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arXiv:1204.4170

Quantum Field Lagrangian model for charge density waves in one-dimensional systems at finite temperature : A Thermofield Dynamics Approach

Luiz Victorio Belvedere, Rubens Gurgel Amaral, Armando Flavio Rodrigues

Published Apr 18, 2012• 0 citations
arXivHTMLPDF

Abstract

We consider the Thermofield Dynamics bosonization to perform a field theory analysis of the effective Lagrangian model for incommensurate charge density waves (ICDW) in one-dimensional systems at finite temperature. The phonon degree of freedom is carryied by a dynamical phase field, contributing to the quantum dynamics and symmetry related features of the ICDW phenomenon. The electron chiral density and the phase of the phonon field condensate as a thermal soliton, carrying the symmetry under the linked electron-phonon $U_e^5(1) \otimes U_{ph}(1)$ global transformations. Using the Gell'Mann-Low formula for finite temperature, the perturbative series of the phonon thermal correlation function is obtained. Due to the electron-phonon charge selection rule we obtain for the thermal vacuum expectation value for the order parameter $<0 (β) |Φ| 0 (β)> = 0$, in accordance with the cluster decomposition property of the corresponding correlation function. This reflects the fact that the quantum description of the ICDW corresponds to a local charge transport through the lattice which is accomplished by an electron-lattice energy redistribution, which accounts for a thermal dynamical mass gap generation. The electron-phonon coupling can be rewritten in terms of a mass operator for the "physical" fermion operator $Ψ$ such that $<0 (β)| \barΨ(x)Ψ(x) | 0 (β)> \neq 0$, without the breakdown of the linked electron-phonon symmetry.

DOI
10.48550/arxiv.1204.4170

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We consider the Thermofield Dynamics bosonization to perform a field theory analysis of the effective Lagrangian model for incommensurate charge density waves (ICDW) in one-dimensional systems at finite temperature. The phonon degree of freedom is carryied by a dynamical phase field, contributing to the quantum dynamics and symmetry related features of the ICDW phenomenon. The electron chiral density and the phase of the phonon field condensate as a thermal soliton, carrying the symmetry under the linked electron-phonon Ue5(1)⊗Uph(1)U_e^5(1) \otimes U_{ph}(1)Ue5​(1)⊗Uph​(1) global transformations. Using the Gell'Mann-Low formula for finite temperature, the perturbative series of the phonon thermal correlation function is obtained. Due to the electron-phonon charge selection rule we obtain for the thermal vacuum expectation value for the order parameter <0(β)∣Φ∣0(β)>=0<0 (β) |Φ| 0 (β)> = 0<0(β)∣Φ∣0(β)>=0, in accordance with the cluster decomposition property of the corresponding correlation function. This reflects the fact that the quantum description of the ICDW corresponds to a local charge transport through the lattice which is accomplished by an electron-lattice energy redistribution, which accounts for a thermal dynamical mass gap generation. The electron-phonon coupling can be rewritten in terms of a mass operator for the "physical" fermion operator ΨΨΨ such that <0(β)∣Ψˉ(x)Ψ(x)∣0(β)>≠0<0 (β)| \barΨ(x)Ψ(x) | 0 (β)> \neq 0<0(β)∣Ψˉ(x)Ψ(x)∣0(β)>=0, without the breakdown of the linked electron-phonon symmetry.
We use the two-dimensional Fermion-Boson mapping to perform a field theory analysis of the effective Lagrangian model for incommensurate charge-density waves (ICDW) in one-dimensional systems. We consider an approach in which both the phase of the complex phonon field and the electron field are dynamical degrees of freedom contributing to the quantum dynamics and symmetry-related features of the ICDW phenomenon. We obtain the bosonized and fermionized versions of the effective electron–phonon Lagrangian. The phase of the phonon field and the phase of the bosonized chiral density of the electron field condense as a soliton order parameter, carrying neither the charge nor the chirality of the electron–phonon system, leading to a periodic sine-Gordon potential. The phonon field is fermionized in terms of a chiral fermionic condensate and the effective model is mapped into the chiral Gross–Neveu (GN) model with two Fermi field species. The linked electron–phonon symmetry of the ICDW system is mapped into the chiral symmetry of the GN model. Within the functional integral formulation, we obtain for the vacuum expectation value of the phonon field < ϕ > = 0 and < ϕ ϕ* > ≠ 0, due to the charge selection rule associated with the chiral electron–phonon symmetry. We show that the two-point correlation function of the phonon field satisfies the cluster decomposition property, as required by the chiral symmetry of the underlying GN model. The quantum description of the ICDW corresponds to charge transport through the lattice, due to the propagation of a "Goldstone mode" carrying the effective charge of the electron–phonon system, is accomplished by an electron–lattice energy redistribution. This accounts for a dynamical Peierls's energy gap generation.
The effective Lagrangian model for charge-density waves interacting with an impurity potential in one-dimensional systems is considered in the dynamical phonon phase approach. Using the fermion–boson mapping we obtain the effective bosonized version of the model. The impurity potential breaks the linked electron–phonon symmetry, the phason field turns out to be in interaction with the electron–phonon condensate and the phonon field develops a non-vanishing vacuum expectation value. The effective fermionized version of the model corresponds to the chiral Gross–Neveu model with quartic self-interaction among a massless and a massive (electron) Fermi fields. The electron–phonon system exhibits a conserved topological charge which is independent of local variations of the phases of the phonon and electron fields. The equation of state of the associated statistical-mechanical system is obtained. For the dynamical phonon phase field the Kosterlitz–Thouless phase transition is suppressed.
Wave function methods have offered a robust, systematically improvable means to study ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogues of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space, which is a formidable task. Besides, excited-state theories are generally not as well studied as ground-state ones. In this mini-review, we present an overview of a finite-temperature wave function formalism based on thermofield dynamics to overcome these difficulties. Thermofield dynamics allows us to map the equilibrium thermal density matrix to a pure state, i.e., a single wave function, albeit in an expanded Hilbert space. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we have developed a procedure to generalize ground-state wave function theories to finite temperatures. As explicit examples, we highlight formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of Fermions in the grand-canonical ensemble. To assess the quality of these approximations, we also show benchmark studies for the one-dimensional Hubbard model, while comparing against exact results. We will see that the thermal methods perform similarly to their ground-state counterparts, while merely adding a prefactor to the asymptotic computational cost. They also inherit all the properties, good or bad, from the ground-state methods, signifying the robustness of our formalism and the scope for future development.
The spontaneous generation of charge-density-wave order in a Dirac fermion system via the natural mechanism of electron-phonon coupling is studied in the framework of the Holstein model on the honeycomb lattice. Using two independent and unbiased quantum Monte Carlo methods, the phase diagram as a function of temperature and coupling strength is determined. It features a quantum critical point as well as a line of thermal critical points. Finite-size scaling appears consistent with fermionic Gross-Neveu-Ising universality for the quantum phase transition and bosonic Ising universality for the thermal phase transition. The critical temperature has a maximum at intermediate couplings. Our findings motivate experimental efforts to identify or engineer Dirac systems with sufficiently strong and tunable electron-phonon coupling.
We present a coupled cluster and linear response theory to compute properties of many-electron systems at nonzero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wave function representation of the thermal density matrix, and extend our recently developed framework ( J. Chem. Phys. 2019, 150, 154109, DOI: 10.1063/1.5089560 ) to parametrize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) and ab initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw{T}_{\mathrm{cdw}}Tcdw​ in two dimensions. Quantum Monte Carlo calculations have determined Tcdw{T}_{\mathrm{cdw}}Tcdw​ for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2d=2d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc{T}_{\mathrm{sc}}Tsc​ for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where ttt is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher TTT. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.