APS March Meeting 2019, K19.00003 Infinite boundary conditions as Adam Iaizzi 1, * a current source for impurity Chung-Yu Lo 2 Pochung Chen 2 conductance in a quantum wire Ying-Jer Kao 1 *iaizzi@bu.edu (presenter) 1 National Taiwan University, Taipei, Taiwan 2 National Tsing Hua University, Hsinchu, Taiwan
Adam Iaizzi - iaizzi@bu.edu Impurity in a LL Wire P H YSICAL REVIEW LETTERS 24 FEBRUARY 1992 VOLUME 68, NUMBER 8 ❖ Luttinger liquid wire G=- 4t~ e~ satisfies ( I+ t2) h t fF dry(2/tag) [coth(Pco/2)( — InP(t) = 1+cosset) ❖ Single impurity — i sintot], where EF is the Fermi energy. This result is similar (but ❖ Theory: Kane-Fisher e 2 not identical) to that obtained et al. [5] who G= — by Devoret G=O g h the elfects of a series resistor a la Cal- studied (modeled deira and Leggett [14]) on a tunnel junction. The boson- ❖ T=0 conductance G ∝ t 2 V 2/ g − 1 of the ic excitations leads described Luttinger-liquid by l It (3) are an explicit of the Caldeira- realization physical l I I Leggett oscillators. In the expression derived by Devoret ❖ g>1 attractive — always I et al. , though, the series lead resistance is set to when FIG. 1. Schematic for 1D interacting How diagram electrons zero, an Ohmic I-V curve follows. In contrast, as we see Here 6 is the conduc- conducts one weakened by fraction t. with link below, (10) and (11) only give an Ohmic I Vcurv-e when tance across the weak link. Perfect reflection is found for repul- (g= I), in the ID leads are not interacting the electrons g & l, and perfect sive interactions, g ) 1. for attractive transmission g = I, are mar- so that the series lead resistance is h/e ❖ g=1 noninteracting interactions, Noninteracting electrons, (10) and (11) at T=0 gives Evaluating a power-law ginal I — ~ t - V- ~ I - V curve: For noninteracting fermions . (g = I ) the expected Ohmic conductance, this gives ❖ g<1 repulsive — always breaks V — 0 for g & l. (t =0) consists of two semi-infinite zeroth-order breaks whereas the expansion down as problem (3) with For g & 1, though, lines, which can be described by the Lagrangian a truly insulating strictly link with At T~O the linear con- the x integration x, re- restricted to positive or negative zero linear conductance is found. Kane & Fisher PRL 68 1220 (1992) law for g & 1: It is again to perform ductance vanishes as a power spectively. convenient a partial � 2 out p(x) for trace (in the p representation), integrating (12) all x away from the weakened We will then obtain link. in terms of the phases p+ (r ) on each an elfective action both T and formula P =(P~ — An approximate interpolation when is I — side of the link. If we further )/2 and define ]. Notice that G p t [Im(T+i V) ~ V are nonzero out 4(r) and obtain 4=(p++p )/2, we may integrate in (12) is not proportional to the square of the tunneling DOS: p(e=T) — of the the effective action terms phase following in T~+'t~ . rele- This is because the difference across the junction: vant DOS for the conductance is that for tunneling into the end of a semi-infinite which varies as Luttinger liquid, s„, =g„ I y(~) I -'. I ~ I (8) p„. „d(e)-c' ~ '. for all gal, p, . „d(e) varies Note that power than the bulk DOS p(c). with a diferent of (6). Note that is precisely the dual this expression For the lattice electron model with one weak link, it is t in terms of y, the perturbation Again, we may express (g= I ) for all t One. of course to calculate the two-terminal conduc- possible operator and the most relevant is case tance for the noninteracting SS — finds G =(e-/h)4t /(I+t ). Thus, in the RG sense, the cos[2 Jap], (9) t 4 z line g= I corresponds to a "fixed line" (see Fig. I). In to hopping an electron across the weak which corresponds of this case, it seems extremely plausible soluble view RG flows for small In this case the leading-order t the RG IIows be- link. [15] that for g&1 one can join together are 8t/t)l=(1 — (I — ')t, which small t). in Fig. 1. Thus, is shown g the two perturbative regimes t and tween once again g =1 is marginal, that 6=0 for but now the perturbation all t&l is g & 1, when This would imply for g & 1. For repulsively electrons irrelevant interacting G =ge /h for all nonzero t when g & 1. whereas with g & 1, an initially scales to zero at low weak hopping by the fact that will be complicated Real experiments to an insulat- below, this corresponds energies. As shown up into wide any one-channel wire must eventually open ing link with strictly zero linear conductance. is applic- Fermi-liquid theory leads, where presumably for the non- This can be seen by deriving an expression scale L or a time scale L/it, able. This defines a length characteristics current-voltage as a perturbation linear will cut off' the infrared associated with divergences which of t. a voltage expansion powers Upon V To study an ideal- in applying this we consider the Luttinger liquid. a vector potential into the across the weak link by adding ized model of an infinite one-channel electron wire with in (9), we can obtain of the cosine an expres- in the "sample" argument lxl & L, interactions with present only "leads, " lxl & L. response to second order in t: sion for the current In the in the (Fermi-liquid) but absent t ' (I — I — absence of the weak link, which we will take to be placed e -t")P(V), - (IO) at x =0, the appropriate La- in the middle of the sample P(t), by (3), but with g depending x, be- of P(V), denoted Fourier on where the transform grangian is given l222
Adam Iaizzi - iaizzi@bu.edu Impurity in a quantum wire ❖ Noninteracting —> Exact solvable R V ❖ Small bias —> static DMRG C I ❖ NonEQ: Experiment quantum simulator for LL wire + impurity Anthore et al. PRX 8 031075 (2018) ❖ Can do finite current 1 µm ❖ Tunable interactions ❖ Little corresponding theory/numerics ❖ Can we improve with Infinite Boundary Conditions? ❖ Measure finite-bias conductance ❖ Improve stability ❖ Important for nano electronic devices � 3
Adam Iaizzi - iaizzi@bu.edu Defining the problem Spinless fermions: ❖ Start with wires � i c i +1 + h.c. + V ( n i − 1 2)( n i +1 − 1 X − tc † H = 2) ❖ Impurity i Spin chain: ❖ Jordan-Wigner H ′ � − ∑ i [ J z ( S z i +1 ) ] i + 1) + J x 2 ( S + Transformation i S z i S − i +1 + S − i S + ❖ Spinless fermions —> S=1/2 XXZ ❖ For now: ❖ t —> J x ❖ NI leads J z =0 ❖ V —> J z ❖ NI impurity ❖ μ —> h � 4
Adam Iaizzi - iaizzi@bu.edu Finite-Size DMRG ❖ Time-dependent DMRG ❖ Measure finite current ❖ Open boundaries ❖ Conservation laws ❖ Finite-size effects—ringing 0.25 0.2 0.15 FIG. 4. � Color online � DMRG results compared to the exact 0.1 results for J � t � / � V obtained using different clusters L and number 0.05 J l (T) of states M , with � V =0.001. � a � L =96, � b � L =64, � c � L =32, and � d � 0 L =16. Note that for L =96 and 64, M =200 shows good qualitative -0.05 agreement and M � 300 even shows good quantitative agreement -0.1 with the exact results. For L =32 and 16, M =200 and 100 already -0.15 show excellent quantitative agreement with the exact results. -0.2 -0.25 0 20 40 60 80 100 120 Time Al-Hassanieh et al. PRB 73 195304 (2006) � 5
Adam Iaizzi - iaizzi@bu.edu Damped BC ❖ Add “Wilson Chains” w/ (a) ∆ V=0.3 0.08 J( τ ) exponentially decaying ∆ V=0.2 0.04 couplings ∆ V=0.1 N=32 Λ =2 N=72 Λ =1 ❖ ‘Soft’ boundary 0 0 10 20 30 τ 2 /h) 1 (b) (c) 0.1 ❖ Steady-state plateaus are d<J> τ /d( ∆ V)(2e <J> τ longer 0.5 0.05 N=32 Λ =2 N=72 Λ =1 ❖ Can we do better? 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 ∆ V ∆ V ❖ Infinite boundary conditions Dias da Silva PRB 78 195317 (2008) � 6
Adam Iaizzi - iaizzi@bu.edu Infinite Boundary Conditions # # # $ $ $ ! | ψ i = # # # # ! $ $ $ $ ❖ Procedure: % % % % % % % % H = & ' ❖ Do iDMRG for bulk wires # ( # ( # ( # ( $ ( $ ( $ ( $ ( ! ❖ Obtain bulk MPO and MPS # # # $ $ $ ❖ Sandwich a finite chain ) ' ) & % % % % % % % % & ' between fixed bulk tensors # ( # ( # ( $ ( $ ( $ ( , , , , ❖ Do finite-size DMRG H = ) + ) & ) ' ) * % % % % % % , ( , ( , ( , ( Phien, Vidal, & McCulloch, PRB 86 245107 (2012) � 7
Adam Iaizzi - iaizzi@bu.edu Background Quantum wire Quantum wire Source Drain V=+ Δ V/2 V=- Δ V/2 Impurity ❖ Source and Drain from IBC ❖ Allow conservation law violations ❖ Open system ❖ Find GS of finite-size system ❖ T=0, turn on field ( Δ V —> Δμ —> Δ h) ❖ Time evolve with TEBD � 8
Adam Iaizzi - iaizzi@bu.edu Charge density 10 -4 L=60, BD=20, IBC Step function charge density L=60, BD=20, IBC 5 0.4 4 0.3 3 0.2 2 (Charge Density)/ Charge Density 0.1 1 0 0 -0.1 -1 -0.2 -2 uniform =0 =0.001 uniform = 0.001 =0.01 -3 Step func. = 0.001 -0.3 =0.1 Impurity Impurity -4 -0.4 0 10 20 30 40 50 60 0 10 20 30 40 50 60 x x � 9
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Next steps ❖ Mitigate reflections at boundaries ❖ Better understand IBC as reservoir ❖ Add interacting impurity/QD ❖ Make wires interacting ❖ More impurities or more complex impurities � 11
Thanks Prof. Pochung Chen Chung-Yu Lo National Tsing Hua University National Tsing Hua University Prof. Ying-Jer Kao National Taiwan University � 12
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