Statistical mechanics approach to the electric polarization and dielectric constant of band insulators

Abstract

We develop a theory for the analytic computation of the free energy of band insulators in the presence of a uniform and constant electric field. The two key ingredients are a perturbation-like expression of the Wannier-Stark energy spectrum of electrons and a modified statistical mechanics approach involving a local chemical potential in order to deal with the unbounded spectrum and impose the physically relevant electronic filling. At first order in the field, we recover the result of King-Smith, Vanderbilt and Resta for the electric polarization in terms of a Zak phase -albeit at finite temperature -and, at second order, deduce a general formula for the electric susceptibility, or equivalently for the dielectric constant. Advantages of our method are the validity of the formalism both at zero and finite temperature and the easy computation of higher order derivatives of the free energy. We verify our findings on two different one-dimensional tight-binding models.

Anomalous dielectric response in insulators with the π Zak phase

“…Nonetheless, an abrupt change of the polarization as a function of the electric field in the SSH model, which we find in this paper, is not studied in Ref. [20]. Differences between the present work and Ref.…”

“…This behavior is dramatically different from that in the other case t 1 > t 2 , where P(ε) linearly increases with a small slope. We note that in the previous work [20] the polarization of the Rice-Mele model, which is the SSH model with a staggered potential added, is calculated with the same formalism, and its relation to the Zak phase is discussed. Nonetheless, an abrupt change of the polarization as a function of the electric field in the SSH model, which we find in this paper, is not studied in Ref.…”

“…In the modern theory of polarization [13][14][15][16], the notion of the bulk polarization is introduced, and it is formulated in term of the Zak phase, in the absence of an electric field. A dielectric response to an electric field is also discussed along the same line [20,26,27]. In these works, the dielectric response is studied as a bulk effect, which is considered to be valid in the thermodynamic limit.…”

Extracting the quantum metric tensor through periodic driving

“…Closely related to the Berry curvature is the quantum metric tensor (or Fubini-Study metric), which is a distinct geometric property of energy eigenstates that reflects the "distance" between different quantum states [20][21][22]. The significance of the quantum metric was recently identified in a wide range of physical phenomena, including the conductivity in dissipative systems [23][24][25][26][27][28], orbital magnetism [29][30][31][32][33], the superfluid fraction [34][35][36], quantum information [37][38][39][40], entanglement and many-body properties [41][42][43][44][45], interference in Bloch states [46], Lamb-shift-like energy shift in excitons [47] and the mathematical construction of maximally-localized Wannier functions in crystals [22,[48][49][50][51]. Despite the importance of the quantum metric in these various contexts, one still lacks a direct experimental measurement of this geometric object.…”

Steady-state Hall response and quantum geometry of driven-dissipative lattices

“…Therefore, it affects physical observables under the fast change of external parameters, or when the system is dissipative [9,12]. It is also known to play important roles in the orbital magnetic susceptibility [13][14][15][16][17], the entanglement and many-body properties of quantum systems [18][19][20][21][22], superfluid density [23,24], and quantum information [25][26][27][28]. Both the Berry curvature and the Fubini-Study metric can be understood from a general framework in terms of the quantum geometric tensor, whose real part gives the Fubini-Study metric while the imaginary part gives the Berry curvature [9].…”