Symmetry-enforced agreement of Kohn--Sham and many-body Berry phases in the SSH--Hubbard chain

This study demonstrates that in the inversion-symmetric gapped regime of the SSH-Hubbard chain, the Kohn-Sham and many-body Berry phases coincide across all interaction strengths not because the electron density encodes the geometric phase, but due to symmetry-enforced matching of their $\mathbb{Z}_2$ sectors despite the many-body wave function exhibiting a nontrivial geometric response.

The Gist

Here is an explanation of the paper using simple language, analogies, and metaphors.

The Big Question: Can a "Shadow" Tell the Whole Story?

Imagine you are trying to understand a complex, 3D sculpture (the Real World). It has intricate curves, hidden depths, and a specific "vibe" or geometric shape.

Now, imagine you only have a 2D shadow of that sculpture projected on a wall (the Density). In the world of physics, specifically Density Functional Theory (DFT), scientists often try to understand the complex 3D object just by looking at its 2D shadow. The theory says: "If the shadow looks the same, the object must be the same."

This paper asks a tricky question: If the shadow (density) stays exactly the same, does the "vibe" or geometric twist (Berry Phase) of the object also stay the same?

Usually, the answer is "No." The shadow can look identical while the object inside twists and turns in completely different ways. However, the authors found a special case where the answer is surprisingly "Yes," but for a very specific reason.

---

The Setup: A Chain of Beads on a Ring

The authors studied a specific model called the SSH-Hubbard chain.

• The Chain: Imagine a necklace of beads (electrons) arranged in pairs (dimers). Some pairs are close together, some are far apart.
• The Ring: The necklace is closed into a circle.
• The "Push" (Interaction $U$): Imagine the beads have a personality.
  • At Low $U$, they are shy and don't mind being near each other.
  • At High $U$, they are extremely grumpy and refuse to sit next to each other (strong repulsion).
• The Twist ($\theta$): The researchers slowly rotate the entire necklace, inserting a "magnetic twist" into the loop. This is like turning a dial from 0 to 360 degrees.

The Experiment: What Happens When You Twist?

The researchers did two things:

1. The Real Calculation (Many-Body): They used a super-powerful computer method (DMRG) to calculate exactly how the grumpy/shy beads move and twist as they turn the dial. This is the Real Object.
2. The Shadow Calculation (Kohn-Sham): They tried to recreate the system using a "fake" version where the beads don't interact at all, but they forced the fake beads to cast the exact same shadow (density) as the real ones. This is the Shadow.

The Surprise Finding

As they turned the dial (increasing the twist) and made the beads grumpier (increasing $U$):

• The Shadow (Density): The shadow on the wall never changed. It looked exactly the same whether the beads were shy or grumpy. The density was "frozen."
• The Real Object (Wave Function): Inside, the real beads were doing a complex dance. Their geometric "vibe" (the Quantum Metric) changed wildly depending on how grumpy they were. When they got very grumpy, their movement froze up.
• The Twist (Berry Phase): Here is the magic. Even though the dance changed, the final twist the necklace made after a full 360-degree turn was identical for both the Real Object and the Shadow.

The "Why": The Rule of Symmetry

You might think, "Oh, so the shadow does tell us everything!"

No, that's not it. The authors explain that this agreement is a coincidence enforced by rules, not a general law.

Think of it like a locked door with a specific keyhole (Symmetry).

• In this specific model, the "lock" is Inversion Symmetry (the chain looks the same if you flip it).
• Because of this lock, the "twist" (Berry Phase) can only be one of two numbers: 0 or 180 degrees (like a light switch being On or Off).
• As long as the chain doesn't break (the energy gap stays open) and the lock stays in place, the twist must be the same. It's mathematically forced to be the same.

So, the "Shadow" (Kohn-Sham) and the "Real Object" (Many-Body) agreed on the twist not because the shadow contained all the geometric secrets, but because the rules of the universe forced them both to pick the same answer.

The "What If": Breaking the Rules

To prove this wasn't a universal law, the authors imagined a different scenario where the "grumpiness" of the beads changed as you twisted the dial.

• In this new scenario, the shadow could still stay frozen (unchanged), but the final twist could become anything.
• In this case, the Shadow and the Real Object would disagree.

This proves that the agreement in their main experiment was a special case caused by Symmetry, not because the density (shadow) is a perfect map of the geometry.

The Takeaway

1. Density isn't everything: Just because two systems look the same on the surface (density), their internal geometric "twists" can be totally different.
2. Symmetry is a powerful boss: In this specific model, the rules of symmetry forced the complex, interacting system to behave exactly like a simple, non-interacting one regarding the final twist.
3. The "Freezing" effect: As the particles got more repulsive (stronger interaction), their ability to move and respond to the twist died down (the Quantum Metric vanished), but the final "On/Off" twist remained locked in place by symmetry.

In short: The paper shows that while a simple model can sometimes mimic a complex one perfectly, it's often because the universe's "rules" (symmetry) left them no other choice, not because the simple model actually understood the complex physics.

Technical Summary

Here is a detailed technical summary of the paper "Symmetry-enforced agreement of Kohn–Sham and many-body Berry phases in the SSH–Hubbard chain" by Kai Watanabe.

1. Problem Statement

The central question addressed is whether a density-matching Kohn–Sham (KS) description in Density Functional Theory (DFT) can accurately reproduce the many-body Berry phase (a geometric/topological invariant) in strongly correlated insulators.

• Context: In non-interacting systems, topological quantities like polarization and Berry phases are determined by occupied Bloch bands and are readily accessible via DFT. However, in strongly correlated systems, the ground state is a genuine many-body wave function, and geometric phases are functionals of this wave function, not just the electron density.
• The Conflict: It is generally expected that a KS system (which reproduces only the ground-state density) might fail to capture interaction-dependent geometric holonomies (Berry connections) because the density alone does not uniquely determine the many-body wave function's phase structure.
• Goal: To determine under what conditions a density-matching KS description succeeds or fails in reproducing the many-body Berry phase, using a controlled 1D interacting topological model.

2. Methodology

The study utilizes the 1D Su-Schrieffer-Heeger (SSH)–Hubbard model on a ring as a minimal testbed for correlated topology.

• Model: A dimerized Fermi–Hubbard chain with $L=2N$ sites, hopping amplitudes $t_{i} = t[1 + (-1)^i\delta]$, and onsite repulsion $U$. The system is studied at half-filling ($N_e = 2N$) with periodic boundary conditions (PBC) to maintain a bulk excitation gap.
• Twist Protocol: A U(1) gauge twist (flux insertion) $\theta$ is introduced by transforming the hopping terms: $c_{j\sigma} \to e^{-i\theta j/2N} c_{j\sigma}$. This defines a cycle $\theta \in [0, 2\pi]$.
• Many-Body Calculation (DMRG):
  • The ground state $|\Psi_0(\theta, U)\rangle$ is computed using the Density Matrix Renormalization Group (DMRG) method (via the TenPy library).
  • The many-body Berry phase ($\gamma_{MB}$) is calculated via the Wilson loop of neighbor overlaps: $\gamma = -\text{Im} \log \prod_j \langle \Psi_0(\theta_j) | \Psi_0(\theta_{j+1}) \rangle$.
  • The quantum metric (local geometric response) is estimated from the magnitude of these overlaps.
  • The excitation gap is monitored to ensure the Berry phase is well-defined (non-degenerate ground state).
• Kohn–Sham Construction:
  • A non-interacting KS system is constructed to exactly reproduce the many-body density $n_i(\theta, U)$ obtained from DMRG.
  • Due to inversion symmetry and half-filling, the density profile is constrained to two sublattice values ($n_A, n_B$). The KS potential reduces to a single parameter $\Delta_{KS}$ (staggered potential).
  • The KS Berry phase ($\gamma_{KS}$) is computed by diagonalizing the resulting non-interacting Hamiltonian with the same twist $\theta$.

3. Key Contributions

1. Density Independence in the SSH-Hubbard Model: The authors demonstrate that in the inversion-symmetric, gapped regime of the SSH-Hubbard model, the ground-state density is constant (independent of both the flux $\theta$ and interaction strength $U$) within numerical accuracy.
2. Collapse to a Free-Fermion Representative: Because the density is constant and symmetric, the density-constrained KS potential collapses to $\Delta_{KS} = 0$. Consequently, the KS reference system becomes identical to the non-interacting SSH model ($U=0$) for all values of $U$.
3. Decoupling of Local Geometry and Global Topology:
   • Local Geometry: The many-body quantum metric (local geometric response) shows strong dependence on $U$. It is significant at weak coupling but is strongly suppressed at large $U$ due to the freezing of charge fluctuations.
   • Global Topology: Despite the suppression of local geometric response and the fact that the KS system is $U$-independent, the Berry phases of the KS and many-body systems coincide exactly across the entire $U$ range.
4. Symmetry-Enforced Agreement: The paper argues that this agreement is not because the density encodes the many-body Berry connection. Instead, it is a result of symmetry-enforced $\mathbb{Z}_2$ sector matching. Both the interacting and non-interacting systems belong to the same symmetry-protected topological (SPT) class, which quantizes the Berry phase to $0$or$\pi$.

4. Key Results

• Numerical Verification: DMRG calculations confirm that the minimum excitation gap remains finite for all $\theta$ and $U$, validating the definition of the Berry phase.
• Quantum Metric Behavior:
  • At small $U$, the quantum metric is large and $\theta$-dependent, reflecting the sensitivity of the Slater determinant to flux.
  • At large $U$ ($U \gg t$), the quantum metric scales as $\sim (t/U)^2$, consistent with the suppression of charge fluctuations in the Mott insulating regime.
  • In contrast, the KS quantum metric remains constant (identical to the $U=0$ case) because the KS Hamiltonian is $U$-independent.
• Berry Phase Agreement:
  • Both $\gamma_{MB}$ and $\gamma_{KS}$ are found to be $0 \pmod{2\pi}$(or$\pi$, depending on the dimerization convention) throughout the gapped regime.
  • The agreement holds even though the KS system fails to capture the $U$-dependent suppression of the local quantum metric.
• Counter-Example Logic: The authors discuss that if the interaction term itself depended on the flux (e.g., $U \to U(\theta)$), the density could remain constant while the Berry phase changes, proving that density matching does not generally guarantee geometric equivalence. The agreement in the SSH-Hubbard case is specific to the symmetry constraints.

5. Significance and Implications

• Clarification of DFT in Topology: The work challenges the assumption that a successful density match in DFT implies a successful reconstruction of all geometric properties. It shows that in symmetric, gapped 1D systems, the agreement of Berry phases is a topological inevitability (due to $\mathbb{Z}_2$ quantization) rather than a proof that the density contains the necessary geometric information.
• Distinction between Local and Global: It highlights a structural hierarchy: local observables (density, quantum metric) can be insensitive to topology or interaction strength, while global topological invariants (Berry phase) remain robust and quantized by symmetry.
• Limitations of Density Constraints: The study suggests that relying solely on density-matching KS descriptions to infer interaction-dependent geometric responses is risky. The agreement observed here is a "symmetry-enforced coincidence" rather than a universal principle.
• Future Directions: The paper calls for testing regimes where protecting symmetries are broken (where Berry phases are not quantized) to see if KS descriptions still fail to capture the correct geometric phase, and exploring other SPT classifications (e.g., $\mathbb{Z}_8$).

In summary, the paper establishes that while density-matching KS descriptions can reproduce the correct topological phase in the SSH-Hubbard model, this is due to symmetry protection rather than the density encoding the many-body geometric structure. The local geometric response (quantum metric) is correctly captured by the many-body wave function but is completely missed by the density-constrained KS approximation in the strong-coupling regime.