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All Local Conserved Quantities of the One-Dimensional Hubbard Model

Kohei Fukai

Abstract

We present the exact expression for all local conserved quantities of the one-dimensional Hubbard model. We identify the operator basis constructing the local charges and find that nontrivial coefficients appear in the higher-order charges. We derive the recursion equation for these coefficients, and some of them are explicitly given. There are no other local charges independent of those we obtained.

All Local Conserved Quantities of the One-Dimensional Hubbard Model

Abstract

We present the exact expression for all local conserved quantities of the one-dimensional Hubbard model. We identify the operator basis constructing the local charges and find that nontrivial coefficients appear in the higher-order charges. We derive the recursion equation for these coefficients, and some of them are explicitly given. There are no other local charges independent of those we obtained.
Paper Structure (21 sections, 2 theorems, 123 equations, 7 figures)

This paper contains 21 sections, 2 theorems, 123 equations, 7 figures.

Table of Contents

  1. S2. Identities of $C^{j,m}_{n,d}\left(\bm{\lambda}\right)$
  2. S3. Proof of conservation law of $Q_k$
  3. Cancellation of connected diagram
  4. Cancellation of non-connected diagram
  5. S4. Proof of identities of $C^{j,m}_{n,d}\left(\bm{\lambda}\right)$
  6. Induction for \ref{['eq:LRswap']}
  7. Induction for \ref{['eq:swap']}
  8. Induction for \ref{['eq:minprop']}
  9. Induction for \ref{['eq:betweenjLR']} and \ref{['eq:betweenj']}
  10. Induction for \ref{['eq:oxo']}
  11. Proof of positivity of $C^{j,m}_{n,d}(\bm{\lambda})$
  12. S5. Proof of identities of $A^{\sigma\mu}_i$
  13. Proof of \ref{['A++']}
  14. Proof of \ref{['A+-']}
  15. Proof of \ref{['AL+']} and \ref{['AR+']}
  16. ...and 6 more sections

Key Result

Theorem 1

For $j\geq 1$, where $\mathcal{S}_{n,d,g}^{k,j,m}$ is the set of $(k-j-2n-d,d)$-connected diagrams $\Psi$ with $l_\Psi = j+1-2m$ and $g_\Psi = g$. $C^{j,m}_{n,d}\left(\lambda_1\ldots\lambda_{l}\right)\in \mathbb{Z}_{>0}$ is invariant under the permutation of $\lambda_i (2\leq i\leq l-1)$, and the exchange of $\lam

Figures (7)

  • Figure 1: Examples of connections of units (a)--(c), and the non-connected diagrams (d). Units on the upper and lower rows of the diagrams in (a)--(c) are connected. The gaps in (c (d)) are indicated by the teal- (orange-) shaded area. The diagram on the bottom right in (d) does not satisfy condition (ii), while the others do not satisfy (i).
  • Figure 2: Structure of $Q_k^j$. $k_j\equiv k-j$. Circles at $(s,d)$ represent $(s,d)$-connected diagrams in $Q_k^j$ ($s>d$). The commutator of diagrams in the circle at $(s,d)$ with $H_0$ generates the diagrams in the crosses at $(s\pm 1,d)$ and $(s,d\pm 1)$, indicated by the solid arrow tip.
  • Figure 3: Structure of $Q_{k}$ for $k=6$. Each plane represents the structure of $Q_{k}^j$ in Fig. \ref{['Qkjfig']}. The commutator of diagrams in the circle at $(s,d)$ in $Q_k^{j-1}$ with $H_\mathrm{int}$ generates diagrams in the cross at $(s,d)$ in $Q_k^{j}$, indicated by the vertical dotted arrow tip. The diagrams generated in the crosses are to be canceled.
  • Figure S1: The general structure of the cancellation of diagrams in $\left[Q_k, H\right]=0$. The upper plane represents $Q_k^{j-1}$ and lower plane represents $Q_k^{j}$. We show only the components of $Q_k$ that are relevant to the cancellation of diagrams in the crosses at $(s+1,d)$ in the plane of $Q_k^{j}$. The coefficients encircled with the orange solid (blue dotted) line are the coefficients of the diagrams in the circles indicated by each arrow tip and contribute to the recursion relation in the form of \ref{['eq:basiceq2']} with the factor of $+1(-1)$.
  • Figure S2: The structure of $Q_{6}^j$ for each $j$ (a) and all the structure of $Q_6$ (b). In (b), each plane represents the structure of $Q_{6}^j$ in (a), and the axis of support and double are omitted. The solid arrow in planes represents the commutator of diagrams with $H_\mathrm{0}$, and the vertical dotted arrow represents the commutator of diagrams with $H_\mathrm{int}$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2