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arXiv:2404.04322

Quantum Spin Chains and Symmetric Functions

Marcos Crichigno, Prakash, Anupam

Published Apr 5, 2024• 1 citation
arXivHTMLPDF

Abstract

We consider the question of what quantum spin chains naturally encode in their Hilbert space. It turns out that quantum spin chains are rather rich systems, naturally encoding solutions to various problems in combinatorics, group theory, and algebraic geometry. In the case of the XX Heisenberg spin chain these are given by skew Kostka numbers, skew characters of the symmetric group, and Littlewood-Richardson coefficients. As we show, this is revealed by a fermionic representation of the theory of "quantized" symmetric functions formulated by Fomin and Greene, which provides a powerful framework for constructing operators extracting this data from the Hilbert space of quantum spin chains. Furthermore, these operators are diagonalized by the Bethe basis of the quantum spin chain. Underlying this is the fact that quantum spin chains are examples of "quantum integrable systems." This is somewhat analogous to bosons encoding permanents and fermions encoding determinants. This points towards considering quantum integrable systems, and the combinatorics associated with them, as potentially interesting targets for quantum computers.

DOI
10.48550/arxiv.2404.04322

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We consider the question of what quantum spin chains naturally encode in their Hilbert space. It turns out that quantum spin chains are rather rich systems, naturally encoding solutions to various problems in combinatorics, group theory, and algebraic geometry. In the case of the XX Heisenberg spin chain these are given by skew Kostka numbers, skew characters of the symmetric group, and Littlewood-Richardson coefficients. As we show, this is revealed by a fermionic representation of the theory of "quantized" symmetric functions formulated by Fomin and Greene, which provides a powerful framework for constructing operators extracting this data from the Hilbert space of quantum spin chains. Furthermore, these operators are diagonalized by the Bethe basis of the quantum spin chain. Underlying this is the fact that quantum spin chains are examples of "quantum integrable systems." This is somewhat analogous to bosons encoding permanents and fermions encoding determinants. This points towards considering quantum integrable systems, and the combinatorics associated with them, as potentially interesting targets for quantum computers.
ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.
Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.
We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of NNN noninteracting identical bosons. In general, the system is described by the NNN-fold symmetric tensor product E⊙N,\mathcal{E}^{\odot N},E⊙N, where E\mathcal{E}E is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of E⊙N.\mathcal{E}^{\odot N}.E⊙N. We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries SSS such that the partition function of the system is SSS-symmetric.
In the present thesis we perform a step towards understanding the effec- tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small excitations: small number of reversed spins or, equivalently, small number of fermions per site. The calculation of the first order normal form Hamiltonian amounts to erasing all the nonresonant interaction terms, and allows to draw interesting conclusions on the dynamics of the system, e.g. to determine ap- proximately conserved quantities. Moreover, we get a good correspondence between our perturbative energy spectrum and the exact one computed via the Bethe ansatz, the approximation being of course the more accurate the smaller is the number of fermions per site. The present method, here tested in an exactly solvable case, can be applied to any nonintegrable system of weakly interacting fermions (and/or bosons).
Kostka, Littlewood-Richardson, Plethysm, and Kronecker coefficients are the multiplicities of irreducible representations in the decomposition of representations of the symmetric group that play an important role in representation theory, geometric complexity, and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups. The conjecture about Kronecker coefficients was disproved by Panova [Polynomial time classical versus quantum algorithms for representation theoretic multiplicities, arXiv:2502.20253] with a classical algorithm which, if optimal, points to a O(n^{4+2k}) vs Ω[over ˜](n^{4k^{2}+1}) polynomial gap in quantum vs classical computational complexity for an integer parameter k.
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...