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arXiv:1210.8344 · Advances in Mathematical Physics

A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

Mark Kelbert, Yurii M. Suhov

Published Jan 1, 2013• 6 citations
arXivHTMLPDF

Abstract

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is ℋ1=L2(M), where M is a d -dimensional unit torus M=ℝd/ℤd with a flat metric. The phase space of k spins is ℋk=L2sym(Mk) , the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H =⊕k=0,1,… ℋk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) -Δ/2 , the minus a half of the Laplace operator on M , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U(1)(x) , x∈M , describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y) , x,y∈M , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y) , x,y∈M , scaled along the graph distance d(i,j) between vertices i,j∈Γ , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on M , represented by a Euclidean space or torus of dimension d'≤d , preserving the metric and the volume in M . Furthermore, we suppose that the potentials U(1) , U(2) , and V are G -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G -invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.

DOI
10.1155/2013/637375

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This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is ℋ1=L2(M), where M is a d -dimensional unit torus M=ℝd/ℤd with a flat metric. The phase space of k spins is ℋk=L2sym(Mk) , the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H =⊕k=0,1,… ℋk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) -Δ/2 , the minus a half of the Laplace operator on M , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U(1)(x) , x∈M , describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y) , x,y∈M , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y) , x,y∈M , scaled along the graph distance d(i,j) between vertices i,j∈Γ , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on M , represented by a Euclidean space or torus of dimension d'≤d , preserving the metric and the volume in M . Furthermore, we suppose that the potentials U(1) , U(2) , and V are G -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G -invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.
In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature T_c T c . The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green’s function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement. The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the \chi χ -sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high- T_c T c cuprate superconductors.
A new approach to the single-band Hubbard model is described in the general context of many-body theories. It is based on enforcing conservation laws, the Pauli principle and a number of crucial sum-rules. More specifically, spin and charge susceptibilities are expressed, in a conserving approximation, as a function of two constant irreducible vertices whose values are found self-consistently. The Mermin-Wagner theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then obtained by a paramagnon-like formula that is consistent with the two-particle properties in the sense that the potential energy obtained from TrΣGTrΣGTrΣG is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. The vertex corrections are included through constant irreducible vertices. The theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. In the two-dimensional renormalized classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase transition. The analogous phenomenon of pairing pseudogap can occur in the attractive model in two dimensions when the pairing fluctuations become critical. Other many-body approaches are critically compared. It is argued that treating the spin fluctuations as if there was a Migdal's theorem can lead to wrong predictions, in particular with regard to the the single-particle pseudogap.
We consider infinite random causal Lorentzian triangulations emerging in quantum gravity for critical values of parameters. With each vertex of the triangulation we associate a Hilbert space representing a bosonic particle moving in accordance with the standard laws of Quantum Mechanics. The particles interact via two-body potentials decaying with the graph distance. A Mermin–Wagner type theorem is proven for infinite-volume reduced density matrices related to solutions to DLR equations in the Feynman–Kac (FK) representation.
In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
Exchange statistics is classified by representations of the symmetric group in three dimensions and of the braid group in two. Both classifications presuppose that the exchanges they represent are physically realizable operations. In fracton phases this premise fails: fractons are immobile, lineons move only along lines, and planons only within planes. We show that the correct replacement is not a new algebraic structure but a computable one. For an excitation whose mobility support is a graph Γ, the group of realizable exchange processes is the graph braid group Bn(Γ)=π1(UCn(Γ)), and abelian statistics form Hom(H1(UCn(Γ)),U(1)). We give an algorithm that derives mobility supports directly from a Pauli stabilizer Hamiltonian by linear algebra over F2, and apply it to the X-cube model, recovering lineon, fracton and fracton-dipole mobility without assuming any of it. We then prove an obstruction: in any lattice model with translation symmetry, the mobility support of a superselection sector is a coset of a subgroup of the translation group, and is therefore a point, a line, a plane, or all of space. No junction can occur, and consequently no translation-invariant fracton phase supports exchange statistics beyond bosons, fermions and ordinary two-dimensional anyons. Breaking translation symmetry evades this: a mobility support with a vertex of degree e≥3 has rankH1(UC2(Γ))=(e−1)(e−2)/2≥1, giving free U(1) exchange phases in three spatial dimensions that are not quantized to ±1. Because π1 is then not the symmetric group, this lies outside the scope of recent three-dimensional no-go theorems, and because no internal index is introduced it is not reducible by the Doplicher–Roberts construction. We exhibit the gauge-invariant signature explicitly: for hard-core particles on a junction the many-body spectrum depends on a statistical parameter θ through a flat, locally undetectable connection on configuration space, whereas on an open line no such parameter exists. All computations are reproducible.