Fluctuations of squeezing fields beyond the Tomonaga--Luttinger liquid paradigm

Abstract

The concept of Tomonaga-Luttinger liquids (TLL) on the basis of the free-boson models is ubiquitous in theoretical descriptions of low-energy properties in one-dimensional quantum systems. In this work, we develop a squeezed-field path-integral description for gapless one-dimensional systems beyond the free-boson picture of the TLL paradigm. In the squeezed-field description, the parameter of the Bogoliubov transformation for the TL Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise to corrections to the free-boson results. We derive an effective nonlinear Lagrangian describing the dispersion relation of the squeezing field, and interactions between the excitations of the TLL and the squeezing modes. Using the effective Lagrangian, we analyze the imaginary-time correlation function of a vertex operator in the non-interacting limit. We show that a side-band branch emerges due to the fluctuation of the squeezing field, in addition to the standard branch of the free-boson model of the TLL paradigm. Furthermore, we perturbatively analyze the spectral function of the density fluctuations for an ultracold Bose gas in one dimension. We evaluate the renormalized values of the phase velocities and spectral weights of the TLL and side-band branches due to the interaction between the TLL and the squeezing modes. At zero temperature, the renormalized dispersion relations are linear in the momentum, but at nonzero temperatures, these acquire a nonlinear dependence on the momentum due to the thermal population of the excitation branches.

Squeezed-field path-integral description of BCS superconductors

“…The second dynamical term, L dyn,2 ≡ k v * k ∂ τ v k , describes the dynamics of the squeezing fields v k and v * k . Furthermore, as an additional feature of squeezed-field path integrals, L dyn includes a nonlinear dynamical term [33,34], given by…”

“…A similar set of dynamical terms also appears in squeezed-field path integrals for bosonic systems, see Refs. [33,34].…”

“…We note that a perturbative analysis of the nonlinear interactions of the squeezing fields has been discussed in Ref. [34].…”

“…Our formalism can be readily applied to more complex orders as well, such as unconventional p-wave superconductors and superfluids [6,35], and strongly-correlated Hubbard-type systems [36]. We demonstrate that the quantized motion of the squeezing parameters, referred to as the squeezing fields [33,34], is described by the classical Anderson pseudospin model [12,37,38], whose Hamiltonian is given by an energy functional of electrons with respect to the BCS variational state. We analyze experimentally-relevant quantities, in particular, the single-particle spectral function, based on the method developed here, and demonstrate that the squeezed-field path integral describes key features of the Higgs modes of BCS superconductors in the weakly-interacting BCS regime.…”

“…In this paper, we develop a generalized Grassmann path-integral approach to describe the fermionic quasi- * knagao@physnet.uni-hamburg.de particles and the bosonic collective excitations on equal footing. Our formulation presented in this work is based on the squeezed-field path-integral description, which has been developed in the context of Bose-Einstein condensates of ultracold gases [33], and more recently, extended to one-dimensional gapless systems described by Tomonaga-Luttinger liquid theory [34]. The squeezedfield path integrals are a field theoretical representation of quantum systems, in which one inserts the completeness relation of the squeezed coherent states, not the coherent states, into the time evolutions of the path integral.…”