Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part I: Formal Developments

Abstract

We develop a quantum field theoretical framework to analytically study the three-body constrained Bose-Hubbard model beyond mean field and non-interacting spin wave approximations. It is based on an exact mapping of the constrained model to a theory with two coupled bosonic degrees of freedom with polynomial interactions, which have a natural interpretation as single particles and two-particle states. The procedure can be seen as a proper quantization of the Gutzwiller mean field theory. The theory is conveniently evaluated in the framework of the quantum effective action, for which the usual symmetry principles are now supplemented with a "constraint principle" operative on short distances. We test the theory via investigation of scattering properties of few particles in the limit of vanishing density, and we address the complementary problem in the limit of maximum filling, where the low lying excitations are holes and di-holes on top of the constraint induced insulator. This is the first of a sequence of two papers. The application of the formalism to the many-body problem, which can be realized with atoms in optical lattices with strong three-body loss, is performed in a related work [14].

Observability of Quantum Criticality and a Continuous Supersolid in Atomic Gases

“…1. The shift is only pronounced for n 1, the reason being that dimer formation and atom criticality approach each other for n → 0 [18].…”

“…Following Ref. [17], we introduce new operators b α,i = (R i ) αβ t β,i (α, β = 0, 1, 2), with a unitary transformation R. The parameters of R are such that b 0,i creates the mean field vacuum, and b 1,i and b 2,i correspond to fluctuations on top of this state, with vanishing expectation values (see [18]). The DSF ground state, for example, corresponds to: b 0,i = cos(θ/2)t 0,i + sin(θ/2) exp(−iφ)t 2,i , b 2,i = cos(θ/2)t 2,i − sin(θ/2) exp(iφ)t 0,i , and b 1,i = t 1,i , where φ is an arbitrary phase and the angle θ ∈ [0, π] is such that 2 sin 2 (θ/2) = n, the density of atoms (on the mean field level for simplicity).…”

“…Application of the formalism -The limit n → 0 gives a stringent check of our formalism, where we recover the nonperturbative Schrödinger equation for the dimer bound state formation (see [18] for details). We now apply our method in the many-body context:…”

“…The other two modes remain massive and do not affect the physics of the phase transition. Integrating out the latter we obtain an effective low energy action for the soft modes [18]:…”

“…In our case, however, the coupling κ vanishes for n → 1. The existence of such a decoupling point can be proven assuming a continuous, monotonic behavior of a particular compressibility, the µ -derivative of the dimer mass term K = −dm 2 d /dµ| n within the DSF phase: it then must have a unique zero crossing, because it is > (<)0 for n = 0(2) [18]. This argument is tied to the existence of a maximum filling, and thus to the three-body constraint.…”