# Spectral Gap and Exponential Decay of Correlations

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## Abstract

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions *D* < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

## Keywords

Degenerate Ground State Connected Correlation Function Fermionic Observable Ground State Sector Exponential Cluster## Preview

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## References

- 1.Haldane, F.D.M. Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett.
**93 A**, 464–468 (1983); Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett.**50**, 1153–1156 (1983)Google Scholar - 2.Koma, T. Spectral gap and decay of correlations in U(1)-symmetric lattice systems in dimensions
*D*< 2. http://arxiv.org/list/math-ph/0505022, 2005Google Scholar - 3.Hastings, M.B. Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B
**69**, 104431(13 pages) (2004)Google Scholar - 4.Hastings, M.B. Locality in quantum and Markov dynamics on lattices and networks. Phys. Rev. Lett.
**93**, 140402(4 pages) (2004)Google Scholar - 5.Fredenhagen K. (1985) A remark on the cluster theorem. Commun. Math. Phys. 97, 461–463zbMATHCrossRefADSMathSciNetGoogle Scholar
- 6.Nachtergaele, B., Sims, R. Lieb-Robinson bounds and the exponential clustering theorem. http://arxiv.org/list/math-ph/0506030, 2005Google Scholar
- 7.Dhar, D. Lattices of effectively nonintegral dimensionality. J. Math. Phys.
**18**, 577–585 (1977); Tasaki, H. Critical phenomena in fractal spin systems. J. Phys. A Math. Gen.**20**, 4521–4529 (1987)Google Scholar - 8.Wreszinski, W.F. Goldstone’s theorem for quantum spin systems of finite range. J. Math. Phys.
**17**, 109–111 (1976); Landau, L., Fernando Perez J., Wreszinski, W.F. Energy gap, clustering, and the Goldstone theorem in statistical mechanics, J. Stat. Phys.**26**, 755–766 (1981); Wreszinski, W.F. Charges and symmetries in quantum theories without locality. Forts. Phys.**35**, 379–413 (1987)Google Scholar - 9.Koma T., Tasaki H. (1995) Classical XY model in 1.99 dimensions. Phys. Rev. Lett. 74, 3916–3919CrossRefADSGoogle Scholar
- 10.Johnson J.D. (1981) A survey of analytic results for the 1-D Heisenberg magnets. J. Appl. Phys. 52, 1991–1992CrossRefADSGoogle Scholar
- 11.Majumdar, C.K., Ghosh, D.K. On next nearest-neighbor interaction in linear chain. I, II. J. Math. Phys.
**10**, 1388–1402 (1969); Majumdar, C.K. Antiferromagnetic model with known ground state. J. Phys. C**3**, 911–915 (1970); Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H. Valence bond ground state in isotropic quantum antiferromagnets. Commun. Math. Phys.**115**, 477–528 (1988); Fannes, M., Nachtergaele, B., Werner, R.F. Finitely correlated states on quantum spin chains. Commun. Math. Phys.**144**, 443–490 (1992)Google Scholar - 12.Kennedy T. (1990) Exact diagonalisations of open spin-1 chains. J. Phys. Cond. Matt. 2, 5737–5745CrossRefADSGoogle Scholar
- 13.Koma T., Tasaki H. (1992) Decay of superconducting and magnetic correlations in one- and two-dimensional Hubbard models. Phys. Rev. Lett. 68, 3248–3251CrossRefADSGoogle Scholar
- 14.Macris N., Ruiz J. (1994) A remark on the decay of superconducting correlations in one- and two-dimensional Hubbard models. J. Stat. Phys. 75, 1179–1184zbMATHCrossRefADSGoogle Scholar
- 15.Lieb E.H., Robinson D.W. (1972) The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257CrossRefADSMathSciNetGoogle Scholar