Core Concepts

The number of physical Bethe states in twisted spin chains can be expressed as an alternating sum of the number of spin configurations in the untwisted or partially twisted limit.

Abstract

The paper presents a counting formula that relates the number of physical Bethe states in integrable models with a twisted boundary condition to the number of states in the untwisted or partially twisted limit.

Key highlights:

- The authors review the algebraic Bethe ansatz and analyze the symmetries preserved by partial twists.
- They formulate the counting of spin configurations combinatorially as a restricted-occupancy problem and derive a generating function for the 3D case.
- The main result is a formula that expresses the multiplicity of the decomposition of the L-fold tensor power of the 2s-symmetric representation of su(r+1) into irreducible representations as an alternating sum of the restricted-occupancy coefficients.
- This formula connects the number of physical solutions in the untwisted spin chain with the number of solutions in the twisted chain, as some of the physical states in the twisted case become descendants of the highest-weight states.
- The completeness of the Hilbert space is guaranteed, as the physical solutions that span the Hilbert space recombine into the highest-weight modules in the untwisted limit.
- The authors also discuss partial twists and present the counting formula for the branching coefficients.
- Generalizations to Kondo-type models and models with Lie superalgebras are also studied.

To Another Language

from source content

arxiv.org

Stats

The number of configurations in the 3D restricted-occupancy problem is given by the coefficient of xM1
1 xM2
2 ... xMr
r in the expansion of the generating function:
gs,L(x1, x2, ..., xr) = [S(2s)(1, x1, ..., x1...xr)]L

Quotes

"The number of physical Bethe states of the untwisted model is related to that of the twisted model as µs,L(M) = Dχ(g)−1cs,L(M)."
"The counting formula tells us how the solutions counted by cs,L(M) in the case where the SU(r+1) symmetry is broken to U(1)r recombine into the highest-weight states of su(r+1) counted by µλ in the untwisted limit."

Key Insights Distilled From

by Hongfei Shu,... at **arxiv.org** 10-01-2024

Deeper Inquiries

The counting formula for Bethe states in spin chains can be extended to accommodate inhomogeneous spins by modifying the algebraic Bethe ansatz framework to account for varying spin representations at each site. In the context of the algebraic Bethe ansatz, the spin at each site can be represented by different representations of the underlying Lie algebra, such as SU(r + 1).
To achieve this, one can define a generalized Lax operator that incorporates the inhomogeneities in the spin representations. The Lax matrix, which is crucial for constructing the transfer matrix, can be expressed as a function of the spin at each site, leading to a modified set of Bethe ansatz equations (BAEs). The BAEs will then reflect the contributions from each site’s spin, allowing for a more complex interaction structure.
The counting formula can be derived by analyzing the modified BAEs and establishing a correspondence between the physical states and the highest-weight representations of the inhomogeneous spin chain. This involves identifying the appropriate Clebsch-Gordan coefficients that account for the different spins, which can be computed using techniques from representation theory. The resulting counting formula will thus express the number of physical Bethe states in terms of the configurations of spins across the chain, taking into account the specific inhomogeneities present.

The Bethe/Gauge correspondence establishes a profound link between the Bethe states of twisted spin chains and the vacua of supersymmetric gauge theories. This correspondence implies that the twisted boundary conditions in the spin chain correspond to specific parameters in the gauge theory, such as the Fayet-Iliopoulos parameter.
As a result, the phase boundaries in the twisted spin chain, where the symmetry is restored or enhanced, correspond to critical points in the gauge theory's moduli space. At these boundaries, the number of physical solutions to the Bethe ansatz equations can change dramatically, reflecting the emergence of new vacua in the gauge theory.
The implications are significant: the presence of phase boundaries indicates regions where the gauge theory exhibits enhanced symmetry, leading to richer physical phenomena. For instance, when the twist angles satisfy certain conditions, the symmetry may restore to a higher rank, such as SU(2) or SU(3), which can lead to the emergence of additional physical states in the spin chain. This interplay between the twisted spin chain and gauge theory provides insights into the nature of phase transitions and the structure of the vacuum in supersymmetric theories.

The insights gained from the twisted spin chain model can significantly enhance our understanding of the structure and singularities of the moduli space in the corresponding gauge theory. The twisted spin chain serves as a simplified model that captures essential features of the gauge theory, particularly in how the physical states correspond to the vacua of the theory.
In the twisted model, the counting of physical Bethe states can be directly related to the Witten index, which counts the number of supersymmetric vacua. This relationship allows for a systematic exploration of how the physical solutions change as one moves through the moduli space, particularly near singularities where the gauge theory may exhibit non-trivial behavior.
Moreover, the twisted boundary conditions can regularize certain singularities in the gauge theory, providing a clearer picture of the vacuum structure. As the twist angles vary, the corresponding changes in the Bethe states can reveal how the gauge theory transitions between different phases, highlighting the nature of the singularities in the moduli space.
By analyzing the twisted spin chain, one can gain insights into the stability of vacua, the emergence of new phases, and the behavior of the gauge theory near critical points. This understanding is crucial for exploring the full landscape of supersymmetric gauge theories and their associated physical implications, such as dualities and the dynamics of gauge fields.

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