Matrix product states and first quantization

This paper introduces a first-quantized Matrix Product State (MPS) approach that reformulates fermionic anti-symmetry to achieve entanglement levels comparable to or even lower than those in second quantization, thereby enabling efficient simulation of fermionic many-body systems like the 1D $t$-$V$ model.

The Gist

Imagine you are trying to organize a massive, chaotic dance party in a long hallway with 100 rooms. You have 50 dancers (particles), and they are all identical twins (fermions). There's a strict rule: no two dancers can ever be in the same room at the same time, and if you swap two dancers, the whole vibe of the party flips upside down (this is the "anti-symmetry" of fermions).

For decades, physicists have used two different ways to describe this party to predict how the dancers move.

The Old Way: The "Second Quantization" Method

The Analogy: Imagine you are a security guard standing at every single door of the hallway. You have a checklist for each room: "Is anyone in here? Yes (1) or No (0)."

• The Good News: This is great for computers. Because the dancers are identical, you only care about the rooms. If you look at half the hallway, the "entanglement" (how much the two halves of the party depend on each other) is usually small. It's like a calm, organized line.
• The Bad News: This method completely ignores which specific dancer is in which room. It treats them as a collective cloud.

The "First Quantization" Method (The Old Problem)

The Analogy: Now, imagine you are trying to track every single dancer by name. You have a list: "Dancer 1 is in Room 3, Dancer 2 is in Room 7..."

• The Problem: Because the dancers are identical, the universe doesn't care who is who. But your list does. To make the math work, you have to account for every possible permutation of the dancers. If you have 50 dancers, there are $50!$ (50 factorial) ways to arrange them. That's a number so huge it breaks computers.
• The Entanglement Explosion: In this view, the "entanglement" is massive. It's like trying to untangle a ball of yarn where every single strand is knotted with every other strand. For a long time, scientists thought, "Forget it. We can't use this method for big systems because the computer memory needed would be larger than the universe."

The New Breakthrough: The "Sorted List" Trick

The authors of this paper, Li and Waintal, found a clever shortcut. They realized they didn't need to track the chaos of who is where. Instead, they decided to force the dancers to line up in a specific order.

The Metaphor:
Imagine you tell the dancers: "Okay, Dancer 1 must be in a room to the left of Dancer 2, who must be to the left of Dancer 3, and so on."

• The Constraint: You only look at the scenarios where the dancers are perfectly sorted by their position.
• The Magic: By doing this, you automatically satisfy the "no two dancers in the same room" rule (Pauli exclusion principle). If Dancer 1 is in Room 5, Dancer 2 must be in Room 6 or higher. You don't need to check every permutation; you only check the one valid, sorted line.

The Result:
When they did this, the "entanglement" (the messiness of the math) dropped from "impossible" to "manageable." It became just as easy to calculate as the old "security guard" method, but with the added benefit of knowing exactly where the particles are.

What They Tested

They tested this new method on a model called the t-V model (a line of fermions that hop around and push each other away).

1. Ground State (The Calm Party): They calculated the lowest energy state of the system. The new method worked just as well as the old method, proving that sorting the dancers doesn't lose any important information.
2. Time Evolution (The Moving Party): They simulated a "domain wall"—imagine all the dancers starting on the left side of the hallway and then suddenly rushing to the right.
   • The Surprise: In the old "security guard" method, the chaos (entanglement) grew very fast as the dancers moved.
   • The New Method: Because the dancers were already sorted, the "mess" grew much slower. It was like watching a wave move through a line of people who are already holding hands in order, rather than a crowd of people running wild.

Why This Matters

This is a big deal because it opens up a new toolbox for physicists.

• Different Problems, Different Tools: Some problems are hard to solve with the "security guard" method but easy with the "sorted list" method (like tracking a moving wall of particles).
• Future Applications: This could help us understand complex materials, like Wigner crystals (where electrons freeze into a rigid grid), which are very hard to study with current methods.

In a nutshell: The authors realized that by simply telling the particles to "line up in order," they could tame the chaotic math of quantum mechanics, making it possible to simulate complex particle movements on a computer without needing a supercomputer the size of a galaxy.

Technical Summary

1. Problem Statement

The paper addresses a fundamental limitation in the application of Matrix Product States (MPS) to fermionic many-body systems.

• The Conventional Wisdom: MPS methods, which rely on low entanglement to efficiently represent quantum states, are almost exclusively formulated in second quantization. In this formalism, the wavefunction is defined over occupation numbers ($n_\ell \in \{0,1\}$) of lattice sites. For 1D systems with short-range correlations, the entanglement entropy obeys an "area law," allowing MPS to work efficiently.
• The First Quantization Barrier: In first quantization, the wavefunction is defined over the positions of $N$ particles ($x_1, \dots, x_N$). Due to the Pauli exclusion principle, fermionic wavefunctions must be anti-symmetric under particle exchange. This anti-symmetry induces a massive "particle entanglement" (scaling as $\ln \binom{N}{P}$), which theoretically suggests that the bond dimension required for an MPS representation would scale exponentially with system size ($e^{S_N} \approx 2^N$). Consequently, first-quantized MPS has been considered computationally intractable.

The Core Question: Can an efficient first-quantized MPS approach be constructed that avoids the exponential entanglement penalty while retaining the benefits of the first-quantization formalism?

2. Methodology

The authors propose a novel formulation that redefines how the wavefunction and constraints are handled in first quantization to suppress entanglement.

A. Wavefunction Reformulation (The $\bar{\psi}$ Approach)

Instead of working with the full anti-symmetric wavefunction $\psi(x_1, \dots, x_N)$, the authors define a restricted wavefunction $\bar{\psi}$ that encodes only one sector of the configuration space where particle positions are strictly ordered:
$$C: 0 < x_1 < x_2 < \dots < x_N < L + 1$$

• $\bar{\psi}$ is non-zero only within this ordered sector.
• The full anti-symmetric wavefunction $\psi$ can be recovered by permuting $\bar{\psi}$ if necessary.
• Key Insight: By restricting the domain to ordered configurations, the "particle entanglement" is drastically reduced. For example, a charge density wave in this ordered sector has zero entanglement entropy ($S_1=0$), whereas the full anti-symmetric version would have $S_1 \approx \ln N$.

B. Variable Transformation

To implement the ordering constraint $C$ naturally within the MPS tensor network, the authors perform a change of variables from particle positions $x_n$ to inter-particle distances $q_n$:

• $q_1 = x_1$
• $q_n = x_n - x_{n-1}$ for $n > 1$
• The constraint $x_n > x_{n-1}$ becomes the simple condition $q_n > 0$. This automatically enforces the Pauli principle without needing explicit anti-symmetrization operators.
• The global constraint that particles remain in the box becomes $\sum q_i \le L$.

C. Matrix Product Operator (MPO) Construction

The authors construct the Hamiltonian $H = H_t + H_V$ (for the $t-V$ model) as an MPO in the $q_n$ coordinates:

1. Interaction Term ($H_V$): Becomes a diagonal sum of projectors on $q_n=1$ (nearest neighbor interaction). This is represented by an MPO with bond dimension $D=2$.
2. Kinetic Term ($H_t$): Hopping a particle $n$ to the right increases $q_n$ by 1 and decreases $q_{n+1}$ by 1. This is represented by an MPO with bond dimension $D=4$.
3. Projection ($P_C$): To enforce the boundary condition $\sum q_i \le L$, a projection operator is introduced.
   • Approximation 1: A Lagrange multiplier $\lambda$ is added to the Hamiltonian ($H' = H + \lambda P_C$) to penalize states outside the valid sector, avoiding the need for a high-rank projection MPO.
   • Approximation 2: A cutoff $Q_{\text{max}}$ is introduced on inter-particle distances. At half-filling, particles are close, so $q_i$ rarely exceeds small values. This keeps the local physical dimension of the MPS finite and manageable.

D. Numerical Implementation

• Ground States: Solved using Density Matrix Renormalization Group (DMRG) with single-site updates.
• Time Evolution: Solved using the Time-Dependent Variational Principle (TDVP) with second-order Trotter decomposition.
• Charge Conservation: In this first-quantized scheme, particle number $N$ is fixed by the length of the MPS chain, making charge conservation automatic (unlike second quantization where $U(1)$ symmetry must be explicitly enforced via block structures).

3. Key Contributions

1. Proof of Concept: Demonstrated that first-quantized MPS is not inherently inefficient. By reordering the configuration space and changing variables to inter-particle distances, the entanglement entropy is reduced to a level comparable to (and in some cases better than) second quantization.
2. Efficient MPO Formulation: Provided a concrete construction for the $t-V$ Hamiltonian in the $q_n$ basis, including the handling of boundary conditions via controlled approximations.
3. Reversal of Entanglement Trends: Showed that for specific problems (like domain wall propagation), first quantization yields lower entanglement growth than second quantization, contrary to the standard belief that first quantization always suffers from high entanglement.

4. Results

The authors benchmarked their method on the 1D $t-V$ model (spinless fermions with nearest-neighbor hopping and density-density interaction).

• Ground State Accuracy (Free Fermions):

  • For non-interacting fermions at half-filling ($N=20$ to $50$), the energy error converged below $10^{-8}$ with bond dimension $D=200$.
  • The convergence rate was comparable to standard second-quantized DMRG.
  • Entanglement: The particle entanglement entropy $S_n$ in the first-quantized MPS was found to be comparable to second quantization and significantly lower than the theoretical lower bound for naive first-quantized particle entanglement ($\ln \binom{N}{n}$).

• Interacting Fermions:

  • The method successfully reproduced the equation of state $\rho(\mu)$, capturing the transition from a gapless Luttinger liquid to a gapped charge-density state at strong interactions ($V/t > 2$).
  • Counter-intuitive Finding: For interacting systems near half-filling, the entanglement entropy in the first-quantized MPS decreased as interaction strength $V$ increased. This contrasts with naive expectations where interactions usually increase entanglement.

• Real-Time Dynamics (Domain Wall):

  • Simulated the relaxation of a domain wall (all particles on the left).
  • Crucial Result: The entanglement entropy growth in first quantization was significantly smaller (by a factor of $\sim 4$ in entropy, implying an exponential reduction in required bond dimension) compared to second quantization.
  • Reasoning: In second quantization, the domain wall starts in the middle of the MPS, causing immediate entanglement growth across the central bond. In first quantization (ordered $x_1 < \dots < x_N$), the domain wall corresponds to the boundary of the MPS chain. Entanglement grows from the boundary inward, delaying the saturation of the central bonds.

5. Significance and Future Outlook

• Paradigm Shift: The paper challenges the dogma that first quantization is unsuitable for MPS methods. It highlights that "entanglement" is formalism-dependent; a problem hard in one picture may be easy in another.
• Specific Advantages: The approach is particularly advantageous for problems involving domain wall propagation and potentially Wigner crystals (where long-range interactions are diagonal in inter-particle distances).
• Optimization Potential: The authors suggest that the current bottleneck (the local physical dimension set by $Q_{\text{max}}$) could be further optimized using "quantics" representations (mapping real-space grids to qubit tuples), a technique previously proposed for quantum computing.
• Generalizability: The framework can be extended to bosons and spin systems, offering a new tool for the quantum many-body toolbox.

In summary, Li and Waintal have successfully constructed a low-entanglement first-quantized MPS framework, demonstrating its viability for both ground state and dynamical simulations, and revealing that for certain physical scenarios, it outperforms the standard second-quantized approach.