Free fermionic and parafermionic quantum spin chains with multispin interactions

Alcaraz, Francisco C.; Pimenta, Rodrigo A.

doi:10.1103/physrevb.102.121101

Cited by 31 publications

(60 citation statements)

References 29 publications

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“…Typically, namely for twist φ / ∈ {0, π}, a primitive state has no roots at infinity either. 8 In the (anti)periodic case φ ∈ {0, π} a primitive state may have n ±∞ = 0 as long as n ∓∞ = 0, see (2.18) and Section 8.1.3.…”

Section: Preliminariesmentioning

confidence: 99%

On the Q operator and the spectrum of the XXZ model at root of unity

2021

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The spin-\frac{1}{2}12 Heisenberg XXZ chain is a paradigmatic quantum integrable model. Although it can be solved exactly via Bethe ansatz techniques, there are still open issues regarding the spectrum at root of unity values of the anisotropy. We construct Baxter’s Q operator at arbitrary anisotropy from a two-parameter transfer matrix associated to a complex-spin auxiliary space. A decomposition of this transfer matrix provides a simple proof of the transfer matrix fusion and Wronskian relations. At root of unity a truncation allows us to construct the Q operator explicitly in terms of finite-dimensional matrices. From its decomposition we derive truncated fusion and Wronskian relations as well as an interpolation-type formula that has been conjectured previously. We elucidate the Fabricius–McCoy (FM) strings and exponential degeneracies in the spectrum of the six-vertex transfer matrix at root of unity. Using a semicyclic auxiliary representation we give a conjecture for creation and annihilation operators of FM strings for all roots of unity. We connect our findings with the `string-charge duality’ in the thermodynamic limit, leading to a conjecture for the imaginary part of the FM string centres with potential applications to out-of-equilibrium physics.

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Section: Preliminariesmentioning

confidence: 99%

On the Q operator and the spectrum of the XXZ model at root of unity

2021

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“…The operative technical insight is that many of the key properties of that model, and its generalizations in Refs. [6,7], are actually special cases of more general recursion relations in the class of models we identify.…”

Section: Definition 3 (Ecf)mentioning

confidence: 98%

“…This family of models generalizes the well-known XY-model and was exactly solved at their critical points in Refs. [6,7]. In the third set of examples, we define a new family of integrable models constructed by combining the equipartition indifference graphs into more complex structures; these may also be solvable if they avoid even holes.…”

Section: Examplesmentioning

confidence: 99%

“…The solution is therefore generic. This solution method has since been reproduced in a family of generalized spin-chain models [6,7], including qudit models, where the system is dual to so-called free parafermions [8]. The generic nature of the free spectrum in these models would suggest that the frustration-graph formalism of Ref.…”

Section: Introductionmentioning

confidence: 99%

“…We next demonstrate how the family of generalized spin chain models included in Refs. [5][6][7] fit into our formalism. These models have a particular one-dimensional structure which makes their asymptotic dispersion relations amenable to the techniques of Toeplitz-matrix analysis.…”

Section: Introductionmentioning

confidence: 99%

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Free Fermions Behind the Disguise

Elman

Chapman

Flammia

2021

Commun. Math. Phys.

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An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anticommutation relations between the Pauli terms in H in a given basis. Specifically, when G is (even-hole, claw)-free, we construct an explicit free-fermion solution for H using only this structure of G, even when no Jordan-Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed "free fermions in disguise." Like Fendley's original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no Jordan-Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

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Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

Leeuw

Paletta

Pribytok

et al. 2021

J. High Energ. Phys.

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In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 [36] satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless and massive AdS3, mixed-flux relativistic AdS3 and massless AdS2. We also attack the class of models akin to AdS5 with our free fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.

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Free fermionic and parafermionic quantum spin chains with multispin interactions

Cited by 31 publications

References 29 publications

On the Q operator and the spectrum of the XXZ model at root of unity

On the Q operator and the spectrum of the XXZ model at root of unity

Free Fermions Behind the Disguise

Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

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