Integral formula for the propagator of the one-dimensional Hubbard model

This paper presents an exact integral formula for the multi-particle propagator of the one-dimensional Hubbard model, derived via the nested Bethe ansatz without the string hypothesis, which facilitates the explicit analysis of nonequilibrium dynamics for arbitrary finite-particle wave functions.

The Gist

Imagine you are watching a crowded dance floor where hundreds of tiny, energetic dancers (electrons) are moving around. They have two special rules:

1. They can't occupy the same spot at the same time (the "exclusion principle").
2. They have a "personality" (spin) that makes them interact with neighbors in a specific way.

This is the Hubbard Model, a famous mathematical puzzle physicists use to understand how materials conduct electricity or become magnets. For decades, scientists could describe the average behavior of these dancers or how they move when the room is empty. But figuring out exactly how a specific group of dancers moves from one side of the room to the other, second by second, was like trying to predict the exact path of every single raindrop in a storm. It was too messy and complex.

This paper by Ishiyama, Fujimoto, and Sasamoto is like handing us a perfect, crystal-clear map for that dance floor.

The Problem: The "String" Trap

In the past, to solve these puzzles, physicists used a trick called the "String Hypothesis." Imagine trying to untangle a knot of headphones. The "String Hypothesis" is like assuming the headphones are neatly coiled in perfect loops so you can just count the loops to know where the ends are. It works well for simple knots, but in the real world, the headphones are often a chaotic mess.

For the Hubbard model, this "neat loop" assumption breaks down. It's too simple for the complex interactions between these electrons. So, scientists were stuck. They could guess the answer, but they couldn't write down the exact formula for how the system evolves over time.

The Solution: A Nested Recipe

The authors didn't try to untangle the whole knot at once. Instead, they realized the knot has a nested structure.

Think of the electrons as having two layers of identity:

1. Where they are (their position on the dance floor).
2. Who they are (their spin, like wearing a red or blue shirt).

The authors used a technique called the Nested Bethe Ansatz. Imagine a Russian nesting doll.

• The outer doll represents the movement of the electrons across the floor.
• Inside that doll is a smaller doll representing the interactions of their "shirts" (spins).
• Inside that is an even smaller doll for the next level of interaction.

By solving the problem layer by layer—starting from the innermost doll and working outward—they found a way to describe the entire system without ever needing to assume the "neat loops" (the string hypothesis) existed.

The Magic Formula: The "Time-Traveling" Integral

The main result of the paper is a multiple contour integral formula.

In simple terms, an "integral" in physics is like a machine that takes all possible paths a particle could take and adds them up to find the final result. Usually, doing this for many particles is impossible because the math explodes into complexity.

The authors' formula is like a super-efficient GPS.

• The Input: You tell the GPS where the dancers started (their initial positions and spins).
• The Process: The formula uses a specific set of "contours" (imaginary loops in a mathematical landscape) to calculate the path.
• The Output: It gives you the exact probability of finding the dancers in any specific configuration at any future time.

Because the formula is "exact," it doesn't rely on approximations. It tells you the truth, down to the last decimal point, for any number of particles on an infinite dance floor.

Why Does This Matter?

You might ask, "Who cares about a math formula for electrons?" Here is the real-world impact:

1. Simulating the Future: Scientists can now use this formula to simulate exactly how quantum materials behave when you suddenly change the temperature or apply a magnetic field. This helps in designing better batteries, superconductors, and quantum computers.
2. The "Open" World: The paper also shows this math works for "open" systems—systems that aren't perfectly isolated. Think of a dance floor where people are constantly leaving or entering, or where the music is glitching. This is crucial for understanding quantum computers, which are very sensitive to noise and errors.
3. Beyond the Lab: The authors mention this math can even describe systems with "imaginary" interactions, which sounds like sci-fi but actually describes real-world scenarios like particles disappearing (decaying) or systems losing energy to their environment.

The Big Picture

Before this paper, trying to calculate the exact motion of interacting electrons was like trying to predict the weather by looking at a single cloud. You could guess, but you couldn't be sure.

This paper provides the satellite imagery. It gives us a complete, exact, and calculable view of how these quantum particles dance through time. It removes the need for "best guesses" (the string hypothesis) and replaces them with a rigorous, mathematical roadmap.

In short: They found the exact instruction manual for the universe's most complex dance floor, allowing us to predict the future of quantum matter with perfect precision.

Technical Summary

1. Problem Statement

The one-dimensional Fermi–Hubbard model is a cornerstone for understanding strongly correlated electron systems, exhibiting phenomena like spin-charge separation and Tomonaga–Luttinger liquid behavior. While the model is integrable, and its equilibrium properties (spectral properties, thermodynamics) are well-understood via the nested Bethe ansatz, the exact analysis of nonequilibrium dynamics remains a formidable challenge.

Existing numerical methods (e.g., Tensor Network Simulations) are limited by entanglement growth, and Generalized Hydrodynamics (GHD) primarily describes average ballistic transport, missing microscopic fluctuations. For simpler integrable systems like the Lieb–Liniger gas or the XXZ spin chain, exact integral representations of the multi-particle propagator (often called the Yudson representation) exist, allowing for the exact calculation of time evolution for finite particles on an infinite lattice without relying on the string hypothesis. However, no such exact integral formula existed for the 1D Fermi–Hubbard model due to the complexity introduced by its internal spin degrees of freedom and the necessity of the nested Bethe ansatz.

2. Methodology

The authors derive an exact multiple contour integral formula for the propagator $\psi_t(x; a|y; b)$ of the 1D Hubbard model on an infinite lattice $\mathbb{Z}$.

• Model Definition: The Hamiltonian includes nearest-neighbor hopping and an on-site interaction $4u$. Notably, the authors allow the interaction strength $u$ to be a complex number ($u \in \mathbb{C} \setminus \{0\}$), which is crucial for applying the results to open quantum systems (e.g., systems with dephasing or particle loss).
• Nested Bethe Ansatz: The derivation relies on the nested Bethe ansatz, which parameterizes the wave function using charge rapidities $z = (z_1, \dots, z_N)$ and spin rapidities $\lambda = (\lambda_1, \dots, \lambda_M)$.
  • The wave function $\phi(x; a|z; \lambda)$ has a nested structure: the spatial part is a standard Bethe wave function, while the scattering amplitude (spin part) corresponds to the Bethe wave function of an inhomogeneous XXX spin chain.
• Integral Representation: The authors propose a formula where the propagator is expressed as a multiple contour integral over the rapidities $z$ and $\lambda$.
  • $z$-contours: Circles $|z_j| = r^{N-j}$ with $r \ll 1$.
  • $\lambda$-contours: Small closed curves $\Gamma_s$ enclosing the set of points $\{s_j + iu\}_{j=1}^N$, where $s_j$ are functions of the charge rapidities.
• Proof Strategy: The proof verifies that the proposed integral formula satisfies two conditions:
  1. Schrödinger Equation: This follows directly from the fact that the integrand involves the eigenstates of the Hamiltonian derived from the nested Bethe ansatz.
  2. Initial Condition: This is the non-trivial part. The authors must prove that at $t=0$, the integral reduces to the determinant of Kronecker deltas (representing the identity operator in the position-spin basis).
     • They exploit the nested structure of the Bethe wave function.
     • They utilize Y-operators ($Y_{j,k}$) which relate scattering amplitudes of different permutations.
     • They perform successive residue calculations on the spin rapidities ($\lambda$) and charge rapidities ($z$).
     • A key technical step involves showing that contributions from non-identity permutations cancel out due to the Yang-Baxter relation and specific symmetries of the Y-operators, leaving only the identity term.

3. Key Contributions

• First Exact Integral Formula for Hubbard Propagator: The paper provides the first exact multiple contour integral representation for the multi-particle propagator of the 1D Fermi–Hubbard model on an infinite lattice.
• Avoidance of String Hypothesis: Unlike thermodynamic limit calculations, this derivation works directly on the infinite lattice for finite particle numbers without relying on the string hypothesis, making it rigorous for non-equilibrium dynamics.
• Generalization of Yudson Representation: It extends the known Yudson representations (from Lieb-Liniger and XXZ models) to the more complex, nested case of the Hubbard model.
• Complex Interaction Strength: The derivation holds for complex $u$, explicitly bridging the gap between closed integrable systems and open quantum systems governed by Lindblad dynamics.

4. Key Results

• Theorem 1 (Main Result): The propagator is given by:
  $$\psi_t(x; a|y; b) = \left[ \prod_{j=1}^N \oint_{|z_j|=r^{N-j}} \frac{dz_j}{2\pi i z_j} z_j^{-y_j} \right] \left[ \prod_{k=1}^M \oint_{\Gamma_s} \frac{d\lambda_k}{2\pi i (\lambda_k - s_{\beta_k} - iu)} \dots \right] e^{-iE(z)t} \phi(x; a|z; \lambda)$$
  where the contours are carefully chosen to avoid singularities and isolate specific poles.
• Initial Condition Verification: The authors rigorously prove (via Lemmas 1 and 2) that the integral collapses to the correct initial state $\det[\delta_{x_j, y_k}\delta_{a_j, b_k}]$ at $t=0$. This involves intricate combinatorial arguments regarding permutations and the cancellation of terms using the Yang-Baxter algebra.
• Application to Open Systems: The formula allows for the exact calculation of time evolution in:
  • Tight-binding chains with dephasing noise: Mapped to a Hubbard model with imaginary interaction.
  • Hubbard models with two-body loss: Mapped to a Hubbard model with complex interaction.
  • This enables the exact calculation of density profiles and full counting statistics for these open systems.

5. Significance

• Foundation for Nonequilibrium Dynamics: This work provides a rigorous microscopic tool to study the time evolution of arbitrary finite-particle wave functions in the Hubbard model, complementing numerical and hydrodynamic approaches.
• Exactness: It allows for the study of phenomena beyond the validity of GHD or tensor networks, such as short-time dynamics, specific correlation functions, and full counting statistics of currents.
• Open Quantum Systems: By accommodating complex interaction parameters, the formula establishes a direct link between integrable closed systems and dissipative open systems, offering a new pathway to analyze non-unitary dynamics exactly.
• Future Directions: The authors suggest this formalism can be extended to the Maassarani model (the $SU(n)$ generalization of the Hubbard model) and used to investigate domain-wall initial conditions and alternating initial states in open systems.

In summary, this paper resolves a long-standing gap in the theory of integrable systems by providing a powerful, exact analytical tool for the nonequilibrium dynamics of the 1D Hubbard model, with immediate applications to both closed and open quantum systems.