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I'm a mathematician who's trying to understand the meaning of Algebraic Bethe Ansatz. What I understood is that when dealing with quantum integrable models (like XXZ Heisenberg spin chain), one is interested in showing that there are conserved quantities (ie operators that commute with the Hamiltonian).

My questions are about the tools that we have to use in order to exhibit them. Can someone explain to me in which way I should think about these objects?

  1. The Lax operator
  2. The $R$ matrix
  3. The monodromy matrix
  4. The transfer matrix

I understand that one defines a Lax operator then show that it verifies a certain commutation relation with the $R$ matrix (which verifies Yang-Baxter equation). One multiplies the Lax operators along all sites and obtain the monodromy matrix, taking its trace yields the transfer matrix. Finally one can show that this last operator can be seen as the generating function of conserved quantites. Do not hesitate to correct me where I'm wrong.

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  • $\begingroup$ This is way too broad. Entire books have been written about these concepts... $\endgroup$ Jan 6 at 17:46
  • $\begingroup$ Short answer: yes, you're right. Long answer: see below. $\endgroup$ Jan 7 at 11:29

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This is a big topic and the most suitable answer depends on your background. Since you're a mathematician I assume that you are familiar with representation theory. I'll try to indicate the key rep-th terms corresponding to the terms in the OP, and hope that helps.

For the basic example one starts from the Yangian of $\mathfrak{gl}_2$. It has a presentation (called Drinfeld 3rd realisation = Faddeev--Reshetikhin--Takhtadzhyan presentation = '$RLL$ presentation') by generators and relations. The generators are often combined into an operator $L(u)$ on an ('auxiliary') vector space (which is 2d for the case of $\mathfrak{gl}_2$) with entries that are formal power series in the 'spectral parameter' $u$ whose coefficients lie in the Yangian. This is the $L$-operator, sometimes called monodromy matrix -- though that name often also/instead denotes its image in a representation, see below -- and also often denoted by $T$ instead of $L$. The generators are subject to quadratic relations called the '$RLL$ relations' because of their symbolic form (and called the 'fundamental commutation relations' by Faddeev), where the (rational) $R$-matrix contains the 'structure constants' of the Yangian. For this to define an associative algebra, the $R$-matrix needs to obey the Yang--Baxter equation.

Any finite dimensional representation of $\mathfrak{sl}_2$ gives rise to a representation of this Yangian called an evaluation representation, with an 'inhomogeneity' parameter that can be either viewed as a complex parameter or an indeterminant. In the basic case we do this for the 2d (spin 1/2) irrep and take trivial value of the inhomogeneity parameter. Physically, this is a single site. The image of the $L$-operator here is sometimes called the (local) Lax operator. Bigger representations can be constructed by taking tensor products to obtain the Hilbert space of the spin chain (multiple spins). The image of the $L$-operator in the resulting space is the (global Lax operator or) monodromy matrix. It still acts on the auxiliary space as well. Taking the trace over this auxiliary space gives the transfer matrix, which is the image of an abelian subalgebra of the Yangian (called the Bethe subalgebra) which, when expanded in the spectral parameter, provides a family of commuting operators (conserved charges) on the spin-chain Hilbert space including the translation operator and the Heisenberg XXX Hamiltonian. Finally, in this language, the algebraic Bethe ansatz is a sort highest-weight construction of the eigenvectors of the transfer matrix (and thus the spin chain) that produces actual eigenvectors provided the spectral parameters involved solve the Bethe-ansatz equations.

NB. For the XXZ chain, start from a quantum affine algebra instead, with trigonometric $R$-matrix, and everything carries over (in the generic case). For the XYZ chain, start from an elliptic quantum group, with elliptic $R$-matrix; then you can still construct a transfer matrix and obtain commuting Hamiltonians, but the construction of eigenvectors is much harder since (depending on the type of elliptic quantum group) there are no highest-weight representations (spin-$z$ is not conserved).

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  • $\begingroup$ Thank you so much for your long answer! If I summarize: - I should think about the monodromy operator and the R-matrix as encoding the generators of the Hopf-algebra called the Yangian - The local Lax operator is the image of the monodromy matrix via the finite dimensional representation of the Lie Alegbra Sl_2. Is it correct ? $\endgroup$ Jan 8 at 20:43
  • $\begingroup$ Also in fact I don't know much about representation theory as I'm more a probabilist, could you recommend a reference about all these tools if it's not annoying ? Thank you so much, it is of big help! $\endgroup$ Jan 8 at 20:44
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    $\begingroup$ @BlueCharlie The monodromy matrix (or rather: $L$-operator) is indeed a matrix whose entries are (generating functions in the spectral parameter for the) generators of an infinite-dimensional Hopf algebra called the Yangian. The $R$-matrix instead contains scalars that appear as coefficients in the defining '$RLL$ relations' for these generators of the Yangian, so they can be thought of as structure constants defining the Yangian. $\endgroup$ Jan 8 at 23:16
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    $\begingroup$ (cont'd) Normally, by 'monodromy matrix' people mean the representation of the $L$-operator on the spin chain. It is indeed a product of 'local Lax operastors' that correspond to a single-site representation that comes from a finite dimensional (usually irreducible) representation of the the Lie algebra $\mathfrak{sl}_2$ $\endgroup$ Jan 8 at 23:17
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    $\begingroup$ For references you could try my lecture notes arxiv.org/abs/1501.06805, although they were aimed at physicists and don't use heavy representation theory. See also p4 therein for more references. A short classic is the book by Jimbo and Miwa, "Algebraic Analysis of Solvable Lattice Models", the first few short chapters might already help. A classic reference about the relevant quantum groups and representation theory is the book of Chari and Pressley, "A Guide to Quantum Groups" $\endgroup$ Jan 8 at 23:21

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