Rigged strings, Bethe Ansatz, and the geometry of the classical configuration space of the Heisenberg magnetic ring

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• 作者：T. Lulek ; B. Lulek ; D. Jakubczyk and P. Jakubczyk
• 关键词：Bethe Ansatz ; Rigged strings ; Paths ; Robinson– ; Schensted and Kerov– ; Kirillov– ; Reshetikhin algorithms
• 刊名：Physica B
• 出版年：2006
• 出版时间：15 June 2006
• 年：2006
• 卷：382
• 期：1-2
• 页码：162-180
• 全文大小：468 K

文摘

Composition of two combinatoric algorithms, Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR), defines the bijection which maps the set of all magnetic configurations of the Heisenberg ring of N nodes with the spin onto that of all exact Bethe Ansatz (BA) eigenstates of the nearest neighbour isotropic Hamiltonian. We point out here that this bijection allows one to predict all quantum numbers of an exact BA eigenstate (encoded as rigged string configurations of KKR) from each magnetic configuration. Moreover, this bijection provides completeness of solutions on each orbit of the symmetric group of N nodes, acting on the set of all magnetic configurations. Each such an orbit can be interpreted as the classical configuration space of the system of r Bethe pseudoparticles—spin deviations which are hard-core objects, moving on the magnetic ring by jumps to non-occupied nearest neighbours. Such an interpretation provides a transparent and combinatorially unique description of all exact BA eigenstates in terms of magnetic configurations—the initial basis for quantum calculations. Within this picture, an l-string introduced by Bethe corresponds to an extended object, consisting of 2l consecutive nodes: first l spin deviations, and then l nodes with the spin projection , all bounded together and put somewhere inside the magnetic chain. Each BA solution is a rigged string configuration, i.e. a distribution of a number q, 0qr, of such objects inside the chain. Allowed distributions are subjected to certain combinatoric restrictions (rules of navigation), expressed in terms of l-holes and riggings, and presented graphically as paths, associated with schemes of consecutive coupling of N spins along the RS algorithm. We discuss here thoroughly some implications of existence of such a bijection. In particular, we demonstrate the way in which structure of orbits of the translation group C_N on the classical configuration space for r Bethe pseudoparticles imposes the corresponding arrangement of rigged string configurations. Essentially, within a single cycle along a C_N-orbit, each l-string moves from the left to the right with increase of its rigging by one unit until reaching the last node N of the ring, then shortens its length to zero, and next arises at the first (leftmost) node and elongates up to its previous length l. Such considerations allow us also to discuss the geography of rigged string configurations on the classical configuration space. In particular, we put emphasis on the fact that—within this combinatoric picture—the l-strings originate from corresponding sizes of islands of consecutive Bethe pseudoparticles in the classical configuration space. Thus, the set of all magnetic configurations with r spin deviations acquires the interpretation of an r-dimensional manifold with F-dimensional boundaries, the integer F, 1Fr, being the number of islands of consecutive Bethe pseudoparticles. Each magnetic configuration belonging to the generic part F