Restoring phase coherence in a one-dimensional superconductor using power-law electron hopping

Abstract

In a one-dimensional (1D) superconductor, zero temperature quantum fluctuations destroy phase coherence. Here we put forward a mechanism which can restore phase coherence: power-law hopping. We study a 1D attractive-U Hubbard model with power-law hopping by Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, non-integer spatial dimensionality d eff . For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppress fluctuations and restore superconducting long-range order for any d eff > 1. A detailed DMRG analysis fully supports these findings. These results are also of direct relevance to quantum magnetism as our model can be mapped onto a S=1/2 XXZ spin-chain with power-law decaying couplings, which can be studied experimentally by cold ion-trap techniques.PACS numbers: 74.78. Na, 75.10.Pq According to the Mermin-Wagner-Hohenberg theorem quantum and thermal fluctuations in low dimensions prevent the spontaneous breaking of a continuous symmetry 1,2 . A paradigmatic example is a onedimensional (1D) superconductor (SC), where fluctuations of the SC order parameter result in quasi long-range order at zero temperature, i.e., the algebraic decay of the order parameter correlation function 3 . By contrast superconducting long-range order (LRO), equivalent to phase coherence in this context, occurs if the correlation function does not decay even for arbitrarily large distances.Therefore one of the main theoretical challenges in the field is to identify mechanisms that are capable to restore phase coherence in 1D. Interestingly, recent theoretical works have shown the possibility to stabilize a 1D SC through a weak coupling to a dissipative environment 4-9 that suppresses fluctuations and restore phase coherence 10 . Experimentally, restoration of phase coherence has been recently observed in thin Zn 11,12 and Al 13 nanowires by increasing the coupling of the wire to dissipative electrodes.The increase of the effective spatial dimensionality is another appealing choice. In the context of noninteracting 1D weakly disordered systems 14 , it is wellknown that power-law hopping ∝ 1/|i − j| α (with α > 1/2) effectively mimics the properties of a system in d eff = 2/ (2α − 1) spatial dimensions with short-range hopping. This effect seems to be robust to the presence of interactions 15 . Similar effects are also well-known in 1D spin chains with ferromagnetic (FM) 16-21 or with non-frustrating antiferromagnetic (AFM)22-24 power-law exchange couplings where LRO can occur at sufficiently low temperatures.In this Letter we study the role of power-law singleparticle hopping in 1D SCs. We focus on the 1D attractive-U Hubbard model with real-valued power-law hoppings t lm ∝ t/|l − m| α , where α is the parameter controlling the decay. We study the quantum phases of the system at zero temperature by analytical (Abelian bosonization and a variational approach) and numerical dens...

Restoring Long-Range Order in One-Dimensional Superconductivity by Power-Law Hopping

“…In the limit of |U| {t, t } the Hamiltonian can be mapped to a hard-core boson model by projecting out singly occupied sites by a unitary transformation [8]:…”

“…Here, θ (x) and φ (x) are bosonic fields which slowly vary on the scale of a and satisfy the canonical commutation relations [∇φ (x) , θ (y)] = iπδ (x − y). In this new representation, the expression of the Hamiltonian (2) at low energies [8] is…”

“…[8] we have studied Eq. (3) by applying the framework of the self-consistent harmonic approximation (SCHA).…”

“…We adopt a Gaussian ansatz S 0 = 1 2βL ∑ q g −1 0 (q) θ * q θ q for the Euclidean action of the system, with q = (k, −ω m ), ω m = 2πT m being the bosonic Matsubara frequencies at temperature T [16], and the functions g −1 0 (q) being unknown variational parameters which must be chosen to minimize the variational free energy F var = F 0 + T S − S 0 0 . Then we obtain a self-consistent equation for g 0 (q) [4,8,17,18] …”

“…[5][6][7] In [8] we have studied the effect of power-law hopping in a 1D SC. Examples of physical systems with interactions obeying a power-law decay in real space include Josephson junction arrays, [9] materials with strong dipolar interactions, [10] and ions in cavities realizing effectively quantum spin chains with long-range exchange interactions.…”