Delta-$T$ noise for weak tunneling in one-dimensional systems: Interactions versus quantum statistics

We study delta-$T$ noise—excess charge noise at zero voltage but finite temperature bias—for weak tunneling in one-dimensional interacting systems. We show that the sign of the delta-$T$ noise is generically determined by the nature of the dominating tunneling process. More specifically, the sign is governed by the leading charge-tunneling operator's scaling dimension. We clarify the relation between the sign of delta-$T$ noise and the quantum exchange statistics of tunneling quasiparticles. We find that, for infinite systems hosting chiral channels with local interactions (e.g., quantum Hall or quantum spin Hall edges), when the delta-$T$ noise is negative, the tunneling particles are boson-like, revealing their tendency towards bunching. However, the opposite is not true: Boson-like particles do not necessarily produce negative delta-$T$ noise. Importantly, the bosonic nature of particles generating the negative delta-$T$ is not necessarily intrinsic, but can be induced by the interactions. This, in particular, implies that negative delta-$T$ noise for tunneling between the edge states cannot serve as a smoking gun for detecting “intrinsic anyons”. We also establish a connection between the delta-$T$ noise and the temperature derivative of the Nyquist-Johnson (thermal) noise in interacting systems, both governed by the same scaling dimensions. As a demonstration of the above statements, we study tunneling between two interacting quantum spin Hall edges. With bosonization and renormalization-group techniques, we find that many-body interactions can generate negative delta-$T$ noise for both direct tunneling through a point contact and in Kondo exchange tunneling via a localized spin. In both these setups, we show that the noise can become negative at sufficiently low temperatures, when interactions renormalize the tunneling to favor boson-like pair-tunneling of electrons rather than single-electron tunneling. Our findings show that delta-$T$ noise can be used to probe the nature of collective excitations in interacting one-dimensional systems.

Figure 1

Two interacting QSH edges with clockwise (solid lines) and counter-clockwise (dashed lines) propagating edge modes. The modes have opposite spin projections because of spin-momentum locking. The couplings $g_{4\parallel }$ and $g_{2\perp }$ determine, respectively, forward (same chirality and, hence, same spin) and dispersive (opposite chiralities and spins) scattering amplitudes with a small momentum transfer. The edges are at the same chemical potential $\mu$ but at temperatures $T_1\ne T_2$ and are bridged by a single tunneling point. This results in a vanishing tunneling current $\langle I\rangle =0$, but finite excess charge noise $S\propto \langle \delta I\delta I\rangle$, called delta-$T$ noise. In panel (a), the particles transfer between the edges is described by the single-electron tunneling amplitude $t$, as well as by the coherent pair tunneling amplitudes $t_1^{\prime }$ and $t_2^{\prime }$ for spin preserving and spin-flipping tunneling, respectively. In panel (b), in addition to the above processes, tunneling is also possible via the Kondo exchange mediated by the spin $\vec{S}$, with backward (intraedge) and forward (interedge) scattering amplitudes $J_1$ and $J_2$, respectively.

Figure 2

Two distinct setups for tunneling between two conventional Luttinger-liquid wires. The interaction strength is quantified by the Luttinger parameter $K$ (dashed arrows). (a) Tunneling between two adjacent wires. Each wire can be described by four independent bosonic fields. The degrees of freedom are doubled in comparison to Fig. 1. Inset: Zoomed-in view of the channel structure. The spinful LL model contains the $g_{1\perp }$ interaction consisting of two coherent subprocesses involving opposite spins (indicated by the two arrows connected by the shaded box). This interaction term can not be diagonalized with bosonization [66]. On the ideal helical QSH edge, this term is forbidden by the time-reversal symmetry. (b) Tunneling across a weak link (between points $0^{-}$ and $0^{+}$) connecting two semi-infinite LL wires. Each semi-infinite wire (thick black lines) hosts particles of both chiralities, but can be transformed to host only a single chirality [see Eq. (90)] by “unfolding” that extends the half-wire to be fully infinite (lower panel). However, in addition to local interactions (black, dashed lines) this unfolding transformation introduces nonlocal interactions (red, wiggly lines).

Figure 3

(a) A two-terminal setup where particles tunnel between two terminals with temperatures $T_1$ and $T_2$ through a structure (wire, single point contact, barrier, or a weak link, etc.) characterized by the transmission probability $D$. The operator $I_{\mathcal{T}}$ (the red arrow) carries the current between two terminals. (b) Hong-Ou-Mandel interferometer with sources $S_1$ and $S_2$ and drains $D_1$ and $D_2$. Transmission and reflection amplitudes are denoted by $t,t^{\prime }$ and $r,r^{\prime }$ for particles emitted from $S_1$ and $S_2$, respectively. The current in the two drains and the tunneling current are denoted correspondingly by $I_1, I_2$, and $I_{\mathcal{T}}$. Current operators before the QPC are instead described by $I_{10}$ and $I_{20}$, respectively.

Figure 4

Sketches of (a) the noise coefficient ${\mathcal{C}}^{\left(2\right)}$ [see Eq. (78)] with varying average temperature $\overline{T}$, (b) the linear charge conductance $G$ (in a log-log form), and (c) The thermal noise $S_{\text{thermal}}$ (red curve) and its derivative over temperature ${\partial }_{\overline{T}}{S}_{\text{thermal}}$ (blue curve). Here, $T_a$ represents the temperature with the lowest conductance, or the transition temperature between regimes with dominating single-electron and coherent two-electron tunnelings. The other two temperatures $T_b$ and $T_c$ denote the characteristic temperature scales where, with decreasing temperature, conductance power laws begin to be determined by coherent pair tunneling and backscattering, respectively. The sketches are drawn for strongly attractive interactions, $K>4$.

Figure 5

Renormalization of the scaling dimension of the interedge Kondo coupling [denoted $\mathrm{\Delta }\left({\tilde{J}}_2\right)$] with decreasing energy (parametrized by the increasing length scale parameter $\ell$). (a) The interaction strength is fixed at $K=1.8$, and the solid, colored curves correspond to different initial values for ${\tilde{J}}_{1,2}$ [see Eq. (105)]. The red-dashed line $\mathrm{\Delta }\left({\tilde{J}}_2\right)=1/2$ marks the transition between positive and negative delta-$T$ noise [see panel (c)]. (b) The initial Kondo coupling strengths are taken as ${\tilde{J}}_{1,2}=0.1$. The solid-colored curves show how $\mathrm{\Delta }\left({\tilde{J}}_2\right)$ renormalizes for different values of $K$. (c) The change in the noise coefficient ${\mathcal{C}}^{\left(2\right)}$ [see Eq. (78)], with renormalized $\mathrm{\Delta }\left({\tilde{J}}_2\right)$ from (b). For $K>1, {\mathcal{C}}^{\left(2\right)}$ changes sign at sufficiently low energy.

Figure 6

Schematic RG flow of the Kondo tunneling model (104) for different interaction strengths $K$. Black arrows depict the flow from the quantum critical point ($K=1$) towards two stable fixed points. For $K<1$, the system flows towards the two-channel Kondo fixed point, while for $K>1$ it flows towards the single-channel Kondo fixed point. In the latter case, we expect a temperature window with negative delta-$T$ noise (${\mathcal{C}}^{\left(2\right)}<0$, in blue) produced by tunneling of emergent quasiparticles.

Figure 7

(a) The noise spectrum $A(\epsilon ,T)$ of free fermions in Eq. (A3). The spectrum is positive for most energies, except for a negative contribution at the lowest energies. (b) A zoom in at low energies. The noise spectrum changes its sign around $\left|\epsilon \right|\approx 2T$. (c) A Gaussian-shape transmission probability $D\left(\epsilon \right)$ [Eq. (A5)], where the width ${\epsilon }_0$ equals the temperature. (d) The free fermion noise as a function of the width (${\epsilon }_0$) of the Gaussian-shape transmission probability, Eq. (A5). The delta-$T$ noise changes sign at ${\epsilon }_0\approx 2.6T$.

Figure 8

(a) Generalized “distribution function” $a_K(T,\epsilon )$ [Eq. (A7)] as a function of the dimensionless energy $\epsilon /T$, for several values of the interaction parameters $K$. (b) The normalized ratio $r_K$ [Eq. (A8)] as a function of $\epsilon /T$. With increasing attractive interaction $K$, particles become more and more condensed around zero energy, leading to negative delta-$T$ noise. (c) The plot of the effective transmission $D_{\text{eff}}$ defined in Eq. (A9). (d) The noise spectrum $A_K(\epsilon ,T)T^{4-2/K}$ [Eq. (A10)] for various interaction strengths $K$. In interacting situations ($K>1$), the condition of weak transmission requires the bare (UV) value of the transmission probability $D_{\text{eff}}(\epsilon \sim 1/{\tau }_0)$ to be extremely small, such that $D_{\text{eff}}\left(\epsilon \right)\ll 1$ for all energies. (e) The coefficient ${\mathcal{C}}^{\left(2\right)}$ determining the delta-$T$ noise as a function of the interaction parameter $K$. The sign change occurs at $K=2$, which corresponds to ${\mathrm{\Delta }}_{\mathcal{T}}=1/2$. In contrast to FQH edges [47], where the effective interaction parameter is fixed by topological properties of the system, here arbitrary interaction parameters are possible.

Figure 9

(a) The thermal noise $S_{\text{thermal}}$ as a function of the interaction parameter $K$. (b) The spectrum $A_{\text{thermal}}^{\left(1\right)}$ of the Nyquist-Johnson noise [Eq. (A13)] for free fermions. The function is positive for all energies. (c) Since the spectral function is positive, ${\partial }_T{S}_{\text{thermal}}$ is positive for all ${\epsilon }_0$, referring to the missing of extreme-value point of $S_{\text{thermal}}$. (d) The second-order derivative ${\partial }_T^2{S}_{\text{thermal}}$ can be negative, with the transition value of ${\epsilon }_0$ close to that of Fig. 7 for the delta-$T$ noise with an energy-dependent transmission probability. (e) The thermal noise spectrum $A_{\text{thermal}}^K$ of interacting cases.

Figure 10

Anyon tunneling between Laughlin edges in a finite size region. The red and blue empty (filled) circles indicate initial (final) positions of two anyons. In panel (a), the top anyon enters the bottom edge from the left side of the scattering area, and propagates in front of the lower anyon. In panel (b), the top anyon instead enters the bottom edge from the right side. Here, the top anyon travels behind the bottom one.

Figure 11

(a) History of the system state $\alpha$ of discrete domains separated by ${\tau }_n$ (“kinks”) in imaginary time $\tau$. The system is the state ${\alpha }_n$ for ${\tau }_n<\tau <{\tau }_{n+1}$. The discretization is such that ${\tau }_{n+1}-{\tau }_n<{\tau }_c$, implying that this kink pair is to be removed or modified in the current RG step. (b) The non-neutral pair situation, where ${\alpha }_{n-1}\ne {\alpha }_{n+1}$. Here the two kinks merge together, adding to the RG flow of the operator ${\widehat{O}}_n^{\prime }={\widehat{O}}_n{\widehat{O}}_{n+1}$. (c) The neutral pair situation where ${\alpha }_{n-1}={\alpha }_{n+1}$. In this case, both kinks are to be removed in the current RG step.