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arXiv:2205.15815 · Journal of Physics A Mathematical and Theoretical

Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine sl2 Gaudin model

Tommaso Franzini, Charles Young

Published Feb 9, 2023• 7 citations
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Abstract

Abstract In this work we will use a general procedure to construct higher local Hamiltonians for the affine s l 2 Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading-order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.

DOI
10.1088/1751-8121/acbacf

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Abstract In this work we will use a general procedure to construct higher local Hamiltonians for the affine s l 2 Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading-order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.
In classical mechanics, an integrable system is a model that admits a complete set of integrals of motion that are in involution. This notion can be extended to include classical integrable field theories. A natural question is how to lift these notions to the quantum setting. Despite the extensive literature on the subject, these models are still not very well understood. As an example, in the first section of this work, we present a new class of integrable quantum field theories which develop an unusual behaviour that may be interpreted as a Hagedorn transition. The principal reason for the enigmatic nature of these models is the absence of a universal mathematical framework to describe them. One potential solution may be provided by quantum affine Gaudin models, which we examine in greater detail in the second part of this work. In particular, we introduce the first non-trivial Hamiltonian of quartic order for the affine sl2 Gaudin model, as well as the next-to-leading order expression for all higher Hamiltonians. Furthermore, we provide new insights into the double-loop version of the Feigin-Frenkel homomorphism, which is expected to be a crucial component in the construction of the Bethe ansatz for these models.
The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not Hermitian. Finally, the inner products and norms of the Bethe states are studied.
We consider quantum integrable systems associated with reductive Lie algebra g l ( n ) and Cartan-invariant non-skew-symmetric classical r -matrices. We show that under certain restrictions on the form of classical r -matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras g l ( n ) ⊃ g l ( n − 1 ) ⊃ . . . ⊃ g l ( 1 ) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r -matrices satisfying such the restrictions there are “twisted” or Z p -graded non-skew-symmetric classical r -matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Z p -graded non-skew-symmetric classical r -matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras g l ( n ) ⊃ g l ( n − n 1 ) ⊃ g l ( n − n 1 − n 2 ) ⊃ g l ( n − ( n 1 + . . . + n p − 1 ) ) , where n 1 + n 2 + . . . + n p = n .
Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the existence of area laws. Although they are a simplified model of quantum computation, the status of the commuting local Hamiltonian problem remains largely unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit completely classical verifiers. Despite intense work, the largest class of commuting local Hamiltonians we can place in NP are those on a square lattice, where each lattice site is a qutrit. Even worse, many of the techniques used to analyze these problems rely heavily on the geometry of the square lattice and the properties of the numbers 2 and 3 as local dimensions. In this work, we present a new technique to analyze the complexity of various families of commuting local Hamiltonians: guided reductions. Intuitively, these are a generalization of typical reduction where the prover provides a guide so that the verifier can construct a simpler Hamiltonian. The core of our reduction is a new rounding technique based on a combination of Jordan's Lemma and the Structure Lemma. Our rounding technique is much more flexible than previous work, and allows us to show that a larger family of commuting local Hamiltonians is in NP, albiet with the restriction that all terms are rank-1. Specifically, we prove the following two results: 1. Commuting local Hamiltonians in 2D that are rank-1 are contained in NP, independent of the qudit dimension. Note that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality. 2. We prove that rank-1, 3D commuting Hamiltonians with qudits on edges are in NP. To our knowledge this is the first time a family of 3D commuting local Hamiltonians has been contained in NP.