Strange metal state near quantum superconductor-metal transition in thin films

Abstract

We develop a theory of quantum T = 0 phase transition (q-SMT) between metal and superconducting ground states in a two-dimensional metal with frozen-in spatial fluctuations δλ(r) of the Cooper attraction constant. When strength of fluctuations δλ(r) exceeds some critical magnitude, usual mean-field-like scenario of the q-SMT breaks down due to spontaneous formation of local droplets of superconducting phase. The density of these droplets grows exponentially with the increase of average attraction constant λ. Interaction between the droplet's order parameters is due to proximity effect via normal metal and scales with distance ∝ 1/r β , with 2 < β ≤ 3. We account for this interaction by means of a real-space strong-disorder renormalization group (RG). Near the q-SMT the RG flow is, formally, a dual equivalent of the Kosterlitz-Thouless RG. The corresponding line of fixed points describes a Griffiths phase of a metal with large fractal clusters of superconducting islands. Typical number of islands in a cluster grows as N δ ∼ 1/δ, where 0 < δ 1 is the distance to the critical point. Superconducting side is described by a runaway of RG trajectories into the strong-coupling region. Close to the transition point on the SC side, 0 < −δ 1, RG trajectories possess an extremum as function of the RG parameter |δ| 1/2 ln(1/T τ ). It results in a wide temperature range where physical properties are nearly T -independent. This observation may be relevant to the understanding of a strange metal state frequently observed near q-SMT.PACS numbers:

Tail states and unusual localization transition in low-dimensional Anderson model with power-law hopping

“…With the choice (7) low-momentum asymptotic of the spectrum is Π(q) ≈ 2|q| α , in agreement with notations used in Ref. [12]. The exponent α is related to the coordinate-space exponent β as α = β − d, and we are interested in the "high dimension" case, 0 < α < d 2 (for usual Schroedinger equation it would be realized at d > 4, see Ref.…”

“…[12] and below in this paper) demonstrates purely exponential decay of ν(E < 0) ∝ e E/E 0 with |E| increase in a wide range of parameters β and W. It was shown in Ref. [12] that such a pure exponential behavior is essential in physical applications: it leads to a power-law distributions of quantum relaxation rates of non-linear localized modes associated with the localized eigenstates in the DOS tail. Therefore we feel it to study DOS in the tail region in more details; in particular, we are going to investigate the dependence of characteristic energy E 0 on the space dimensions d and exponent β, and to elucidate the reasons for the existence of a purely exponential behavior in a broad range of ν(E) variation.…”

“…Eigenfunction problem with deterministic power-law hopping amplitudes J(r) ∝ −r −β and on-site disorder was studied in numerous papers [1,2,3,4,5,6,10,7,8,9], see also recent review [11]. In the recent paper [12] two of us demonstrated that the same mathematical framework is relevant, in particular, for a quantum superconductor-metal transition in 2D disordered metals.…”

“…On the other hand, our own numerical analysis (present in Ref. [12] and below in this paper) demonstrates purely exponential decay of ν(E < 0) ∝ e E/E 0 with |E| increase in a wide range of parameters β and W. It was shown in Ref. [12] that such a pure exponential behavior is essential in physical applications: it leads to a power-law distributions of quantum relaxation rates of non-linear localized modes associated with the localized eigenstates in the DOS tail.…”

Non-Anderson critical scaling of the Thouless conductance in 1D

“…They have later been demonstrated 13,17,18 to occur in all systems with the power-law quasiparticle dispersion ∝ k α in high dimensions 13,18 d > 2|α|. These systems include, but are not limited to, arrays of trapped ultracold ions which exhibit power-law hopping of excitations [19][20][21][22][23][24][25][26][27][28] , high-dimensional semiconductors, quantum kicked rotors 13 and certain disordered supercoductive systems 29 . Apart from non-Anderson universality classes, such systems display critical scaling of the density of states (which does not exist for Anderson transitions), unconventional behaviour of Lifshitz tails [30][31][32] , energylevel statistics and ballistic-transport properties 13 .…”