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edited Jun 15, 2020 at 7:27

The Bethe equations and Bethe ansatz are the central tools to study quantum integrable systems and there are thousands papers devoted to them. I cannot pretend to know substantial part of this research, so let me write some remarks which I am aware of.

You ask "... is there logic to their roots ... "

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).

Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive imaginary part. Then the l.h.s of (26) is exponentially large with N. To achieve this large value on the r.h.s. there should be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using the reality of solution of the Bethe Ansatz [36], we conclude that the Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.

String hypothesis in its strong form states that all solutions of the Bethe Ansatz equations can be represented as a collection of strings, and that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N precision. In its strong form the string hypothesis is wrong. However there is an evidence that its weaker version is correct if the proper thermodynamic limit is taken. The weaker version states that most of the Bethe roots are organized into strings with exponential in N precision, and that the fraction of solutions which signiﬁcantly diﬀer from (27) decreases to 0 when N → ∞. We discuss in more details applicability of the string hypothesis in appendix A.

You ask "... is there logic to their roots ... "

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).

Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive imaginary part. Then the l.h.s of (26) is exponentially large with N. To achieve this large value on the r.h.s. there should be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using the reality of solution of the Bethe Ansatz [36], we conclude that the Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.

String hypothesis in its strong form states that all solutions of the Bethe Ansatz equations can be represented as a collection of strings, and that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N precision. In its strong form the string hypothesis is wrong. However there is an evidence that its weaker version is correct if the proper thermodynamic limit is taken. The weaker version states that most of the Bethe roots are organized into strings with exponential in N precision, and that the fraction of solutions which signiﬁcantly diﬀer from (27) decreases to 0 when N → ∞. We discuss in more details applicability of the string hypothesis in appendix A.

You ask "... is there logic to their roots ... "

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).

Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive imaginary part. Then the l.h.s of (26) is exponentially large with N. To achieve this large value on the r.h.s. there should be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using the reality of solution of the Bethe Ansatz [36], we conclude that the Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.

String hypothesis in its strong form states that all solutions of the Bethe Ansatz equations can be represented as a collection of strings, and that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N precision. In its strong form the string hypothesis is wrong. However there is an evidence that its weaker version is correct if the proper thermodynamic limit is taken. The weaker version states that most of the Bethe roots are organized into strings with exponential in N precision, and that the fraction of solutions which signiﬁcantly diﬀer from (27) decreases to 0 when N → ∞. We discuss in more details applicability of the string hypothesis in appendix A.

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created Jul 13, 2013 at 15:46

Alexander Chervov

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You ask "... is there logic to their roots ... "

Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).

Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive imaginary part. Then the l.h.s of (26) is exponentially large with N. To achieve this large value on the r.h.s. there should be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using the reality of solution of the Bethe Ansatz [36], we conclude that the Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.

String hypothesis in its strong form states that all solutions of the Bethe Ansatz equations can be represented as a collection of strings, and that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N precision. In its strong form the string hypothesis is wrong. However there is an evidence that its weaker version is correct if the proper thermodynamic limit is taken. The weaker version states that most of the Bethe roots are organized into strings with exponential in N precision, and that the fraction of solutions which signiﬁcantly diﬀer from (27) decreases to 0 when N → ∞. We discuss in more details applicability of the string hypothesis in appendix A.