ONE SYK SINGLE ELECTRON TRANSISTOR

1. Introduction

The recent transformation [1–4] of the nearly three-decade old Sachdev–Ye (SY) model [5–8] into the novel Sachdev–Ye–Kitaev (SYK) one has bolstered vigorous activity on a number of important topics. In particular, the asymptotically soluble SYK model is believed to provide an important case study for a genuine holographic (albeit not exactly one-to-one) correspondence and a toy picture of quantum black holes [1–4]. Also, its various higher dimensional generalizations, including granular arrays of the coupled SYK systems and their hybrids with other subsystems, can serve as controlled examples of non-Fermi liquids (non-FL) [9–14] and associated phase transitions between such states and more conventional (disordered) Fermi liquids or (many-body) insulators [15–30]. Besides, the studies of the SYK model rekindled the field of quantum chaos, including the quest for its calculable quantifiers and real-time chaotic dynamics [3, 4].

The problem of two interacting SYK systems has also received much attention from the perspective of black hole physics (‘traversable wormhole’) [31–34]. As to the more down-to-Earth context of many-body quantum systems, a few recent works [35–37] addressed the problem of charge transport in the SYK ‘quantum dot’ attached to the ordinary FL leads. In the present note we further extend this analysis to the case of a tunnel junction between two (possibly different) SYK systems, one (or both) of which might be in the FL state.

To that end there has been a number of proposals of realizing the SYK physics in the quantum dot/wire environment [38–42]. In particular, it was argued that the SYK model could serve as a viable description of an irregularly shaped graphene flake in perpendicular magnetic field forcing all the electrons to occupy the highly degenerate lowest Landau level.

The actual SET layout may include additional leads and/or capacitively coupled gates (e.g. single vs two point contact geometry), as well as multiple modes of transmission. Nevertheless, its effective theory can often be reduced to that of a single tunnel junction by virtue of projecting out the decoupled linear combinations of the fermion modes.

2. Hamiltonian

A sufficiently general Hamiltonian of such a tunnel junction reads

\begin{array}{l} H = \sum_{α = L, R} (\frac{J_{i ... j}^{α}}{N_{α}^{(q_{α} 1) / 2}} Ψ_{α i}^{†} ... Ψ_{α j} - μ_{α} Ψ_{α i}^{†} Ψ_{α i}) + \\ \sum_{α, β = L, R} \frac{U_{α β}}{2} (Ψ_{α i}^{†} Ψ_{α i} - Q_{α}) + (Ψ_{β j}^{†} Ψ_{β j} - Q_{β}) + (1) \\ \frac{t}{{(N_{L} N_{R})}^{1 / 4}} \sum_{n}^{N_{L}} \sum_{m}^{N_{R}} (Ψ_{L i}^{†} Ψ_{R j} + Ψ_{R j}^{†} Ψ_{L i}), \end{array}

where the complex fermion operators ψ_αi correspond to the states localized in the left/right dot and q_α are even integers. The Coulomb charging energy is represented by the (cross)interaction terms U_αβ and also includes the offset charges Q_α on the capacitively coupled gates (if any).

The salient features of the (complex) SYK model can be brought about by treating the amplitudes $J_{i ... j}^{α β}$ of the all-to-all q_α-fermion couplings as Gaussian random variables with the time- and state-independent variances

〈 J_{i ... j}^{α} J_{i' ... j'}^{β} 〉 = δ^{α β} J_{α}^{2} δ_{i i} ... δ_{j j'} . (2)

Averaging Eq. (1) over such distribution results in introducing time-independent 2q-fermion SYK-type couplings to the action.

In what follows, we consider coupled SY K_q models with (possibly different) indices q_L and q_R, one (or both) of which can be in the FL regime formally corresponding to q = 2. In that regard, it is worth noting that a random q = 2 term in Eq. (1) would result in the same averaged SY K₂ coupling regardless of whether it represents a random diagonal on-site potential (Σ_iϵ_iψ^†iψ_i) or an off-diagonal kinetic energy (Σ_ijϵ_ijψ^†iψ_j).

When Eq. (1) is complemented with the SY K₂ ϵ-couplings, the SYK correlations are set in at ϵ²/J < T ≲ j, whereas at lower temperatures the system undergoes a crossover to the disordered FL regime [15–30, 43, 44].

Lastly, the t-term in Eq. (1) describes single-particle tunnelling between the dots. Treating it as fixed is customary in the standard theory of SET. However, one might also consider its amplitude to be random, as in the SYK-lattice models studied in Refs. [15–30, 43, 44].

The Hamiltonian (1) reveals a number of energy/temperature scales, the list of which includes J_α, U_αβ, t²/J_α, J_α/N_α, as well as the average single-particle level spacing δ₁ ~ J/N and its many body counterpart δ_N ~ Je^–O⁽^N⁾ [1–4].

In the previous work on the conventional SET [45–47] it was found that the onset of the Coulomb blockade (CB) would in general compete with the formation of a single-level (for T ≲ δ₁) or multi-level (for T ≲ δ_N) Kondo resonances (KR) with the characteristic Kondo temperatures T_K1 ~ Je^−U^/Γ and T_K_N ~ Je^−U^/^N^Γ, respectively (for Γ < U < J, where Γ ~ t²/J is the width of the KR).

It was also predicted that the two phenomena (CB and KR) can develop in either order depending on the system’s parameters, thus giving rise to a variety of the potential scenarios. Among them is the ‘direct’ one where CB would be followed by KR upon lowering the temperature. By contrast, in the ‘inverse’ scenario the multi-level Kondo scale T_K_Ncould exceed the Coulomb one and the onset of the Kondo effect can then precede that of the CB. Also, a 2-stage Kondo might occur when the Coulomb scale falls right within the range of temperatures between T_K1 and T_K_N.

3. Tunnelling as perturbation

From Eq. (1) one can readily obtain equations for the fermion propagators $G_{α β} = \sum_{i}^{N_{α}} \sum_{j}^{N_{β}} 〈 Ψ_{α i} Ψ_{β j}^{†} 〉 \sqrt{N_{α} N_{β}}$ obeying the equations

\begin{array}{l} (\partial_{τ} - μ_{L / R} - \sum_{L L / R R} *) G_{L L / R R} - t * G_{R L / L R} = δ (τ), \\ (\partial_{τ} - \sum_{L L / R R} *) G_{L L / R R} - t * G_{R L / L R} = 0, (3) \end{array}

where the familiar SYK self-energies [1–4] read

\sum_{L L / R R} = J_{L / R}^{2} G_{L L / R R}^{q_{L / R} - 1} (4)

and G_LR_,_RL stands for the ‘anomalous’ correlators. Notably, for q_L = q_R a development of such an off-diagonal order parameter breaks the Z₂symmetry of the corresponding ‘SYK-duplex’ model, as in the wormhole scenario of Refs. [31–34].

In the absence of the (random) inter-dot SYK correlations (⟨J^LJ^R⟩ = 0, as per Eq. (2)), the ‘anomalous’ propagators can be readily found in the form

G_{L R / R L} \approx G_{L L / R R} * t * G_{R R / L L} (5)

and the first equation in (3) for the diagonal propagator G_LL_/_RR = G_L_/_R takes a closed form

G_{L R} = {(\partial_{τ} - μ - \sum_{L / R})}^{- 1} (6)

in terms of the effectively diagonal self-energy

\sum_{L / R} = J_{L / R}^{2} G_{L / R}^{q_{L / R}^{- 1}} + t * G_{R / L} * t . (7)

The diagonal propagators can then be viewed as a saddle point of the effective theory where action includes the double time integral

\begin{array}{l} S_{t u n} = \frac{t^{2}}{2} \sum_{i l}^{N_{R}} \sum_{j k}^{N_{L}} \int_{τ_{1}, τ_{2}} Ψ_{R i}^{†} (τ_{1}) Ψ_{L j} (τ_{1}) Ψ_{L k}^{†} (τ_{2}) \\ \times Ψ_{R l} (τ_{2}), (8) \end{array}

which replaces the original tunnelling term in (1). This approximation can be further improved, thereby systematically recovering all the (even) higher-order processes, the next one in line (~t⁴) being that of (inelastic) co-tunnelling.

4. Conductance: direct tunnelling

The charge transport properties of the junction can be assessed by computing the current

I = \pm \frac{d Q_{L, R}}{d τ} = i t \sum_{i}^{N_{L}} \sum_{j}^{N_{R}} (Ψ_{L i}^{†} Ψ_{R j} - Ψ_{R j}^{†} Ψ_{L i}) (9)

with its ensemble average yielding the conductivity g(T) = dI/dV|_V→₀ due to the first-order (direct tunneling) processes

\begin{array}{l} g_{2} (T) = N_{L} N_{R} t^{2} \\ \times \int d ω \frac{\partial n (ω)}{\partial ω} Im G_{L} (ω) Im G_{R} (- ω) . (10) \end{array}

In what follows, we first neglect the charging energy in Eq. (1) and treat the tunnelling term perturbatively while focusing on the mean-field effects of the entangling SYK interactions. Such analysis holds for temperatures satisfying the conditions U_αβ, t²/J_α, J_α/N_α < T < J_α, where the zero-temperature mean-field fermion propagators read

G_{α} (τ \pm 0) = \pm \frac{A_{α}}{τ^{2 Δ_{α}}} e^{\pm π ε_{α}} . (11)

Here Δ_α = 1/q_α while the dimensionless parameters ℰ_α control the fermion occupation numbers in the dots, and the prefactor A_α is a known function of q_α and ℰ_α [1–4].

A finite-temperature counterpart of Eq. (11) can be obtained by the standard conformal transformation τ → sin(πτT)/πT, although Eq. (11) would still suffice for estimating the exponents in any (approximate) power-law dependences. Computing (10) for J_α/N_α < T < J_α one finds the linear conductance to the lowest (second) order in t

g_{2} (T) \sim g_{0} {(\frac{T}{\sqrt{J_{L} J_{R}}})}^{η_{q_{L}, q_{R}}}, (12)

where g₀ = πN_LN_Rt²/(J_LJ_R) and the exponent

η_{q_{L}, q_{R}} = 2 (Δ_{L} + Δ_{R}) - 2 (13)

stems from the product of the (vanishing at ω → 0, except for q_α = 2) densities of states in the coupled dots.

In particular, if both quantum dots are in the FL regime, the conductance is constant (η_2,2 = 0). In the case of a junction between FL and SY K₄, the exponent in the power–law T-dependence is η_2,4 = –1/2, such value being in agreement with the results of Refs. [35–37]. However, a junction between two SY K₄ models would feature the exponent η_4,4 = –1.

5. SYK fluctuations

As regards the effect of the SYK fluctuations about the mean-field solution (11), in the no-tunnelling/zero-Coulomb limit (t,U_αβ→ 0) the Hamiltonian (1) possesses a whole manifold of nearly degenerate states which are continuously connected to Eq. (11) by virtue of the arbitrary diffeomorphisms of the thermodynamic time variable τ → f_α(τ), obeying the boundary conditions f_α(τ + β) = f_α(τ) + β, combined with the U(1) phase rotations

\begin{array}{l} G_{α} (τ_{1}, τ_{2}) = A_{α} e^{{iΦ}_{α} (τ_{1}) - {iΦ}_{α} (τ_{2})} \\ \times {(\frac{\partial_{τ} f_{α} (τ_{1}) \partial_{τ} f_{α} (τ_{2})}{{(f_{α} (τ_{1}) - f_{α} (τ_{2}))}^{2}})}^{Δ_{q}} . (14) \end{array}

In addition to being spontaneously broken by a particular choice of the mean-field solution (11) down to the subgroup formed by the Mobius transformations SL(2, R), the reparametrization symmetry is also explicitly violated by the temporal gradients as well as the tunnelling and Coulomb terms present in Eq. (1).

At finite temperatures the dynamics of an SYK reparametrization mode is governed by the time integral of the so-called Schwarzian derivative $S c h {\tan \frac{π f}{β}, τ}$ defined according to the formula $S c h {y, x} \frac{y^{‴}}{y^{'}} - \frac{3}{2} {(\frac{y^{″}}{y^{'}})}^{2}$ and obeying the differential ‘chain rule’ Sch{F(y), x}Sch{F(y), y}y^'2 + Sch{y, x} [1–4].

Upon the change of variables ∂_τ f_α = e^ϕ_α this part of the overall action takes the form

S_{S Y K} = \sum_{α} \frac{γ_{α} N_{α}}{J_{α}} \int_{τ} (\frac{1}{2} {(\partial_{τ} ϕ_{α})}^{2} - {(\frac{2 π}{β})}^{2} e^{2 ϕ_{α}}), (15)

where the q_α-dependent prefactor γ_αwas computed numerically (see Refs. [3, 4] and references there-in). In the FL case of q_α = 2 one has γ_α = 0.

Thus, the fluctuations of the soft ϕ_α modes develop at energies below J_α/N_α, so at higher temperatures their effect can be neglected. In contrast, for T < J_α/N_α the ‘gravitational dressing’ of any product of the vertex operators e^ϕ_α(τ) can be performed in the basis of the eigenstates of the underlying quantum mechanical Hamiltonian of the Liouville theory deformed with the ‘quench’ potential acting during the time intervals between consecutive insertions of such operators [43, 44, 48, 49]. As the result, an arbitrary power p of the two-point propagator of an isolated SYK system develops a universal asymptotic behaviour

〈 G_{α}^{p} (τ) 〉 \sim 1 / {(J_{α} τ)}^{3 / 2} (16)

for all positive integer p and q_α > 2 [43, 44, 48, 49].

Applying this result to Eq. (12) one finds that as long as the CB effects remain negligible the lowest order conductivity develops a linear temperature dependence (η_qL_,_qR =1) for all q_α> 2 in the entire range E_c < T < J_α/N_α). However, for q_L = 2 the corresponding exponent is η_2,4 = 1/2.

It is worth mentioning, though, that an apparent agreement between the above exponent and the value obtained for q_L = 2, q_R = q in Ref. [35] for the non-linear conductance g(V) in the range t²/J, J/N < V < J is no more than an accident. In fact, the latter pertains to the limit of strong tunnelling where the propagators G_L_/_R in Eq. (10) would have to be replaced with G_L_/_R/(1 ± ig₀G_R_/_L).

Consequently, the conductance would turn out to be proportional to 1/g₀, thereby inverting the Fourier transform of the long-τ asymptotic (11), while accounting for neither the SYK fluctuations nor the effects of CB.

Nonetheless, the conductance g(V) computed in [48, 49] demonstrates an interesting duality, changing from η_2,_q= 2/q – 1 at weak tunnelling, (t²^q/J⁴)^1/(2^q–⁴⁾ < V, to η_2,_q = 1 – 2/q in the complementary strong tunnelling regime.

Lastly, in the general case of a large disparity between the (both non-vanishing) γ_LJ_L/N_L and γ_RJ_R/N_R the exponent (13) approaches η_qL, qR = 2/q_L_/_R – 1/2 for J_L_/_R/N_L_/_R < T < J_R_/_L/N_R_/_L since the SYK fluctuations affect only one of the two dots.

Moreover, with increasing strength of the coupling between the dots an independent application of two different Diff(S¹) transformations would conflict with the tunnelling term. Such a strong symmetry-breaking effect could be minimized, though, if the L/R reparametrizations were locked into one common transformation f_L = f_R, thereby resulting in the universal behaviour of the averaged products of an arbitrary number of G_L and G_R alike, .

6. Charging effects

In addition to the Schwarzian modes, for a sizable CB energy E_c > J_α/N_α the mean-field results become strongly affected by the charge fluctuations. According to the earlier studies of the conventional SET and other quantum devices comprised of the total of up to four dots and/or leads [45–47, 50–55], the analysis of such fluctuations can be facilitated by introducing the standard representation of the fermion operator as a product of its energy and charge degrees of freedom ψ_αn = χ_αne^iΦ_α. In this way, the fermionic particle–hole excitations in the dots become separated from the collective (‘plasmon’) degrees of freedom.

While the ‘fractionalized’ fermionic degrees of freedom χ_αn can still be traded for the bi-local collective variables G_α and Σ_α, those are now coupled to the phase fields Φ_α. The relevant part of the action then takes the form S = S_SYK + S_Φ, where the two terms describe (approximately) uncoupled fluctuations of the fields ϕ_α and Φ_α, respectively.

In particular, in the regime dominated by CB, J_α/N_α < T < E_c, the SYK fluctuations remain frozen, while the dynamics of Φ_α is described in terms of the (Euclidean) ‘phase-only’ effective action for the antisymmetric (out-of-phase) combination Φ = Φ_L – Φ_R (while the symmetric one decouples from the fields ϕ_α) and the concomitant parameters ΔQ, Δℰ

\begin{array}{l} S_{Φ} = \frac{1}{2 E_{c}} {\int_{τ} (\partial_{τ} Φ - i \frac{E_{c} Δ Q}{e} + 2 π i T Δ ε)}^{2} \\ + \frac{1}{2} \int_{τ_{1}} \int_{τ_{2}} K (τ_{1} - τ_{2}) \cos [Φ (τ_{1}) - Φ (τ_{2})] . (17) \end{array}

The effective Coulomb energy E_c includes a contribution stemming from the time derivatives in the action of the complex SYK model which explicitly breaks the U(1) symmetry (alongside the Diff(S¹) one) and is proportional to the inverse compressibility κ of the multi-fermion SYK system, E_c = U + κ^–¹ [56, 57].

As previously mentioned, the last term in Eq. (17) is the leading one in the systematic expansion of the effective bosonic action in powers of t². Alternatively, in the case of a random inter-dot hopping amplitude t it results from averaging over the Gaussian distribution, ⟨t²⟩ = t².

The corresponding kernel

K (τ) = t^{2} G_{L} (τ) G_{R} (- τ) D (- τ) (18)

represents the dissipative medium of the SYK particle–hole excitations. It is to be determined self-consistently in terms of the phase field propagator D(τ) = ⟨e^iΦ(^τ⁾e^–^iΦ(0)⟩ given by the formula

D (τ) = {(\partial_{τ}^{2} / E_{c} + λ + K)}^{- 1}, (19)

where the Lagrange multiplier λ enforces the normalization condition D(0) = 1.

It is worth mentioning that in the regime of interest a strongly non-Gaussian nature of the action (17) prevents one from computing the correlator D(τ) simply as the Debye–Waller factor exp(–⟨Φ(τ)Φ(0)⟩/2).

Also, the functional integral with the action (17) is to be evaluated on the field distributions subject to the boundary condition Φ(τ + β) = Φ(τ) + 2πn which reflects the compactness of the phase variable Φ. Therefore, for large phase fluctuations it is essential to account for the effect of the topological ‘θ-term’ [58, 59]

S_{top} = - i (Δ \tilde{Q} / e) \int_{τ} \partial_{τ} Φ (20)

embedded in Eq. (17) which is purely imaginary and proportional to the difference Δ = ΔQ – 2πeTΔℰ/E_c between the offset charges on the gates capacitively coupled to the dots (if any), corrected for the T-dependent excess charges on the dots themselves (notably, 2πℰ_α = –∂μ_α/∂T|_T→₀ [1, 2, 56, 57]).

7. Phase dynamics

Importantly, in the regime where the kernel demonstrates an algebraic behaviour

K (τ) = g_{0} / τ^{1 + s} (21)

governed by the exponent

s = Δ_{L} + Δ_{R}, (22)

it appears to be generically sub-ohmic (s < 1), as long as, at least, one of q_L_/_R is greater than 2.

This should be contrasted against the ordinary (FL) SET where attaining such a regime would only be possible in the presence of sufficiently strong excitonic enhancement in the final state. Otherwise, the behaviour of the kernel (21) would turn out to be super-ohmic due to the competing effect of orthogonality catastrophe [50–55]).

The earlier analyses of the sub-ohmic phase-only model with the kernel (21) demonstrated that at the critical coupling the conductivity takes its maximal (temperature independent) value which was estimated in Refs. [50–55] as

g_{c} \approx \frac{1}{2 π^{2} (1 - s)} . (23)

At this critical point the system undergoes a second order quantum phase transition from the disordered (⟨cosΦ⟩ = 0) CB-governed phase for g < g_c to a dissipation-driven ordered (⟨cosΦ⟩) ≠ 0) conducting one for g > g_c.

In the critical regime the system of coupled equations (6) and (7), where the term t * G_R_/_L * t now contains the extra factor D(τ)D(–τ), and Eq. (19) permits a scaling-invariant solution. Namely, G_L_/_R retain their SYK behaviour (11) while dragging along the phase correlator featuring a power–law asymptotic D(τ) = B/τ^2ΔΦ with the exponent

Δ_{Φ} = \frac{1}{2} (1 - s) (24)

and the prefactor B ~ (t²A_LA_R)^–¹. Correspondingly, the amplitudes A_L_,_R get reduced, as per the equation

In the ordered current-carrying resistive regime developing for g > g_c the phase field gets condensed and its propagator reaches a finite limit D(τ → ∞) = const corresponding to the vanishing effective charging energy E ^* c, consistent with the slower than 1/τ² power-law decay of the kernel (21).

By contrast, in the insulating phase the field Φ is disordered (while the charge dual to Φ tends to become quantized, ΔQ = n), its fluctuations are gapped, and their correlator decays exponentially

D (τ) = e^{- E_{c}^{*} / 2 T} F (T τ, ε), (25)

where the factor F = Σ_ne^–Ecn2/2+2πnE−nTτ stems from the infinite sum over the arbitrary winding numbers.

The renormalized Coulomb gap can then be estimated from the normalization condition D(0) = 1 which yields

E_{c}^{*} = 〈 λ 〉 = E_{c} {(1 - g / g_{c})}^{ν}, (26)

where ν = 1/(1 – s) for s > 1/2 and ν = 1/s for s < 1/2 [50–55]. For comparison, in the marginal case s = 1 the renormalized charging energy would be reduced, yet remains finite ( = 2πg²E_ce^–πg and = E_c(1–4g/π + ...) for g << 1 and g >> 1, respectively [50–55]).

Also, the behaviour of (hence the conductance) strongly depends on the offset gate charges. In the previous works [35–37] this dependence was not addressed, as if ΔQ assumed the default value of zero.

8. Conductance: inelastic co-tunnelling

At the transition points between the charge quantization plateaus (Q/e = n+1/2) the bare gap vanishes and the conductance computed in the ‘phase-only’ theory (17) appears to increase indefinitely upon lowering temperature. However, by invoking the Friedel sum rule one might argue that the conductance attains a finite value given by the total number of transmission channels (which may depend on the configuration of setup) and corresponding to the unitarity limit [45–47, 50–55].

In contrast, on the charge quantization plateaus ΔQ = n the renormalized Coulomb energy is maximal, so in the CB regime J_α/N_α < T < E_c the averaged current acquires the extra D²(τ) factor. Accordingly, the conductance (12) gets suppressed by an additional activation-like factor, g₂(T) ~ g₀e^–/T (T/J)^η.

The leading temperature dependence will then be dictated by the next (fourth) order contribution corresponding to the co-tunnelling processes which contribute to the current as

g_{4} (T) = N_{L} N_{R} t^{4} \int d ω \frac{\partial n (ω)}{\partial ω} Im {[K D]}_{ω} Im {[K D]}_{- ω} . (27)

Converting the integral (27) into the time domain one can see that it escapes the activation-like suppression due to the integration over two short (~1/, as opposed to the typically longer, ~1/T) times. It then dominates the power–law behaviour of the conductivity

g_{4} (T) \sim g_{0}^{2} \frac{J_{L} J_{R}}{E_{c}^{2}} {(\frac{T}{\sqrt{J_{L} J_{R}}})}^{η_{q_{L} . q_{R}}^{'}} (28)

with the exponent

{n^{'}}_{q_{L}, q_{R}} = 4 (Δ_{L} + Δ_{R}) - 2. (29)

In the case of q_L = 2, q_R = 4, one obtains the linear T dependence (η′_2.4 = 1) applicable for J_R/N_R < T < E_c (the parameters J_L, N_L of the dot in the FL state do not affect the range of applicability) which then dominates over the CB-suppressed direct tunnelling (12).

At still lower temperatures, T < J_R/N_R, this behaviour will crossover to another asymptotic with η′_2.4 = 3/2. Both of the above regimes were predicted in Ref. [37]. Either power-law would then be markedly different from that in the ordinary (FL) quantum dots with the standard η′_2.2 = 2 behaviour, independent of J_α.

It is worth mentioning, though, that while the complementary elastic counterpart of the co-tunnelling conductance is sensitive to the details of the actual fermion motion through the dots (hence, the phase of its wave function) it may result in a T-independent contribution, akin to the FL case, where the conductance saturates at g₄ ~ g²₀δ₁/E_c [60].

Moreover, at low temperatures (T < J_α/N_α) the ‘gravitational dressing’ with the ϕ_α fluctuations responsible for the above hallmarks of the SYK physics may eventually be forestalled by the developing KR [45–47].

In the FL case, the single-level KRs start to form upon lowering the temperature past the average single particle level spacing δ₁. Incidentally, the Schwarzian fluctuations begin to develop at the same T ~ δ₁ (possibly, up to a factor ~ 1/lnN [3, 4, 48, 49]). Regardless of whether or not the single-level FL scenario of the KR remains intact in the SYK systems, it can still develop below the characteristic multi-level Kondo scale T_K_N, where coherent transport through the SET can be supported by a many-body resonance close to the Fermi level. In this regime the conductance, too, tends to the unitarity limit as T → 0.

Given such an interplay of the competing CB, KR, topological term (20), and sub-ohmic ‘dissipation’ manifested by the kernel (17), the conductance g(T) would generally be non-monotonic and showing several regimes, ranging from an activation type of behaviour to an algebraic decay (or increase, depending).

9. Discussion

The overall picture of charge transport in the coupled SYK dots can then be described as follows. At high temperatures (T > J, E_c) the entire spectrum of an isolated SYK quantum dot appears nearly N-fold degenerate with an effective width of order J. The fermion spectral weight can then be approximated by a broadened peak, thereby resulting in the behaviour akin to that in the multi-level Kondo effect. The conductance gradually increases with decreasing T.

However, once the temperature drops below J but remains above E_c, the SYK physics starts to set in. For as long as the phase Φ fluctuations remain suppressed, it manifests itself in the power-law conductance (12) which continues to rise.

At lower temperatures, J/N ≲ T ≲ E_c, the sub-ohmic phase dynamics gets activated and the behaviour of the conductance starts to non-trivially depend on the coupling strength g₀. However, the reparametrization mode remains frozen and the mean-field SYK description can still be used.

In particular, at g₀ < g_c with g_c given by Eq. (23) the system is in the CB regime with the exponentially decaying g₂(T). At g₀> g_c, however, gets renormalized all the way down to zero, the phase variable remains condensed, and the conductivity continues to increase while the dependence on Q can only be observed at some intermediate temperatures.

At still lower temperatures, T ≲ J/N, the fluctuations of the reparametrization mode become fully developed, thereby renormalizing the arbitrary products of G_L_/_R in the universal manner. Regardless of whether it happens on just one or both sides of the junction, the effective kernel (17) becomes super-ohmic (s > 1) and then CB ensues for all g₀, except for the transitions between the plateaus at Q = n + 1/2, where vanishes.

Eventually, the CB regime gets arrested below T_K, where the formation of KRs takes over, the conductivity once again reversing its behaviour. It is possible, though, that such a recovery gets delayed until much lower temperatures set by the many-body level spacing δ_N, rather than its FL single-level counterpart δ₁, due to the presence of the strong SYK correlations.

A further analysis of this convoluted interplay between the SYK, charging, Kondo, and tunnelling effects will be presented elsewhere. It would also be of interest to extend it to the full counting statistics and its properties, as in Ref. [35], as well as the other physically relevant observables.

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VADINAMASIS SYK VIENO ELEKTRONO TRANZISTORIUS

D.V. Khveshchenko

Šiaurės Karolinos universitetas, Čepel Hilas, JAV