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arXiv:0902.2225

Persistent current of Luttinger liquid in one-dimensional ring with weak link: Continuous model studied by configuration interaction and quantum Monte Carlo

R. Németh, Martin Moško, Roman Krčmár, Andrej Gendiar, K. M. Indlekofer, Luboš Mitáš

Published Feb 12, 2009• 1 citation
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Abstract

We study the persistent current of correlated spinless electrons in a continuous one-dimensional ring with a single weak link. We include correlations by solving the many-body Schrodinger equation for several tens of electrons interacting via the short-ranged pair interaction V(x - x'). We solve this many-body problem by advanced configuration-interaction (CI) and diffusion Monte Carlo (DMC) methods. Our CI and DMC results show, that the persistent current (I) as a function of the ring length (L) exhibits for large L the power law typical of the Luttinger liquid, $I \propto L^{-1-α}$, where the power $α$ depends only on the electron-electron (e-e) interaction. For strong e-e interaction the previous theories predicted for $α$ the formula $α= {(1 + 2 α_{RG})}^{1/2} - 1$, where $α_{RG} = [V(0)-V(2k_F)]/2π\hbar v_F$ is the renormalisation-group result for weakly interacting electrons, with V(q) being the Fourier transform of V(x-x'). Our numerical data show that this theoretical result holds in the continuous model only if the range of V(x - x') is small (roughly $d \lesssim 1/2k_F$, more precisely $4d^2k_F^2 << 1$). For strong e-e interaction ($α_{RG} > 0.25$) our CI data show the power law $I \propto L^{-1-α}$ already for rings with only ten electrons, i.e., ten electrons are already enough to behave like the Luttinger liquid. The DMC data for $α_{RG} > 0.25$ are damaged by the so-called fixed-phase approximation. Finally, we also treat the e-e interaction in the Hartree-Fock approximation. We find the exponentially decaying I(L) instead of the power law, however, the slope of log(I(L)) still depends solely on the parameter $α_{RG}$ as long as the range of V(x - x') approaches zero.

DOI
10.48550/arxiv.0902.2225

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The persistent current $(I)$ of the spinless-electron Luttinger liquid is calculated by numerical microscopic methods in the continuous one-dimensional ring containing a weak link with transmission amplitude ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}⪡1$. The electrons interact via a screened interaction of range $d$. If $d\ensuremath{\lesssim}1/2{k}_{F}$, the interaction modifies the transmission as ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{N}^{\ensuremath{-}\ensuremath{\alpha}}$ and the current as $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $N$ is the electron number, $L$ the ring length, $N/L$ a fixed density, and $\ensuremath{\alpha}$ depends on the interaction in accord with the renormalization-group (RG) theory. Unlike our results, the RG theory predicts for a finite $d$ the power laws ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ and $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$, both depending on two interaction parameters, $d$ and $\ensuremath{\alpha}$. To explain why our results depend solely on $\ensuremath{\alpha}$, we show analytically that the interaction-matrix elements depend only on $\ensuremath{\alpha}$ (not on $d$) when $d\ensuremath{\lesssim}1/2{k}_{F}$. As $d$ exceeds $1/2{k}_{F}$, our result for $LI/e{v}_{F}$ starts to depend on $d$ as well and likely approaches the result $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ in the limit $L⪢d⪢1/2{k}_{F}$, not feasible by our numerical methods.

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arXiv:cond-mat/0209588•Physical review. B, Condensed matter•Jan 15, 2003•49 citations
We study the persistent current of correlated spinless electrons in a continuous one-dimensional ring with a single weak link. We include correlations by solving the many-body Schrodinger equation for several tens of electrons interacting via the short-ranged pair interaction V(x - x'). We solve this many-body problem by advanced configuration-interaction (CI) and diffusion Monte Carlo (DMC) methods. Our CI and DMC results show, that the persistent current (I) as a function of the ring length (L) exhibits for large L the power law typical of the Luttinger liquid, I∝L−1−αI \propto L^{-1-α}I∝L−1−α, where the power ααα depends only on the electron-electron (e-e) interaction. For strong e-e interaction the previous theories predicted for ααα the formula α=(1+2αRG)1/2−1α= {(1 + 2 α_{RG})}^{1/2} - 1α=(1+2αRG​)1/2−1, where αRG=[V(0)−V(2kF)]/2πℏvFα_{RG} = [V(0)-V(2k_F)]/2π\hbar v_FαRG​=[V(0)−V(2kF​)]/2πℏvF​ is the renormalisation-group result for weakly interacting electrons, with V(q) being the Fourier transform of V(x-x'). Our numerical data show that this theoretical result holds in the continuous model only if the range of V(x - x') is small (roughly d≲1/2kFd \lesssim 1/2k_Fd≲1/2kF​, more precisely 4d2kF2<<14d^2k_F^2 << 14d2kF2​<<1). For strong e-e interaction (αRG>0.25α_{RG} > 0.25αRG​>0.25) our CI data show the power law I∝L−1−αI \propto L^{-1-α}I∝L−1−α already for rings with only ten electrons, i.e., ten electrons are already enough to behave like the Luttinger liquid. The DMC data for αRG>0.25α_{RG} > 0.25αRG​>0.25 are damaged by the so-called fixed-phase approximation. Finally, we also treat the e-e interaction in the Hartree-Fock approximation. We find the exponentially decaying I(L) instead of the power law, however, the slope of log(I(L)) still depends solely on the parameter αRGα_{RG}αRG​ as long as the range of V(x - x') approaches zero.
The persistent current (I)(I)(I) of the spinless-electron Luttinger liquid is calculated by numerical microscopic methods in the continuous one-dimensional ring containing a weak link with transmission amplitude ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}⪡1$. The electrons interact via a screened interaction of range ddd. If $d\ensuremath{\lesssim}1/2{k}_{F}$, the interaction modifies the transmission as ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{N}^{\ensuremath{-}\ensuremath{\alpha}}$ and the current as $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where NNN is the electron number, LLL the ring length, N/LN/LN/L a fixed density, and $\ensuremath{\alpha}$ depends on the interaction in accord with the renormalization-group (RG) theory. Unlike our results, the RG theory predicts for a finite ddd the power laws ${t}_{{k}_{F}}\ensuremath{\simeq}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ and $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$, both depending on two interaction parameters, ddd and $\ensuremath{\alpha}$. To explain why our results depend solely on $\ensuremath{\alpha}$, we show analytically that the interaction-matrix elements depend only on $\ensuremath{\alpha}$ (not on ddd) when $d\ensuremath{\lesssim}1/2{k}_{F}$. As ddd exceeds 1/2kF1/2{k}_{F}1/2kF​, our result for LI/evFLI/e{v}_{F}LI/evF​ starts to depend on ddd as well and likely approaches the result $LI/e{v}_{F}\ensuremath{\simeq}|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}}_{{k}_{F}}|{(L/d)}^{\ensuremath{-}\ensuremath{\alpha}}$ in the limit L⪢d⪢1/2kFL⪢d⪢1/2{k}_{F}L⪢d⪢1/2kF​, not feasible by our numerical methods.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study the persistent current of the interacting electron gas in a one-dimensional continuous ring containing a single δδδ barrier. We calculate the persistent current as a function of the ring circumference, magnetic flux threading the ring, barrier strength, etc. We compare our results with the results of correlated models like the Luttinger liquid model and the Hubbard model solved by means of the renormalization group. A good agreement is found. First, the persistent current decays with the increasing ring circumference (LLL) faster than L−1L^{-1}L−1 and eventually like L−α−1L^{-α-1}L−α−1, where α>0α>0α>0 depends only on the electron-electron interaction. Second, the persistent current is a sine-shaped function of magnetic flux. This sine-like dependence and in particular the universal power law L−α−1L^{-α-1}L−α−1 have sofar been believed to arise only in the correlated many-body models. Observation of these features within the Hartree-Fock model is a surprising result.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we calculate the persistent current of the interacting electron gas in a one-dimensional ring containing a single δ barrier.Our results agree with correlated models like the Luttinger liquid model and lattice model with nearest-neighbor interaction.The persistent current is a sine-like function of magnetic flux.It decays with the ring length (L) faster than L -1 and eventually like L -α-1 , where α > 0 is universal.
Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.
Charge persistent currents in single-channel ballistic rings are investigated in various models of strongly correlated electrons (Wigner crystal, Luttinger liquid, Hubbard model, t–J model, and the fermion representation of the Heisenberg–Ising model). The primary objective of the survey is to compare the properties of Aharonov–Bohm oscillations for weakly and strongly correlated electrons and to formulate a set of persistent current characteristics that will unambiguously support the anomalous (departing from a Fermi liquid) behavior of one-dimensional electron systems. For this purpose the onset of the parity effect (dependence of the system response on whether the number of particles in the ring is odd or even) and various properties of the persistent current are analyzed in detail for model spinless fermions and real electrons. The influence of solitary impurities on the value and temperature dependence of the amplitude of the Aharanov–Bohm oscillations is also investigated. It is shown that the formation of a Wigner crystal (which is possible for weakly shielded Coulomb interaction) leads to a very pronounced anomalous (nonmonotonic) temperature dependence of the current in a nonideal ring. The properties of the charge current near the Mott metal–insulator transition are altered to the greatest degree by local types of electron–electron interaction.
The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than 1/N.1/N.1/N. Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.