Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints

This paper utilizes matrix product states to classify one-dimensional modulated symmetry-protected topological phases, demonstrating their equivalence to internal SPT phases via the crystalline equivalence principle and Lyndon-Hochschild-Serre spectral sequence, while deriving new Lieb-Schultz-Mattis constraints that forbid symmetric short-ranged entangled ground states for systems with modulated and non-invertible symmetries.

The Gist

The Big Picture: A Dance with a Twist

Imagine a crowded dance floor where everyone is following a specific set of rules. In physics, these rules are called symmetries. Usually, the rules are simple: "Everyone clap at the same time" (Internal Symmetry) or "Everyone move one step to the left" (Spatial Symmetry/Translation).

This paper is about a special, more complicated dance called a Modulated Symmetry. Here, the rules change depending on where you are standing on the dance floor.

• If you are at the front, the rule might be "Clap twice."
• If you are at the back, the rule might be "Clap three times."
• The rule isn't just about what you do, but how the rule itself shifts as you move across the room.

The authors ask: What happens to the "topological" patterns (the hidden, unbreakable knots in the dance) when the rules themselves are shifting like this?

The Toolkit: The "Matrix Product State" (MPS)

To figure this out, the authors use a mathematical tool called Matrix Product States (MPS).

• The Analogy: Imagine the dance floor is a long line of people holding hands. To understand the whole line, you don't need to look at everyone at once. You just need to look at how Person A holds hands with Person B, and how Person B holds hands with Person C.
• MPS is like a "local instruction manual" for each person. It tells you how to connect with your neighbor. If the whole line is in a special "Topological Phase," these local instructions create a hidden, global pattern that can't be easily untangled.

The Main Discovery: The "Crystal Mirror"

The paper proves a concept called the Crystalline Equivalence Principle.

• The Analogy: Imagine you have a magic mirror. If you look at a dance where the rules change as you move (Modulated Symmetry), the mirror shows you a different dance where the rules are fixed, but the dancers are "twisted" in a specific way (Internal Symmetry).
• The Result: The authors show that these two dances are actually the same thing. You can solve the hard problem (shifting rules) by solving the easier problem (fixed rules with a twist). They proved that the number of possible "knots" (Topological Phases) in the shifting-rule dance is exactly the same as in the twisted-rule dance.

The Two Types of "Knots" (Classification)

The authors found that these special phases come in two flavors, which they call Strong Indices and Weak Indices.

1. Strong Indices (The Unbreakable Edge):

   • Analogy: Imagine a long rope. If you cut the rope in the middle, the ends might have a special "glow" or "fraying" that only happens because of the knot in the middle.
   • Physics: These are phases where the edges of the system behave strangely (like having extra energy levels) no matter how you look at them. They are "strong" because they are robust and protected by the shifting rules.

2. Weak Indices (The Hidden Charge):

   • Analogy: Imagine every dancer is wearing a badge. If you walk around the room, the badges change color based on your position. A "Weak" phase is like a pattern where the total number of badges of a certain color is balanced, but if you take a step, the balance shifts slightly.
   • Physics: These phases don't have "glowing" edges. Instead, they are defined by how the symmetry charges are distributed across the lattice. They are "weak" because they are more subtle and depend on the specific size or shape of the system.

The "Lieb-Schultz-Mattis" (LSM) Constraint: The "No-Go" Rule

One of the most exciting parts of the paper is applying this to the Lieb-Schultz-Mattis (LSM) theorem.

• The Old Rule: In normal physics, if you have a certain number of dancers and a specific rule, the system cannot sit still in a calm, quiet state. It must either break the rule (chaos) or keep dancing forever (gapless/excited).
• The New Discovery: The authors found that for these "shifting rule" dances, the "No-Go" rule is more nuanced.
  • Sometimes, the shifting rules make it impossible to have a calm state (just like the old rule).
  • But, sometimes, the shifting rules allow a calm state only if the dancers are holding hands in a very specific, complex, "entangled" way.
  • The Metaphor: It's like saying, "You can't sit still unless you are holding a secret handshake with your neighbor that no one else knows." If you try to sit still without that secret handshake, the system breaks.

Real-World Examples: Exponential and Dipole Dances

To prove their theory, they built two specific "dance models":

1. Exponential Symmetry: The rule changes exponentially as you move (e.g., 1, 2, 4, 8...). They showed how to build a lattice (a grid of atoms) that follows these rules and creates the "knots" they predicted.
2. Dipole Symmetry: This is like a rule where the "charge" of a dancer depends on their position relative to a neighbor. They showed how these create "dipole" knots.

The "Non-Invertible" Surprise

Finally, they looked at a weird type of symmetry called Non-invertible Kramers-Wannier symmetry.

• The Analogy: Imagine a magic trick where you can turn a "Clap" into a "Spin," but you can't turn the "Spin" back into a "Clap" perfectly. It's a one-way street.
• The Finding: They proved that if you have this one-way street symmetry combined with the shifting rules, the system is "anomalous." It's like a car that has a flat tire and a broken engine; it simply cannot exist in a calm, stable state. It must be chaotic or gapless.

Summary

In short, this paper is a masterclass in organizing chaos.

1. It takes a confusing situation (rules that change as you move) and says, "Hey, this is actually the same as a simpler situation (fixed rules with a twist)."
2. It categorizes the possible "knots" in these systems into "Strong" (edge effects) and "Weak" (charge distribution).
3. It uses this map to predict when a physical system is forced to be chaotic or entangled, providing a new "No-Go" list for physicists designing quantum materials.

It's like finding a universal translator for the language of quantum dance, allowing us to predict exactly how the dancers will move, even when the music is changing tempo every step of the way.

Technical Summary

1. Problem Statement

The paper addresses the classification of Symmetry-Protected Topological (SPT) phases in one spatial dimension (1+1D) protected by modulated symmetries.

• Modulated Symmetries: These are internal symmetries ($G_{int}$) that act non-uniformly in space due to their non-trivial intertwining with spatial symmetries ($G_{sp}$), such as lattice translations or reflections. The total symmetry group is a semidirect product $G = G_{int} \rtimes G_{sp}$.
• The Gap: While SPT phases with purely internal symmetries are well-understood (classified by group cohomology $H^2(G_{int}, U(1))$), and crystalline SPTs (protected by spatial symmetries) are understood via the Crystalline Equivalence Principle, a systematic microscopic classification for phases where internal and spatial symmetries are mixed (modulated) has been incomplete.
• Specific Challenge: It was unclear how standard classification schemes generalize when the internal symmetry action depends on the spatial position (e.g., $U(a)$ at site $n$ differs from site $n+1$ via a translation map).

2. Methodology

The authors employ Matrix Product States (MPS) as the primary theoretical framework to derive the classification microscopically.

• MPS Formalism: They utilize the fundamental theorem of MPS, which states that any gapped 1D ground state can be approximated by an injective MPS. They analyze how symmetry operators act on the MPS tensors ($A$).
• Symmetry Action: For a modulated symmetry $U(a)$, the action on the MPS tensor at site $n$ involves a position-dependent map $t^n(a)$ (for translation) or $r(a)$ (for reflection).
• Push-Through Relations: The authors derive "push-through" equations where the symmetry operator is moved through the MPS tensor, generating virtual operators ($V_n$) and phase factors ($\theta_n$) on the virtual bond space.
• Cohomological Analysis: By enforcing consistency conditions (associativity and group multiplication) on these virtual operators, they derive constraints on the projective phases (2-cocycles) and 1-cocycles.
• Lyndon–Hochschild–Serre (LHS) Spectral Sequence: The authors provide a novel MPS-based derivation of the LHS spectral sequence for the group cohomology $H^2(G_{int} \rtimes G_{sp}, U(1))$. This mathematical tool relates the cohomology of the total group to the cohomology of the subgroups, allowing them to map modulated SPTs to internal SPTs.

3. Key Contributions

A. Classification of Modulated SPT Phases

The paper establishes that modulated SPT phases in 1D are classified by the second group cohomology of the total symmetry group:
$$H^2(G_{int} \rtimes G_{sp}, U(1))$$
This result confirms the Crystalline Equivalence Principle for modulated symmetries: SPT phases protected by spatially modulated symmetries are in one-to-one correspondence with internal SPT phases protected by the same group (viewed as an internal symmetry), provided orientation-reversing elements are mapped to anti-unitary symmetries.

The classification naturally decomposes into two distinct indices:

1. Strong Indices: Correspond to translation-invariant projective representations of the internal symmetry group. These are elements of $H^2(G_{int}, U(1))$ that are invariant under the spatial symmetry action (e.g., $T^*[\omega] = [\omega]$). These manifest as symmetry-protected edge modes and degeneracy in the entanglement spectrum.
2. Weak Indices: Correspond to symmetry charges attached to unit cells (1-cocycles in $H^1(G_{int}, U(1))$) that are defined modulo the equivalence relation induced by the spatial symmetry action. These do not affect boundary projectivity but can be detected by inserting spatial defects (e.g., translation defects) and measuring the resulting symmetry charge or momentum shift.

B. Derivation of the LHS Spectral Sequence via MPS

The authors derive the conditions for lifting a $G_{int}$ SPT to a $G_{int} \rtimes G_{sp}$ SPT directly from MPS constraints. They show that these conditions (involving 2-cocycles $\omega$, 1-cocycles $\phi$, and their compatibility with spatial actions) exactly match the differentials and convergence conditions of the LHS spectral sequence. This provides a physical, microscopic interpretation of the spectral sequence terms (e.g., domain wall decorations).

C. Lieb-Schultz-Mattis (LSM) Constraints

The classification is applied to derive LSM-type theorems for modulated symmetries:

• Standard LSM: If the local symmetry operators form a projective representation that is incompatible with the strong indices of the modulated SPT classification (i.e., the projective phase cannot be "trivialized" by the spatial symmetry), the system cannot have a symmetric, short-range entangled (SRE) ground state. It must be gapless or spontaneously break symmetry.
• SPT-LSM Constraint: If the projective representation is compatible (allowing an SRE ground state), the ground state is still constrained. If the local projective phase is non-trivial, the resulting SPT ground state must possess non-trivial entanglement (degeneracy in the entanglement spectrum), even if it is short-range entangled in the bulk.

D. Non-Invertible Symmetries

The authors apply their framework to non-invertible Kramers-Wannier reflection symmetries in systems with exponential modulated symmetries. They prove that such non-invertible symmetries are anomalous in the presence of modulated SPT phases, implying that any system preserving both must be gapless or break symmetry.

4. Results and Examples

The paper validates the classification with two concrete lattice models:

1. Exponential SPT Phases:

   • Symmetry: $G_{int} = \mathbb{Z}_N \times \mathbb{Z}_N$ with translation acting as $(g_1, g_2) \to (a g_1, b g_2)$.
   • Classification: $H^2 \cong \mathbb{Z}_{(ab-1, N)} \times \mathbb{Z}_{(a-1, N)} \times \mathbb{Z}_{(b-1, N)}$.
   • Realization: Constructed using controlled-phase gates. The strong index corresponds to boundary projective phases, while the weak index corresponds to charges carried by translation defects.

2. Dipolar SPT Phases:

   • Symmetry: $G_{int} = \mathbb{Z}_N \times \mathbb{Z}_N$ (charge and dipole) with translation acting as $(g_0, g_D) \to (g_0, g_D + g_0)$.
   • Classification: $H^2 \cong \mathbb{Z}_N \times \mathbb{Z}_N$.
   • Realization: Constructed using a specific cluster-state-like Hamiltonian. The authors also provide a field theory perspective using foliated gauge fields, showing the topological response action matches the lattice classification.

5. Significance

• Unification: The work unifies the understanding of crystalline and internal SPT phases, demonstrating that the Crystalline Equivalence Principle holds robustly even for complex, spatially modulated symmetries.
• Microscopic Foundation: By deriving the LHS spectral sequence from MPS, the paper bridges the gap between abstract algebraic topology and concrete many-body physics, offering a practical tool for classifying new phases.
• New Constraints: The generalized LSM theorems provide powerful diagnostic tools for identifying gapless phases or symmetry breaking in systems with exotic symmetries (e.g., fractons, dipole conservation, and non-invertible symmetries).
• Experimental Relevance: The classification of weak indices via translation defects offers a concrete protocol for detecting these phases in experiments or numerical simulations (e.g., DMRG) by analyzing the response to lattice dislocations.

In summary, this paper provides a comprehensive framework for understanding topological phases in systems where internal symmetries are spatially modulated, establishing a rigorous link between group cohomology, matrix product states, and physical observables like edge modes and LSM anomalies.