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1 oi:.38/nature4 Contents S-. Overview PART I: EXPERIMENT VS. THEORY S-. Our experimental ata vs. theory A. Experimental setup B. Fermi liqui properties C. Top-ate tunin of effective ss -factor 3 S-3. Other experimental ata vs. frg 7 A. Spin-resolve conuctance, shot noise 7 B. Compressibility an chare susceptibility 8 PART II: THEORETICAL DETAILS S-4. Moels use for barrier shape A. Hamiltonian, chemical potential, U B. Moel I C. Bare local ensity of states (LDOS) D. Moel II 4 E. Comparison: bare LDOS of QPC an QD 5 S-5. The low-enery scale B 6 A. Exponential Ṽc-epenence of B 6 B. Effects of interactions on B an T 8 C. Geometry-epenence of interaction U S-6. Functional renormalization roup A. Observables B. frg stratey an approximations C. frg Flow equations D. frg for non-uniform systems E. Zero-temperature limit 3 F. Static frg 4 S-7. Secon-orer perturbation theory 5 A. Equilibrium SOPT 5 B. Nonequilibrium SOPT 5 C. B-, T -, an Ṽ s -epenence of (Ṽc) 6 D. SOPT artefact arisin for increasin U 7 References 8 S-. OVERVIEW The followin supplementary material provies aitional information relate to various aspects of the main article. Its sections can be rea inepenently an in arbitrary orer. They are roupe into two parts: Part I (Secs. S- an S-3) is evote to experiments an their comparison with theory; Part II (Secs. S-4 to S-7 C) provies further theoretical etails. Section S- ives supplementary information about our measurements iscusse in the main article. Section S- A escribes the experimental setup. In Sec. S- B we present the raw ata on which the experimental tests of Fermi liqui preictions in the main article are base, toether with corresponin results obtaine by the functional renormalization roup (frg) (Fi. S). We also present aitional ata (Fi. S3) illustratin the ate-voltae epenence of the crossover scales in manetic fiel, temperature an source-rain-voltae, B, T an V s, toether with corresponin calculations usin secon-orer perturbation theory (SOPT). Sec. S- C explains in etail how the effective -factor ss is extracte from the transconuctance for lare fiels (Fi. S5), an offers some comments on the much-iscusse scenario that the.7-anomaly is ue to spontaneous spin polarization in the QPC reion. Sec. S-3 presents further T = frg results (Fis. S6 an S7) that emonstrate qualitative areement with shot noise an compressibility measurements by other roups. These frg results, an those in Sec. S-5, were calculate usin static frg, which iffers from the ynamic frg approach use in the main text by nelectin the frequency epenence of the self-enery an all vertices (see Sec. S-6 F). Static frg yiels results that are very similar to those of ynamic frg (see Fi. S5), while bein numerically cheaper by a factor 3. Section S-4 escribes our theoretical moel in etail. We have use two slihtly ifferent parametrizations of the QPC barrier shape, calle moel I an moel II, which both escribe parabolic barrier tops an hence ive essentially equivalent results for QPC properties. They are efine in Secs. S-4 B an S-4 D, respectively (the main article uses only moel II). Sections S-4 C an S- 4E explain how the effects of eometry are encoe in the bare local ensity of states (LDOS), focussin in particular on the van Hove rie of a QPC, which is key to unerstanin the.7-anomaly. Section S-5 focuses on the low-enery scale B (Ṽc) for a QPC: it shows that its exponential Ṽc-epenence has a purely eometric oriin, an that the strenth of its U- epenence likewise epens on the shape of the barrier. Sections S-6 an S-7 iscuss etails of the two theoretical methos use here to incorporate the effect of interaction: the functional Renormalization Group (frg) an secon orer perturbation theory (SOPT), respectively. Section S-7 C is evote to a etaile escription of our

2 SOPT results for finite temperature or finite source-rain voltae, offerin a summary of the features of the.7- anomaly which SOPT oes or oes not capture qualitatively. Finally, Sec. S-7 D iscusses an SOPT artefact that arises with increasin U. Equation an fiure an section numbers from the main article or the supplementary material are preface below by A (for article ) vs. S (for supplementary ), respectively, e.. Eq. (A), Fi. Af, Sec. A- vs. Eq. (S), Fi. Sb, Sec. S-4. As in the main article, we use tiles to istinuish theory parameters from those use in experiment, writin, e.. T = kb T, B = el µ B B, an Ṽc,s e V c,s. PART I: EXPERIMENT VS. THEORY S-. OUR EXPERIMENTAL DATA VS. THEORY A. Experimental setup The ate layout of our sample, shown in Fi. Aa for a ummy sample whose layout is ientical to that of the actual sample, provies a particularly hih tunability of the central constriction reion (CCR). The ates can be use to laterally efine either a quantum point contact (QPC) or a quantum ot (QD) in the two-imensional electron system (DES) 85 nm beneath the surface of a GaAs/AlGaAs heterostructure. In this work, we focus exclusively on the QPC eometry; a stuy of the crossover from QD to QPC will be publishe elsewhere. More information about the experimental conitions is provie in the methos summary section of the main article. In our two-terminal transport measurements the current I s flows throuh the nanostructure between ohmic contacts marke by source an rain in Fi. Aa, an we measure the ifferential conuctance = (I s /V s )/G Q (henceforth simply calle conuctance) usin lockin methos. In all measurements iscusse in this paper we apply a neative voltae V c to both center ates an a neative voltae V s to all four sie ates. This epletes the DES in the vicinity of the ates, so that propaation between source an rain throuh the CCR is confine to a narrow channel, leain to the quantization of transverse moes. (Further variations of iniviual ate voltaes allow aitional control of the lateral symmetry properties of the CCR, but such stuies are not inclue in this work.) Moreover, our sample also contains a lobal top ate (see Fi. Aa). In this work, we focus on ate voltaes such that transport is carrie solely by the first subban, corresponin to the lowest transverse moe. Its behavior can be escribe by a one-imensional effective moel for motion in the lonituinal (say x-) irection. The shape of the effective potential V eff (x) in the CCR can be chane by tunin V c, V s an V t. Increasin the top ate voltae V t increases the carrier ensity of the DES in the contacts of the CCR an hence the chemical potential, thereby eepenin (w. r. t. µ) the trenches between the reions of hih potential enery cause by the central an sie ates 4. This chanes not only the shape of V eff (x), but also causes the transverse wave functions to be more localize an hence increases the effective one-imensional on-site interaction strenth U within the CCR. For future reference, we summarize this tren as follows: The effective interaction strenth U can be increase experimentally by increasin V t. (S) For a QPC eometry, increasin V t has an aitional effect: ue to the eepene trenches in the potential lanscape, the enery spacin of the transverse eienmoes increases, resultin in an increase subban spacin 5. This tren is emonstrate in Fi. S base on measure pinch-off curves of our QPC for varyin top-ate voltaes. It can be use, in principle, to quantify the V t - inuce increase in U in terms of the V t -inuce increase in subban spacin 6, as will be elaborate in Sec. S-5 C below. B. Fermi liqui properties Fis. Sb an Sc show the raw experimental ata for the measure linear response conuctance of our QPC (a constant lea resistance has alreay been subtracte for all ata). They show how the pinch-off curves epen on manetic fiel an temperature, respectively. For comparison, Fi. Sa shows corresponin frg ata calculate for zero temperature as a function of the manetic fiel B. Both calculate an measure ata exhibit the expecte transition from a weak kink at.7 at small T an B to a pronounce.7-anomaly if either manetic fiel (measure an calculate ata) or temperature (measure ata) is substantially increase. The raw ata from Fis. Sb an Sc unerly the experimental results presente in Fis. Ae-h of the main article. Fis. S3a- shows aitional ata sets, plotte in the same way as in Fis. A an Ah, but isplayin ata not shown there for lack of space. Toether, these ata confirm the Fermi-liqui behavior expecte theoretically for sufficiently low fiels an temperatures: Fiures A an S3a,b show that at sufficiently low temperatures, T T (in our measurements T = T DES 3 mk), the leain manetic fielepenence of the linear conuctance is quaratic, (B)/() = (B/B ), B B, (Sa)

3 3 [mev ] subban spacin a b.4 Experiment(V s =.7V, B =, T= 3mK ) V s =.7V V t =.8V V c [V].6 V t =.7V V t [V] Fiure S: V t-epenence of subban spacin. a, Measure pinch-off curves (V c) of our QPC for a series of top-ate voltaes in the rane.7v V t.8 V. As V t is ecrease the carrier ensity also becomes smaller which, in turn, results in a larer pinch-off voltae V c an, clearly, in more narrow plateaus at inteer. The steep increase of (V c) inepenent of V t at V c.5 V inicates the transition from D to D transport as the DES irectly below the center ates is no loner eplete. b, Enery spacin between the lowest two D subbans as a function of V t. The ata points were evaluate from finite-v s measurements (raw ata not shown), usin a proceure escribe in Refs.,3, whose uncertainty is inicate by the error bars. The resultin subban spacin is approximately proportional to the with of the first conuctance plateau in a. As expecte from a simple capacitive moel, it is also proportional to V t (the ashe straiht line is a uie for the eye). as expecte from Eq. (A). Similarly, Fis. Ah an S3c, show that at zero fiel (B = ), the leain temperature epenence is likewise quaratic, (T )/(T )= (T/T ), T T. (Sb) Fittin Eqs. (Sa) an (Sb) to the ata in Fis. Sb an Sc, respectively, yiels the low-enery scales B (V c ) an T (V c ) use in Fis. A an Ah an epicte by colore symbols in Fis. Ae an Af, respectively (an similarly for Fis. S3a-). The scale conuctance curves isplaye in Fis. S3a- are plotte only in the restricte ranes (B)/().8 an (T )/(T ).8, respectively. For smaller conuctances, where the conitions B B or T T no loner hol, the measure B- an T -epenences of the conuctance eviate from quaratic behavior by benin upwar, tenin towar saturation (as shown in Fis. A,h ). The fit parameters B an T are compare in the half-loarithmic presentation in Fi. Af as functions of the center ate voltae V c. For convenience, this ata is shown aain in Fi. S3f, toether with the low-enery source-rain voltae scale V s. The latter was obtaine by eterminin the curvature of the nonlinear conuctance curves nl (V s ) (shown in Fi. A3i) at V s = : nl (V s )/ nl () = (V s /V s ), V s V s. (Sc) Compare to our eterminations of B an T from linear-response ata, those for V s have rather larer error marins, since for technical reasons the non-linear conuctance ata was measure with a smaller sinal-tonoise ratio. As mentione in the main article, SOPT makes two preictions for the Ṽc-epenence of the crossover scales B, T an Ṽs in the V c -rane where : first, all three scales epen exponentially on V c (Fi. S3e); an secon, the ratios B / T an Ṽs / T are, to within a few %, inepenent of Ṽc (as illustrate in Fi. S4 for a rane of U-values). The experimental results for B, T an V s shown in Fi. S3f confirm both preictions. This emonstrates that at low eneries a QPC shows Fermi-liqui behavior, as arue in etail in the main article. C. Top-ate tunin of effective ss-factor In a QPC eometry, interactions are known to enhance the effective electronic -factor 7 9. For lare fiels (B B ), an effective -factor, say ss, can be extracte from the transconuctance /V c, by exploitin the fact that the measure fiel-inuce subban splittin of the first conuctance step, say E, increases linearly with fiel, E = ss B. In previous experiments with in-plane fiels (B in the DES plane), ss -values have been observe exceein the bulk value ( GaAs.45) by up to a factor of 6 9,, an increase that was attribute by Koop et al. to interaction effects 9. In Fi. S5 we present the results of frg calculations an measurements of the transconuctance an the topate epenence of ss that confirm this interpretation. We numerically euce the transconuctance G/V c (G/Ṽc) from both the measure an calculate conuctance ata. Typical experimental results are plotte in Fi. S5a for the rane << as a function of V c. They show two peaks whose splittin E increases linearly for lare fiels, as E ss B + hfo (Fi. S5b), where both the slope ss an the hih-fiel offset hfo are foun to increase with top-ate voltae V t (Fi. S5c). 3

4 frg(u =.5τ, N =, Ω x =.4τ) Experiment ( V s =.4V, T = 3mK) a b c Experiment( V s =.4V, B = T) Ṽ V c [V] V c [V] B[T ].6 c [τ] T [K].4 B[τ ] Fiure S: QPC theory versus experiment, raw ata: a, frg ata (moel II) for the normalize conuctance = G/G Q, calculate at T = an fixe sie ate voltae Ṽs =.75τ as function of center ate voltae Ṽc an manetic fiel B. b, c Experimental ata for the normalize linear response conuctance = (I/V s )/G Q (lea resistance subtracte), measure at fixe sie an top ate voltaes, V s =.4 V an V t =.8 V. b at T DES = T = 3 mk, measure as function of center ate voltae V c an inplane manetic fiel B aline alon the narrow constriction. c, at B =, measure as function of V c an temperature T. The ata presente here are the raw ata use for Fis. e-h in the main article an in Fis. S3a- below. For better visibility, the pinch-off curves at minimal an maximal manetic fiel / temperature have been hihlihte by thick black lines, servin as uies for the eyes. The best sinal-to-noise ratio was achieve by slowly sweepin B at constant V c in b, an by sweepin V c at constant T in c. (B)/().8 - T = T B = T T = T B = T Vc.4 Experiment (V t =.8V, T = 3mK, V s =.4V ) a b B[T] µb B [mev ] c (T)/(T ) - - B[T] T = T B = T SOPT ( U =.35τ, N =, Ω x =.4τ ) e B=T= f V.5 =.4V B Ṽ c / Ω x Ṽ s - T Experiment (V t =.8V, V s =.4V ) B=, T=3mK Ẽ [τ] - (T)/(T ).8 B = T T = T Vc.4 kbt [mev ] T [K] -(T)/(T ) T [K] µ B B k B T.4 V c = V c V.5 [V] ev s E [mev] - Fiure S3: a-, Experiments; Fermi-liqui behavior. a, (B)/() on a lin-lo-scale an b, (B)/() on a lolo-scale. c an, as in a an b but for temperature instea of manetic fiel epenence. These ata supplement similar ata shown in Fis. A an Ah, not repeate here. Black lines are fits of the form (X)/() = X /X an express the leain quaratic ecrease in both temperature an manetic fiel. Insets emonstrate the exponential epenence of the scalin eneries µ BB an k BT (extracte from the fits) on V c, respectively. Colore symbols in the main plots an corresponin insets have the same V c-values. e, f, Comparison of low-enery scales from theory an experiment: e SOPT results for moel II, for the conuctance (thick black line) an the low-enery scales B, T an Ṽs (thin rey, black an brown lines, respectively), as functions of Ṽc. f Corresponin experimental results for, µbb an kbt (ata repeate from insets of a an c) an ev s as functions of V c = V c V.5, where V.5 is the ate voltae for which the conuctance at B = an T = T is (V.5) =.5. V s has been extracte from the leain quaratic epenence of the ZBP (as in a- for X = V s ). Note the strikinly oo qualitative areement between the SOPT results in e an the experimental results in f, for the ate-voltae epenence of all three low-enery scales. 4

5 V s * /T * B * /T * 4 SOPT ( U =.35τ, N =, Ω x =.4τ ) V s * /T * B * /T * V c /Ω x U[τ] Fiure S4: The ratios Ṽs / T (soli lines) an B / T (ashe lines) as functions of Ṽc/Ωx for several ifferent U- values, calculate by SOPT for moel II, usin the same QPC barrier shape as use for Fis. Ac,. The effect of increasin V t can be mimicke in our moel by increasin U (for reasons explaine in Supplementary Sec. S- A). Inee, the results of our frg calculations, shown in Fi. S5-f, qualitatively match the trens shown by the experimental ata in Fi. S5a-c. This establishes several important points. First, interactions are the reason why the -factor extracte from E(B) is anomalously lare. Secon, the effective interaction strenth can be tune experimentally via a top ate voltae. Thir, the experimental observation of hfo can be unerstoo without aoptin the spontaneous spin polarization scenario that is often avocate 7,9,3 to explain it. Let us now elaborate these points in more etail. We theoretically stuie the U-epenence of ss by usin frg to calculate pinch-off curves for parabolic QPC barrier shapes such as that of Fi. Ab, for a rane of fiels B an interaction strenths U. Fi. S5 plots the transconuctance, i. e. the erivative (Ṽc)/Ṽc as function of Ṽc (varie over a rane corresponin to ), for a lare number of ifferent B-fiels, at U =.5τ. In such a plot the fiel-inuce spin splittin of the conuctance step manifests itself as a pair of local maxima 7 9,. The Ṽc-separation of their peaks, say E, is proportional to the effective B-inuce subban splittin. Eviently E increases with B. Fi. S5e shows E( B) vs. B for six values of U, incluin the ata extracte from Fi. S5. For lare fiels ( B B ) we fin a linear relation, E( B) ( ss / el ) B + hfo, (S3) where hfo represents the hih-fiel offset as efine by Koop et al. 9, i. e. the linear extrapolation of the hihfiel behavior to B =. Fi. S5f an its inset show that both the slope an the offset increase with U, implyin that both ss an hfo serve as measures of the effective interaction strenth. Koop et al. have reporte a stron enhancement of the -factor as the spacin ω between the electronic subbans of the QPC is increase 9. Our theory nicely explains this finin: an increase in ω correspons to a smaller transverse channel with, implyin an enhance interaction strenth (as arue at the en of section S- A) an hence an increase in ss (see Fi. S5f). This interpretation is confirme by the experimental ata shown in Fi. S5a-c. This ata was measure usin a secon sample ( sample ), of similar esin than that use to stuy the Fermi-liqui properties of Fis. Ae-h iscusse in the main text ( sample ). For sample, we measure E(B) =a V c (for the values of the conversion factor a see table in Fi. S5b) as function of top ate voltae V t, which correspons to tunin the effective interaction strenth. Accorin to our theoretical consierations, increasin V t causes increasin U [see Eq. (S)] an hence increasin ss (by Fi. S5f). Fi. S5a-c present experimental results corresponin to the preictions in Fi. S5-f (usin V t instea of U). They qualitatively confirm our numerical results, especially that both ss an hfo increase with V t an, therefore, the interaction strenth. (In contrast to us, Koop et al. i not observe a systematic correlation between ss an hfo. A possible reason is that their stuy varie the shape of the QPC potential by varyin the with an lenth of the QPC, whereas we varie V t. Our stuies thus iffer from theirs in the etaile shape of the D potential lanscape. The effective interaction strenth is very sensitive towars the latter, as iscusse in more etail in Sec. S-5 C.) We conclue our iscussion on E(B) with an important comment on the hih-fiel offset hfo. In several experimental stuies of the.7-anomaly 7 9, the observation of a nonzero value for hfo was interprete as evience that there is a possible spin polarization of the D electron as in zero manetic fiel (the quote is from Thomas et al. 7 ). Our frg results show that this interpretation is not compellin, since we obtain hfo without any spontaneous spin polarization. hfo simply implies that the B = conuctance step (Ṽc) is somewhat skewe (see Fis. Ak, Al, Aa), so that the peak in the transconuctance is not symmetric (as seen in Fi. S5); as shown here, this can be achieve with a manetization that is strictly zero. Inee, our frg approach assumes from the outset that the manetization per site, m = (n n ), is strictly zero at B = (see blue line in Fi. A, an introuction of Sec. S-6). This a priori assumption is ustifie a posteriori by the oo qualitative areement between frg an experiment foun throuhout this work, an in Fi. S5 in particular. Moreover, this assumption is a prerequisite for unerstanin the low-enery Fermi-liqui properties of the.7-anomaly iscusse in the main text, an the resultin analoies between the.7-anomaly an the Kono effect: for the latter, there is zero spin polarization at 5

6 a -/V c [V - ] V t =.8V b E [mev] 5 c 4 ǀ ss ǀ V c [V] V c channel splittin ss - factor B[T].8.5 hfo [mev] V t[v] a[mev/v] V t [V] 9 B [T] Ω x =.4τ U=.5τ -/V c [Ω x - ] 6.5 E [Ω x ] U/τ V c [Ω x ] E channel splittin ss - factor - B min,.5.5 B [Ω x ].5 B[Ω x ].6 ǀ ss ǀ/ el.8 hfo [Ω x] U[τ].5..5 U [τ] Fiure S5: Determination of the subban-splittin - factor ss. a-c. Results from experimental measurements on a sample ( sample ) of similar esin as that iscusse in the main text ( sample ). -f, Corresponin results from frg calculations. a,, The transconuctance, i. e. the erivative of the conuctance with respect to ate voltae (V c in a, Ṽc in ), plotte as a function of ate voltae, for several manetic fiels. An increasin manetic fiel lifts the spin eeneracy, causin the conuctance step to split into two spin-resolve sub-steps an ivin rise to two local maxima in a, (marke by blue ots). In, Bmin,.5 (re square) min stans for B at U =.5. b,e. The peak istance E, etermine by fittin a pair of Gaussians (shown by ray lines in a) to the peak pairs in a,, is plotte as function of manetic fiel, in b for three ifferent top ate voltaes, an in e for seven ifferent values of the on-site interaction U. Linear least-square fits to such curves in the rane of lare fiels, usin E ssb + hfo, yiel the effective -factor ss an hih-fiel offset hfo. Errors, s. e. m. (n = 5-7). (To convert V c in a to E in b, we use the V t-epenent conversion factors a = E/ V c liste in the leen of b, obtaine approximately from nonlinear transport measurements 7,9.) c,f, ss (an in insets, hfo ), plotte as a function of V t (in c) or U (in f). The re straiht line in c is a error-weihte least square fit. Both theory an experiment show the same tren, namely that ss an hfo increase with the effective interaction strenth U (which increases with V t in our sample eometry. e f B = T =, because lea electrons screen the local spin into a spin sinlet. It is noteworthy, thouh, that the linear increase in E( B) in Fi. S5e sets in alreay at rather small fiels, of orer O( B ) an similarly for E(B) in Fi. S5b. The reason is that at small fiels the spin polarization rapily rows with fiel, since the spin susceptibility is lare. It is lare because it is stronly enhance by interactions (Fi. A), as reconize an emphasize by Thomas et al. 7, an because the effects of interactions are further enhance by the van Hove rie in the QPC, as iscusse in the main article. Accorin to our analysis, the lare spin susceptibility oes han in han with a stron interaction-inuce enhancement in the inverse scale /B ( χ tot ) [Fi. A], as iscusse in the main article, an also in Sec. S-5 B below. The scale B overns the strenth of the.7- anomaly, in that the conuctance is sinificantly reuce once B or T increase past B. In an alternative moel propose by Reilly et al., one of the avocates of spontaneous spin polarization, the strenth of the.7- anomaly is overne by the size of the spin ap. This moel was use successfully, for example, to moel the shot noise measurements of Ref. 3. The Reilly moel assumes that the spin ap increases stronly with ecreasin V c, i. e. with increasin ensity in the QPC-reion. Note that this Ṽc-epenence of the propose spin ap shows the same tenency as that shown by the Hartreeenhancement of the barrier size in our work, which likewise increases linearly with increasin ensity as Ṽc is mae more neative. (The ensity near the CCR center also increases as temperature or source-rain voltae is increase, an becomes stronly spin-asymmetry as B increases.) In this sense, our work shes liht on why the Reilly moel is phenomenoloically successful at lare eneries: it makes qualitatively correct assumptions about the V c -epenence of the effective barrier heiht that overns the strenth of the conuctance s B- or T - epenence. That havin been sai, we emphasize once more that our Hartree-shift in barrier heiht is not a spin ap, an that our scenario iffers eciely from that of the Reilly moel for eneries below B : there we assert the appearance of Fermi-liqui behavior that is not compatible with spontaneous spin polarization. In our theory, a spin splittin sets in only once spin symmetry is broken by finite B (thouh a spin-symmetric Hartreeshift in barrier heiht is present even at B = ). The spin ap preicte by our theory for B = oes increase with the ensity in the QPC, as in the Reilly moel, since it arises from Hartree contributions to the self-enery (see Eq. (S4) in our frg scheme, or the first two iarams in Eq. (S53) when oin perturbation theory). 6

7 S-3. OTHER EXPERIMENTAL DATA VS. FRG This section presents aitional frg results on the zero-temperature behavior of the conuctance, the shot noise, an the chare susceptibility. Their Ṽc- an B-epenence is foun to be in qualitative areement with that observe experimentally by other roups (Di- Carlo et al. for the shot noise 3, Smith et al. for the compressibility 4. The frg results presente below were obtaine usin static frg, a simplifie version of the ynamic frg scheme use in the main text. Static frg nelects the frequency epenence of the self-enery an all vertices (see Sec. S-6 F). This simplification reuces computational costs by a factor of 3. Nevertheless, for the moel stuie here the results of static frg turn out to be qualitatively very similar to those of ynamic frg (see Fi. S5 below). Hence we have opte to use static frg for the results presente in Secs. S-3 an S-5. A. Spin-resolve conuctance, shot noise This subsection presents a etaile iscussion of the spin-resolve conuctance. It is base on calculations usin moel I (efine in Sec. S-4 B), but the results are fully analoous to those shown in Fis. Aa, b for moel II (efine in Sec. S-4 D). The role of interactions for the manetoconuctance of a QPC at zero temperature can be very clearly reveale by stuyin the spin-resolve conuctance σ = T σ an the shot noise factor 5 N = σ ( σ ). (S4) σ Fi. S6 shows these quantities toether with the full conuctance = +, all calculate at T = as functions of Ṽc, for various fiels. To hihliht the effect of interactions, we also show corresponin results for the bare (U = ) moel, which we iscuss first. We bein with some elementary observations: First, the bare transmission probability Tσ (Ṽc, ) at zero fiel, stuie as function of Ṽc, is antisymmetric w. r. t. the point Tσ (, )=.5 [cf. Eq. (S3) below]: T σ (Ṽc, )= T σ ( Ṽc, ). (S5) A finite fiel B shifts the bare potential in opposite irections for opposite spins, δṽ = σ B (with σ = ± for, ). Thus the bare spin-resolve transmission probability at finite B is equal to that at B = but for a shifte value of Ṽc: T σ T σ (Ṽc, B) =T σ (Ṽc σ B,). (S6) This implies that B inuces a shift (but not a chane in shape) for the spin-resolve conuctance step in σ by σ B (see Fis. S6b-c). Nevertheless, since Eqs. (S6) an (S5) toether imply T σ (Ṽc, B) = T σ ( Ṽc, B), (S7) the full conuctance remains antisymmetric w. r. t. the point T σ (, B) =.5 even for finite B (see Fi. S6a): (Ṽc, B) = ( Ṽc, B). (S8) Eq. (S7) also implies that the bare shot noise, N, is symmetric w. r. t. Ṽc =, or =.5 (see Fi. S6). The above antisymmetry of (Ṽc) w. r. t. Ṽc = is broken in the presence of interactions, in a manner that becomes increasinly more pronounce with increasin fiel, see Fis. S6 an S6, for U/τ =. an.45, respectively. In the latter case the broken antisymmetry is visible alreay at zero fiel, in that the frg conuctance curve shows a sliht.7-shouler, in areement with experiment (cf. Fi. Ae). This shouler at B = T = occurs because the interaction-inuce increase of the effective potential barrier is enhance by the van Hove rie in the local ensity of states (LDOS) an hence is nonuniform in Ṽc (see the main article for a etaile explanation). The breakin of Ṽc-antisymmetry increases with B because (exchane) interactions amplify the fiel-inuce asymmetry in the population of spin-up an -own electrons in the CCR, in particular near the top of the barrier: a small B-inuce surplus of spin-up electrons leas to a sinificantly increase Hartree barrier, an more so for spin-own electrons than for spin-up electrons (ue to the Pauli principle), causin a stron ecrease of relative to. This effect, whose strenth increases with U (compare n an 3r columns of Fi. S6) results in the fiel-inuce strenthenin of the.7-shouler that is characteristic of the.7-anomaly, an its evolution into a ouble step for lare fiels. The increasin Ṽc-asymmetry (i. e. eparture from perfect antisymmetry) in σ (Ṽc) as B increases is also reflecte in the shot noise factor N () [Eq. (S4)], see Fis. S6k an S6l, for U/τ =. an.45, respectively. For zero applie fiel, N () is symmetric w. r. t. =.5; this follows irectly from the form of Eq. (S4) (which hols whenever a Fermi-liqui escription applies), an our assumption that there is no spontaneous breakin of spin symmetry at B =, implyin =. With increasin fiel, N () evelops an -asymmetry w. r. t. =.5, bein somewhat suppresse in the rane >.5 relative to its values in the rane <.5. This fiel-inuce -asymmetry is in oo qualitative areement with the experimental measurements of the noise factor by Di- Carlo et al., cf. Fi. 4() of Ref. 3. Note, thouh, that the measure noise factor shows an -asymmetry even at zero fiel, in contrast to our frg preictions; we believe that this remnant -asymmetry is a finite-temperature effect that will raually isappear if the experimental temperature is lowere further. Reproucin this behavior explicitly by a finite-temperature calculation of the 7

8 .5.5 a b c bare moel: U = B[ 3 τ] U = V s =.5τ, Ω x =.6τ, N=3.5 frg: U =.τ frg: U =.45τ e f h i N.5 N.5 k l U =.τ U =.45τ N V c /Ω x V c /Ω x V c /Ω x Fiure S6: Comparison of results for moel I, for its bare U = version (first column), or treate usin static frg for U/τ =. an.45 (secon an thir columns, respectively). The top, mile an bottom rows show, respectively, the full QPC conuctance = + an its spin-resolve contributions an, all plotte as functions of Ṽc/Ωx for several values of manetic fiel B. The fourth column shows a similar comparison for the shot noise factor N [Eq. (S4)], plotte as function of. noise factor for our moel is left as a task for future stuy. B. Compressibility an chare susceptibility Recently, Smith et al. 4 have experimentally stuie the compressibility of the electron as of a QPC. In particular, they measure the V c -epenence of the compressibility in the vicinity of the.7-anomaly an stuie its evolution with increasin temperature an manetic fiel. The compressibility is a measure of the ensity of states at the chemical potential. In a QPC eometry, its V c -epenence is thus overne by that of the LDOS maxima at the bottom of the D ban, i. e. by the van Hove rie iscusse in etail in the main article an in Sec. S-4 C below (see the yellow ries in Fi. Ab an Fi. S); an its B-epenence is overne by the spin splittin of this van Hove rie. Within our moel, the compressibility can be associate with the chare susceptibility of the CCR, χ µ = n CCR tot µ, n tot = n σ, σ (S9) where n tot is the total chare in the CCR an µ the chemical potential. Fis. S7a an S7b show zero-temperature frg results for the conuctance (Ṽc) an the chare susceptibility χ µ (Ṽc), respectively. The results exhibit a number of features, enumerate below, that are qualitatively consistent with features observe by Smith et al. 4. Consier first the noninteractin case, U = (black ashe lines for an χ µ): When Ṽc is lowere past, the bare chare susceptibility χ µ(ṽc) in Fi. S7b traverses a sinle broa peak, aline with the center of the corresponin conuctance step in Fi. S7a. This peak arises because the bare chare susceptibility equals the bare total ensity of states at the chemical potential [cf. 8

9 χ µ Ω x 5 Ω x =.6τ, V s =.5τ, N = 3, U =.45τ a min B/B * b χµ = A tot () V c /Ω x c A () Ωx τ Fiure S7: Chare Susceptibility. Static frg results (moel I) for (a) the conuctance (Ṽc) an (b) the chare susceptibility χ µ(ṽc) [Eq. S9] as function of Ṽc, calculate for six values of B at a fixe Ṽs an T =. Black ashe lines in a an b show the bare (U =, B = ) curves, an χ µ = A tot(), respectively. Vertical ashe lines are a uie for the eyes an mark the weak shouler or secon maximum of χ µ(ṽc). c, The full (U = ) LDOS at the chemical potential, A (), as function of ate voltae Ṽc an site inex. Eq. (S39)],.3.. χ µ = CCR σ A σ() = A tot(), (S) which traverses a peak when the spin-eenerate van Hove rie is lowere past µ. For nonzero U but still B = (black soli lines), χ µ (Ṽc) is reuce, since interactions ten to counteract the (infinitesimal) increase in chare inuce by an (infinitesimal) increase in µ [Eq. (S9)]. This reuction occurs in such a way that (i) χ µ (Ṽc) retains a ominant peak, with (ii) a weak shouler evelopin on its riht (even thouh B = ), rouhly aline with the roll-over of (Ṽc) towars the first conuctance plateau. This shouler arises because when Ṽc ecreases into the open-channel reime, the van Hove rie apex rops so far below µ that A (), the LDOS at µ, ecreases rapily (Fi. S7c). As a result, its interaction-enhancin effects, an hence also the Coulomb-blockae reuction in χ µ, weaken rapily, resultin in a shouler in χ µ. The colore lines in Fi. S7 show the evolution of the conuctance ( B) an chare susceptibility χ µ (Ṽc) with manetic fiel for U =.45τ. While the conuctance step evolves into the familiar spin-split ouble step with increasin fiel (Fi. S7a), (iii) the ominant peak in χ µ (Ṽc) (Fi. S7b) remains aline with the center of the first conuctance step, while (iv) the shouler in χ µ (Ṽc) evelops into a weak peak that shifts towars the riht, remainin rouhly aline with the roll-over to the secon conuctance plateau (as inicate by ashe colore lines between Fis. S7a an S7b). This reflects the fiel-inuce spin-splittin of the van Hove rie into two spin-resolve sub-ries, which et lowere past µ at ifferent Ṽc-values. As a result, (v) χ µ (Ṽc) evelops a weak minimum between the two peaks. Features (i)-(v) can also be foun, on a qualitative level, in Fis. an 3(a) of Smith et al.. Their measure sinal, calle V s /V mi there, has minima when the compressibility has maxima, an vice versa. In their Fi. (a), the re curve shows a stron ip at V mi =.4 V an a very weak minimum at. V. We associate these, respectively, with the ominant peak (i) an the weak shouler (ii) in χ µ (Ṽc) iscusse above. In their Fi. (b), the two ips in the re curve at V mi =. V an.9 V, correspon, respectively, to the two maxima mentione in (iii) an (iv) above. An in their Fi. 3(a), the peak marke by an arrow correspons to the ip mentione in (v). We thus conclue that the measure compressibility maxima accompanyin the conuctance steps are inee ue to maxima in the ensity of states at the ban bottom, as sueste by Smith et al. themselves (an in Ref. 6). This supports our contention that van Hove ries play a central role in the physics of the.7-anomaly. By implication it also confirms the presence of the quasi-boun states avocate by Meir an collaborators 7 9, provie that we ientify their quasi-boun states with our van Hove ries as arue in Sec. S-4 E below, both names refer to the same peake structures in the LDOS. This ientification was not clear at the time of writin of Ref. 4, however. Instea, Smith et al. arue that they see no evience of the formation of the quasiboun state preicte by the Kono moel. This statement was base on a comparison of their B = ata for V s /V mi to simulations 6 usin ensity-functional theory (DFT) combine with the local spin ensity approximation (LSDA). These ata an the simulation results are shown, respectively, as black an re curves in Fi. 4(b) of Ref. 4. The simulations yiele an aitional stron ip [inicate by an arrow in Fi. 4(b)], 9

10 aline with the onset of the conuctance plateau, that ha no counterpart in the measure ata. We suspect that this aitional stron ip miht be an artefact of the tenency of DFT+LSDA calculations, when initialize usin a small nonzero manetic fiel 9,, to yiel a nonzero spin polarization in reions where the spin susceptibility is lare (as is the case in the QPC). We assert, however, that at B = the spin polarization is strictly zero (in contrast to views expresse in Refs. 7,8,4), since this is a prerequisite for unerstanin the Fermi-liqui properties iscusse in the main article. Our frg calculations for B = thus assume zero spin polarization from the outset. Remarkably they yiel, instea of the stron aitional peak foun by DFT+LSDA, only the weak shouler (ii) mentione above, which is consistent with the compressibility ata of Smith et al. Further aruments in support of the absence of spontaneous spin polarization at zero fiel are offere at the en of Sec. S- C. PART II: THEORETICAL DETAILS S-4. MODELS USED FOR BARRIER SHAPE In the course of our stuies of the.7-anomaly, we have explore many ifferent parametrizations of smooth, symmetric QPC barrier shapes. We foun that as lon as the barrier top is parabolic, characterize by a barrier heiht Ṽc (w. r. t. to the chemical potential) an a curvature parameter Ω x, the etails of the parametrization of the barrier o not matter. In this section we present the etails of two ifferent parametrizations for parabolic barriers, to be calle moel I an moel II, whose results for QPC properties are fully equivalent when expresse as functions of Ṽc an Ω x. Both moels use the same Hamiltonian, choice of chemical potential an local interaction strenth U, specifie in Sec. S-4 A, but iffer in their choices for the hoppin amplitue τ (which is -inepenent for Moel I but not for Moel II) an the on-site potential E. Moel I is presente in Sec. S-4 B: its hoppin amplitue is -inepenent, τ = τ, an the barrier shape is specifie by parametrizin E in terms of a central ate voltae Ṽc an a sie ate voltae Ṽs. It is esine to allow a theoretic stuy of the crossover between a Kono quantum ot (KQD) an a QPC by continuously eformin the D potential from a ouble-barrier to a sinle-barrier shape (see Fis. S9c an S9 below, respectively). (The results of a corresponin stuy will be publishe elsewhere.) Here we use moel I to calculate numerous QPC properties presente in various parts of the supplementary material (Fis. S6, S7, S, S, S3 an S4). Moreover, moel I allows instructive insihts into the similarities an ifferences between the bare ensity of states of a QD an a QPC, which are key to unerstanin the similarities an ifferences between the Kono effect an the.7-anomaly, as briefly iscusse in Sec. S-4 C. For moel II, presente in Sec. S-4 D, τ epens nontrivially on, an the barrier shape is specifie solely in terms of a central ate voltae Ṽc an the barrier curvature Ω x (auste via the lenth N of the CCR, but without reference to a sie ate voltae). Compare to moel I, moel II has technical avantaes when treate usin SOPT (as explaine below). For clarity, moel II was use for all numerical results (both from frg an SOPT) presente in the main article. It was also use for Fis. S, S4, S, S6 in the supplementary material. We emphasize that the results obtaine usin moels I an II are qualitatively consistent. To conclue our introuctory comments on the moels use here, we remark that the iea of stuyin the.7- anomaly usin an effective D moel with a smoothly varyin QPC potential an local interactions has of course been pursue previously by numerous authors. For example, a moel with local exchane interactions was stuie in Refs. an, a moel with an unscreene Coulomb interaction in Ref. 6, an a moel with a point like interaction restricte to the center of the QPC potential in Ref. 3. Our work is similar in spirit to these, but our use of frg allows us to treat the effects of interactions more systematically than Refs. an 6, an for loner chains than Ref., which also i not have access to the limit T. Works base on D or 3D ensity-functional theory calculations 6 treat the potential lanscape more realistically than we o, but at the expense of not treatin correlation effects as accurately as frg oes. In particular, our frg treatment allows accurate preictions for the conuctance at zero temperature, which is beyon the scope of all previous treatments. Moreover, our SOPT calculations at finite source-rain voltaes are first to ive a etaile escription of the oriin of the ZBP. A. Hamiltonian, chemical potential, U The moel Hamiltonian efine in the main article, Ĥ = [ ] E σˆn σ τ ( + σ σ +h.c.) + σ U ˆn ˆn, (S) with E σ = E σ B, is epicte schematically in Fi. S8. It shows a tiht-binin chain ivie into two semiinfinite, non-interactin, uniform leas on the left an riht, connecte to the central constriction reion (CCR),

11 = N = = N τ B left lea central constriction reion (CCR) riht lea Fiure S8: Schematic epiction of the one-imensional moel of Eq. (S) (for a QPC barrier shape). It represents an infinite tiht-binin chain with hoppin matrix element τ (ray); the prescribe local potential E (blue) an on-site interaction U (re) are nonzero only within a central constriction reion (CCR) of N =N + sites. The CCR is connecte to two semi-infinite non-interactin leas on the left an riht. A homoeneous Zeeman manetic fiel B (orane) can be switche on alon the whole chain. consistin of an o number N =N + of sites centere on =. The lattice oes not represent actual atomic sites, but instea is merely use to obtain a iscrete, coarse-raine escription of transport in the lowest subban. The position-epenent parameters U an E, nonzero only within the CCR, are taken to vary slowly on the scale of the lattice spacin a. (We set a = in our calculations.) Choice of µ: Since the chemical potential is a property of the bulk, we bein by consierin our moel for E = U = an τ = τ, representin a bulk tihtbinin chain (infinite, homoeneous). The eieneries ɛ k corresponin to wave number k have ispersion E U ɛ k = τ cos(ka) [ τ,τ], (S) plotte in Fi. S9a. To escribe the phenomena of present interest, the chemical potential µ shoul lie somewhere within this ban, not too close to its ees; the precise value oes not matter. All our numerical calculations (frg an SOPT) use µ =, implyin half-fille leas; but for the sake of enerality, we keep µ arbitrary below, particularly in Fis. S9a,b an Sa,b.) The enery ifference between the chemical potential an the bulk ban bottom efines the bulk Fermi enery, ε F =τ + µ (> ). (S3) Choice of U : In choosin a purely on-site interaction in Eq. (S), we implicitly assume that screenin is stron enouh to rener the interaction short-rane. (A more realistic treatment of screenin is beyon the scope of this work.) We set the on-site interaction U equal to U throuhout the CCR, except near its ees, where it rops smoothly to zero to avoi spurious backscatterin effects (Fi. S9e):, [ N, U = U exp ( ] N )6 N ( (S4). N ) U is to be reare as an effective parameter, whose value is influence by the transverse moes not treate explicitly in our moel. In particular, the effect of increasin ɛ k /τ ( E ɛ F )[τ] ( E ɛ F )[τ] U [τ] a ω µ max ω bulk min ɛ F ɛ F = ω bulk π π 4 6 ka πτ A bulk (ω).5 c e 5 ɛ F ɛ F V c =.6τ V s =.6τ V c =.3τ V s =.3τ QD QPC U =.5τ s = 6 s N = 5 N = 5 b 5 V S µ V c Fiure S9: a, Dispersion relation ɛ k vs. k [Eq. (S)] for a bulk non-interactin tiht-binin chain without manetic fiel (infinite, homoeneous, E σ = U = ). The fillin factor in the leas is controlle by the lobal chemical potential µ (blue ashe line); it is here rawn at µ = for enerality, althouh our frg calculations use µ =. b, The corresponin -inepenent bulk LDOS [Eq. (S9)], shown both as A bulk(ω) (on x-axis) versus ω = ɛ k µ (on y-axis), an usin a color scale. The istance from the chemical potential to the bulk ban bottom ωbulk min is ε F =τ + µ = ωbulk min (> ). c an, Moel I: The one-imensional potential E of Eq. (S5) (thick ashe black line) for a QD potential (Ṽs > Ṽc) an a QPC potential (Ṽc > Ṽs), respectively. In the outer reion of the CCR ( N ), E is escribe by quartic polynomial, in the inner reion ( < ) by a quaratic one (thin re an blue lines, respectively, shown only for >.) For iven N, s, Ṽ s an Ṽc, the parameters an Ωx are auste such that the resultin potential E epens smoothly on throuhout the CCR. e, The on-site interaction U of Eq. (S4). V c µ the top ate voltae V t can be mimicke by increasin U [Eq. (S)], as will be iscusse in more etail in Sec. S- 5C. We typically take U to be somewhat smaller than the maximum value of the inverse bare LDOS, since if U max[a (ω)], is too lare, the frg calculations o not convere. We remark that we have also explore the option of takin U to be proportional to E, or of takin the rane of sites where U = U to be several times larer than that where E. Such moifications chane etails of the results, such as the precise shape of the con- V S ω /τ

12 uctance (Ṽc, Ṽs) as function of Ṽc or Ṽs, but not the qualitative trens iscusse in the main article, as lon as U rops smoothly to zero near the ees of the CCR. B. Moel I For moel I, we choose the the hoppin amplitue to be -inepenent, τ = τ, while the on-site potential E escribes a reflection-symmetric barrier within the CCR. Its shape is tunable between a ouble barrier escribin a QD (Fi. S9c) an a sinle barrier escribin a QPC (Fi. S9) (thick ashe black lines). We have parametrize it as follows:, N, [ ( ) (Ṽs + ε F ) N ( ) ] E = s N N 4 s N, N, Ṽ c + ε F + Ω x 4τ sn(ṽs Ṽc), <. (S5) The sites ± ivie the CCR into two outer reions, where the potential is a quartic polynomial in, an an inner reion, where it is quaratic in. In the latter, the manitue of the curvature is overne by the parameter Ω x ( ), which has units of enery. (The quaratic term for the inner reion was chosen to have the form mω xx use in Ref. 5, with ω x = Ω x /, x = a an m = /(τa ) corresponin to the effective mass at the bottom of a tiht-binin chain.) The shape of E is controlle by four inepenent parameters: (i) N, which sets the halfwith of the CCR; (ii) s, which overns the with of the outer flanks of the potential; (iii) Ṽs an (iv) Ṽc, which ive the potential s heiht w. r. t. ε F at the sites = ± s an, respectively: E ±s = Ṽs + ε F ; E = = Ṽc + ε F. (S6) Once the four parameters N, s, Ṽ s an Ṽc have been specifie, the epenent parameters an Ω x are chosen such that E is a smooth function of at the bounaries ± between the inner an outer reions. An electron incient at the chemical potential has enery ε F w. r. t. to the bulk ban bottom an hence sees a relative potential of heiht E ε F at site. For Ṽs >Ṽc, the relative potential escribes a QD potential with two maxima of heiht Ṽs at = ± s an a local parabolic minimum of heiht Ṽc at =. For Ṽc > Ṽs (the case of present interest), it escribes a QPC potential with a sinle parabolic maximum at =, of heiht Ṽc. The crossover point between QD an QPC lies at Ṽs = Ṽc (for which Ω x = ). Eviently Ṽc an Ṽs respectively mimick the role of the voltaes applie to the central ates (V c ) an sie ates (V s ) in the experiment (with Ṽ c,s e V c,s ). Fiure N Ṽ c[τ] Ṽ s[τ] Ω x s Fi. S to Fi. S to Fi. S9a-b Fi. S9c Fi. S Fi. Sa Fi. Sb Fi. Sc Fi. S Fi. S Fi. S to Fi. S to Table I: Parameters use for moel I [efine in Eq. (S5)] for the frg results shown in various fiures of the supplementary information. We emphasize that the QPC barriers stuie in this work are all parabolic near the top. For quantitative stuies of the.7-anomaly usin moel I, we fix N, s an V s, an tune the QPC from close to open by lowerin Ṽc past, at which the bare (U = ) conuctance equals.5. The with of the conuctance step [see Fi. Ak, an Eq. (S3)] is overne by the curvature parameter at this point, Ω x = Ω x Ṽc=, which we will simply call curvature henceforth. (Ω x itself chanes slihtly urin this crossover, but for the barrier shapes use in this work this chane is typically less than % between Ṽc = ±Ω x.) The curvature Ω x also overns the exponential Ṽc-epenence of B [Eq. (S35a)]. Note that formulas such as Eqs. (S3) an (S35a) woul chane for non-parabolic QPC barriers, e.. barriers with a flat top. Stuyin the.7-anomaly for such situations woul be an interestin extension of the present work, which we leave for the future 4. C. Bare local ensity of states (LDOS) In the main article we have arue that eometry stronly influences the.7-anomaly, via its effect on the local ensity of states (LDOS) an the van Hove rie of the latter. Here we elaborate this in etail, by iscussin the eometry-epenence of the noninteractin LDOS (for moel I). We o so not only for the QPC barrier shape of present interest, but also for a QD barrier shape. This lays the roun for a subsequent comparison, presente in Sec. S-4 E below, of the LDOS structures of a QPC an a QD, which shes liht on the similarities an ifferences between the.7-anomaly an the Kono effect. The LDOS per spin species σ at enery ω (measure

13 ω /τ ω /τ 4 4 V c =.5τ V s =.5τ ɛ F V c =.5τ V s =.5τ ɛ F U =, N = 3, B = E ω min = E ɛ F E ω min = E ɛ F V c QD QPC a b V S µ V c V c µ V S A (ω) τ. ω /Ω x ω /Ω x 5 5 µ QD V c =.5τ V s = ω min QPC V c =.8τ V s =.5τ U =, N = 3, B = Ω x =.5τ Ω x =.6τ ω min 5 5 c µ V c V c µ A (ω) τ. Fiure S: Moel I: Noninteractin zero-fiel LDOS per spin species of a,c, a QD, an b,, a QPC, for potential shapes shown by thin black lines (marke by black arrows) for ω min = E ε F. (The loarithmic color scale shows A (ω) smeare by a Lorentzian of with δ =.τ, in orer to rener very sharp structures visible.) Panels c, focus on the central reion of the CCR an eneries close to µ (black ashe lines); the Ṽc- an Ṽs-choices iffer from those use in a,b. Thin reen ashe lines in c, inicate the shape of the LDOS ries iscusse in the text. For the KQD, they enclose an area in the -ω plane on which the corresponin LDOS rie has weiht ; for the QPC, they trace a contour alon which A,QPC (ω) =.7. relative to the chemical potential µ) is efine as A σ (ω) = π ImGσR (ω), (S7) where Gi σr (ω) is the Fourier transform of the retare T = propaator 5, Gi σr (t) = iθ(t) G { iσ (t), σ ()} G, (S8) where G is the moel s roun state. In this subsection we will iscuss only the spin-eenerate case of zero fiel ( B = ) an zero interaction (U = ). We thus rop the spin inex σ (as in the main article) an instea put a superscript on A (ω) to enote the bare LDOS. For E =, representin an infinite, homoenous, bulk tiht-binin chain, the LDOS of Eq. (S7) is - inepenent an equal to the D bulk LDOS, A bulk(ω) = a [ ] k. (S9a) π ɛ k ɛ k =ω+µ This is nonzero only for ω min bulk <ω<ωmax bulk, where ω min bulk = ε F, ω max bulk = ε F +4τ, (S9b) enote the bottom an top of the ban, measure w. r. t. µ, respectively. Within these limits, it has the form A bulk(ω) =, π (ωbulk max ω)(ω ωmin bulk ) (S9c) shown in Fi. S9b, featurin square-root van Hove sinularities near the ban ees (yellow frines in Fi. S9b). While the upper van Hove sinularity (of unoccupie states) may be viewe as an artefact of escribin the lowest subban usin a tiht-binin chain, the lower one is realistic for effective one-imensional eometries; it woul also arise, e.., when usin a free-electron moel. Now consier a nonzero potential E that is smooth on the scale of the lattice spacin, moellin a QD or QPC in the CCR, as shown by the thick black lines in Fis. Sa-. The color scale in these fiures inicates the corresponin -epenent LDOS, A (ω). The latter has an ω-epenence that, for fixe, is reminiscent of the bulk case, but with several ifferences, cause by the spatial structure in E. First, the ban ees now are -epenent an follow the shape of the potential, with ω min = E ε F, ω max = E ε F +4τ. (S) In particular, the ban bottom at the CCR center, =, is iven by ω min = Ṽc. Secon, A (ω) exhibits narrow frines (visible clearly in Fis. Sa-), ue to the fact that the electronic wave functions form stanin wave patterns. In the central part of the QD potential (Fi. Sa), an in the central part of the QPC potential for eneries ω>ωbulk max (Fi. Sb), these stanin waves correspon to boun state wave functions. (For the case of the QPC these boun states are artefacts an they are avoie in moel II.) In the outer reions of both QD an QPC potentials they correspon to Frieel oscilla- 3

14 .4 A (ω) τω x.8 QPC: V c =, V s =.5τ, Ω x =.6τ, N = 3 ω = ω min ω min ɛ F V c = ω = ω min ω min Ω x 4 bulk ω/τ QPC V c =.τ N =, Ω x =.4τ, U = ω min 5 5 ω max A (ω)τ V c. (ω ω min )/Ω x Fiure S: Inset: The ban bottom ω min (black line) as function of, for a 3-site CCR with a parabolic QPC barrier (moel I) with curvature Ω x an heiht Ṽc = ωmin =. Main plot: Enery epenence of the LDOS near the ban bottom, showin A bulk(ω) (ashe), an A,QPC (ω) (soli) for three -values near the center, all plotte as functions of (ω ω min )/Ω x (blue line correspons to Fi. in Ref. 3). Arrows of matchin colors in the inset inicate the corresponin values of (namely, an 4). The short, heavy, colore vertical lines in the main panel inicate where the enery coincies with the barrier top, ω = ω min ; the corresponin values of the x-coorinate (ω min ω min )/Ω x (namely,.6 an 6.4) ive the remainin barrier heiht as seen from site. In the bulk, ω min = ε F. The peak of A,QPC (ω) lies at an enery ω H = ω min + O(Ω x). For = it lies at ω H =.Ω x an has heiht.8/ τω x (otte blue lines). Note that A,QPC (ω) matches A bulk(ω) once the enery ω lies above ω min by more than O(Ω x), corresponin to free propaation above the barrier. tions. Thir, the van Hove sinularities are somewhat smeare out on the outer flanks of the QD, an throuhout the entire QPC, in the latter case on a scale set by Ω x (see Fi. S). For the rest of this subsection, we focus on the QPC barrier of Fis. Sb,. (The QD barrier of Fis. Sa,c is revisite in Sec. S-4 E below, where we compare its LDOS to that of a QPC.) For a QPC, A,QPC (ω) epens smoothly on ω an near the center of the CCR, its weiht bein concentrate alon a curve, broa van Hove rie (frame by the reen ashe lines in Fi. S). This rie oriinates from a van Hove sinularity ust above the ban bottom that has been pushe upwar by the QPC potential barrier. The van Hove rie has limite spatial extent when traverse at constant ω, reflectin the limite spatial size of the QPC. At the outsie flanks of the CCR barrier, the tails of the rie split up into iscrete frines, representin Frieel oscillations associate with stanin waves that buil up near the barrier (as also seen in Fis. Sa,b). For iven, the ω-epenence of the van Hove rie, shown in Fi. S, is asymmetric w. r. t. to its maximum, with a steep, exponentially ecayin flank below the maximum, an above it a lon tail, whose envelope ecays as Fiure S: Noninteractin zero-fiel LDOS per spin species, A (ω), shown on a loarithmic color scale, for the QPC moel II efine by Eqs. (S) an (S3). The thin black line (marke by black arrow) inicates the lower ban ee, ω min [Eq. (S4)]. The curvature of the lower an upper ban ees is, respectively, neative an positive throuhout the CCR, ensurin that no boun states occur. [τ(ω ω min )] /, reflectin the ω-asymmetry of the bulk van Hove sinularity of Eq. (S9). The iverence of the latter is cut off here, ue to the absence of translational invariance, on a scale set by the barrier curvature. Inee, the maximum value taken by the van Hove peak in A,QPC (ω) occurs at an enery, say ω H, that lies above the lower ban ee by an amount of orer Ω x, ω H = ω min + O(Ω x ). (S) For example, for a purely parabolic barrier top, the van Hove peak in A,QPC = (ω), the LDOS at the center of the QPC, lies at ω H = ω min +.Ω x. In that case, the van Hove peak lies precisely at the chemical potential, ω H =, when Ṽc =.Ω x. Eq. (S) implies not only that the van Hove peak enery epens on Ω x, but also that its heiht (i. e. the maximum value of the LDOS) scales as / τω x. As a consequence, all local quantities that epen on A,QPC (ω), such as the local manetic susceptibility χ, epen on Ω x, too. In this way they acquire an explicit epenence on the shape of the QPC barrier. D. Moel II In this section we escribe moel II, use for all numerical results (frg an SOPT) presente in the main article. For moel II, esine to moel exclusively a QPC, we have moifie the choice of E an τ in two minor ways relative to moel I of Sec. S-4 B, which turn out to facilitate SOPT calculations. The two chanes, escribe below, are esine (i) to allow usin a shorter CCR while maintainin a small curvature Ω x at the barrier top, an (ii) to avoi the occurrence of artificial boun states in the bare ensity of states of the QPC (such as those seen in Fi. Sb near the upper ban ee, for eneries ω>ωbulk max). 4