Universal low-temperature crossover in two-channel Kondo models

Signatures of the NFL to FL crossover on the scale of $T^{*}$ should appear in all physical quantities. In 2CK or 2IK quantum dot devices which could access this physics,Potok et al. (2007); Zaránd et al. (2006); Mitchell et al. (2012) the quantity of interest is the $dI/dV$ conductance $G_{c}^{\alpha}(V,T)\equiv(2e^{2}h^{-1}G_{0}^{\alpha})\tilde{G}_{c}^{\alpha}(V,T)$ through channel $\alpha=L$ or $R$. Here, $G^{\alpha}_{0}=4\Gamma^{\alpha}_{s}\Gamma^{\alpha}_{d}/(\Gamma^{\alpha}_{s}+\Gamma^{\alpha}_{d})^{2}$ describes the relative strength of coupling to source and drain leads (see Fig. 2 for an illustration of the setup). At zero-bias $V=0$, the conductance is given exactly by,Meir and Wingreen (1992)

$$\tilde{G}^{\alpha}_{c}(0,T)=\tfrac{1}{2}\sum_{\sigma}\int^{\infty}_{-\infty}\textit{d}\omega\frac{-\partial f(\omega,T)}{\partial\omega}t_{\sigma\alpha}(\omega,T),$$

(5)

where $f(\omega,T)=[e^{\omega/T}+1]^{-1}$ is the Fermi function, and $t_{\sigma\alpha}(\omega,T)$ is the energy-resolved local density of states (or ‘spectrum’),

$$t_{\sigma\alpha}(\omega,T)=-\pi\nu\leavevmode\nobreak\ \text{Im}\mathcal{T}_{\sigma\alpha,\sigma\alpha}(\omega,T),$$

(6)

itself related to the t matrixHewson (1993) $\mathcal{T}_{\sigma\alpha,\sigma\alpha}(\omega,T)$, describing scattering of a $\sigma=\uparrow$ or $\downarrow$ electron within channel $\alpha=L$ or $R$, with bare lead density of states per spin $\nu$. Note that for equal hybridization to source and drain leads, $\Gamma^{\alpha}_{s}=\Gamma^{\alpha}_{d}$, $G^{\alpha}_{0}=1$ is maximal; while in the asymmetric limit $\Gamma^{\alpha}_{s}\ll\Gamma^{\alpha}_{d}$, $G^{\alpha}_{0}\ll 1$. In the latter case, the weakly-coupled source lead probes the system perturbatively, so the system remains near equilibrium, even at finite bias $V>0$. The resulting conductance is then simply,asy

$$\tilde{G}^{\alpha}_{c}(V,T)=\tfrac{1}{2}\sum_{\sigma}\int^{\infty}_{-\infty}\textit{d}\omega\frac{-\partial f(\omega-V,T)}{\partial\omega}t_{\sigma\alpha}(\omega,T),$$

(7)

where $t_{\sigma\alpha}$ is the equilibrium (zero-bias) spectrum.

The t matrix itself must show signatures of the NFL to FL crossover since scattering is purely inelastic at the NFL FP,Affleck and Ludwig (1993); CFT but inelastic scattering must cease at energies $\ll T^{*}$, where the impurity degrees of freedom are fully quenched.bor Thus the crossover also shows up in conductance. Our goal here is to calculate the full crossover t matrix, and hence conductance, at finite temperature for the 2CK and 2IK models in the presence of symmetry-breaking perturbations described by Eqs. 1–4.

In the next sections we derive an exact expression for the desired t matrix, describing the universal crossover from NFL to FL behavior in the 2CK and the 2IK models at finite temperature — and which as such generalize the results of our previous work in Ref. Sela et al., 2011. Here we pre-empt the full derivation, and present our key results.

The NFL to FL crossover is characterized by a low-energy scale $T^{*}$ arising due to the presence of symmetry-breaking perturbations to the 2CK and 2IK models. It is given generically bynot

$$T^{*}=\lambda^{2},$$

(8)

where $\lambda^{2}=\sum_{j=1}^{8}\lambda_{j}^{2}$. The eight contributions correspond to relevant perturbations which have distinct symmetry at the NFL FP. Two perturbations which have the same symmetry correspond to the same $\lambda_{j}$. The perturbations given in Eqs. 3 and 4 are classified viz,

$c_{1},c_{V},c_{B}=\mathcal{O}(1)$ are fitting parametersnot which depend on the model and on $J$, and $T_{K}\propto e^{-\frac{1}{\nu J}}$ is the Kondo temperature, characterizing RG flow from the high-energy free fermion FP to the NFL FP.Mitchell et al. (2012) We do not discuss the high-energy crossover in the present work.

The various perturbations described by Eqs. 3 and 4 describe very different physical processes — but the resulting crossover scale Eq. 8, has a simple form due to an emergent $SO(8)$ symmetry of the effective NFL FP Hamiltonians, as discussed in the following sections.

The main result of this paper is the NFL to FL crossover t matrix, given by,

$\displaystyle 2\pi i\nu\mathcal{T}_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}(\omega,T)=\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}-S_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}\mathcal{G}\left(\frac{\omega}{T^{*}},\frac{T}{T^{*}}\right)\leavevmode\nobreak\$

(9)

where $S_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}$ is the scattering S matrix, which is an $\omega=0$ and $T=0$ quantity characterizing the FL FP. For the 2CK model it is given by,

$$S^{2CK}_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}=\left[-\delta_{\sigma\sigma^{\prime}}(\vec{\lambda}_{f}\cdot\vec{\tau}_{\alpha\alpha^{\prime}})+i(\vec{\lambda}_{B}\cdot\vec{\sigma}_{\sigma\sigma^{\prime}})\delta_{\alpha\alpha^{\prime}}\right]/\lambda,$$

(10)

with $\vec{\lambda}_{f}=\{\lambda_{2},\lambda_{3},\lambda_{1}\}$. For the 2IK model it is,

$$\begin{split}S^{2IK}_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}=&\Big{[}-\lambda_{1}\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}+i\delta_{\sigma\sigma^{\prime}}(\lambda_{2}\tau^{x}_{\alpha\alpha^{\prime}}+\lambda_{3}\tau^{y}_{\alpha\alpha^{\prime}})\\ &+i(\vec{\lambda}_{B}\cdot\vec{\sigma}_{\sigma\sigma^{\prime}})\tau^{z}_{\alpha\alpha^{\prime}}\Big{]}/\lambda.\end{split}$$

(11)

The single function $\mathcal{G}$ describes the crossover due to a generic combination of relevant perturbations in both 2CK and 2IK models. It does not depend on details of the model or the particular perturbations present, except through the resulting crossover scale $T^{*}$. Thus $\mathcal{G}(\tilde{\omega},\tilde{T})$ is a universal function of rescaled energy $\tilde{\omega}=\omega/T^{*}$ and temperature $\tilde{T}=T/T^{*}$. Our exact result at finite temperature is,

$$\begin{split}&\mathcal{G}\left(\tilde{\omega},\tilde{T}\right)=\frac{\frac{-i}{\sqrt{2\pi^{3}\tilde{T}}}}{\tanh\frac{\tilde{\omega}}{2\tilde{T}}}\frac{\Gamma\left(\frac{1}{2}+\frac{1}{2\pi\tilde{T}}\right)}{\Gamma\left(1+\frac{1}{2\pi\tilde{T}}\right)}\times\\ &\int_{-\infty}^{\infty}dx\frac{e^{\frac{ix\tilde{\omega}}{\pi\tilde{T}}}}{\sinh x}{\rm{Re}}\left[{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1+\frac{1}{2\pi\tilde{T}},\frac{1-\coth x}{2}\right)\right],\end{split}$$

(12)

where $\Gamma$ is the Gamma function, and ${}_{2}F_{1}(a,b,c,z)$ is the Gauss hypergeometric function.Abr At $T=0$, Eq. 12 reducesnot to the result of Ref. Sela et al., 2011,

$$\mathcal{G}(\tilde{\omega},0)=\frac{2}{\pi}K\left[-i\tilde{\omega}\right],$$

(13)

where $K[z]$ is the complete elliptic integral of the first kind, yielding asymptotically $\mathcal{G}(\tilde{\omega},0)=1-i\tilde{\omega}/4-(3\tilde{\omega}/8)^{2}+\mathcal{O}(\tilde{\omega}^{3})$ for $\tilde{\omega}\ll 1$; and $\mathcal{G}(\tilde{\omega},0)=\frac{\sqrt{i}}{2\pi}(\pi-2i\log[16\tilde{\omega}])\tilde{\omega}^{-1/2}+\mathcal{O}(\tilde{\omega}^{0})$ for $\tilde{\omega}\gg 1$.

Below we consider the local density of states (spectrum) $t_{\alpha\sigma}(\omega,T)$, from which conductance can be calculated (see Eqs. 5 and 7). It is related to the t matrix via Eq. 6, and is thus given exactly along the NFL to FL crossover by Eqs. 9–12:

$$t_{\sigma\alpha}(\omega,T)=\tfrac{1}{2}-\tfrac{1}{2}\text{Re}\left[S_{\sigma\alpha,\sigma\alpha}\mathcal{G}\left(\tilde{\omega},\tilde{T}\right)\right],$$

(14)

where the required diagonal elements of the full S matrix (Eqs. 10 and 11) are more simply expressed as,

$$S^{2CK}_{\sigma\alpha,\sigma\alpha}=(-\alpha\lambda_{1}+i\sigma\lambda_{B}^{z})/\lambda=\alpha S^{2IK}_{\sigma\alpha,\sigma\alpha}$$

(15)

with $\sigma=\pm 1$ for spins $\uparrow/\downarrow$ and $\alpha=\pm 1$ for channel $L/R$ (and we use $\vec{\lambda}_{B}\parallel\hat{z}$ for simplicity). For $\lambda_{f}^{x}=\lambda_{f}^{y}=0$ and $\lambda_{B}^{x}=\lambda_{B}^{y}=0$ scattering preserves channel and spin, and the FL phase shift $\delta_{\sigma\alpha}$ then follows from $S_{\sigma\alpha,\sigma\alpha}=\exp[2i\delta_{\sigma\alpha}]$.

These exact results for the crossover are compared with finite-temperature NRG calculations in Sec. VII, with excellent agreement.

We now examine the generic behavior of the spectral function at finite temperatures in the crossover regime. Although we consider explicitly $L$-channel spectra $t_{\sigma L}(\omega,T)$ in the following, note from Eq. 15 that $t_{\sigma L}(\omega,T)\leftrightarrow t_{\sigma R}(\omega,T)$ upon exchanging $\Delta_{z}\leftrightarrow-\Delta_{z}$ in the 2CK model, or $\vec{B}_{s}\leftrightarrow-\vec{B}_{s}$ in the 2IK model. Also, $t_{\uparrow\alpha}(\omega,T)\leftrightarrow t_{\downarrow\alpha}(\omega,T)$ on reversing the magnetic field, $\vec{B}\leftrightarrow-\vec{B}$ (and in the zero-field case, $\sigma=\uparrow$ and $\downarrow$ spectra are of course identical).

In Fig. 3, we take the representative case of finite channel anisotropy $\Delta_{z}$ in the 2CK model, or finite detuning $(K-K_{c})$ in the 2IK model, and plot $t_{\sigma L}(\omega,T)$ as a full function of $\omega/T^{*}$ for different temperatures $T/T^{*}$. Since only $\lambda_{1}$ acts in either case, $S_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}=\pm\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}$ is diagonal (see Eqs. 10, 11), meaning that an electron in channel $\alpha$ scatters elastically at low energies, and stays in channel $\alpha$. By Eq. (14), the spectrum $t_{\sigma\alpha}(\omega,T)$ then probes the real part of the universal function $\mathcal{G}$ because $S_{\sigma\alpha,\sigma\alpha}$ is real.

General scaling arguments suggest that RG flow stops on an energy scale given by the temperature. As seen from Fig. 3, this is indeed the case, with the spectrum $t_{\sigma L}(\omega,T)\simeq t_{\sigma L}(0,T)$ essentially constant for $|\omega|\ll T$. Mutatis mutandis, for $T\ll T^{*}$ one obtains $t_{\sigma L}(\omega,T)\simeq t_{\sigma L}(\omega,0)$, corresponding to the $T=0$ limit considered previously.Sela et al. (2011) At $T=0$ and $\omega=0$, Eq. 14 yields $t_{\sigma\alpha}(0,0)=\tfrac{1}{2}-\tfrac{1}{2}\text{Re}S_{\sigma\alpha,\sigma\alpha}$, which is determined solely by the S matrix and hence the phase shift associated with the stable FL FP. When only $\lambda_{1}$ acts, the spectrum is thus $t_{\sigma\alpha}(0,0)=0$ or $1$ only (with corresponding phase shifts $0$ or $\pi/2$). In particular, the Kondo phase is characterized by unitarity $t_{\sigma\alpha}(0,0)=1$, obtained in the more strongly coupled channel for the 2CK model, and in both channels for $K<K_{c}$ in the 2IK model.

In the opposite limit $T\gg T^{*}$ ($\ll T_{K}$) RG flow to the FL FP is completely cut off, and inelastic scatteringbor at the NFL FP results in $t_{\sigma\alpha}(\omega,T)\simeq\tfrac{1}{2}$ for all $|\omega|\ll T_{K}$.

The generic RG picture illustrated in Fig. 1 and supported by Fig. 3, suggests an approximate complementarity between $\omega$ and $T$. This is explored further in Fig. 4, where we compare the zero-frequency value of the spectrum as a function of temperature, with the zero-temperature spectrum as a function of frequency. As immediately seen, there is a striking similarity. Indeed, a classic signature of FL physics (arising at low-energies/temperatures $|\omega|,T\ll T^{*}$) is the common quadratic dependence of the spectrum on both frequency and temperature,

$$t_{\sigma\alpha}(\omega,T)\leavevmode\nobreak\ \leavevmode\nobreak\ \overset{FL}{\sim}\leavevmode\nobreak\ \leavevmode\nobreak\ t_{\sigma\alpha}(0,0)+a\bigg{(}\frac{\omega}{T^{*}}\bigg{)}^{2}+b\bigg{(}\frac{T}{T^{*}}\bigg{)}^{2},$$

(16)

with $a$ and $b$ constants $\mathcal{O}(1)$ that depend on the particular model under consideration. Perturbation theory with respect to the FL FP in the spirit of Nozières,Noz yields $a/b=9/(7\pi^{2})$ for the 2IK model (see Appendix A). This same ratio also follows directly from the limiting behavior of the full crossover function, Eq. 12, which yields $|a|=9/128\approx 0.07$ and $|b|=7\pi^{2}/128\approx 0.54$; and as such provides a stringent check of our results.

But the exact symmetry between $\omega$ and $T$ in Eq. 16 is a special property of the FL FP itself, and does not in general apply at higher energies; although as seen from Fig. 4, the qualitative behavior over the full crossover is in fact rather similar. In the vicinity of the NFL FP (arising for $T^{*}\ll\max(\omega,T)\ll T_{K}$), Eq. 12 gives asymptotically,

	$\displaystyle t_{\sigma L}(\omega,T=0)$	$\displaystyle\overset{NFL}{\sim}\tfrac{1}{2}\pm\left\{\beta^{\prime}+\delta\log\frac{\omega}{T^{*}}\right\}\left(\frac{\omega}{T^{*}}\right)^{-\tfrac{1}{2}}$		(17a)
	$\displaystyle t_{\sigma L}(\omega=0,T)$	$\displaystyle\overset{NFL}{\sim}\tfrac{1}{2}\pm\beta^{\prime\prime}\left(\frac{T}{T^{*}}\right)^{-\tfrac{1}{2}},$		(17b)

with $\pm$ for $\lambda_{1}\gtrless 0$; and $\beta^{\prime}=\frac{(1+\pi)(2\gamma-\pi+\log 64)}{2^{5/2}\pi}\approx 0.5061$ (where $\gamma$ is Euler’s constant), $\delta=-\frac{1}{2^{3/2}\pi}\approx-0.1125$, and $\beta^{\prime\prime}\approx 0.4925$. Terms of the form $(\omega/T^{*})^{-1/2}$ and $(T/T^{*})^{-1/2}$ in Eq. 17 signal the scaling dimension $1/2$ of the relevant perturbation. Whereas such powerlaws occur in both the frequency and temperature dependence, additional logarithmic corrections appear in the frequency-dependence only. This difference can be understood by comparing the full $T=0$ result (Eq. 13) with the high-$T$ behavior captured by perturbation theory around the NFL FP (see Ref. Sela and Mitchell, 2012 and Appendix B.2). The full dependence on $\omega$ and $T$ described by Eq. 12 naturally leads to more subtle behavior when $|\omega|$ and $T$ are of the same order, as shown in Fig. 3.

When some degree of inter-channel charge transfer is also present, the NFL to FL crossover is generated by the combination of relevant perturbations $\lambda_{1}$ and $\lambda_{2}$. In the 2CK model, this corresponds to finite channel anisotropy $\Delta_{z}$ and impurity-mediated tunneling $\Delta_{x}$ (see Eq. 3); while for 2IK it corresponds to finite detuning $(K-K_{c})$ and direct tunneling $V_{LR}$ (see Eq. 4). The resulting behavior in the 2CK model can be simply understood because the perturbations $\Delta_{z}$ and $\Delta_{x}$ are related by a ‘flavor’ rotation of the bare Hamiltonian, as discussed further in Sec. VI.1. The 2IK model does not possess such a flavor symmetry, although an emergent symmetryAffleck and Ludwig (1993); CFT of the NFL FP Hamiltonian can be exploited when $T^{*}\ll T_{K}$. In fact, as shown in Sec. VI.3, this symmetry allows all the relevant perturbations in either 2CK or 2IK models to be simply related,Sela et al. (2011) implying the existence of a single crossover function.

The rotation $(\lambda_{1},\lambda_{2})\rightarrow\tilde{\lambda}_{1}$ can be used to relate systems where both $\lambda_{1}$ and $\lambda_{2}$ act, to those in which $\lambda_{1}$ alone acts (as in Fig. 3). But $t_{\sigma\alpha}(\omega,T)$ probes the system in the original unrotated basis, and hence the spectra undergo a rescaling when finite $\lambda_{2}$ is included: $t_{\sigma\alpha}(\omega,T)\rightarrow\tfrac{1}{2}+\left|\tfrac{\lambda_{1}}{\lambda}\right|[\tilde{t}_{\sigma\alpha}(\omega,T)-\tfrac{1}{2}]$. In particular, the spectral function is totally flattened the limit $|\lambda_{2}|\gg|\lambda_{1}|$, with $t_{\sigma\alpha}\simeq\tfrac{1}{2}$ at both FL and NFL FPs. At the FL FP, electrons thus scatter elastically between $\alpha=L$ and $R$ channels; and the corresponding S matrix $|S_{\sigma\alpha,\sigma^{\prime}\alpha^{\prime}}|=\delta_{\sigma\sigma^{\prime}}(1-\delta_{\alpha\alpha^{\prime}})$ is purely off-diagonal when inter-channel charge transfer dominates (see Eqs. 10, 11). Thus, no crossover shows up in the spectrum or conductance, although $T^{*}$ is of course finite (see Eq. 8); and the crossover can still appear in other physical quantities.Mitchell et al. (2011a)

When a uniform (2CK) or staggered (2IK) magnetic field acts (finite $\lambda_{B}^{z}$ only), $S_{\sigma\alpha,\sigma\alpha}=\pm i$ is pure imaginary (with phase shifts $\delta_{\sigma\alpha}=\pm\pi/4$), and again we obtain $t_{\sigma\alpha}(0,0)=\tfrac{1}{2}$ at the FL FP. However, $t_{\sigma\alpha}(\omega,T)$ now probes the imaginary part of the universal function $\mathcal{G}$ (see Eq. 14); and so the full spectrum along the NFL to FL crossover due to $\lambda_{B}^{z}$ is simply the Hilbert transform of the spectrum due to $\lambda_{1}$ — compare Figs. 3 and 5.

A spectral feature in consequence appears on the intermediate scale of $T^{*}$ for finite $\lambda_{B}^{z}$, even though $t_{\sigma\alpha}=\tfrac{1}{2}$ at both NFL and FL FPs; as shown in Fig. 5. The existence of such a feature can be understood physically from the impurity magnetization $M\sim B^{z}$ arising for small applied field $B^{z}$ in the 2CK model (or staggered magnetization $M_{s}\sim B_{s}^{z}$ due to a staggered field in the 2IK model). Since the magnetization $M(T)\propto\int_{-\infty}^{\infty}d\omega\leavevmode\nobreak\ f(\omega,T)[t_{\uparrow\alpha}(\omega,T)-t_{\downarrow\alpha}(\omega,T)]\neq 0$ is finite for finite applied field, $t_{\uparrow\alpha}(\omega,T)\neq t_{\downarrow\alpha}(\omega,T)$. A ‘pocket’ thus opens between the ‘up’ and ‘down’ spin spectra at $|\omega|\sim T^{*}$, whose area is proportional to the magnetization. In particular, the temperature-dependence of the magnetization can be extracted from the universal function, Eq. 12, viz

$$M(T)\propto\int_{-T_{K}}^{T_{K}}d\omega\leavevmode\nobreak\ f(\omega,T)\leavevmode\nobreak\ \text{Im}\mathcal{G}\left(\frac{\omega}{T^{*}},\frac{T}{T^{*}}\right),$$

(18)

valid for small perturbations $\lambda_{B}^{z}$, such that $T^{*}\ll T_{K}$ as usual (the high-frequency cutoff $|\omega|\sim T_{K}$ then being justified since $t_{\uparrow\alpha}(\omega,T)\simeq t_{\downarrow\alpha}(\omega,T)$ for $|\omega|\gtrsim T_{K}$, as confirmed directly from NRG).

Using Eq. 13, the zero-temperature magnetization depends on $T_{K}$ via,

$$M(0)=c\sqrt{T_{K}T^{*}}\log\left(\frac{T_{K}}{T^{*}}\right),$$

(19)

and since $T^{*}\sim(B^{z})^{2}/T_{K}$ (see Eq. 8), it follows that $M(0)\sim B^{z}\log(T_{K}/B^{z})$. The full temperature dependence $M(T)$ vs $T$ is shown in Fig. 6, demonstrating scaling collapse for different Kondo scales $T_{K}$. The asymptotic behavior in the vicinity of the FL and NFL FPs follows as,

	$\displaystyle\frac{1}{\sqrt{T_{K}T^{*}}}\left[M(T)-M(0)\right]\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\$	$\displaystyle\overset{T/T^{*}\ll 1}{\sim}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -d\left(\frac{T}{T^{*}}\right)^{2},$		(20a)
		$\displaystyle\overset{T/T^{*}\gg 1}{\sim}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -c\log\left(\frac{T}{T^{*}}\right),$		(20b)

yielding in particular $M(T)\sim B^{z}\log(T_{K}/T)$ when $T^{*}\ll T\ll T_{K}$. This asymptotic behavior can again be understood from perturbation theory around the FL and NFL FPs. Furthermore, since $\chi_{\text{imp}}(T)=\lim_{B^{z}\rightarrow 0}M(T)/B^{z}$, when $T^{*}\ll B^{z}\ll T\ll T_{K}$, one obtains

$$\chi_{\text{imp}}(T)\sim\log(T_{K}/T),$$

(21)

for the uniform (staggered) magnetic susceptibility of the 2CK (2IK) model. This diverging susceptibility is a classic signature of the NFL FP, known for example from the Bethe ansatz solution of the 2CK model,Andrei and Jerez (1995) or from CFT for the 2IK model.CFT However, at lower temperatures $T\ll B^{z}$, $M(T)/B^{z}$ does not correspond to the magnetic susceptibility: here the NFL to FL crossover itself is being probed. Indeed, for $T\ll T^{*}\ll B^{z}$, we obtain a quadratic $(T/T^{*})^{2}$ temperature-dependence of magnetization, Eq. 20a, characteristic of the FL FP.

Finally, we turn to conductance $\tilde{G}_{c}^{\alpha}(V,T)$, obtained from the spectrum $t_{\sigma\alpha}(\omega,T)$ by combining Eqs. 7, 12 and 14. It follows as

$$\begin{split}&\tilde{G}_{c}^{\alpha}(V,T)=\frac{1}{2}-\frac{\Gamma(\tfrac{1}{2}+\tfrac{1}{2\pi\tilde{T}})}{(8\pi\tilde{T})^{3/2}\Gamma(1+\tfrac{1}{2\pi\tilde{T}})}\sum_{\sigma}\text{Im}\Bigg{\{}S_{\sigma\alpha,\sigma\alpha}\times\\ &\int_{-\infty}^{\infty}dx\leavevmode\nobreak\ I(V,T,x){\rm{Re}}\left[{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1+\frac{1}{2\pi\tilde{T}},\frac{1-\coth x}{2}\right)\right]\Bigg{\}},\end{split}$$

(22)

where the integral over $\omega$ can be evaluated using contour methods,

$$\begin{split}I(V,T,x)=&\int_{-\infty}^{\infty}d\tilde{\omega}\leavevmode\nobreak\ \frac{\exp\left({\frac{ix\tilde{\omega}}{\pi\tilde{T}}}\right)\text{sech}^{2}\left(\frac{\tilde{\omega}-\tilde{V}}{2\tilde{T}}\right)}{\sinh(x)\tanh\left(\frac{\tilde{\omega}}{2\tilde{T}}\right)}\\ =&2\pi i\tilde{T}\leavevmode\nobreak\ \text{csch}^{2}(x)\leavevmode\nobreak\ \text{sech}^{2}\left(\tfrac{\tilde{V}}{2\tilde{T}}\right)\Bigg{[}\cosh(x)\\ &\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -\exp\left(\tfrac{ix\tilde{V}}{\pi\tilde{T}}\right)\left(1+\tfrac{ix}{\pi}\sinh\left(\tfrac{\tilde{V}}{\tilde{T}}\right)\right)\Bigg{]},\end{split}$$

(23)

with rescaled $\tilde{V}=V/T^{*}$, $\tilde{T}=T/{T^{*}}$, $\tilde{\omega}=\omega/T^{*}$ as before. In particular at zero-bias,

$$I(V=0,T,x)=\frac{\pi i\tilde{T}}{\cosh^{2}\left(\tfrac{x}{2}\right)}.$$

(24)

A color plot of conductance $\tilde{G}_{c}^{\alpha}(V,T)$ along the NFL to FL crossover is shown in Fig. 7, for the representative case of $\lambda_{1}>0$. The black lines connect regions of equal conductance. In the FL regime $V,T\ll T^{*}$, the quadratic form of Eq. 16 thus yields a simple ellipse; while in the NFL regime there is a pronounced $V$-$T$ anisotropy. The detailed behavior of Eq. 22 is seen on taking cuts through Fig. 7 at constant $V$ and $T$, as shown in Fig. 8.

Returning to the 2CK quantum dot system of Ref. Potok et al., 2007, we comment now on the possible strength of symmetry-breaking perturbations present in the experiment. Due to the Coulomb blockade physics of the quantum box, inter-channel charge-transfer was effectively suppressed. In the absence of a magnetic field, the dominant perturbation is thus channel anisotropy, $\Delta_{z}$ (see Eq. 3). The experiment showedPotok et al. (2007) 2CK scaling of conductance around $T_{K}$, but no FL crossover at $T^{*}$. This implies $T^{*}\ll T\ll T_{K}$ — see for example Fig. 3 for $T/T^{*}=100$, which shows little sign of the crossover. Since $T/T_{K}\approx 0.1$ in the experiment and taking $T/T^{*}>100$, from Eq. 8 and Table 1 it follows that $c_{1}\nu\Delta_{z}$ could be at most $\sim 0.03$ ($c_{1}=\mathcal{O}(1)$ depends on details of the model/device setup). The observed 2CK physics is thus an impressive testament to the tunability and control available in such quantum dot devices.

Having presented our main results and discussed their physical implications, we turn in the following sections to the formal derivation.

The structure of the NFL fixed point Hamiltonian of the 2IK model allows for an elegant description of the NFL to FL crossover.Sela et al. (2011) Before presenting that derivation, we discuss first some relevant preliminaries which will be of later use. In particular, we consider now the representation of the fixed point Hamiltonians within CFT, and the structure of the corresponding Green functions.

Our starting point is a description of the free conduction electron Hamiltonian $H_{0}$ in terms of chiral Dirac fermions. A 1D quadratic dispersion relation $\epsilon(\vec{k})\equiv\epsilon_{k}=k^{2}/2m-\epsilon_{F}$ can be linearized near the Fermi points $k=\pm k_{F}$, $\epsilon_{k}\simeq\pm v_{F}(k\mp k_{F})$ (with $v_{F}$ the Fermi velocity). This is the standard caseHewson (1993) and applies to arbitrary dimension within the assumption that the bare density of states is flat at low energies.Affleck (1995) Conduction electron operators can be Fourier transformed and expanded near the Fermi points, focusing on states within width $2D\ll v_{F}k_{F}$ around $\epsilon_{F}$,

	$\displaystyle\Psi_{\sigma\alpha}(x)$	$\displaystyle=$	$\displaystyle\sum_{k}e^{ikx}\psi_{k\sigma\alpha}$
		$\displaystyle\cong$	$\displaystyle e^{ik_{F}x}\sum_{k=k_{F}-D/v_{F}}^{k_{F}+D/v_{F}}\psi_{k\sigma\alpha}e^{i(k-k_{F})x}+(k_{F}\to-k_{F}).$

We thus define left and right movers,Affleck (1995)

$$\psi_{(l,r)\sigma\alpha}(x)=\sum_{k=-D/v_{F}}^{D/v_{F}}\leavevmode\nobreak\ e^{ikx}\psi_{k\mp k_{F},\sigma\alpha},$$

(26)

defined for $x\geq 0$, with $x$ the distance from the impurities located at the ‘boundary’ $x=0$. With a boundary condition $\psi_{r,\sigma\alpha}(0)=\psi_{l,\sigma\alpha}(0)$ we introduce a single left-moving chiral Dirac fermion for all $x$, $\psi_{\sigma\alpha}(x)=\theta(x)\psi_{l,\sigma\alpha}(x)+\theta(-x)\psi_{r,\sigma\alpha}(-x)$. Since the new operators satisfy the usual fermionic anticommutation relations $\{\psi^{\dagger}_{\sigma\alpha}(x_{1}),\psi^{\phantom{\dagger}}_{\sigma^{\prime}\alpha^{\prime}}(x_{2})\}=\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}\delta(x_{1}-x_{2})$, the free Hamiltonian $H_{0}$ can be expressed as a chiral Hamiltonian,Affleck (1995)

$$H_{0}=v_{F}\sum_{\sigma,\alpha}\int_{-\infty}^{\infty}\textit{dx}\leavevmode\nobreak\ \psi^{\dagger}_{\sigma\alpha}i\partial_{x}\psi^{\phantom{\dagger}}_{\sigma\alpha}.$$

(27)

Hereafter, we set $v_{F}\equiv 1$ [and the Fermi level density of states is then $\nu=1/(2\pi v_{F})\equiv 1/(2\pi)$].

The free electron Green function then follows as,von. Delft and Scholler (1998)

$$\begin{split}\langle\psi_{\sigma\alpha}^{\phantom{\dagger}}(\tau,x_{1})\psi_{\sigma^{\prime}\alpha^{\prime}}^{\dagger}(0,x_{2})\rangle_{0}=&\frac{\frac{1}{2\pi}\frac{\pi}{\beta}\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}}{\sin[\frac{\pi}{\beta}(\tau+ix_{1}-ix_{2})]}\\ \equiv&\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}G^{0}(\tau,x_{1}-x_{2}),\end{split}$$

(28)

where $\tau$ is imaginary time. The corresponding Matsubara Green function is then defined by the transformations

$$\begin{split}&G^{0}(\tau,x_{1}-x_{2})=\frac{1}{\beta}\sum_{n}e^{-i\omega_{n}\tau}\mathcal{G}^{0}(x_{1}-x_{2},i\omega_{n})\\ &\mathcal{G}^{0}(x_{1}-x_{2},i\omega_{n})=\int_{-\beta/2}^{\beta/2}\textit{d}\tau\leavevmode\nobreak\ e^{i\omega_{n}\tau}G^{0}(\tau,x_{1}-x_{2}),\end{split}$$

(29)

with Matsubara frequencies $\omega_{n}=\tfrac{\pi}{\beta}(1+2n)$ for integer $n$, and where $\beta=1/k_{B}T$ is inverse temperature. Direct evaluation of Eq. 29 using Eq. 28 then yields simply,

$$\begin{split}&\mathcal{G}^{0}(x_{1}-x_{2},i\omega_{n})=ie^{\omega_{n}(x_{1}-x_{2})}\times\\ &\delta_{\sigma\sigma^{\prime}}\delta_{\alpha\alpha^{\prime}}[\theta(\omega_{n})\theta(x_{2}-x_{1})-\theta(-\omega_{n})\theta(x_{1}-x_{2})].\end{split}$$

(30)

Of course, the interesting behavior arises when coupling to the impurities is switched on. As usual,Hewson (1993) the full Green function is related to the t matrix via

$$\begin{split}\bm{\mathcal{G}}(x_{1},x_{2},i\omega_{n})=\bm{\mathcal{G}}^{0}(x_{1}-x_{2},i\omega_{n})\\ +\bm{\mathcal{G}}^{0}(x_{1},i\omega_{n})\bm{\mathcal{T}}(i\omega_{n})\bm{\mathcal{G}}^{0}(-x_{2},i\omega_{n}),\end{split}$$

(31)

where $\bm{\mathcal{G}}$, $\bm{\mathcal{G}}^{0}$ and $\bm{\mathcal{T}}$ are $4\times 4$ matrices with indices taking the values $\sigma\alpha=\uparrow L,\downarrow L,\uparrow R,\downarrow R$. In particular, it should be noted that the t matrix is local. Eqs. LABEL:G0 and 31 also imply that if $x_{1}$ and $x_{2}$ have equal sign then the full Green function reduces to the free Green function, reflecting the chiral nature of the Dirac fermion, Eq. 27. Since all such fermions are now left-moving, information about scattering from the impurities located at the boundary $x=0$ — and hence the t matrix — is obtained from the full Green function with $x_{1}$ and $x_{2}$ located on opposite sides of the boundary.

The free Green function Eq. 28 is naturally obtained at FPs where the free boundary condition pertains. But at any conformally invariant FP, the powerful machinery of boundary CFT gives nonperturbative information about the Green function. Specifically, when the boundary condition is obtained from fusing with some primary field $a$, then correlation functions are given generically byCardy and Lewellen (1991)

$$\langle\mathcal{O}^{\phantom{\dagger}}_{d}(\tau,ix_{1})\mathcal{O}^{\dagger}_{d}(0,ix_{2})\rangle=\frac{\frac{S_{a}^{d}/S_{0}^{d}}{S_{a}^{0}/S_{0}^{0}}}{[\tau+ix_{1}-ix_{2}]^{2d}},$$

(32)

where $d$ is the scaling dimension of the primary field $\mathcal{O}$, and $S_{j}^{a}$ are elements of the modular S matrix.Cardy and Lewellen (1991) Using the conformal mapping from the plane to the cylinder with circumference $\beta$, one obtainsCardy and Lewellen (1991) the generalization to finite temperature $T=\beta^{-1}$,

$$\langle\mathcal{O}^{\phantom{\dagger}}_{d}(\tau,ix_{1})\mathcal{O}^{\dagger}_{d}(0,ix_{2})\rangle=\frac{\frac{S_{a}^{d}/S_{0}^{d}}{S_{a}^{0}/S_{0}^{0}}}{(\tfrac{\beta}{\pi}\sin[\tfrac{\pi}{\beta}(\tau+ix_{1}-ix_{2})])^{2d}}.$$

(33)