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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.07229 |
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| _version_ | 1866915740972482560 |
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| author | Frahm, Holger Klümper, Andreas Wagner, Dennis Zhang, Xin |
| author_facet | Frahm, Holger Klümper, Andreas Wagner, Dennis Zhang, Xin |
| contents | The XXX spin-$\frac{1}{2}$ Heisenberg chain with non-diagonal boundary
fields represents a cornerstone model in the study of integrable systems
with open boundaries. Despite its significance, solving this model exactly
has remained a formidable challenge due to the breaking of $U(1)$
symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set
of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum
of the model.
For $U(1)$ symmetric spin-$\frac{1}{2}$ chains such NLIEs involve two
functions $a(x)$ and $\bar{a}(x)$ coupled by an integration kernel with
short-ranged elements. The solution functions show characteristic features
for arguments at some length scale which grows logarithmically with system
size $N$.
For the non $U(1)$ symmetric case, the equations involve a novel third
function $c(x)$, which captures the inhomogeneous contributions of the
$T$-$Q$ relation. The kernel elements coupling this function to the
standard ones are long-ranged and lead for the ground-state to a winding
phenomenon. In $\log(1+a(x))$ and $\log(1+\bar a(x))$ we observe a
sudden change by $2π$i at a characteristic scale $x_1$ of the
argument. Other features appear at a value $x_0$ which is of order $\log N$.
These two length scales, $x_1$ and $x_0$, are independent: their ratio
$x_1/x_0$ is large for small $N$ and small for large $N$. Explicit
solutions to the NLIEs are obtained numerically for these limiting cases,
though intermediate cases ($x_1/x_0 \sim 1$) present computational
challenges.
This work lays the foundation for studying finite-size corrections and
conformal properties of other integrable spin chains with non-diagonal
boundaries, opening new avenues for exploring boundary effects in quantum
integrable systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields Frahm, Holger Klümper, Andreas Wagner, Dennis Zhang, Xin Strongly Correlated Electrons High Energy Physics - Theory The XXX spin-$\frac{1}{2}$ Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of $U(1)$ symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For $U(1)$ symmetric spin-$\frac{1}{2}$ chains such NLIEs involve two functions $a(x)$ and $\bar{a}(x)$ coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size $N$. For the non $U(1)$ symmetric case, the equations involve a novel third function $c(x)$, which captures the inhomogeneous contributions of the $T$-$Q$ relation. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In $\log(1+a(x))$ and $\log(1+\bar a(x))$ we observe a sudden change by $2π$i at a characteristic scale $x_1$ of the argument. Other features appear at a value $x_0$ which is of order $\log N$. These two length scales, $x_1$ and $x_0$, are independent: their ratio $x_1/x_0$ is large for small $N$ and small for large $N$. Explicit solutions to the NLIEs are obtained numerically for these limiting cases, though intermediate cases ($x_1/x_0 \sim 1$) present computational challenges. This work lays the foundation for studying finite-size corrections and conformal properties of other integrable spin chains with non-diagonal boundaries, opening new avenues for exploring boundary effects in quantum integrable systems. |
| title | Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields |
| topic | Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2502.07229 |