Study of local conserved quantities in the one-dimensional Hubbard model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914678760800256 |
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| author | Fukai, Kohei |
| author_facet | Fukai, Kohei |
| contents | In this thesis, I study the local charges $\{Q_k\}$ in the one-dimensional Hubbard model, which is an integrable system used for theoretical studies of non-perturbative effects in strongly correlated electron systems. I obtained their explicit expressions in a closed form. An expression of the $k$-local charge $Q_k$ for $k > 5$, where $k$ denotes the support, had been unknown in previous studies. I find that $Q_k$ is a linear combination of a particular kind of products of the hopping terms. I introduce a diagrammatic notation to represent these products efficiently, which enables the prediction of the general formula of the local charges. These linear combinations have non-trivial coefficients for $k > 5$, and some of the coefficients were proved to be identical to the generalized Catalan numbers, which also appear in the local charges of the Heisenberg chain. I derive the recursion equation for these coefficients. I also prove that the obtained local charges are exhaustive. Thus, any local charge is written as a linear combination of $\{Q_k\}$. Although the completeness of the local charges is a natural conjecture in the transfer-matrix formulation, its proof has not been presented for almost all cases. Lastly, I emphasize that this is the first study demonstrating general explicit expressions of the local charges for an electron system that lacks the boost operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Study of local conserved quantities in the one-dimensional Hubbard model Fukai, Kohei Statistical Mechanics Strongly Correlated Electrons Mathematical Physics Exactly Solvable and Integrable Systems In this thesis, I study the local charges $\{Q_k\}$ in the one-dimensional Hubbard model, which is an integrable system used for theoretical studies of non-perturbative effects in strongly correlated electron systems. I obtained their explicit expressions in a closed form. An expression of the $k$-local charge $Q_k$ for $k > 5$, where $k$ denotes the support, had been unknown in previous studies. I find that $Q_k$ is a linear combination of a particular kind of products of the hopping terms. I introduce a diagrammatic notation to represent these products efficiently, which enables the prediction of the general formula of the local charges. These linear combinations have non-trivial coefficients for $k > 5$, and some of the coefficients were proved to be identical to the generalized Catalan numbers, which also appear in the local charges of the Heisenberg chain. I derive the recursion equation for these coefficients. I also prove that the obtained local charges are exhaustive. Thus, any local charge is written as a linear combination of $\{Q_k\}$. Although the completeness of the local charges is a natural conjecture in the transfer-matrix formulation, its proof has not been presented for almost all cases. Lastly, I emphasize that this is the first study demonstrating general explicit expressions of the local charges for an electron system that lacks the boost operator. |
| title | Study of local conserved quantities in the one-dimensional Hubbard model |
| topic | Statistical Mechanics Strongly Correlated Electrons Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2402.08924 |