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arXiv:cond-mat/9709208 · Advances in Theoretical and Mathematical Physics

The complete set of ground states of the ferromagnetic XXZ chains

Tohru Koma, Bruno Nachtergaele

Published Jan 1, 1998• 35 citations
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Abstract

We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-5 ferromagnetic XXZ chains with Hamiltonian H = -EJS^S^ 4-Si 2) S% + ASi^S^l for all A > 1, and all 5 € |N.For the isotropic model (A = 1) we show that all ground states are translation invariant.For the proof of these statements we propose a strategy for demonstrating completeness of the list of the pure infinite-volume ground states of a quantum many-body system, of which the present results for the XXX and XXZ chains can be seen as an example.The result for A > 1 can also be proved by an easy extension to general S of the method used in [T.Matsui, Lett.Math.Phys.37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with A > 1.However, our proof is e-print archive:

DOI
10.4310/atmp.1998.v2.n3.a4

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We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-5 ferromagnetic XXZ chains with Hamiltonian H = -EJS^S^ 4-Si 2) S% + ASi^S^l for all A > 1, and all 5 € |N.For the isotropic model (A = 1) we show that all ground states are translation invariant.For the proof of these statements we propose a strategy for demonstrating completeness of the list of the pure infinite-volume ground states of a quantum many-body system, of which the present results for the XXX and XXZ chains can be seen as an example.The result for A > 1 can also be proved by an easy extension to general S of the method used in [T.Matsui, Lett.Math.Phys.37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with A > 1.However, our proof is e-print archive:
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We begin by providing a brief introduction to quantum spin systems including the de nitions and outlining the framework in which we'll be working for the rest of the thesis. We de ne and provide examples of F-functions and reproduce the proof of the Lieb-Robinson bound. [1] We then consider the Heisenberg XXZ spin-1/2 chain model and prove the form of the ground state space with kink and anti- kink boundary conditions. With the addition of a single pinning field, we demonstrate how this selects a single state in the ground state space to be the new unique ground state.
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We prove that for the spin-1/2 Heisenberg XXZ chain in the range Δ∈[−1,1/2]Δ\in[-1,1/2]Δ∈[−1,1/2], the ground state on the torus of length LLL converges to an infinite volume ground state ⟨⋅⟩\langle\cdot\rangle⟨⋅⟩ as L→∞L\to\inftyL→∞, and that the spin-spin correlation ⟨S0(1)Sx(1)⟩\langle S_0^{(1)}S_x^{(1)}\rangle⟨S0(1)​Sx(1)​⟩ decays polynomially fast in xxx. In the range Δ∈[−1,0]Δ\in[-1,0]Δ∈[−1,0] we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that LLL and β=1/Tβ=1/Tβ=1/T can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that ⟨S0(1)Sx(1)(t)⟩\langle S_0^{(1)}S_x^{(1)}(t)\rangle⟨S0(1)​Sx(1)​(t)⟩ decays to zero in ∣(x,t)∣|(x,t)|∣(x,t)∣, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points Δ=0,−1Δ=0,-1Δ=0,−1), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.