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

Taiki Ishiyama, Kazuya Fujimoto, Tomohiro Sasamoto

Abstract

We present an exact integral formula for the multi-particle propagator of the one-dimensional Fermi--Hubbard model on an infinite lattice. The proof is based on the nested Bethe ansatz without relying on the string hypothesis. Our formula enables an explicit integral representation of the time evolution of arbitrary finite-particle wave functions and thereby provides a foundation for the exact analysis of nonequilibrium dynamics in the Hubbard model. It can further be applied to related open quantum models.

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

Abstract

We present an exact integral formula for the multi-particle propagator of the one-dimensional Fermi--Hubbard model on an infinite lattice. The proof is based on the nested Bethe ansatz without relying on the string hypothesis. Our formula enables an explicit integral representation of the time evolution of arbitrary finite-particle wave functions and thereby provides a foundation for the exact analysis of nonequilibrium dynamics in the Hubbard model. It can further be applied to related open quantum models.
Paper Structure (15 sections, 4 theorems, 109 equations, 6 figures)

This paper contains 15 sections, 4 theorems, 109 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Setup: the 1D Fermi--Hubbard model
  3. Nested Bethe ansatz
  4. Integral formula for the propagator
  5. Proof of Lemma \ref{['lem1']}
  6. Proof of Lemma \ref{['lem2']}
  7. Conclusion
  8. Open quantum models related to the Hubbard model
  9. Tight-binding chain with dephasing noise
  10. Hubbard model under two-body loss
  11. Inhomogeneous XXX structure of the scattering amplitude
  12. Technical details in the proof of Lemma \ref{['lem2']}
  13. Derivation of \ref{['eq:A_P_n']}
  14. Derivation of \ref{['eq:triangle']}
  15. Derivation of \ref{['eq:(n,m)(n,l)']}

Key Result

Theorem 1

The propagator eq:def_pro can be expressed as where $E(\bm{z})$ and $\phi(\bm{x};\bm{a}\vert \bm{z};\bm{\lambda})$ are given in eq:energy and eq:wave, respectively. $\beta_k$ denotes the position of the $k$-th down spin in $(b_1,\cdots,b_N)$. All integration contours in eq:formula are oriented counterclockwise. We set $r\ll 1$ so that the poles

Figures (6)

  • Figure 1: The integration contour $\Gamma_{\bm{s}}$ in the complex $\lambda$-plane, indicated by blue circles with arrows. (a) Inhomogeneous case, where $s_j$ are distinct. (b) Homogeneous case ($s_j=0$). The situation in Theorem \ref{['thm']} corresponds to the inhomogeneous case (a), since $s_j$ is defined in \ref{['eq:s_j']} and the choice of the $z$-contour, $|z_j|=r^{N-j}$ with $r\ll 1$, implies that $s_j$ are well-separated. In subsection \ref{['sub:spin']}, however, we treat $\bm{s} = (s_1,\cdots,s_N)$ as general complex numbers subject only to the condition $s_j -s_k\neq 2iu$ for any $j,k$, which includes the homogeneous case (b).
  • Figure 2: Graphical representations of $A^{(\bm{s})}_P(\bm{a}\vert \bm{b})$ for (a) $P=\Pi_{1,2}$. (b) $P=\Pi_{1,3}$. From Lemma \ref{['lem1']}, one has $A^{(\bm{s})}_{\Pi_{1,2}}(\bm{a}\vert \bm{b}) = \langle \bm{a} \vert Y_{1,2}(s_1-s_2)\vert \bm{b} \rangle$ and $A^{(\bm{s})}_{\Pi_{1,3}}(\bm{a}\vert \bm{b}) = \langle \bm{a} \vert Y_{1,2}(s_2-s_3) Y_{2,3}(s_1-s_3) Y_{1,2}(s_1-s_2)\vert \bm{b} \rangle$.
  • Figure 3: Schematic illustration of $\mathop{\mathrm{Res}}\limits_{s_n = s_m+2iu}[A^{(\bm{s})}_P (\bm{a} \vert \bm{b})]$. The blue circles and diamonds represent the factors given in \ref{['eq:(l,m)']} and \ref{['eq:(l,n)']}, respectively, while the red square corresponds to the pole-contributing factor in the $z_n$-integration. The dashed triangle represents the factor given in \ref{['eq:triangle']}.
  • Figure 4: Schematic illustration of \ref{['eq:(n,m)(n,l)']}. The red square and blue diamonds represent the pole-contributing factors in the $z_n$- and $z_m$-integrations, respectively. The intersection between the lines $l$ and $m$ corresponds to the $Y$-operator $Y(s_l-s_m)$. Note that this operator reduces to the identity operator when $s_l=s_m$.
  • Figure 5: Graphical representations of the $L$-matrix and monodromy matrix. (a) $L$-matrix $L_n(\lambda-\mu)$, (b) Monodromy matrix $T(\lambda\vert \bm{z}P)$ for the case of $N=4$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Remark 4
  • Proposition 1
  • Remark 5