Violation of the string hypothesis and the Heisenberg<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>XXZ</mml:mi></mml:math>spin chain

Ilakovac, Amon; Kolanović, Marko; Pallua, S.; Prester, Predrag Dominis

doi:10.1103/physrevb.60.7271

Cited by 23 publications

(34 citation statements)

References 24 publications

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“…The square root behavior as (∆ − ∆ e ) 1/2 reflects the transformation from real to pure imaginary of the momentum k 1 , i.e., the edge-locking transformation. After substituting (29) in the Bethe equations, one obtains thatk, ∆ e are determined by a set of coupled equations as…”

Section: The Exceptional Pointsmentioning

confidence: 99%

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

Alba¹,

Saha²,

Haque³

2013

J. Stat. Mech.

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At large values of the anisotropy ∆, the open-boundary Heisenberg spin-1 2 chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this 'edge-locking' phenomenon in the entire ∆ > 1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors.Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large ∆ there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as ∆ is lowered below a certain value.For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, and both free either as single magnons or as bound composites corresponding to 'string' solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of ∆ ('exceptional points'), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy ∆, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (∆ = 1) and edge-locking becomes stable in the whole ∆ > 1 region. arXiv:1306.2666v3 [cond-mat.str-el]

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Section: The Exceptional Pointsmentioning

confidence: 99%

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

Alba¹,

Saha²,

Haque³

2013

J. Stat. Mech.

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“…However it has been shown in [15] that some of the 2-string rapidities λ ± , behave as Re(λ ± ) ∼ N , Im(λ ± ) ∼ ± √ N for large length N spin chains. Even, there are some 2-strings which for large N and large Bethe quantum numbers deform back to two real rapidities [16][17][18][19], which is observed numerically. Despite these drawbacks, the string hypothesis has been very helpful in numerical analysis in the iteration method to obtain good initial guess for the finite length chains.…”

Section: Introductionmentioning

confidence: 99%

Non self-conjugate strings, singular strings and rigged configurations in the Heisenberg model

Deguchi¹,

Giri²

2015

J. Stat. Mech.

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We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N = 12, where the central rapidity of some of the odd-length strings becomes complex making not all the strings self-conjugate individually. We show that there are at most (N − 2)/2 singular solutions for M = 4, M = 5 down-spins and at most (N 2 − 6N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ≥ 2M . Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.

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“…Although making use of the string hypothesis [14] one can estimate the total number of Bethe eigenstates, its certain assumptions do not always hold for any given finite length spin chain. For example, as the length of the chain increases, some of the two string solutions deform back to form two real distinct rapidities [15][16][17] and some of the two strings have much larger rapidities [18] for very large length spin chains, which are a violation of the string hypothesis. It is therefore necessary to look into the detailed analysis of the Bethe ansatz solutions.…”

Section: Introductionmentioning

confidence: 99%

Singular eigenstates in the even(odd) length Heisenberg spin chain

Giri¹,

Deguchi

2015

J. Phys. A: Math. Theor.

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We study the implications of the regularization for the singular solutions on the even(odd) length spin-1/2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions {λα} are invariant under the sign changes of their rapidities {λα} = {−λα}, then the Bethe ansatz equations are reduced to a system of (M − 2)/2((M − 3)/2) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N = 3 (2k + 1) with k = 1, 2, 3, · · · . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin-1/2 XXX chains.

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Violation of the string hypothesis and the HeisenbergXXZspin chain

Abstract: In this paper we count the numbers of real and complex solutions to Bethe constraints in the two particle sector of the XXZ model. We find exact number of exceptions to the string conjecture and total number of solutions which is required for completeness.

Cited by 23 publications

References 24 publications

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

Non self-conjugate strings, singular strings and rigged configurations in the Heisenberg model

Singular eigenstates in the even(odd) length Heisenberg spin chain

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