Universal subdiffusive behavior at band edges from transfer matrix exceptional points

Abstract

Understanding symmetries and transitions in non-Hermitian matrices and their physical consequences is of tremendous interest in the context of open quantum systems. The spectrum of one dimensional tight-binding chain with periodic on-site potential can be obtained by casting the problem in terms of 2 × 2 transfer matrices. We find that these non-Hermitian matrices have a symmetry akin to parity-time symmetry, and hence show transitions across exceptional points. The exceptional points of the transfer matrix of a unit cell correspond to the band edges of the spectrum. When connected to two zero temperature baths at two ends, this consequently leads to subdiffusive scaling of conductance with system size, with an exponent 2, if the chemical potential of the baths are equal to the band edges. We further demonstrate the existence of a dissipative quantum phase transition as the chemical potential is tuned across any band edge. Remarkably, this feature is analogous to transition across a mobility edge in quasi-periodic systems. This behavior is universal, irrespective of the details of the periodic potential and the number of bands of the underlying lattice. It, however, has no analog in absence of the baths.

Environment assisted superballistic scaling of conductance

“…The band-edges therefore correspond to the EPs of the transfer matrix. As has been recently shown [26], this behavior is a consequence of an antilinear symmetry of transfer matrices of nearestneighbour tight-binding chains, and holds in general even for multi-band cases. As a consequence, in absence of the probes, conductance shows a universal sub-diffusive scaling G 0 (ω b ) ∼ N −2 at every band-edge [26].…”

“…Note that, at the band edge, the first term provides a subdiffusive 1/N 2 scaling in conductance [26]. We therefore focus on the second term…”

“…The determinants ∆ 1,N (ω), ∆ 1, −1 (ω) and ∆ N −p,N (ω) that appears in Eq. (S21) can be computed following the transfer matrix approach [26]. The equations which solve the deter-minants are following,…”

“…The surrounding environments are modelled by Büttiker voltage probes, as commonly done [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. We show that the superballis-tic scaling of conductance at every band-edge arises from an interplay of the incoherent effects from the Büttiker probes and exceptional points (EPs) of the system's transfer matrix that occurs at every band-edge [26]. The transfer matrix is a non-Hermitian matrix that appears in scattering theory.…”

“…where T 0 (ω)= ω 2 (I 2 + σ z ) − iσ y is the transfer matrix of the tight-binding chain, I 2 is 2 × 2 identity matrix and σ x,y,z are the Pauli matrices. The transfer matrix determines the band structure of the chain in the thermodynamic limit in absence of the baths via the relation Tr[T 0 (ω)] = 2 cos k, where k is the Bloch wave-vector [26][27][28][29]. This, in our simple single-band setting, gives the standard dispersion relation ω = 2 cos k. The bandedges correspond to k = 0, ±π, where ω ≡ ω b = ±2.…”