International Conference on Strongly Correlated Electron Systems (SCES)

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

Masaki Tezuka, Alejandro M. Lobos, Antonio M. Garcı́a-Garcı́a

Published Jun 16, 2014• 0 citations

Abstract

Quantum fluctuations destroy phase coherence and thus long-range order in one-dimensional (1D) superconductivity as long as all interactions are short-ranged.Here we discuss the possibility of restoring phase coherence by power-law hopping.A 1D attractive-U Hubbard model with powerlaw hopping having the power-law exponent α is studied by Abelian bosonization and density-matrix renormalization group (DMRG).For 1/2 < α < 3/2, the system has an effective spatial dimensionality d eff > 1.We demonstrate that long-range superconducting order is restored for 1/2 < α < α c , with 5/4 < α c < 3/2.This conclusion is supported by a DMRG analysis of the model.

DOI: 10.7566/jpscp.3.016004

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International Conference on Strongly Correlated Electron Systems (SCES)

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

Masaki Tezuka, Alejandro M. Lobos, Antonio M. Garcı́a-Garcı́a

Published Jun 16, 2014• 0 citations

Publisher

Abstract

DOI: 10.7566/jpscp.3.016004

Related papers

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

Alejandro M. Lobos, Masaki Tezuka, Antonio M. Garcı́a-Garcı́a

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 using Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, noninteger spatial dimensionality ${d}_{\mathrm{eff}}$ . For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppresses fluctuations and restores superconducting long-range order for any ${d}_{\mathrm{eff}}>1$ , at zero temperature. 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 an $S=1/2$ XXZ spin chain with power-law decaying couplings, which can be studied experimentally with cold-ion-trap techniques.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 using Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, noninteger spatial dimensionality ${d}_{\mathrm{eff}}$. For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppresses fluctuations and restores superconducting long-range order for any ${d}_{\mathrm{eff}}>1$, at zero temperature. 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 an $S=1/2$ XXZ spin chain with power-law decaying couplings, which can be studied experimentally with cold-ion-trap techniques.

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Interacting one-dimensional bosons with long range hopping decaying as a power law $r^{-α}$ with distance $r$ are considered with the renormalization group and the self-consistent harmonic approximation. For $α\ge 3$ , the ground state is always a Tomonaga-Luttinger liquid, whereas for $α<3$ , a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to $α<2$ , and for $2<α<3$ , a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.Interacting one-dimensional bosons with long range hopping decaying as a power law $r^{-α}$ with distance $r$ are considered with the renormalization group and the self-consistent harmonic approximation. For $α\ge 3$, the ground state is always a Tomonaga-Luttinger liquid, whereas for $α<3$, a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to $α<2$, and for $2<α<3$, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.

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Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.

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We study the superfluid--Bose-glass transition in a one-dimensional lattice boson model with power-law decaying hopping amplitude ${t}_{i\ensuremath{-}j}\ensuremath{\sim}1/{|i\ensuremath{-}j|}^{\ensuremath{\alpha}}$, using bosonization and the nonperturbative functional renormalization group (FRG). When $\ensuremath{\alpha}$ is smaller than a critical value ${\ensuremath{\alpha}}_{c}<3$, the U(1) symmetry is spontaneously broken, which leads to a density mode with nonlinear dispersion and dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}1)/2$; the superfluid phase is then stable for sufficiently weak disorder, contrary to the case of short-range hopping where disorder is a relevant perturbation when the Luttinger parameter is smaller than $3/2$ . In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion ( $z=1$ ) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.We study the superfluid--Bose-glass transition in a one-dimensional lattice boson model with power-law decaying hopping amplitude ${t}_{i\ensuremath{-}j}\ensuremath{\sim}1/{|i\ensuremath{-}j|}^{\ensuremath{\alpha}}$, using bosonization and the nonperturbative functional renormalization group (FRG). When $\ensuremath{\alpha}$ is smaller than a critical value ${\ensuremath{\alpha}}_{c}<3$, the U(1) symmetry is spontaneously broken, which leads to a density mode with nonlinear dispersion and dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}1)/2$; the superfluid phase is then stable for sufficiently weak disorder, contrary to the case of short-range hopping where disorder is a relevant perturbation when the Luttinger parameter is smaller than $3/2$. In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion ($z=1$) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.

arXiv:2404.16959Physical Review ASep 16, 20244 citations

U

Hubbard model with power-law hopping using Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, noninteger spatial dimensionality

{d}_{\mathrm{eff}}

. For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppresses fluctuations and restores superconducting long-range order for any

{d}_{\mathrm{eff}}>1

, at zero temperature. 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 an

S=1/2

XXZ spin chain with power-law decaying couplings, which can be studied experimentally with cold-ion-trap techniques.

Interacting one-dimensional bosons with long range hopping decaying as a power law

r^{-α}

with distance

r

are considered with the renormalization group and the self-consistent harmonic approximation. For

α\ge 3

, the ground state is always a Tomonaga-Luttinger liquid, whereas for

α<3

, a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to

α<2

, and for

2<α<3

, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.

3/2

. In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion (

z=1

) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.