93-292 Tohru Koma, Hal Tasaki

Obscured Symmetry Breaking and Low-Lying Excited States (130K, LaTeX) Nov 11, 93

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Abstract. We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. In the corresponding finite system, the symmetry breaking is usually ``obscured'' by ``quantum fluctuation'' and one gets a symmetric ground state with a long range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose energy per site converges to the ground state energy per site as the system size increases. When the system has a continuous symmetry, we prove that the number of independent low-lying eigenstates grows faster than any given small order of the system size. We show that a translation invariant low-lying state converges to a ground state in the infinite volume limit. We also construct infinite volume ground states with explicit symmetry breaking by taking linear combinations of the (finite-volume) ground state and the low-lying states, and then taking infinite volume limits. We conjecture these infinite volume ground states to be pure. Our general theorems do not only shed light on the nature of symmetry breaking in quantum many-body systems, but provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples.

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