Quantum Hall Systems (PDF)
Quantum Hall Systems talk by B. I. Halperin at the Simons Symposium on Quantum Physics Beyond Simple Systems St. John, U.S.V.I., February 1, 2012
Outline 1. Survey of important issues in quantum Hall physics 2. Theory of Interferometer Experiments
Quantum Hall Systems Two-‐dimensional electron systems in a strong magneMc field (broken Mme reversal symmetry) (Low temperatures) Many varieMes: different Landau level filling factors (0.1 – 100); different levels and types of disorder; different materials. Why are the problems hard? For a parMally full Landau level, system is highly degenerate in the absence of interacMons. So electron-‐ electron interacMons, and electron impurity interacMons have a huge effect. If interacMons are weak compared to the cyclotron energy, can project onto a single Landau level, kineMc energy is quenched, and can be set equal to zero. But not like a classical system at T=0; a\er projecMon, different Fourier components of density operator do not commute with each other.
Issues in Quantum Hall Systems Universal issues: e.g. Topological classificaMon. CharacterisMcs which can take on only a discrete set of values, robust to small perturbaMons. Non-‐universal issues: Depend conMnuously on details. Necessary to understand real materials and experiments.
Universal properMes Ideal QuanMzed Hall States: Characterized by a precise filling factor. Energy gap for charged excitaMons in the bulk. Topological classificaMon of states, determined by Hamiltonian in the bulk, but low-‐energy consequences only manifest at edges or defects. Edges of quanMzed Hall states must support low-‐energy charge-‐carrying excitaMons. Certain (minimal) properMes of edges determined by Hamiltonian in the bulk. Others can be affected by edge reconstrucMons. A discrete set of finite-‐energy defects (quasiparMcles) can exist in the bulk. Their possible charges and their quantum staMsMcs are dictated by the nature of the defect-‐free quantum Hall ground state. Universal properMes are robust to small changes in the Hamiltonian, weak disorder. Can only change if the system undergoes a phase transiMon: energy gap for extended charge carrying excitaMons goes to zero in the bulk; or first order transiMon.
Universal properMes of edges Universal properties include quantized Hall conductance GH = ν e2/h , quantized thermal Hall conductance KH = κ π2 T/ 3h. (Kane and Fisher, 1996). (ν and κ are rational numbers.) Minimal edges have a specified number of edge states traveling in specified directions. Number can change if surface is reconstructed, but ν and κ must be conserved. In general: κ = NR-NL . (Majorana modes count ½). To compute ν: edge modes must be weighted by “charges”. Example: Integer QHE with f = N. Minimal model has N integer edge states, one for each Landau level, all moving in the same direction: ν = κ = N. If electron density varies slowly at the edge, no disorder, could get reconstruction with additional pairs of left and right moving edge states. But ν = κ = NR-NL = f is unchanged. In presence of disorder, extra pairs will become (at least eventually) localized, and only N right-movers remain extended, as in the minimal model.
Q1
QuesMon (Das Sarma): Are you talking about real edges or ideal edges? Answer: I was about to get to this quesMon. There are certain properMes that are universal, and cannot be affected by any reconstrucMons that occur at the edge.
Other properMes can be affected by edge reconstrucMons.
Q2
QuesMon: (Marcus) How can thermal conductance be zero or negaMve? Answer: Le\ movers carry heat to the le\ and right-‐movers carry heat to the right. What we are talking about here is the thermal Hall conductance. You apply a temperature gradient in the x-‐direcMon and you measure a heat flow in the y-‐ direcMon. (Imagine a strip oriented along the y axis, with a temperature gradient in the x-‐direcMon, so the two edges are a different temperatures. The edges will then carry a net energy current in the y-‐direcMon.) How you can measure this effect is another quesMon. There are many complicaMons in pracMce, for example due to the large parallel thermal conducMon by phonons. Also, one must be careful about boundary condiMons for the electron system, because you may be measuring a combinaMon of longitudinal and Hall components, and if there is electric current flow, you may mix in thermoelectric effects.
FracMonal QHE edges Fractional QHE states can be more complicated. Minimal edges may have only co-propagating modes (“right movers”), or may have both right and left movers (counter-propagating edge modes). Edge modes can carry fractional charge, make different contributions to ν and κ . Reconstructions can change charges as well as number of modes. Non-Abelian states can have additional modes, such as Majorana modes, which have charge zero and contribute ½ to κ . Note: Low energy modes also occur at an interface between two different QHE states.
Q3
Comment: (Fisher): There are other universal properMes in bulk systems, such as associated with transiMons between different quanMzed Hall states. Reply: Yes, I will come to that later in my talk.
Non-‐universal bulk properMes: Nature of the Ground State Several types of QHE states can exist at a given filling factor. ParMcularly if system has addiMonal low energy degrees of freedom: spin or pseudospin (mulMple valleys, bilayers, sublafces). QuesMon: Given a microscopic Hamiltonian and filling fracMon: what is the resulMng ground state? Answer depends on details. Only a limited number of QHE states will occur in a given system. Ground state at a general (irraMonal) filling fracMon f will generally consist of localized quasiparMcles on top of a nearby quanMzed state with raMonal filling factor f0 . QuasiparMcles may be localized by disorder or by interacMons (e.g. Wigner crystal, lafce of bubbles, stripes). Hall conductance determined by f0.
Q4
QuesMon: Why can’t there be an infinite number of quanMzed Hall plateaus? Answer: It may be that in principle there could be a quanMzed Hall state at any raMonal filling fracMon, given a suitably engineered Hamiltonian. In pracMce however, this is not what we expect. There will be a certain number of stable states with relaMvely large gaps, which can be the lowest energy underlying state over a relaMvely large range of filling factors. These states will then swallow up all other possible quanMzed Hall states for fillings in this range. More precisely, consider what happens at some arbitrary irraMonal filling factor f. Depending on the details of the system, including the amount of disorder present, the system will generally fall in the domain of airacMon of some quanMzed Hall state, at a raMonal filling f0 which is not too far away. The ground state at fracMon f will therefore consist of the ideal ground state at filling f0, together with a finite density of quasiparMcles or holes as necessary to achieve the required electron density for filing factor f. In general, we expect the added quasiparMcles or quasiholes to be localized at low temperatures, due to some combinaMon of disorder or Coulomb interacMons, so they do not contribute to transport at very low temperatures. Thus, one obtains vanishing longitudinal resistance, and one sees the Hall conducMvity of the parent state, f0. In most cases we may expect to see sharp transiMons from one Hall plateau to another as one varies the filling factor f. However, in some cases, as in the vicinity of f=1/2 for a single=layer system, we find that there exists a metallic state, with conMnuously varying Hall resistance, and a finite longitudinal resistance, whose value depends on the sample. •
Q5
QuesMon: Das Sarma: Can the plateau you are on depend on temperature? Answer: Strictly speaking, the plateaus are only rigorously defined at zero temperature. However, there can be approximate plateaus , which can exist at some finite temperature over some range of filling factors where there is no plateau, or a different plateau, zero temperature. Such things have been seen.
TransiMons between quanMzed Hall plateaus when f is varied. TransiMons between integer plateaus for non-‐interacMng electrons in the presence of disorder have been studied extensively. Believed to have universal properMes (e.g. criMcal exponents) that have been calculated numerically. Applicability in the presence of interacMons not clear. For interacMng electrons when disorder is small, fracMonal states intervene, can get transiMons between fracMonal states, or between integers and fracMons. Can also have metallic states at certain filling factors (e.g. f=1/2) with σxx finite (sample-‐dependent) down to lowest reasonable temperatures. Metallic states have no energy gap, finite compressibility, no plateau in σxy. Universal properMes predicted for transport, excitaMon spectrum, etc. in limit of vanishing disorder.
Q6
QuesMon: (Das Sarma). Don’t you have to have a compressible state between
any two incompressible quanMzed Hall states?
Answer: In some cases, as near f=1/2, as previously menMoned, there can be a “compressible” metallic state, with finite resisMvity, over a range of filling factors. However, this is probably the excepMon, rather than the rule. For non-‐interacMng electrons, in the presence of disorder, we believe there will be a direct transiMon between integer Hall states, which is sharp at zero temperature. This is based on the renormalizaMon group arguments of Pruisken, and various numerical simulaMons, etc. At finite temperature, the Hall resistance will change conMnuously as one varies the filling factor, but the range of f over which the Hall resistance changes its value becomes narrower as the temperature is lowered, vanishing in the limit of T=0. (For the strictly non-‐interacMng case, the longitudinal conductance is always zero, but in the presence of phonons or other mechanisms for inelasMc scaiering, there should be a finite longitudinal resistance in the range of f where the Hall resistance is changing.) We believe that similar sharp transiMons between integer plateaus will occur when electron-‐electron interacMons are taken into account, provided there is sufficient disorder to suppress any fracMonal quanMzed Hall states. For smaller values of disorder, where fracMonal quanMzed Hall states are present, there can be transiMons between quanMzed Hall states that are qualitaMvely similar to the integer transiMons, though they might have different criMcal exponents, etc.
Q7
Comment: Fisher: A direct transiMon is always possible between a mother state and a daughter state, as between f=1/3 and f=2/5, but this may not be possible in other cases, such as 1/3 to 2/3. Reply: I am not sure what you mean by a direct transiMon. I would think that at least a first-‐order transiMon between any two fracMons might be possible in principle. [At least in the case of short-‐range interacMons between the parMcles]. Fisher: The quesMon is whether you can have a direct second-‐order transiMon between say 1/3 and 4/5. Reply: We should probably talk about this more, off line. [Added remarks: The situaMon becomes more subtle when one takes into account effects of the long-‐range Coulomb interacMon in addiMon to disorder and short-‐range interacMons. When direct transiMons are disfavored, there can be re-‐entrant phases, where the Hall conductance is a non-‐monotonic funcMon of the electron density. ]
An open problem of great current interest: FQHE in Graphene Lowest Landau Level pseudopotenMals are the same as in GaAs. But Valley + Spin Degeneracy -‐> SU(4) Symmetry (Approximate). New states are possible. Experiments show many fracMons, but some are missing. Why? ExplanaMons require understanding of excitaMon energies as well as ground states.
Q8
QuesMon: (Das Sarma): How much can we trust the experiments on fracMonal Hall states in graphene? Are we confident that the reported fracMons are correctly idenMfied? Are we confident that the missing fracMons are really missing? Answer: At least in the case of Amir’s compressibility measurements, I find the data very convincing. As he can produce color plots on the plane of filling factor and electron density, one can see very clearly the existence of straight verMcal lines at idenMfiable filling fracMons, which are clearly associated with fracMonal quanMzed Hall states. Of course, we can’t say that the missing fracMons are truly missing. Perhaps they will appear in beier samples at lower temperatures. But we do not understand why some are seen in the experiments and some not. For example, Amir does not see an incompressible state at filling fracMon f=1/3 (corresponding to Hall conductance -‐5/3), while he does see clear fracMons at f=2/3, 4/3, and 5/3. In fact, I would be preiy sure that a quanMzed Hall state of the Laughlin type should be the true ground state at filling fracMon f=1/3 in a perfect graphene layer. However, the energy to produce charged excitaMons, such as valley skyrmions, might be sufficiently low that the plateau is easily wiped out by disorder and/or finite temperature. Perhaps Amir will show us data on this in his talk.
QuanMzed Hall State at ν=5/2 Possible ground states: Pfaffian state (Pf), proposed by Moore and Read. Elementary charges have charge e/4 and non-Abelian statistics. Electrons in second Landau level are fully spin polarized Anti-Pfaffian state (APf): (Particle-hole conjugate of Pfaffian). Bulk properties are similar to Pf; quasiparticles have similar charge and statistics to Pf; but edge states are different. APf edge has counter-propagating neutral mode; minimal Pf edge has no counter-propagating mode. (Lee, Ryu, Nayak and Fisher; Levin, Halperin, and Rosenow 2007). Other possible ground states are not completely ruled out.
Experimental tests Necessary characterisMcs of either Pf or APf state at 5/2: State should have charge e/4 quasiparMcles. State should be maximally spin-‐polarized. Recent measurements of charging events using an SET (Yacoby group) point strongly to charge e/4. Recent NMR measurements (Murata group, Bar-‐Joseph group) support full spin polarizaMon.
Experimental observaMons of counter-‐ propagaMng neutral modes Heiblum group has seen compelling evidence for a counter-‐ propagaMng neutral edge mode in 5/2 state, (as well as for 2/3, 5/3). Experiments study shot-‐noise in a geometry with a QPC and extra contacts arranged to prevent charge mode from reaching the detector. Results at 5/2 (along with previous studies of non-‐linear conductance at a constricMon by Marcus group), suggests APf rather than Pf. But reconstructed Pf edge could also have counter-‐propagaMng modes. Ground states completely different than Pf and APf cannot be ruled out.
Q9
QuesMon: Didn’t the counter-‐propagaMng neutral mode measurements depend on shot noise? Since shot noise measurements have given varying values for the charge, can we trust the neutral mode results? Answer: The neutral mode experiments do not depend on precisely what is tunneling between the edges, as long as something is tunneling, so I think the conclusions are preiy robust. QuesMon: What about edge reconstrucMon? What if the edge is very gradual Answer: Yes, it is true that if there is enough edge reconstrucMon, any bulk fracMon can acquire counter-‐propagaMng modes at the edge. However, the Weizmann experiments did see counter-‐propagaMng modes at fracMons where they are expected and did not see counter-‐propagaMng modes at various other fracMons where they would not be expected in a minimal edge.
Non-‐universal pracMcal quesMon What determines strength of disorder in actual samples? How do various type of disorder limit conducMvity in metallic state at f=1/2, or the number of fracMonal states seen. Only partly correlated with zero field mobility, or with quantum scaiering rate, measured in weak-‐field Shubnikov-‐de Haas experiments.
Other non-‐universal bulk properMes Energies to produce various forms of charged excitaMons. (QuasiparMcles can have different spin states, could be skyrmions) Energy spectra for neutral excitaMons (spin waves, excitons) Transport properMes at finite temperatures. Effects of disorder (long wave-‐length and short) formaMon of puddles of quasiparMcle or quasihole. Transport “acMvaMon energies”: reducMon from ideal values because of disorder. (cf. d’Ambrumenil, Halperin, Morf, PRL 2011, effects of tunneling through saddle points in the impurity potenMal)
Q10
QuesMon: (Marcus) Can you conclude that if we go to low enough temperatures, in
clean enough samples, the there will be a quanMzed Hall state at f=1/2, like at f=5/2?
Answer: No. Just like in normal metals, some metals become superconductors at low temperatures and some do not, at least at any experimentally reachable temperature, I would expect that the 5/2 state has an energy gap and the ½ does not. However, we do not know the answer for sure. The difference between the quanMzed Hall state and the metallic state is in fact analogous to the occurrence or non-‐occurrence of superconducMvity in an ordinary metal at zero magneMc field. The existence of superconducMvity in ordinary metals depends on the nature of the effecMve interacMons between the electrons; in the quantum Hall case, the occurrence of a gap depends on the interacMons between composite fermions. These are different in the first and second Landau levels, because the form factors for the electron states are different. In the case of zero magneMc field, there is a Kohn-‐Lufnger theorem which states that even for repulsive interacMons, there should always be superconducMvity at very low temperatures (perhaps ridiculously low). As far as I understand it, this theorem should not apply to the composite fermions at f=1/2 because the system is spin polarized , and pairing in even angular momentum states is excluded. However, we cannot rule out the possibility or pairing at some very low temperature.
Tools for addressing non-‐universal properMes Exact diagonalizaMon of small systems. VariaMonal calculaMons (e.g. Jain’s trial wave funcMons) Some exactly soluble special models. Approximate methods. (e.g. “Hamiltonian method” of Murthy and Shankar) Other methods?
Q11
QuesMon: What about DMRG calculaMons? Answer: This is what I was thinking of when I said that renormalizaMon group methods. Have been used, and I was thinking of the work in Das Sarma’s group. However, so far this has only been able to extend the range of exact diagonalizaMon to systems with a few more electrons. Comment (Das Sarma). We were able to go from 22 to 26 electrons at filling ½. It is hard to imagine one could go much further.
Quantum Hall interferometers Interference measurements are potenMally an important tool for studying various types of quanMzed Hall states. Several types of interferometers; transport through edge states and constricMons. Fabry-‐Perot and Mach-‐Zehnder geometries, have been implemented in quanMzed Hall systems. (MZ only implemented in integers.) Also two-‐parMcle interferometers (Hanbury-‐Brown Twiss). Interference between two electrons from two independent sources. We will focus here on the Fabry-‐Perot.
Fabry Perot interferometer in a quanMzed Hall system: 1
I
2
t1
t2
I
A Hall bar with two constricMons, (quantum point contacts) narrow enough so that parMcles can scaier from one edge to the other. Look for interference between two constricMons.
QuanMzed Hall system with no backscaiering
1
I
2
t1
t2
I
If no backscaiering (t1=t2=0) there will be no voltage drop between contacts 1 and 2. Contacts on opposite edges of the Hall bar will measure the quanMzed Hall voltage, VH =I h / ν e2.
Fix gate voltage at point contacts. Vary area A by varying voltage on side gate. Measure resistance V12/I. Expect oscillaMons in the resistance as a funcMon of A
1
I
2
t1
t2
Side Gate
I
Other Variables Oscillations should also occur if one varies the magnetic field B, keeping the gate voltage constant. What are the periods of the oscillations that result? What are their amplitudes? Can also get oscillations by applying voltage to a back gate or top gate, which vary the electron densities in different regions. Note: actual side gates will couple to the electron density as well as to the area of the interferometer.
Non-‐interacMng electrons ν = 1. Weak backscaiering at constricMons
Basic interference process: R ∝ |t1 + t2eiϕ|2 , Where φ = 2π BA /Φ0
Φ0 = h/|e| = “flux quantum”.
Dependence on field: When B is varied, if A is fixed, ϕ changes by 2π when δΒ A = Φ0 . Field period δΒ = Φ0 /A
Dependence on area: if B is fixed, ϕ changes by 2π when δA B = Φ0 . Area period δA = Φ0 /B = area that encloses one magnetic flux quantum.
Prediction: Lines of constant interference phase should have negative slope in the B -VG plane, assuming dA/dVG > 0 . (Similar predicMons for integer states with ν > 1.)
Predicted Aharonov-‐Bohm behavior Measure in a 2D plane of B and VG Lines of constant phase have negaMve slope. .
Yiming Zhang et al., 2009: Experimental results
Two types of behavior Measure in a 2D plane of B and VG “Coulomb Dominated”
“Aharonov-‐Bohm”
CD stripes have posiMve slope, and field period may be different than AB
Checker-‐board paiern -‐-‐ Experiments From Ofek et al., PNAS 2011
Real quantum Hall interferometers are not simple systems. Need to take into account Coulomb interacMons and disorder, interacMon between edge-‐states and localized quasi-‐parMcles in the interior of the dot, small but finite mobility of localized quasi-‐parMcles that allows their distribuMon to relax and screen Coulomb interacMons on a laboratory Mme scale. Total net number of quasiparMcles in localized states is constrained to be an integer.
TheoreMcal framework for discussing Fabry-‐ Perot Quantum Hall Interferometers Described in work with Bernd Rosenow, Ady Stern, and Izhar Neder. (PRB 2011). Motivated by experiments at Harvard by Yiming Zhang, Doug McClure, Angela Kou, and C. M. Marcus, who also contributed to the theoretical picture. Also, by experiments in Heiblum lab at Weizmann. [See article by N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Mahalu (PNAS, 2010)] Partly based on earlier work: B. Rosenow and B. I. Halperin,
PRL 2007.
And many other antecedents in the literature, dating back to the late 1980s.
Historical remarks Fabry-‐Perot interference experiments in the integer regime date back to late 1980s: van Wees, et al; Simmons et al.; etc. Many technical improvements since then. Include use of back gates to vary density by Goldman group. Recent introducMon of 2D color maps of resistance in B – VG plane. TheoreMcal work, recognizing the importance of Coulomb interacMons in interference experiments , and of fracMonal staMsMcs for FQHE, also date back to late 1980’s and early 1990’s: Jain, Kivelson, Patrick Lee, Goldman and Su. Important paper by Chamon, Freed, Kivelson, Su, and Wen (1997). Current work is a refinement of these ideas.
Example from a talk by C. Marcus, experiments by McClure et al. Varying magneMc field over a wide range, sees oscillaMons at various plateaus. Focus on low field edges – strong backscaiering regime.
Behavior at a constricMon (QPC) Example: Integer QHE .
ConstricMon Example 0 νout = 1 νin = 2 ν=3 2 1 0 1 < νc < 2 ; red curve is partially back-scattered 3±ε
νbulk =
Weak Back-‐Scaiering ConstricMon 0 νout = 1 νin =
2 ν=3 2 1 0 νc = 2-ε ; red edge is weakly back-scattered. High field side of νc = 2 plateau.
Strong Back-‐Scaiering ConstricMon = Weak Forward Scaiering 0 νout = 1 νin = 2 ν=3 2 1 0 νc = 1+ε ; red edge is mostly back-scattered. High field side of the νc = 1 plateau.
Behavior at a ConstricMon (2) Only extended edge states were shown. Generally there will be many addiMonal states corresponding to localized electrons or holes in the various Landau levels, with energies slightly above or below the Fermi energy. At non-‐zero T, these states will give rise to a finite conducMvity, which may be small, but enough to screen potenMal variaMons on laboratory Mme scale.
Alternate tunneling path Example: system with 2 constricMons at ν = 2 (Rosenow and BIH, 2007) .
Different paths can contribute in different regimes of filling fracMon. Density in constricMons may be different than in bulk.
Origin of CD period – weak back-‐scaiering regime. Area Ae of interfering edge state maybe wriien as Ae = A0 + δAe , where A0 is a smooth funcMon of VG and B, while δAe is small but oscillates on the scale of one flux a quantum or of the addiMon of one electron. OscillaMons occur because of Coulomb interacMon between edge state area and the (integer) number of localized charges in the interior of the dot. OscillaMons in δAe give rise to oscillaMons in the resistance with a different period than AB period. (We neglect dependence of A0 on B)
Periodicity of energy funcMon and interference phase Let b = B A0 / Φ0 be the number of flux quanta in area A0. Claim: if b is increased by 1, interference signal returns to original value. Reason: we can keep the areas of the partially reflected edge state and all fully transmitted edge states fixed, and decrease NLoc by νin = νout+1. Then total charge of the island is unchanged, position of each edge state is unshifted, and total energy will be the same as before. Interference phase φ = 2π BAe / Φ0 is increased by 2π , and e iφ is unchanged. Fundamental field period is the AB period. But φ will not just increase linearly in the interval. Increase in φ need not be monotonic. Fourier transform of e iφ may be stronger at harmonic (- νout) than at fundamental. Because of Coulomb interactions, area will tend to shrink as B is increased, then will increase again each time NLoc changes by -1.
Area Ae and phase for νC = 2 -‐ε, νout= 1 φ = 2π BAe / Φ0
If Coulomb coupling is strong, Ae may shrink enough so that φ iniMally decreases as B is increased Φ0/2 AB Slope A0 δ B Jump Δφ = 2π keL / ke .
Extreme CD Limit
In extreme CD limit, where energy tries to keep total charge fixed. When NLoc changes by -‐1, we find a jump Δφ = 2π. Has no effect on eiφ, no effect on interference signal.
Phase for νC = 2 -‐ε, νout = 1; CD Limit φ = 2π BAe / Φ0
CD slope
Φ0/2 AB Slope A0 δ B Apparent slope = -‐ 2π νout / Φ0 . Flux period = -‐ Φ0 / νout
Results for field period – Integer QHE For weak back-scattering, find CD period is A0 ΔB = Φ0 / m , where m = - ν out Period is unchanged if we increase back scattering, all the way to strong back scattering regime. In strong backscattering regime, CD period may be understood as Coulomb blockade physics. (AB period could also exist in strong backscattering regime for weak Coulomb interactions.) Periods should only change when you cross a Hall plateau, not along a riser
Problem We do not have a good understanding of when a parMcular system will be in the AB regime and when it will be in the CD regime.
FracMonal QuanMzed Hall States Methods can also be applied to fractional quantized Hall edge states. (Either ν out or ν in or both may be fractions.) Must take account of fractional charges and fractional statistics (i.e, effective magnetic flux associated with addition of quasiparticles). Do bookkeeping carefully. For field period, always find A0 ΔB = Φ0 / m , where m is an integer. Never find period bigger than Φ0 as one might naively expect for quasiparticles with fractional charge. Prediction for Jain fractions with no counter-flowing modes: Flux period in CD regime is given by m = - ν out / e*out . (AB regime can also exist, results are more complicated.) Predictions agree with most experiments. (Experiments by Goldman group at Stony Brook, reporting period larger than Φ0 are not understood.)
Q12
QuesMon: What if you have dirty edges? Answer: Except in the case of the third process which I menMoned, I have assumed that you can neglect scaiering between co-‐propagaMng edge modes along the perimeter of the interferometer, and that tunneling only takes place in the constricMons between two modes from opposite edges, which move in opposite direcMons. If there is a lot of scaiering between several parallel edge states, I expect that would tend to weaken or destroy the interference signal, but we have not considered this situaMon in detail.
Edges with counter-‐propagaMng modes? Example: boundary between ν = 2/3 and ν = 0. Situation is more complicated; not completely sorted out.. Experiments have not seen interference in Fabry-Perot geometry in situation where (say) νout= 0, νin = 2/3 . Interference has been seen in situation where νout = 2/3, νin = 1, by McClure and collaborators (Marcus lab). This is a simple boundary, one edge mode. Results show CD stripes, with expected field period, m = νout / e*out = 2.
Tunneling through an AnM-‐Dot at ν=2/3 Experiments by Angela Kou et al. (Marcus Lab) have observed oscillations in tunneling through an anti-dot, when the filling factor outside the dot is on the high-field side of the ν=2/3 plateau . (ν = 2/3 – ε ). This corresponds to νout=2/3, νin=0, edge with counterpropagating modes. Experimental results: stripes have positive slope in plane of B and VG (voltage on center gate that produces anti-dot). Field period corresponds to addition of flux Φ0 . Gate period corresponds to addition of charge 2e/3 . (Note: AB stripes would have positive slope, as well as CD stripes, in the anti-dot configuration.) Experiments are not well understood.
Q13
•
Comment (Marcus). I am not quite ready to buy into your disMncMon between “Coulomb Dominated” and “Coulomb Blockade”.
•
Answer: I feel that there are really two different phenomena here, and it is confusing to call them by the same name. I would say that the Coulomb Dominated period could be the dominant one even for the case of very weak back scaiering, where the interfering edge state is almost perfectly transmifng, and the conducMon is in no sense blockaded. I would reserve the work “blockade” to refer to the strong back-‐scaiering case, where one has an almost closed edge state, with a definite parMcle number, and transmission maxima occur at points where edge state has equal energies for N and N+1 parMcles. I would call this Coulomb blockaded if Coulomb interacMons dominate the energy spacings. However, the nearly-‐closed edge state would contain a discrete number of electrons even for non-‐interacMng electrons. In that case we would say that the dot is quantum-‐blockaded rather than Coulomb blockaded. As one climes the riser between two plateaus in RD, one goes conMnuously from weak-‐back scaiering to strong back scaiering. In principle, interferometer oscillaMons may be seen all along the riser, and the period need not change. The period could be characterisMc of the CD regime or cold be characterisMc of the AB regime. For weakly interacMng electrons one could see AB periods even in the strong back-‐scaiering, blockaded regime.
Interference experiments might also provide a way of detecMng non-‐abelian staMsMcs at f=5/2
Suggestions by: Das Sarma, Freedman and Nayak (PRL 2005) Stern and Halperin (PRL 2006) Bonderson, Kitaev, and Shtengel (PRL 2006)
Weak back-‐scaiering: V12∝⏐t1 + t2 e2πiΩ⏐2 , with δΩ=δA B/4Φ0 ,-‐-‐ only if the qh number is even.
1
I
2 +
ν= 5/2
+
t1
t2 +
ν= 5/2
I
+
If interference path contains an odd number of localized quasiholes, quasiparticle path tunneling at point t2 changes the state of enclosed zero-energy modes, and cannot interfere with path tunneling at t1.
If central region contains an odd number of localized quasiparticles, the lowest order interference term is absent. Then leading interference term varies as Re [t1* t2 e2πiΩ ]2 . (Period corresponds to an area containing two flux quanta, rather than four.) Similar period could result from tunneling of doubly charged (e/2) quasiparticle. If area is varied by a large amount, number of enclosed charged quasiparticle may change; period may alternate between (e/4) and (e/2).
AlternaMon of two periodiciMes has been observed Seen in experiments on ν=5/2 system by Willei and coworkers. But there are important problems with the interpretaMon: Predicted behavior should only apply if localized quasiparMcles are very well isolated from edges. This criterion is not met in experiments. Possibly, behavior could be explained if e/4 quasiparMcles are almost always bound into pairs.
Recent development: “Telegraph noise” in a Fabry Perot interferometer at ν=5/2 and ν=7/3 Experiments by An et al (W. Kang lab) arXiv :1112.3400 Theory by Rosenow and Simon (November. 2011) Evidence for non-‐Abelian staMsMcs in size of phase jumps. (Requires several assumpMons).
Conclusions Quantum Hall systems are not simple. There are many open quesMons.
Q14
QuesMon (Das Sarma): Do you understand Willei’s data at filling 5/2? Answer: No. One could speculate that some type of pairing or strong coupling between the quasiparMcles restores the e/4 period in some intervals of gate voltage and not in others, but I do not have a convincing theory. The recent preprint by Rosenow and Simon has some proposals about this, primarily in the context of the telegraph noise experiments of An et al; but there are a number of assumpMons in the paper which, the authors admit, are rather speculaMve, and we will have to see how this works out.
Q15
QuesMon: The theoreMcal analysis of Rosenow and Simon invokes a rather involved argument about screening of Coulomb interacMons in order to explain the telegraph noise results at f=7/3. Can’t one understand these without recourse to Coulomb interacMons? Answer: I think one could understand the results without Coulomb interacMons if one believes the experiments at 7/3 are in an AB regime. However, Rosenow and Simon do not want to make that assumpMon because the same experiment at f=2, seem to be clearly in the CD regime.
Q16
QuesMon: How do we know that the measurements are really at 5/2? Answer: I think we know that one is very close to 5/2 at least in the constricMons, because (at least in Willei’s experiments) they are looking at small deviaMons in the diagonal resistance from the plateau value corresponding to 5/2, which they see in the measurements. The density in the bulk could be different.
Q17
QuesMon: In thermal transport measurements, can one subtract out the contribuMon of phonons by appropriate experiments? Answer. Possibly. It is also possible to reduce the effects of phonons by various tricks, such as Jim Eisenstein has done, by going to a thin sample with a rough back surface. Whether this can go far enough to be able to reliably subtract out the phonon contribuMons to sufficient accuracy, I do not know. One is helped by the fact that the phonon contribuMon by itself would not give an off-‐diagonal component to the thermal conducMon, but the there are sMll complicaMons from the coupling between phonons and electrons, and as I menMoned, one would have to be careful to separate the diagonal and off-‐diagonal components in a real experiment. QuesMon: What about thermopower? Answer: Of course, thermoelectric effects and thermal conductance will be Med together when the transport is due to charge carriers in an edge state. If the temperature is too high, or if one is not on a well-‐established plateau, there can also be thermal and charge transport through the bulk, which is more complicated.
Outline 1. Survey of important issues in quantum Hall physics 2. Theory of Interferometer Experiments
Quantum Hall Systems Two-‐dimensional electron systems in a strong magneMc field (broken Mme reversal symmetry) (Low temperatures) Many varieMes: different Landau level filling factors (0.1 – 100); different levels and types of disorder; different materials. Why are the problems hard? For a parMally full Landau level, system is highly degenerate in the absence of interacMons. So electron-‐ electron interacMons, and electron impurity interacMons have a huge effect. If interacMons are weak compared to the cyclotron energy, can project onto a single Landau level, kineMc energy is quenched, and can be set equal to zero. But not like a classical system at T=0; a\er projecMon, different Fourier components of density operator do not commute with each other.
Issues in Quantum Hall Systems Universal issues: e.g. Topological classificaMon. CharacterisMcs which can take on only a discrete set of values, robust to small perturbaMons. Non-‐universal issues: Depend conMnuously on details. Necessary to understand real materials and experiments.
Universal properMes Ideal QuanMzed Hall States: Characterized by a precise filling factor. Energy gap for charged excitaMons in the bulk. Topological classificaMon of states, determined by Hamiltonian in the bulk, but low-‐energy consequences only manifest at edges or defects. Edges of quanMzed Hall states must support low-‐energy charge-‐carrying excitaMons. Certain (minimal) properMes of edges determined by Hamiltonian in the bulk. Others can be affected by edge reconstrucMons. A discrete set of finite-‐energy defects (quasiparMcles) can exist in the bulk. Their possible charges and their quantum staMsMcs are dictated by the nature of the defect-‐free quantum Hall ground state. Universal properMes are robust to small changes in the Hamiltonian, weak disorder. Can only change if the system undergoes a phase transiMon: energy gap for extended charge carrying excitaMons goes to zero in the bulk; or first order transiMon.
Universal properMes of edges Universal properties include quantized Hall conductance GH = ν e2/h , quantized thermal Hall conductance KH = κ π2 T/ 3h. (Kane and Fisher, 1996). (ν and κ are rational numbers.) Minimal edges have a specified number of edge states traveling in specified directions. Number can change if surface is reconstructed, but ν and κ must be conserved. In general: κ = NR-NL . (Majorana modes count ½). To compute ν: edge modes must be weighted by “charges”. Example: Integer QHE with f = N. Minimal model has N integer edge states, one for each Landau level, all moving in the same direction: ν = κ = N. If electron density varies slowly at the edge, no disorder, could get reconstruction with additional pairs of left and right moving edge states. But ν = κ = NR-NL = f is unchanged. In presence of disorder, extra pairs will become (at least eventually) localized, and only N right-movers remain extended, as in the minimal model.
Q1
QuesMon (Das Sarma): Are you talking about real edges or ideal edges? Answer: I was about to get to this quesMon. There are certain properMes that are universal, and cannot be affected by any reconstrucMons that occur at the edge.
Other properMes can be affected by edge reconstrucMons.
Q2
QuesMon: (Marcus) How can thermal conductance be zero or negaMve? Answer: Le\ movers carry heat to the le\ and right-‐movers carry heat to the right. What we are talking about here is the thermal Hall conductance. You apply a temperature gradient in the x-‐direcMon and you measure a heat flow in the y-‐ direcMon. (Imagine a strip oriented along the y axis, with a temperature gradient in the x-‐direcMon, so the two edges are a different temperatures. The edges will then carry a net energy current in the y-‐direcMon.) How you can measure this effect is another quesMon. There are many complicaMons in pracMce, for example due to the large parallel thermal conducMon by phonons. Also, one must be careful about boundary condiMons for the electron system, because you may be measuring a combinaMon of longitudinal and Hall components, and if there is electric current flow, you may mix in thermoelectric effects.
FracMonal QHE edges Fractional QHE states can be more complicated. Minimal edges may have only co-propagating modes (“right movers”), or may have both right and left movers (counter-propagating edge modes). Edge modes can carry fractional charge, make different contributions to ν and κ . Reconstructions can change charges as well as number of modes. Non-Abelian states can have additional modes, such as Majorana modes, which have charge zero and contribute ½ to κ . Note: Low energy modes also occur at an interface between two different QHE states.
Q3
Comment: (Fisher): There are other universal properMes in bulk systems, such as associated with transiMons between different quanMzed Hall states. Reply: Yes, I will come to that later in my talk.
Non-‐universal bulk properMes: Nature of the Ground State Several types of QHE states can exist at a given filling factor. ParMcularly if system has addiMonal low energy degrees of freedom: spin or pseudospin (mulMple valleys, bilayers, sublafces). QuesMon: Given a microscopic Hamiltonian and filling fracMon: what is the resulMng ground state? Answer depends on details. Only a limited number of QHE states will occur in a given system. Ground state at a general (irraMonal) filling fracMon f will generally consist of localized quasiparMcles on top of a nearby quanMzed state with raMonal filling factor f0 . QuasiparMcles may be localized by disorder or by interacMons (e.g. Wigner crystal, lafce of bubbles, stripes). Hall conductance determined by f0.
Q4
QuesMon: Why can’t there be an infinite number of quanMzed Hall plateaus? Answer: It may be that in principle there could be a quanMzed Hall state at any raMonal filling fracMon, given a suitably engineered Hamiltonian. In pracMce however, this is not what we expect. There will be a certain number of stable states with relaMvely large gaps, which can be the lowest energy underlying state over a relaMvely large range of filling factors. These states will then swallow up all other possible quanMzed Hall states for fillings in this range. More precisely, consider what happens at some arbitrary irraMonal filling factor f. Depending on the details of the system, including the amount of disorder present, the system will generally fall in the domain of airacMon of some quanMzed Hall state, at a raMonal filling f0 which is not too far away. The ground state at fracMon f will therefore consist of the ideal ground state at filling f0, together with a finite density of quasiparMcles or holes as necessary to achieve the required electron density for filing factor f. In general, we expect the added quasiparMcles or quasiholes to be localized at low temperatures, due to some combinaMon of disorder or Coulomb interacMons, so they do not contribute to transport at very low temperatures. Thus, one obtains vanishing longitudinal resistance, and one sees the Hall conducMvity of the parent state, f0. In most cases we may expect to see sharp transiMons from one Hall plateau to another as one varies the filling factor f. However, in some cases, as in the vicinity of f=1/2 for a single=layer system, we find that there exists a metallic state, with conMnuously varying Hall resistance, and a finite longitudinal resistance, whose value depends on the sample. •
Q5
QuesMon: Das Sarma: Can the plateau you are on depend on temperature? Answer: Strictly speaking, the plateaus are only rigorously defined at zero temperature. However, there can be approximate plateaus , which can exist at some finite temperature over some range of filling factors where there is no plateau, or a different plateau, zero temperature. Such things have been seen.
TransiMons between quanMzed Hall plateaus when f is varied. TransiMons between integer plateaus for non-‐interacMng electrons in the presence of disorder have been studied extensively. Believed to have universal properMes (e.g. criMcal exponents) that have been calculated numerically. Applicability in the presence of interacMons not clear. For interacMng electrons when disorder is small, fracMonal states intervene, can get transiMons between fracMonal states, or between integers and fracMons. Can also have metallic states at certain filling factors (e.g. f=1/2) with σxx finite (sample-‐dependent) down to lowest reasonable temperatures. Metallic states have no energy gap, finite compressibility, no plateau in σxy. Universal properMes predicted for transport, excitaMon spectrum, etc. in limit of vanishing disorder.
Q6
QuesMon: (Das Sarma). Don’t you have to have a compressible state between
any two incompressible quanMzed Hall states?
Answer: In some cases, as near f=1/2, as previously menMoned, there can be a “compressible” metallic state, with finite resisMvity, over a range of filling factors. However, this is probably the excepMon, rather than the rule. For non-‐interacMng electrons, in the presence of disorder, we believe there will be a direct transiMon between integer Hall states, which is sharp at zero temperature. This is based on the renormalizaMon group arguments of Pruisken, and various numerical simulaMons, etc. At finite temperature, the Hall resistance will change conMnuously as one varies the filling factor, but the range of f over which the Hall resistance changes its value becomes narrower as the temperature is lowered, vanishing in the limit of T=0. (For the strictly non-‐interacMng case, the longitudinal conductance is always zero, but in the presence of phonons or other mechanisms for inelasMc scaiering, there should be a finite longitudinal resistance in the range of f where the Hall resistance is changing.) We believe that similar sharp transiMons between integer plateaus will occur when electron-‐electron interacMons are taken into account, provided there is sufficient disorder to suppress any fracMonal quanMzed Hall states. For smaller values of disorder, where fracMonal quanMzed Hall states are present, there can be transiMons between quanMzed Hall states that are qualitaMvely similar to the integer transiMons, though they might have different criMcal exponents, etc.
Q7
Comment: Fisher: A direct transiMon is always possible between a mother state and a daughter state, as between f=1/3 and f=2/5, but this may not be possible in other cases, such as 1/3 to 2/3. Reply: I am not sure what you mean by a direct transiMon. I would think that at least a first-‐order transiMon between any two fracMons might be possible in principle. [At least in the case of short-‐range interacMons between the parMcles]. Fisher: The quesMon is whether you can have a direct second-‐order transiMon between say 1/3 and 4/5. Reply: We should probably talk about this more, off line. [Added remarks: The situaMon becomes more subtle when one takes into account effects of the long-‐range Coulomb interacMon in addiMon to disorder and short-‐range interacMons. When direct transiMons are disfavored, there can be re-‐entrant phases, where the Hall conductance is a non-‐monotonic funcMon of the electron density. ]
An open problem of great current interest: FQHE in Graphene Lowest Landau Level pseudopotenMals are the same as in GaAs. But Valley + Spin Degeneracy -‐> SU(4) Symmetry (Approximate). New states are possible. Experiments show many fracMons, but some are missing. Why? ExplanaMons require understanding of excitaMon energies as well as ground states.
Q8
QuesMon: (Das Sarma): How much can we trust the experiments on fracMonal Hall states in graphene? Are we confident that the reported fracMons are correctly idenMfied? Are we confident that the missing fracMons are really missing? Answer: At least in the case of Amir’s compressibility measurements, I find the data very convincing. As he can produce color plots on the plane of filling factor and electron density, one can see very clearly the existence of straight verMcal lines at idenMfiable filling fracMons, which are clearly associated with fracMonal quanMzed Hall states. Of course, we can’t say that the missing fracMons are truly missing. Perhaps they will appear in beier samples at lower temperatures. But we do not understand why some are seen in the experiments and some not. For example, Amir does not see an incompressible state at filling fracMon f=1/3 (corresponding to Hall conductance -‐5/3), while he does see clear fracMons at f=2/3, 4/3, and 5/3. In fact, I would be preiy sure that a quanMzed Hall state of the Laughlin type should be the true ground state at filling fracMon f=1/3 in a perfect graphene layer. However, the energy to produce charged excitaMons, such as valley skyrmions, might be sufficiently low that the plateau is easily wiped out by disorder and/or finite temperature. Perhaps Amir will show us data on this in his talk.
QuanMzed Hall State at ν=5/2 Possible ground states: Pfaffian state (Pf), proposed by Moore and Read. Elementary charges have charge e/4 and non-Abelian statistics. Electrons in second Landau level are fully spin polarized Anti-Pfaffian state (APf): (Particle-hole conjugate of Pfaffian). Bulk properties are similar to Pf; quasiparticles have similar charge and statistics to Pf; but edge states are different. APf edge has counter-propagating neutral mode; minimal Pf edge has no counter-propagating mode. (Lee, Ryu, Nayak and Fisher; Levin, Halperin, and Rosenow 2007). Other possible ground states are not completely ruled out.
Experimental tests Necessary characterisMcs of either Pf or APf state at 5/2: State should have charge e/4 quasiparMcles. State should be maximally spin-‐polarized. Recent measurements of charging events using an SET (Yacoby group) point strongly to charge e/4. Recent NMR measurements (Murata group, Bar-‐Joseph group) support full spin polarizaMon.
Experimental observaMons of counter-‐ propagaMng neutral modes Heiblum group has seen compelling evidence for a counter-‐ propagaMng neutral edge mode in 5/2 state, (as well as for 2/3, 5/3). Experiments study shot-‐noise in a geometry with a QPC and extra contacts arranged to prevent charge mode from reaching the detector. Results at 5/2 (along with previous studies of non-‐linear conductance at a constricMon by Marcus group), suggests APf rather than Pf. But reconstructed Pf edge could also have counter-‐propagaMng modes. Ground states completely different than Pf and APf cannot be ruled out.
Q9
QuesMon: Didn’t the counter-‐propagaMng neutral mode measurements depend on shot noise? Since shot noise measurements have given varying values for the charge, can we trust the neutral mode results? Answer: The neutral mode experiments do not depend on precisely what is tunneling between the edges, as long as something is tunneling, so I think the conclusions are preiy robust. QuesMon: What about edge reconstrucMon? What if the edge is very gradual Answer: Yes, it is true that if there is enough edge reconstrucMon, any bulk fracMon can acquire counter-‐propagaMng modes at the edge. However, the Weizmann experiments did see counter-‐propagaMng modes at fracMons where they are expected and did not see counter-‐propagaMng modes at various other fracMons where they would not be expected in a minimal edge.
Non-‐universal pracMcal quesMon What determines strength of disorder in actual samples? How do various type of disorder limit conducMvity in metallic state at f=1/2, or the number of fracMonal states seen. Only partly correlated with zero field mobility, or with quantum scaiering rate, measured in weak-‐field Shubnikov-‐de Haas experiments.
Other non-‐universal bulk properMes Energies to produce various forms of charged excitaMons. (QuasiparMcles can have different spin states, could be skyrmions) Energy spectra for neutral excitaMons (spin waves, excitons) Transport properMes at finite temperatures. Effects of disorder (long wave-‐length and short) formaMon of puddles of quasiparMcle or quasihole. Transport “acMvaMon energies”: reducMon from ideal values because of disorder. (cf. d’Ambrumenil, Halperin, Morf, PRL 2011, effects of tunneling through saddle points in the impurity potenMal)
Q10
QuesMon: (Marcus) Can you conclude that if we go to low enough temperatures, in
clean enough samples, the there will be a quanMzed Hall state at f=1/2, like at f=5/2?
Answer: No. Just like in normal metals, some metals become superconductors at low temperatures and some do not, at least at any experimentally reachable temperature, I would expect that the 5/2 state has an energy gap and the ½ does not. However, we do not know the answer for sure. The difference between the quanMzed Hall state and the metallic state is in fact analogous to the occurrence or non-‐occurrence of superconducMvity in an ordinary metal at zero magneMc field. The existence of superconducMvity in ordinary metals depends on the nature of the effecMve interacMons between the electrons; in the quantum Hall case, the occurrence of a gap depends on the interacMons between composite fermions. These are different in the first and second Landau levels, because the form factors for the electron states are different. In the case of zero magneMc field, there is a Kohn-‐Lufnger theorem which states that even for repulsive interacMons, there should always be superconducMvity at very low temperatures (perhaps ridiculously low). As far as I understand it, this theorem should not apply to the composite fermions at f=1/2 because the system is spin polarized , and pairing in even angular momentum states is excluded. However, we cannot rule out the possibility or pairing at some very low temperature.
Tools for addressing non-‐universal properMes Exact diagonalizaMon of small systems. VariaMonal calculaMons (e.g. Jain’s trial wave funcMons) Some exactly soluble special models. Approximate methods. (e.g. “Hamiltonian method” of Murthy and Shankar) Other methods?
Q11
QuesMon: What about DMRG calculaMons? Answer: This is what I was thinking of when I said that renormalizaMon group methods. Have been used, and I was thinking of the work in Das Sarma’s group. However, so far this has only been able to extend the range of exact diagonalizaMon to systems with a few more electrons. Comment (Das Sarma). We were able to go from 22 to 26 electrons at filling ½. It is hard to imagine one could go much further.
Quantum Hall interferometers Interference measurements are potenMally an important tool for studying various types of quanMzed Hall states. Several types of interferometers; transport through edge states and constricMons. Fabry-‐Perot and Mach-‐Zehnder geometries, have been implemented in quanMzed Hall systems. (MZ only implemented in integers.) Also two-‐parMcle interferometers (Hanbury-‐Brown Twiss). Interference between two electrons from two independent sources. We will focus here on the Fabry-‐Perot.
Fabry Perot interferometer in a quanMzed Hall system: 1
I
2
t1
t2
I
A Hall bar with two constricMons, (quantum point contacts) narrow enough so that parMcles can scaier from one edge to the other. Look for interference between two constricMons.
QuanMzed Hall system with no backscaiering
1
I
2
t1
t2
I
If no backscaiering (t1=t2=0) there will be no voltage drop between contacts 1 and 2. Contacts on opposite edges of the Hall bar will measure the quanMzed Hall voltage, VH =I h / ν e2.
Fix gate voltage at point contacts. Vary area A by varying voltage on side gate. Measure resistance V12/I. Expect oscillaMons in the resistance as a funcMon of A
1
I
2
t1
t2
Side Gate
I
Other Variables Oscillations should also occur if one varies the magnetic field B, keeping the gate voltage constant. What are the periods of the oscillations that result? What are their amplitudes? Can also get oscillations by applying voltage to a back gate or top gate, which vary the electron densities in different regions. Note: actual side gates will couple to the electron density as well as to the area of the interferometer.
Non-‐interacMng electrons ν = 1. Weak backscaiering at constricMons
Basic interference process: R ∝ |t1 + t2eiϕ|2 , Where φ = 2π BA /Φ0
Φ0 = h/|e| = “flux quantum”.
Dependence on field: When B is varied, if A is fixed, ϕ changes by 2π when δΒ A = Φ0 . Field period δΒ = Φ0 /A
Dependence on area: if B is fixed, ϕ changes by 2π when δA B = Φ0 . Area period δA = Φ0 /B = area that encloses one magnetic flux quantum.
Prediction: Lines of constant interference phase should have negative slope in the B -VG plane, assuming dA/dVG > 0 . (Similar predicMons for integer states with ν > 1.)
Predicted Aharonov-‐Bohm behavior Measure in a 2D plane of B and VG Lines of constant phase have negaMve slope. .
Yiming Zhang et al., 2009: Experimental results
Two types of behavior Measure in a 2D plane of B and VG “Coulomb Dominated”
“Aharonov-‐Bohm”
CD stripes have posiMve slope, and field period may be different than AB
Checker-‐board paiern -‐-‐ Experiments From Ofek et al., PNAS 2011
Real quantum Hall interferometers are not simple systems. Need to take into account Coulomb interacMons and disorder, interacMon between edge-‐states and localized quasi-‐parMcles in the interior of the dot, small but finite mobility of localized quasi-‐parMcles that allows their distribuMon to relax and screen Coulomb interacMons on a laboratory Mme scale. Total net number of quasiparMcles in localized states is constrained to be an integer.
TheoreMcal framework for discussing Fabry-‐ Perot Quantum Hall Interferometers Described in work with Bernd Rosenow, Ady Stern, and Izhar Neder. (PRB 2011). Motivated by experiments at Harvard by Yiming Zhang, Doug McClure, Angela Kou, and C. M. Marcus, who also contributed to the theoretical picture. Also, by experiments in Heiblum lab at Weizmann. [See article by N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Mahalu (PNAS, 2010)] Partly based on earlier work: B. Rosenow and B. I. Halperin,
PRL 2007.
And many other antecedents in the literature, dating back to the late 1980s.
Historical remarks Fabry-‐Perot interference experiments in the integer regime date back to late 1980s: van Wees, et al; Simmons et al.; etc. Many technical improvements since then. Include use of back gates to vary density by Goldman group. Recent introducMon of 2D color maps of resistance in B – VG plane. TheoreMcal work, recognizing the importance of Coulomb interacMons in interference experiments , and of fracMonal staMsMcs for FQHE, also date back to late 1980’s and early 1990’s: Jain, Kivelson, Patrick Lee, Goldman and Su. Important paper by Chamon, Freed, Kivelson, Su, and Wen (1997). Current work is a refinement of these ideas.
Example from a talk by C. Marcus, experiments by McClure et al. Varying magneMc field over a wide range, sees oscillaMons at various plateaus. Focus on low field edges – strong backscaiering regime.
Behavior at a constricMon (QPC) Example: Integer QHE .
ConstricMon Example 0 νout = 1 νin = 2 ν=3 2 1 0 1 < νc < 2 ; red curve is partially back-scattered 3±ε
νbulk =
Weak Back-‐Scaiering ConstricMon 0 νout = 1 νin =
2 ν=3 2 1 0 νc = 2-ε ; red edge is weakly back-scattered. High field side of νc = 2 plateau.
Strong Back-‐Scaiering ConstricMon = Weak Forward Scaiering 0 νout = 1 νin = 2 ν=3 2 1 0 νc = 1+ε ; red edge is mostly back-scattered. High field side of the νc = 1 plateau.
Behavior at a ConstricMon (2) Only extended edge states were shown. Generally there will be many addiMonal states corresponding to localized electrons or holes in the various Landau levels, with energies slightly above or below the Fermi energy. At non-‐zero T, these states will give rise to a finite conducMvity, which may be small, but enough to screen potenMal variaMons on laboratory Mme scale.
Alternate tunneling path Example: system with 2 constricMons at ν = 2 (Rosenow and BIH, 2007) .
Different paths can contribute in different regimes of filling fracMon. Density in constricMons may be different than in bulk.
Origin of CD period – weak back-‐scaiering regime. Area Ae of interfering edge state maybe wriien as Ae = A0 + δAe , where A0 is a smooth funcMon of VG and B, while δAe is small but oscillates on the scale of one flux a quantum or of the addiMon of one electron. OscillaMons occur because of Coulomb interacMon between edge state area and the (integer) number of localized charges in the interior of the dot. OscillaMons in δAe give rise to oscillaMons in the resistance with a different period than AB period. (We neglect dependence of A0 on B)
Periodicity of energy funcMon and interference phase Let b = B A0 / Φ0 be the number of flux quanta in area A0. Claim: if b is increased by 1, interference signal returns to original value. Reason: we can keep the areas of the partially reflected edge state and all fully transmitted edge states fixed, and decrease NLoc by νin = νout+1. Then total charge of the island is unchanged, position of each edge state is unshifted, and total energy will be the same as before. Interference phase φ = 2π BAe / Φ0 is increased by 2π , and e iφ is unchanged. Fundamental field period is the AB period. But φ will not just increase linearly in the interval. Increase in φ need not be monotonic. Fourier transform of e iφ may be stronger at harmonic (- νout) than at fundamental. Because of Coulomb interactions, area will tend to shrink as B is increased, then will increase again each time NLoc changes by -1.
Area Ae and phase for νC = 2 -‐ε, νout= 1 φ = 2π BAe / Φ0
If Coulomb coupling is strong, Ae may shrink enough so that φ iniMally decreases as B is increased Φ0/2 AB Slope A0 δ B Jump Δφ = 2π keL / ke .
Extreme CD Limit
In extreme CD limit, where energy tries to keep total charge fixed. When NLoc changes by -‐1, we find a jump Δφ = 2π. Has no effect on eiφ, no effect on interference signal.
Phase for νC = 2 -‐ε, νout = 1; CD Limit φ = 2π BAe / Φ0
CD slope
Φ0/2 AB Slope A0 δ B Apparent slope = -‐ 2π νout / Φ0 . Flux period = -‐ Φ0 / νout
Results for field period – Integer QHE For weak back-scattering, find CD period is A0 ΔB = Φ0 / m , where m = - ν out Period is unchanged if we increase back scattering, all the way to strong back scattering regime. In strong backscattering regime, CD period may be understood as Coulomb blockade physics. (AB period could also exist in strong backscattering regime for weak Coulomb interactions.) Periods should only change when you cross a Hall plateau, not along a riser
Problem We do not have a good understanding of when a parMcular system will be in the AB regime and when it will be in the CD regime.
FracMonal QuanMzed Hall States Methods can also be applied to fractional quantized Hall edge states. (Either ν out or ν in or both may be fractions.) Must take account of fractional charges and fractional statistics (i.e, effective magnetic flux associated with addition of quasiparticles). Do bookkeeping carefully. For field period, always find A0 ΔB = Φ0 / m , where m is an integer. Never find period bigger than Φ0 as one might naively expect for quasiparticles with fractional charge. Prediction for Jain fractions with no counter-flowing modes: Flux period in CD regime is given by m = - ν out / e*out . (AB regime can also exist, results are more complicated.) Predictions agree with most experiments. (Experiments by Goldman group at Stony Brook, reporting period larger than Φ0 are not understood.)
Q12
QuesMon: What if you have dirty edges? Answer: Except in the case of the third process which I menMoned, I have assumed that you can neglect scaiering between co-‐propagaMng edge modes along the perimeter of the interferometer, and that tunneling only takes place in the constricMons between two modes from opposite edges, which move in opposite direcMons. If there is a lot of scaiering between several parallel edge states, I expect that would tend to weaken or destroy the interference signal, but we have not considered this situaMon in detail.
Edges with counter-‐propagaMng modes? Example: boundary between ν = 2/3 and ν = 0. Situation is more complicated; not completely sorted out.. Experiments have not seen interference in Fabry-Perot geometry in situation where (say) νout= 0, νin = 2/3 . Interference has been seen in situation where νout = 2/3, νin = 1, by McClure and collaborators (Marcus lab). This is a simple boundary, one edge mode. Results show CD stripes, with expected field period, m = νout / e*out = 2.
Tunneling through an AnM-‐Dot at ν=2/3 Experiments by Angela Kou et al. (Marcus Lab) have observed oscillations in tunneling through an anti-dot, when the filling factor outside the dot is on the high-field side of the ν=2/3 plateau . (ν = 2/3 – ε ). This corresponds to νout=2/3, νin=0, edge with counterpropagating modes. Experimental results: stripes have positive slope in plane of B and VG (voltage on center gate that produces anti-dot). Field period corresponds to addition of flux Φ0 . Gate period corresponds to addition of charge 2e/3 . (Note: AB stripes would have positive slope, as well as CD stripes, in the anti-dot configuration.) Experiments are not well understood.
Q13
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Comment (Marcus). I am not quite ready to buy into your disMncMon between “Coulomb Dominated” and “Coulomb Blockade”.
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Answer: I feel that there are really two different phenomena here, and it is confusing to call them by the same name. I would say that the Coulomb Dominated period could be the dominant one even for the case of very weak back scaiering, where the interfering edge state is almost perfectly transmifng, and the conducMon is in no sense blockaded. I would reserve the work “blockade” to refer to the strong back-‐scaiering case, where one has an almost closed edge state, with a definite parMcle number, and transmission maxima occur at points where edge state has equal energies for N and N+1 parMcles. I would call this Coulomb blockaded if Coulomb interacMons dominate the energy spacings. However, the nearly-‐closed edge state would contain a discrete number of electrons even for non-‐interacMng electrons. In that case we would say that the dot is quantum-‐blockaded rather than Coulomb blockaded. As one climes the riser between two plateaus in RD, one goes conMnuously from weak-‐back scaiering to strong back scaiering. In principle, interferometer oscillaMons may be seen all along the riser, and the period need not change. The period could be characterisMc of the CD regime or cold be characterisMc of the AB regime. For weakly interacMng electrons one could see AB periods even in the strong back-‐scaiering, blockaded regime.
Interference experiments might also provide a way of detecMng non-‐abelian staMsMcs at f=5/2
Suggestions by: Das Sarma, Freedman and Nayak (PRL 2005) Stern and Halperin (PRL 2006) Bonderson, Kitaev, and Shtengel (PRL 2006)
Weak back-‐scaiering: V12∝⏐t1 + t2 e2πiΩ⏐2 , with δΩ=δA B/4Φ0 ,-‐-‐ only if the qh number is even.
1
I
2 +
ν= 5/2
+
t1
t2 +
ν= 5/2
I
+
If interference path contains an odd number of localized quasiholes, quasiparticle path tunneling at point t2 changes the state of enclosed zero-energy modes, and cannot interfere with path tunneling at t1.
If central region contains an odd number of localized quasiparticles, the lowest order interference term is absent. Then leading interference term varies as Re [t1* t2 e2πiΩ ]2 . (Period corresponds to an area containing two flux quanta, rather than four.) Similar period could result from tunneling of doubly charged (e/2) quasiparticle. If area is varied by a large amount, number of enclosed charged quasiparticle may change; period may alternate between (e/4) and (e/2).
AlternaMon of two periodiciMes has been observed Seen in experiments on ν=5/2 system by Willei and coworkers. But there are important problems with the interpretaMon: Predicted behavior should only apply if localized quasiparMcles are very well isolated from edges. This criterion is not met in experiments. Possibly, behavior could be explained if e/4 quasiparMcles are almost always bound into pairs.
Recent development: “Telegraph noise” in a Fabry Perot interferometer at ν=5/2 and ν=7/3 Experiments by An et al (W. Kang lab) arXiv :1112.3400 Theory by Rosenow and Simon (November. 2011) Evidence for non-‐Abelian staMsMcs in size of phase jumps. (Requires several assumpMons).
Conclusions Quantum Hall systems are not simple. There are many open quesMons.
Q14
QuesMon (Das Sarma): Do you understand Willei’s data at filling 5/2? Answer: No. One could speculate that some type of pairing or strong coupling between the quasiparMcles restores the e/4 period in some intervals of gate voltage and not in others, but I do not have a convincing theory. The recent preprint by Rosenow and Simon has some proposals about this, primarily in the context of the telegraph noise experiments of An et al; but there are a number of assumpMons in the paper which, the authors admit, are rather speculaMve, and we will have to see how this works out.
Q15
QuesMon: The theoreMcal analysis of Rosenow and Simon invokes a rather involved argument about screening of Coulomb interacMons in order to explain the telegraph noise results at f=7/3. Can’t one understand these without recourse to Coulomb interacMons? Answer: I think one could understand the results without Coulomb interacMons if one believes the experiments at 7/3 are in an AB regime. However, Rosenow and Simon do not want to make that assumpMon because the same experiment at f=2, seem to be clearly in the CD regime.
Q16
QuesMon: How do we know that the measurements are really at 5/2? Answer: I think we know that one is very close to 5/2 at least in the constricMons, because (at least in Willei’s experiments) they are looking at small deviaMons in the diagonal resistance from the plateau value corresponding to 5/2, which they see in the measurements. The density in the bulk could be different.
Q17
QuesMon: In thermal transport measurements, can one subtract out the contribuMon of phonons by appropriate experiments? Answer. Possibly. It is also possible to reduce the effects of phonons by various tricks, such as Jim Eisenstein has done, by going to a thin sample with a rough back surface. Whether this can go far enough to be able to reliably subtract out the phonon contribuMons to sufficient accuracy, I do not know. One is helped by the fact that the phonon contribuMon by itself would not give an off-‐diagonal component to the thermal conducMon, but the there are sMll complicaMons from the coupling between phonons and electrons, and as I menMoned, one would have to be careful to separate the diagonal and off-‐diagonal components in a real experiment. QuesMon: What about thermopower? Answer: Of course, thermoelectric effects and thermal conductance will be Med together when the transport is due to charge carriers in an edge state. If the temperature is too high, or if one is not on a well-‐established plateau, there can also be thermal and charge transport through the bulk, which is more complicated.
