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Nanosystems Physics Chemistry Mathematics

The Lieb–Mattis theorem revisited

Wojciech Florek

Published Apr 12, 2018• 2 citations
Publisher

Abstract

Simple formulation and a straightforward proof of the Lieb-Mattis theorem (LMT) do not indicate how powerful a tool it is.For more than fifty years, this theorem has been mainly discussed in its 'strong' form and applied mainly to many type of infinite spin lattices.It can be easily proved that in such cases, geometrically frustrated systems have to be excluded.However, it has been recently shown that the so-called 'weak'or general form of the LMT can be exploited considering some small, geometrically frustrated quantum spin systems.Moreover, many systems, which do not satisfy assumptions of the LMT, show identical features, including the level order characteristic for bipartite spin systems.It yields a question about possible generalizations or modifications of this theorem to involve a larger class of problems.To begin, algebraic aspects have to be understood with the invaluable role of the Perron-Frobenius theorem.The latter theorem is investigated, discussed, modified etc. in immense number of works.Many of them are important in physical applications and ways of reasoning they present can be exploited in different approaches to the LMT.

DOI
10.17586/2220-8054-2018-9-2-196-205

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Simple formulation and a straightforward proof of the Lieb-Mattis theorem (LMT) do not indicate how powerful a tool it is.For more than fifty years, this theorem has been mainly discussed in its 'strong' form and applied mainly to many type of infinite spin lattices.It can be easily proved that in such cases, geometrically frustrated systems have to be excluded.However, it has been recently shown that the so-called 'weak'or general form of the LMT can be exploited considering some small, geometrically frustrated quantum spin systems.Moreover, many systems, which do not satisfy assumptions of the LMT, show identical features, including the level order characteristic for bipartite spin systems.It yields a question about possible generalizations or modifications of this theorem to involve a larger class of problems.To begin, algebraic aspects have to be understood with the invaluable role of the Perron-Frobenius theorem.The latter theorem is investigated, discussed, modified etc. in immense number of works.Many of them are important in physical applications and ways of reasoning they present can be exploited in different approaches to the LMT.
Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407 ) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q . But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii–Moriya type interactions or other nonsymmetric models.
The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. Independently, Kolb considered the relation between the wave number qqq and the twisting boundary condition, and he obtained a similar result as LSM. However, in frustrating cases it is known that there exist several exceptions for the assumption of the unique lowest state for the finite size, which is important in the traditional LSM theorem. In our previous paper, without the assumption of the uniqueness, we have extended the LSMK theorem for frustrating and non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work.
We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.
A generalization of the Lieb-Schultz-Mattis theorem to higher-dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.
The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of s=12s=\frac{1}{2}s=21​ spins per unit cell. Here, the author presents a new family of Lieb-Schultz-Mattis type theorems for fermionic systems that have no analog in spin systems. An example is a particle-hole symmetric system of spinless fermions on a lattice with an odd number of fermionic modes per unit cell. The author shows that a symmetric gapped ground state must be topologically ordered, and the lattice translation symmetries must permute anyon types, resulting in non-Abelian zero modes bound to lattice dislocations. These results can be interpreted as quantum anomalies on the surface of three-dimensional fermionic symmetry-protected phases.