are allowed for a given modulated symmetry structure.

• LSM Constraints:
  The framework is extended to cases where the physical degrees of freedom carry projective representations (characterized by a cocycle $\nu$). The authors derive a consistency condition between the physical projective data ($\nu$) and the virtual projective data ($\omega$):
  $$\omega_j(g, h) = \nu_j(g, h) \omega_{j-1}(g, h)$$
  If no solution exists for the virtual cocycles $\omega$ that satisfies both the translation constraint and the physical representation, the system cannot host a unique, gapped, symmetric ground state (an LSM obstruction).

3. Key Contributions

A. Generalized Push-Through Theorem

The paper provides a rigorous proof (using transfer matrix arguments and injectivity) that for any injective MPS with a modulated symmetry, the symmetry action on the physical leg must be absorbed into site-dependent unitaries on the virtual legs. This generalizes the standard "push-through" lemma to non-uniform symmetry actions.

B. Classification of Modulated SPT Phases

Using the derived constraints on the 2-cocycles, the authors classify 1D SPT phases for specific modulated symmetries:

• Exponential Symmetries: For symmetries generated by $U_g = \bigotimes_j U_g^{b^{j-1}}$, the classification of SPT phases is determined by the condition $\omega(g, h)^{b^2-1} = 1$. For a group $G = \mathbb{Z}_N \times \mathbb{Z}_N$, the SPT phases are classified by $\mathbb{Z}_{\gcd(b^2-1, N)}$.
• Mixed Symmetries: The framework handles systems with multiple modulated symmetries (e.g., a global charge symmetry and an exponential symmetry), deriving constraints on mixed cocycles.
• Explicit MPS Construction: The authors construct explicit MPS tensors representing these SPT phases, verifying that the physical symmetry action is linear while the virtual bonds carry the required projective representation.

C. Generalized LSM and SPT-LSM Theorems

The paper formulates LSM constraints within the MPS framework for modulated symmetries:

• LSM Obstruction: If the physical site carries a projective representation that is incompatible with the translation-invariant virtual bond structure (i.e., the consistency equation for $\omega$ has no solution), the system cannot be a trivial gapped phase.
• SPT-LSM Theorem: Even if a solution exists, if all valid solutions for the virtual cocycle are non-trivial (i.e., $k \neq 0 \mod N$), the system is forced into a non-trivial SPT phase. This prevents a trivial gapped ground state.
• Examples:
  • Exponential Symmetry: For a $\mathbb{Z}_N$ chain with exponential symmetries $U_x = \prod X_j^{a^{j-1}}$ and $U_z = \prod Z_j^{n b^{j-1}}$, an LSM constraint arises unless $n$ is divisible by $\gcd(ab-1, N)$.
  • Non-Abelian Case: The authors analyze a system with a partially modulated Dihedral group $D_{2N}$, showing that for even $N$, an LSM obstruction exists due to the incompatibility of the site-dependent projective phase with the translation constraint.

D. Lattice Models

The authors propose explicit lattice Hamiltonians (e.g., Eq. 15 and Eq. 19) that realize these symmetries and demonstrate the predicted LSM constraints (e.g., forcing gaplessness or ground state degeneracy).

4. Results

• Unified Framework: The work successfully unifies the classification of SPT phases and the derivation of LSM theorems for arbitrary discrete modulated symmetries under a single MPS formalism.
• New Classification Groups: The paper derives specific classification groups for SPTs protected by exponential and mixed symmetries, recovering and extending previous results obtained via cellular chain complexes or defect networks.
• Connection to Conventional LSM: The authors demonstrate that certain modulated symmetry constraints can be mapped to conventional LSM theorems via local unitary transformations (e.g., charge conjugation on odd sites), providing a physical intuition for the obstructions.
• Weak vs. Strong SPTs: The framework distinguishes between strong SPTs (characterized by 2-cocycles) and weak SPTs (characterized by 1-cocycles/linear representations that vary with translation), providing a classification for the latter in modulated settings.

5. Significance

• Theoretical Unification: This work bridges the gap between the algebraic classification of fractonic/modulated phases and the tensor network language, offering a powerful tool for analyzing 1D quantum matter with complex conservation laws.
• Predictive Power: By providing a systematic method to check for LSM obstructions, the paper allows researchers to predict whether a proposed lattice model with modulated symmetries can host a trivial gapped phase or must be gapless/topological.
• Foundation for Future Work: The authors outline extensions to continuous symmetries, non-injective MPS (for symmetry breaking), and higher-dimensional generalizations (PEPS), setting the stage for a broader understanding of modulated symmetries in quantum matter.
• Experimental Relevance: The results are directly applicable to experimental platforms like cold atoms in tilted optical lattices and quantum Hall systems, where modulated symmetries naturally arise.

In summary, this paper establishes a rigorous "push-through" calculus for modulated symmetries in MPS, enabling the systematic classification of SPT phases and the derivation of no-go theorems (LSM) for a wide class of 1D quantum systems previously difficult to analyze within the tensor network framework.