The Bethe Ansatz

Models and eigenstates: the Coordinate Bethe Ansatz
Further spin chains
The planar Heisenberg chain (XXZ with $- 1 < Δ < 1$ )
String state classification

String state classificationc.sc.p.s

Following Takahashi and Suzuki 1972.Takahashi.PTP.48, we take $ζ / π$ to be a real number between $0$ and $1$ , which is expressed as a continued fraction of real positive integers as

\frac{ζ}{π} = \frac{1 |}{| ν_{1}} + \frac{1 |}{| ν_{2}} + . . . \frac{1 |}{| ν_{l}}, ν_{1}, . . ., ν_{l - 1} \geq 1, ν_{l} \geq 2.

For large $N$ , rapidities congregate to form strings centered either on the real line (for positive parity strings) or on the axis $i π / 2$ (for negative parity strings),

\begin{matrix} (p.sh) & λ_{α}^{n_{j}, a} = λ_{α}^{n_{j}} + i \frac{ζ}{2} (n_{j} + 1 - 2 a) + i \frac{π}{4} (1 - v_{j}) + i δ_{α}^{n_{j}, a}, a = 1, . . ., n_{j} \end{matrix}

where the allowable lengths $n_{j}$ and parities $v_{j} = \pm 1$ are to be determined. In a string configuration, the parameters $δ_{α}^{n_{j}, a}$ are exponentially suppressed with system size.

The classification of allowable string types in the thermodynamic limit proceeds according to the following algorithm. First, the positive integer series $y_{- 1}, y_{0}, y_{1}, \dots, y_{l}$ and $m_{0}, m_{1}, \dots, m_{l}$ are defined as

\begin{aligned} y_{- 1} = 0, y_{0} = 1, y_{1} = ν_{1}, and y_{i} = y_{i - 2} + ν_{i} y_{i - 1}, i = 2, \dots, l, \\ m_{0} = 0, m_{i} = \sum_{k = 1}^{i} ν_{k} . \end{aligned}

Lengths and parities are then given by (our conventions here have the advantage of giving a proper ordering of string lengths, $n_{j} > n_{k}$ , $j > k$ )

n_{j} = y_{i - 1} + (j - m_{i}) y_{i}, v_{j} = (- 1)^{⌊ (n_{j} - 1) ζ / π ⌋}, m_{i} \leq j < m_{i + 1} .

The total number of possible strings is $N_{s} = m_{l} + 1$ , and the index $j$ runs over the set $1, . . ., N_{s}$ . The real parameters $λ_{α}^{n_{j}}$ represent the centers of strings with length $n_{j}$ and parity $v_{j}$ , and are hereafter noted as $λ_{α}^{j}$ , $α = 1, . . ., M_{j}$ , where $M_{j}$ is the number of strings of length $n_{j}$ in the eigenstate under consideration. We therefore have the constraint $\sum_{j = 1}^{N_{s}} n_{j} M_{j} = M$ .

In a string configuration, many factors appearing in the Bethe equations become of the indeterminate form $δ / δ$ . Remultiplying p.be for each member of a particular string gets rid of these factors, and allows one to rewrite the whole set of Bethe equations in terms of a reduced set involving only the string centers $λ_{α}^{j}$ . Doing this, one finds (for $N$ even) the reduced set of Bethe-Takahashi equations 1972.Takahashi.PTP.48

\begin{matrix} (p.bgt) & {\bar{e}}_{j}^{N} (λ_{α}^{j}) = (- 1)^{M_{j} - 1} \prod_{(k, β) \neq (j, α)} {\bar{E}}_{j k} (λ_{α}^{j} - λ_{β}^{k}), \end{matrix}

where

\begin{matrix} (p.ej) & {\bar{e}}_{j} (λ) = - v_{j} \frac{\sinh (λ + i \frac{π}{4} (1 - v_{j}) + i n_{j} ζ / 2)}{\sinh (λ + i \frac{π}{4} (1 - v_{j}) - i n_{j} ζ / 2)}, \end{matrix}

and

\begin{matrix} (p.ejk) & {\bar{E}}_{j k} (λ) = {\bar{e}}_{| n_{j} - n_{k} |}^{1 - δ_{n_{j} n_{k}}} (λ) {\bar{e}}_{| n_{j} - n_{k} | + 2}^{2} (λ) . . . {\bar{e}}_{n_{j} + n_{k} - 2}^{2} (λ) {\bar{e}}_{n_{j} + n_{k}} (λ) . \end{matrix}

For the state classification and computation, it is preferable to work with the logarithmic form

\begin{matrix} (p.bgtl) & N ϕ_{j} (λ_{α}^{j}) - \sum_{k = 1}^{N_{s}} \sum_{β = 1}^{M_{k}} Φ_{j k} (λ_{α}^{j} - λ_{β}^{k}) = 2 π I_{α}^{j} \end{matrix}

where $I_{α}^{j}$ is integer if $M_{j}$ is odd, and half-integer if $M_{j}$ is even. The dispersion kernels and scattering phases appearing here are

\begin{aligned} ϕ_{j} (λ) & = 2 v_{j} atan [(\tan n_{j} ζ / 2)^{- v_{j}} \tanh λ], \\ Φ_{j k} (λ) & = (1 - δ_{n_{j} n_{k}}) ϕ_{| n_{j} - n_{k} |} (λ) + 2 ϕ_{| n_{j} - n_{k} | + 2} (λ) + \dots + 2 ϕ_{n_{j} + n_{k} - 2} (λ) + ϕ_{n_{j} + n_{k}} (λ) . \end{aligned}

The energy and momentum of a string are given by

\begin{aligned} E_{α}^{j} = - J \frac{\sin ζ \sin n_{j} ζ}{v_{j} \cosh 2 λ_{α}^{j} - \cos n_{j} ζ}, \\ (p.eps) & p_{0} (λ_{α}^{j}) = i \ln [\frac{\sinh (λ_{α}^{j} + i \frac{π}{4} (1 - v_{j}) + i n_{j} ζ / 2)}{\sinh (λ_{α}^{j} + i \frac{π}{4} (1 - v_{j}) - i n_{j} ζ / 2)}] \end{aligned}

so the total momentum is again expressible in terms of the string quantum numbers as

\begin{matrix} (p.p) & P = π T_{v = +} - \frac{2 π}{N} \sum_{j = 1}^{N_{s}} \sum_{α = 1}^{M_{j}} I_{α}^{j} mod 2 π \end{matrix}

in which $T_{v = +}$ is the total number of excitations with positive parity.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24

String state classification c.sc.p.s

String state classificationc.sc.p.s