to the overall single-crystal orientation of theinitial microcube. CBED analysis performed onthe sample showed largemisorientations betweenadjacent grains that exhibit high contrast (Fig. 4Band table S7) (20) and a gradually varying CBEDpattern within each of those grains, consistentwith a structure at a lower length scale containingdefects and small subgrain misorientations.Upon impact, the pristine single-crystal single-

grain Ag microcube undergoes an extreme defor-mation trajectory of micro- and nanostructuralchanges due to the high-strain-rate deformation:a highly deformed external geometry and creationof nanoscale grains and strong spatial gradientsin grain size along the height of the sample. SuchaGNG structure has been shown to result in a newgradient-plasticity strengthening and tougheningmechanism in metals with spatial gradients instress and strain under uniform overall deforma-tions (6). Because of the correlation between theyield strength and the grain size, application ofa uniform external load to amaterial with GNGstructure still leads to different strain distribu-tions in different-sized grains, resulting in a spatialgradient of strain and stress (6). This strengtheningand tougheningmechanism,which is distinct fromthe commonly known strain-gradient plasticity,prevents catastrophic failure through progressiveyielding and strain hardening (4–6).Our observations of recrystallization driven by

the stored elastic energy suggest amicrostructuralevolution path that progresses from nanocrystal-line toward single crystal. The recrystallizationprocesses also soften the hard and brittle nano-crystalline material that results from dynamic de-formation (31). Therefore, the GNG-structuredmetal with intermediate states of recrystallizationshould have desirable strength and toughness formechanical applications requiring high fatigue lifeand survivability in extreme environments suchas automobile and aircraft crashes, sport-relatedcollisions, and body and vehicle armors. The re-crystallization process can be retarded by theaddition of alloying elements that preferentiallysegregate to grain boundaries for thermodynamicnanostructure stabilization (32).TheGNG-structuredmaterialsmade by a single-

step high-strain-rate process, particularly inmetallicalloy systems that have the ability to retard con-tinuous recrystallization even at high temperatures(32), will be useful for applications requiring ul-trahigh strength and toughness. Our studies alsodemonstrate that controlling the impact orienta-tionwill provide additional control over tailoringthe GNG structure and hence themechanical prop-erties of the resultant material. Additionally, ourfindings suggest important roles played by boththe intrinsic crystal symmetries and the extrinsicsample geometries, inspiring further fundamen-tal investigations to understand the interplay ofintrinsic and extrinsic symmetries.

REFERENCES AND NOTES

1. M. D. Uchic, D. M. Dimiduk, J. N. Florando, W. D. Nix, Science305, 986–989 (2004).

2. J. R. Greer, J. T. M. De Hosson, Prog. Mater. Sci. 56, 654–724(2011).

3. T. H. Fang, W. L. Li, N. R. Tao, K. Lu, Science 331, 1587–1590(2011).

4. K. Lu, Science 345, 1455–1456 (2014).5. X. Wu, P. Jiang, L. Chen, F. Yuan, Y. T. Zhu, Proc. Natl. Acad.

Sci. U.S.A. 111, 7197–7201 (2014).6. Z. Zeng, X. Li, D. Xu, L. Lu, H. Gao, T. Zhu, Extrem. Mech. Lett.

2015 (2015); www.sciencedirect.com/science/article/pii/S2352431615001261.

7. J. D. Reid, Thin-walled Struct. 24, 285–313 (1996).8. T. W. McAllister et al., Ann. Biomed. Eng. 40, 127–140 (2012).9. P. Hazell, Armour: Materials, Theory, and Design (CRC Press,

2015).10. S. Dai, Y. Zhu, Z. Huang, Vacuum 125, 215–221 (2016).11. C. Montross, Int. J. Fatigue 24, 1021–1036 (2002).12. F. Findik, Mater. Des. 32, 1081–1093 (2011).13. R. C. Dykhuizen et al., J. Therm. Spray Technol. 8, 559–564

(1999).14. E. Grossman, I. Gouzman, R. Verker, MRS Bull. 35, 41–47 (2010).15. T. J. Carter, Eng. Fail. Anal. 12, 237–247 (2005).16. M. J. Burek et al., Mater. Sci. Eng. A 528, 5822–5832 (2011).17. G. Youssef et al., J. Appl. Phys. 113, 084309 (2013).18. H. A. Colorado et al., J. Appl. Phys. 114, 233510 (2013).19. J.-H. Lee, P. E. Loya, J. Lou, E. L. Thomas, Science 346,

1092–1096 (2014).20. Materials and methods are available as supplementary

materials on Science Online.21. S.-J. Jeon, J.-H. Lee, E. L. Thomas, J. Colloid Interface Sci. 435,

105–111 (2014).22. E. B. Tadmor, N. Bernstein, J. Mech. Phys. Solids 52,

2507–2519 (2004).23. S. Buzzi, M. Dietiker, K. Kunze, R. Spolenak, J. F. Löffler, Philos.

Mag. 89, 869–884 (2009).

24. W. Heye, G. Wasserman, Scr. Metall. 2, 205–207 (1968).25. H. Paul, J. H. Driver, C. Maurice, A. Piatkowski, Acta Mater. 55,

575–588 (2007).26. I. J. Beyerlein, L. S. Tóth, C. N. Tomé, S. Suwas, Philos. Mag.

87, 885–906 (2007).27. D. R. Smith, F. R. Fickett, J. Res. Natl. Inst. Stand. Technol. 100,

119 (1995).28. A. M. Meyers, Dynamic Behavior of Materials (Wiley, 1994).29. J. A. Hines, K. S. Vecchio, Acta Mater. 45, 635–649

(1997).30. R. D. Doherty et al., Mater. Today 1, 14–15 (1998).31. H. Paul, J. H. Driver, C. Maurice, A. Piatkowski, Acta Mater. 55,

833–847 (2007).32. T. Chookajorn, H. A. Murdoch, C. A. Schuh, Science 337,

951–954 (2012).

ACKNOWLEDGMENTS

This work was financially supported by the George R. BrownSchool of Engineering, Rice University. We thank D. Pham for hishelp in constructing 3D illustrations. We also acknowledge theuseful discussions with E. Ringe, Electron Microscopy Center,Rice University.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/354/6310/312/suppl/DC1Material and MethodsFigs. S1 to S7Tables S1 to S7References

19 May 2016; accepted 22 September 201610.1126/science.aag1768

SOLID-STATE PHYSICS

Observation of a nematic quantumHall liquid on the surface of bismuthBenjamin E. Feldman,1* Mallika T. Randeria,1* András Gyenis,1* Fengcheng Wu,2

Huiwen Ji,3 R. J. Cava,3 Allan H. MacDonald,2 Ali Yazdani1†

Nematic quantum fluids with wave functions that break the underlying crystallinesymmetry can form in interacting electronic systems. We examined the quantum Hallstates that arise in high magnetic fields from anisotropic hole pockets on the Bi(111)surface. Spectroscopy performed with a scanning tunneling microscope showed that acombination of single-particle effects and many-body Coulomb interactions lift thesix-fold Landau level (LL) degeneracy to form three valley-polarized quantum Hall states.We imaged the resulting anisotropic LL wave functions and found that they have a differentorientation for each broken-symmetry state. The wave functions correspond to thoseexpected from pairs of hole valleys and provide a direct spatial signature of a nematicelectronic phase.

Nematic electronic states represent an in-triguing class of broken-symmetry phasesthat can spontaneously form as a result ofelectronic correlations (1, 2). They are char-acterized by reduced rotational symmetry

relative to the underlying crystal lattice and haveattracted considerable interest in systems suchas two-dimensional electron gases (2DEGs) (3–5),

strontium ruthenate (6), and high-temperature su-perconductors (7–12). The sensitivity of electronicnematic phases to disorder results in short-rangeordering and the formation of domains, makingthem difficult to study using global measurementsthat average overmicroscopic configurations. Theeffect of perturbations, such as crystalline strain,may be used to show a propensity for nematicorder—that is, to provide evidence that vestigesof nematic behavior survive even in the presence ofmaterial imperfections (1). However, it is difficultto quantitatively correlate the experimental evidenceof ordering with a microscopic description of theelectronic states and the interactions responsiblefor nematic behavior. To put the study of nematicelectronic phases on more quantitative ground,

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1Joseph Henry Laboratories and Department of Physics,Princeton University, Princeton, NJ 08544, USA.2Department of Physics, University of Texas, Austin, TX78712, USA. 3Department of Chemistry, Princeton University,Princeton, NJ 08544, USA.*These authors contributed equally to this work. †Correspondingauthor. Email: yazdani@princeton.edu

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it is therefore important not only to perform localmeasurements, but also to find a material systemfor which theory can fully characterize the un-derlying broken-symmetry states and the elec-tronic interactions.Multivalley 2DEGswith anisotropic band struc-

ture have been anticipated as a model platformto explore nematic order in the quantum Hall re-gime (13–17). The key idea is that Coulomb inter-actions can spontaneously lift the valley degeneracyinmaterials with low disorder and thereby breakrotational symmetry. In contrast to previouslystudied metallic nematic phases, this leads to agapped nematic state with quantized Hall con-ductance. We examined such a 2DEG on the sur-face of single crystals of bismuth (Bi), which isone of the cleanest electronic systems, with a bulkmean free path reaching 1 mmat low temperatures(18). Interest in Bi has recently been rekindled bybulk measurements showing phase transitionsand anisotropic behavior, possibly related to ne-matic electronic phenomena, in the presence oflarge magnetic fields (19–22). We focus here onthe (111) surface of Bi, for which strong Rashbaspin-orbit coupling results in a rich 2DEG con-sisting of spin-split surface states that producemultiple electron and hole pockets (23, 24). Scan-ning tunnelingmicroscope (STM) images (Fig. 1A)show that the in situ cleaved Bi(111) surface haslarge (>200 nm× 200 nm) atomically ordered ter-

races that are separated by steps oriented alonghigh-symmetry crystallographic directions (25).Angle-resolvedphotoemissionspectroscopy (ARPES)measurements (23, 24, 26, 27) of this surface showthat its Fermi surface consists of a hexagonal elec-tron pocket at the G point, three additional elon-gated electron pockets around the M points, andsix anisotropic hole pockets along the G-M direc-tions (Fig. 1B, inset). The multiply degenerate ani-sotropic valleys and the low disorder of the Bi(111)surface make it an ideal system to search for nem-atic electronic behavior using the STM.In the absence of magnetic field, spectroscopic

measurements of the Bi(111) surface with the STM(Fig. 1B) show features in the tunneling conduct-anceG that are related to vanHove singularitiesof the density of states (DOS), such as the sharppeak at energy E = 220meV and the abrupt dropat 33meV. These features correspond to the upperband edges of the surface states along the G-Mdirection (25, 28, 29). In the presence of a largemagnetic field B, the electron and hole states ofthe Bi(111) surface are quantized into Landaulevels (LLs), each with degeneracy geB/h, where eis the electron charge, h is Planck’s constant, andg accounts for the degeneracy arising from thevalley degree of freedom (g = 6 for holes). At highmagnetic field, the STM spectra show a series ofsharp peaks (Fig. 1B) whose evolution with mag-netic field can be used to distinguish between

electron- and hole-like LLs, which disperse inenergy with positive or negative slopes, respec-tively, as a function of magnetic field (Fig. 1, Cand D). They do not exhibit avoided crossings,and the total conductance is additive when theycross, which suggests independent tunneling intoeach LL. LL spectroscopy on thin Bi(111) filmswasrecently reported (28) but did not show evidence ofsymmetry breaking,which is the focus of ourwork.Our first key observation is that the surface-

state LLs do not disperse linearly withmagneticfield. Instead, they are pinned to the Fermi leveluntil they are fully occupied, as is clearly shown forthe hole states in Fig. 1D. Such behavior is rarelyobserved in LL spectroscopy of ungated samplesperformed using a STM (30), and it indicates thatthe surface charge density is held constant in oursystem. Electron LLs exhibit pinning only whenthere are no proximal hole states, whereas theyotherwise cross straight through the hole LLs atthe Fermi level. This difference in behavior signalsan intriguing competition between electron- andhole-like states in a magnetic field, and suggestscharge rearrangement between pockets (29). Wefocus below on the hole states, for which the or-bital index Nh is straightforward to assign, withthe highest-energy peak corresponding toNh = 0closely matched to the zero-field drop in con-ductance at 33 meV. Using the values of the fieldand filling factor at which LLs cross the Fermi

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Fig. 1. Landau levels (LLs) of the Bi(111) surface states. (A) A typicalcleaved Bi(111) surface, with crystallographic axes labeled. The data in (E)and (G) are an average of spectra measured along the blue line, and theconductance maps in Fig. 3 were performed in the area denoted by theblack box. Surface defects are circled in purple; the inset shows a zoom-inon one defect (inset z height scale, 1.3 Å). (B) Conductance G as a functionof energy E at magnetic field B = 0 (blue) and at 14 T (red). The curves areoffset by 0.5 for clarity. At B = 0, the data are taken at temperature T ≈ 4 K.All other data throughout the manuscript are measured at 250 mK. Theinset is a diagram of the Bi(111) first Brillouin zone, showing the electron(purple) and hole (blue) Fermi pockets of the surface states. (C) Landaufan diagram of G(E, B) that shows crossing electron- and hole-like LLs.Thedata are averaged over a 20-nm line, with individual spectra showing

almost no spatial variation on this energy scale. Select orbital indices Ne

and Nh of the respective electron and hole LLs are labeled. (D) Higher-energy resolution measurement of G(E, B) that clearly shows Fermi-levelpinning of each hole LL. (E) High-resolution measurement of G(E, B) in aregion where the LLs corresponding to Nh = 3, 4, and 5 each show splittinginto a two-fold degenerate and a four-fold degenerate LL peak. Data areaveraged over the blue line in (A). (F) Line cut of spectra showing strain-induced splitting of the six-fold degenerate Nh = 3 LL into two or three peaks,depending on position. Numbers in parentheses denote the degeneracy ofeach broken-symmetry state. (G) Zoom-in on G(E, B) in the same location asin (E). The four-fold degenerate peak further splits into two distinct LLs as itcrosses the Fermi level, indicating broken symmetry states arising fromexchange interactions. Arrows mark Dstrain and Dexch.

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Fig. 2. Rotational symmetry breaking and localdomains of a nematic electronic phase. (A) Averageconductance spectrum at 12.9 T, measured over a100-nm line cut (which exhibits little spatial depen-dence) near the start of the line cut in Fig. 1F, show-ing three broken-symmetry hole LLs, two of whichare split by exchange interactions at the Fermi level.(B to D) Spatial maps of conductance normalized byits average value G=G at energies corresponding to thethree split hole LL peaks. Ellipses of reduced conduct-ance are centered on surface defects, with differentorientations at each energy. (E) Average conductancespectrum at 14 T, measured in the same location asin (A).The spectrum shows restored symmetry of theexchange-split LLs in (A) to produce a four-fold dege-

nerate LL. (F) Spatial map of G=G at the energy of thefour-fold degenerate LL peak, which shows ellipses

with two orientations. (G) Spatial map of G=G at theenergy of the two-fold degenerate LL peak that is splitfrom the four-fold degenerate peak by strain, showingthe same unidirectional behavior as in (D). The spatialmaps in (F) and (G) are measured in the same area as(B) to (D). (H) Average conductance spectrum (mea-sured over a 100-nm line cut that exhibits little spatialdependence) at 12.9 T in a location about 1 mm awayfrom the region shown in (A) to (G). (I to K) Spatial

maps of G=G in the new location at energies corres-ponding to the three split hole LL peaks. The ener-getic order of the three directions is different, withthe first two orientations switched, demonstrating thepresence of domains. For all conductance spectra,the electron LLs are labeled, and the hole LL degeneracy is denoted in parentheses near each peak.

Fig. 3. Isolated anisotropic cyclotron orbits and theoretical modeling.

(A to E) Spatial maps of G=G at 14 T in the area denoted by the black boxin Fig. 1A, at energies corresponding to the strain-induced broken-symmetryhole LL for orbital indices Nh = 0 to 4. Isolated anisotropic cyclotron orbitsare present around surface defects. (F) Spatial map of G=G in the samearea at the energy of the Ne = 8 LL, showing circular rings of suppressedconductance (black arrows) that occur around the same surface defects.The weak elliptical feature around the lower defect is related to a missingcyclotron orbit from the Nh = 3 LL at a nearby energy. The trapezoidal

feature in the background conductance results from the shape of theterrace because the LLvisibility is suppressed near step edges. (G) Amplitude

2πl2Bjφ4;4ðrÞj2 of the m = Nh = 4 cyclotron orbit wave function. (H to K)Simulated maps of the expected conductance, 1−2πl2BjφN;NðrÞj2, with individualcyclotron orbits centered on the surface defects circled in Fig. 1A. The size andshape of the simulated conductance are a good match to the data in (B) to (E).(L) Semimajor axis size of the cyclotron orbits for Nh = 4 (blue) and ring size ofthose from electron LLs near the Fermi level (red) as a function of magneticfield. Dashed lines are fits to the field dependence of the extracted sizes.

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level, we determine the hole surface density to bep ≈ 7.1 × 1012 cm−2 (29).High-resolution spectroscopic measurements

provide an indication that both single-particleeffects and electron-electron interactions breakthe six-fold symmetry of the hole LLs. Evidenceof symmetry breaking can be seen inFig. 1E,whichshows the field evolution of the conductance spec-tra in one region of the sample where the LLs cor-responding toNh = 3, 4, and 5 are each split intotwo peakswith different amplitudes, indicating alifting of the six-fold valley degeneracy of eachlevel to form two- and four-fold degenerate LLs.The fact that the splitting (characterized by a gapDstrain) occurs away from the Fermi level indicatesthat it is a single-particle effect. The very weakdependence of Dstrain onmagnetic field and orbitalindex and the fact that we observe different mag-nitude gaps in different regions of the samplesuggest that local strain underlies this partial sym-metry breaking (29). As an illustration of the spa-tial dependence of this behavior, we show in Fig.1F a spectroscopic line cut from a region of thesample in which the six-fold degeneracy of theNh = 3 LL is lifted to produce either two or threebroken-symmetry states, depending on locationwithin the sample.

Electron-electron interactions further lift the LLdegeneracy and are manifested in spectroscopicmeasurements by the appearance of energy gapswhen theLLs cross the Fermi level. Figure 1G showsa high-resolutionmeasurement of the Fermi-levelcrossing of theNh = 4 LL (in the same area as inFig. 1E), where over a range of 0.5 T, the four-folddegenerate peak develops an exchange energy gap(Dexch = 450 meV) that is coincidentwith the Fermilevel. Although there are spatial variations in theexactmagnitude of the gaps between the broken-symmetry LLs, exchange interactions consistentlyenhance gaps between LLs that are already splitby strain and induce a gap between previouslydegenerate levels when they cross the Fermi level.The magnitude of the exchange gap is consistentwith that estimated theoretically for the hole poc-kets of Bi(111), and it is not related to an Efros-Shlovskii Coulomb gap (29). These observationsdemonstrate that a combination of a single-particleeffect, likely strain, and many-body interactionslift the six-fold valley degeneracy of the hole LL toproduce three broken-symmetry states.We performed spectroscopic mapping with the

STM to directly visualize the underlying quantumHall wave functions and to demonstrate the break-ing of crystalline symmetry in these phases. Con-

ductance maps at energies corresponding to eachof the three broken-symmetry hole LLs show an-isotropic ellipse-like features that point along high-symmetry crystal axes, with relative angles rotatedby 120° with respect to each other (Fig. 2, A to D).The elliptical features are centered on atomic-scalesurface defects, and the samedefects produce ringsin all three directions. This suggests that ellipseorientation is not associatedwith symmetry break-ing from the defect itself, which is further con-firmed by atomic-resolution topographs (29). Aswe showbelow, the three different directionalitiesarise from cyclotron orbits in pairs of hole valleysthat are elongated in the same direction. More im-portant, such spatially resolved measurementsenable us to directly visualize the spontaneousbreaking of the LL degeneracy by electron-electroninteractions. By tuning the magnetic field to ad-just the occupancy of two of the three brokensymmetry states, we can contrast spatial maps ofthe LLswith andwithout exchange splitting. Themeasurements in Fig. 2, E to G, obtained in thesame region as those in Fig. 2, A to D, show thatthe elliptical features in the conductancemaps canoccur as a superposition of two different orienta-tions, indicating that the symmetry between thesetwo orientations is not broken in the absence of

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Fig. 4. Energy shift of the cyclotron orbits. (A to I) Spatial maps of G=G around an isolated impurity at B = 10 T with energy spaced by 100 meVthroughout one broken-symmetry Nh = 4 LL peak. These maps show the shift to lower energy of the m = N cyclotron orbit. (J) Correspondingconductance spectrum (averaged over a 12 nm × 2.5 nm area centered about 5 nm underneath the defect) marked with colored circles for each mappedenergy. (K) Oscillations of G=G along the semiminor axis, averaged over 100 and 200 meV (blue) and over 400 and 500 meV (red), respectively,highlighting the contrast reversal in the maps.

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an exchange gap. A comparison of Fig. 2F withFig. 2, B and C, clearly shows that unidirectionalelliptical features emerge as the exchange gapopens, providing a directmanifestation of nematicvalley-polarized states on the Bi(111) surface.Another key feature of a nematic electronic

phase without long-range order is the presenceof domains, which we observe in our system byperforming spatially resolved spectroscopy withthe STM. We find that the sequence in energy ofthe three broken-symmetry hole LL states canchange depending on the location within thesample. An example of this behavior can be seenby contrasting the spectrum and correspondingconductance maps in Fig. 2, A to D, to those mea-sured about 1 mm away, shown in Fig. 2, H to K.These data reveal that the orientations of the twobroken-symmetry states corresponding to the firsttwo peaks in the spectra have switched betweenthe two locations on theBi surface. Thus, our STMmeasurements not only show that electron-electroninteractions drive nematic behavior, but also illus-trate the formation of local nematic domains.We show below that the elliptical features in

our STM conductance maps arise from cyclotronorbit wave functions of the broken-symmetryquantum Hall phases that are pinned by surfacedefects. To characterize these features in detail,we studied them in an area with few surface de-fects (box in Fig. 1A) and examined their depen-dence on orbital index at a constant magnetic field(14 T) around the same defects (circled in Fig. 1A).The conductance maps shown in Fig. 3, A to E,were obtained at the energies of the strain-induced broken-symmetry LLs for Nh = 0 to 4,and they revealed concentric ellipses of suppressedconductance similar to those in Fig. 2, with aconsistent orientation for all the orbital indices.The size of the outermost ring increased with in-creasing orbital index, as did the number of con-centric rings of suppressed conductance. Aroundthese same surface defects, we observed approx-imately circular rings in conductance maps mea-sured at the nearby electron LL peak (Fig. 3F),which further confirms that the defects them-selves do not break rotational symmetry.The rings of suppressed conductance for both

electron and hole LLs can be understood as aconsequence of cyclotron orbits that are shiftedin energy because of the sharp potential producedby the atomic surface defects. In the symmetricgauge, the cyclotron orbits of each LL can belabeled by a second orbital quantum numberm(31, 32). Only them=N cyclotron orbit hasweightat the defect, so it is the only state whose energy isshifted by the defect potential, which wemodeledas a delta function (29).Without the defect, conduc-tance maps measured at the LL peak would in-clude DOS contributions from all cyclotron orbits,and no spatial variation would be expected. How-ever, because the m = N orbit is shifted to adifferent energy by the defect, it becomes visibleas a decreased conductance in the shape of thewave function when measurements are per-formed at the unperturbed LL energy.A theoretical model of cyclotron orbit wave

functions for the surface states of Bi(111) can be

used to capture the elliptical features in the STMconductance maps near individual defects withexcellent accuracy. The anisotropy of the surfacestate hole pockets is reflected in their cyclotronorbitwave function, as exemplified by them=Nh=4 state, whose amplitude 2pl2Bjφ4;4ðrÞj2 (wherelB =

ffiffiffiffiffiffiffiffiffiffiℏ=eB

pis the magnetic length) is plotted

in Fig. 3G. The number of elliptical features in thesewave functions increases with orbital index and is areflection of the spatial oscillations of them = Nh

wave function, which is proportional to a Laguerrepolynomial with Nh + 1 peaks (29). Using the de-fects marked in Fig. 1A as the centers of suchcyclotron orbits, we simulated the expected con-ductance pattern by subtracting 2pl2BjφN;N ðrÞj2from a uniform background (Fig. 3, H to K). Thesimilarity to the experimental data in Fig. 3, B toE, for differentNh states is remarkable, especiallygiven that the only adjustable fit parameter is theanisotropy of the hole pocket effective mass. Weextract a ratio of 5 for the hole pocket anisotropy,in good agreement with previous ARPES mea-surements (23, 26, 27) and calculations (33). Ourmodel also captures the field dependence of thecyclotron orbit size of the hole LL for Nh = 4, aswell as that of the electron LLs near the Fermilevel. Figure 3L shows the experimentally mea-sured size of the outermost rings for both sets oforbits. They follow the expected 1=

ffiffiffiB

por 1/B

scaling for hole and electron LLs, respectively,which reflects the dependence of the cyclotronorbit wave functions on magnetic length andorbital index (29).On the basis of the model described above, we

anticipate that the suppression we have detectedin the conductance maps at the LL peaks shouldbe accompanied by an enhanced conductancerelative to the background at other energies. Anexample of such contrast reversal is shown inFig. 4, A to I, which displays conductance mapsnear an isolated defect over a range of energieswithin one broken-symmetry LLpeakwith orbitalindexNh = 4 (Fig. 4J). The maps measured at theLL peak and at higher energies show ellipses ofsuppressed conductance that correspond to amissing cyclotron orbit, whereas at lower ener-gies, such maps show ellipses of higher conduct-ance that indicate the lower energy to which thisorbit has been shifted by the defect potential.This reversal of the contrast is clearly illustratedby the energy-averaged line cuts shown in Fig.4K, which demonstrate that the cyclotron orbitenergy has been lowered by about 300 mV by thisparticular defect. Examining different defects, wehave found evidence for both attractive and re-pulsive potentials from the contrast reversal in theconductance maps (29).Our measurements are in the clean regime

where signatures of isolated cyclotron orbits arevisible around individual defects, in contrast toprevious studies of DOS modulations from driftstates moving along equipotential lines in thedisordered limit (34–36). Cyclotron orbits thatare shifted in energy by an isolated defect havebeenexplored ingraphene (32), andothermeasure-ments have indirectly probed the size and shapeof cyclotron orbits (36–38) by examining LL spatial

dependence caused by potential modulations. Weperformed direct two-dimensional mapping ofisolated cyclotron orbits, which enabled us to vi-sualize nematic order on theBi(111) surface,wherethe anisotropic hole mass leads to anisotropiccyclotron orbits.The Bi(111) 2DEG represents an interesting

venue to explore electron-electron interactionswithin anisotropic valleys. The ability to bringthe lowest hole-like LL to the Fermi level, eitherby external gating or doping,may allow for directvisualization of fractional quantum Hall statesand Wigner crystallization with a STM. In addi-tion, the boundaries between different nematicdomains are expected to harbor low-energy edgemodes that are analogous to topologically protec-ted states (13). The ability to generate a valley-polarized nematic phase that can be externallytuned with strain makes Bi(111) surface statesideally suited for controlled engineering of aniso-tropic physical properties. The predicted semimetal-to-semiconductor transition with decreasingthickness in bulk Bi (18) means that the transportproperties of thin Bi(111) crystalswill be dominatedby the surface states, yielding further prospects forintegration into devices that exploit the uniquephysical properties reported here.

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ACKNOWLEDGMENTS

We thank D. A. Abanin, S. A. Kivelson, S. A. Parameswaran,S. L. Sondhi, and A. Yacoby for helpful discussions. Work atPrinceton was supported by the Gordon and Betty MooreFoundation as part of the EPiQS initiative (GBMF4530) and bythe U.S. Department of Energy (DOE) Office of Basic Energy Sciences.Facilities at the Princeton Nanoscale Microscopy Laboratory aresupported by NSF grant DMR-1104612, NSF-MRSEC programs throughthe Princeton Center for Complex Materials (DMR-1420541, LPS andARO-W911NF-1-0606), ARO-MURI program W911NF-12-1-0461, andthe Eric and Wendy Schmidt Transformative Technology Fund atPrinceton. Also supported by a Dicke fellowship (B.E.F.), a NSFGraduate Research Fellowship (M.T.R.), and DOE Division of Materials

Sciences and Engineering grant DE-FG03-02ER45958 and Welchfoundation grant F1473 (F.W. and A.H.M.).

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/354/6310/316/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S8Tables S1 and S2References (39–43)

18 May 2016; accepted 23 September 201610.1126/science.aag1715

GROUP DYNAMICS

Network science on belief systemdynamics under logic constraintsNoah E. Friedkin,1* Anton V. Proskurnikov,2,3 Roberto Tempo,4 Sergey E. Parsegov5

Breakthroughs have been made in algorithmic approaches to understanding howindividuals in a group influence each other to reach a consensus. However, whathappens to the group consensus if it depends on several statements, one of whichis proven false? Here, we show how the existence of logical constraints on beliefsaffect the collective convergence to a shared belief system and, in contrast,how an idiosyncratic set of arbitrarily linked beliefs held by a few may becomeheld by many.

Converse [(1), p. 207] defined a belief sys-tem as “a configuration of ideas and at-titudes in which the elements are boundtogether by some form of constraint orfunctional interdependence.” The existence

of belief systems is widely accepted and a subjectof interest in the scientific community (2–4), butthere are still unresolved puzzles. According tocognitive consistency theory, inconsistent beliefscause tension that individuals seek to resolve(5, 6). Thus, if an individual’s certainty of beliefon the truth of one statement is altered, the al-teration may propagate changes of the individ-ual’s certainties of beliefs on the truth of otherstatements. Individual-level, independent adjust-ments of certainties of belief (7–14) do not suf-fice to explain the existence of shared beliefs ina population of individuals. Some additional,natural, social control and coordination mecha-nism is required. Public dispute on global warm-ing is a prominent case in which individualshave varying certainties of belief on the truthvalues of a logically interdependent set of state-ments, which has implications for reaching a

conclusion that collective action is required tomitigate global warming. Debates in econom-ics on appropriate macroeconomic policy, anddebates in politics on acceptable legislation, arealso examples of interpersonal influences modify-ing individuals’ certainties of belief on multipleinterdependent statements. A critical open prob-lem is the theoretical integration of theory oncognitive consistency and theory on interpersonalinfluence systems. We report a generalization ofthe Friedkin-Johnsen model (15–17) that achievesthis integration. When individuals’ beliefs on mul-tiple statements are being influenced, the Friedkin-Johnsen model assumes that a change of be-lief on one statement does not affect beliefs onother statements. We develop and apply a morerealistic model on the dynamics of belief sys-tems in which individuals’ certainties of beliefon a set of interdependent true or false state-ments are being changed by network mecha-nisms of interpersonal influence.A shared logic constraint structure on a set

of truth statements (e.g., if X is true, then Yand Z are true) does not imply belief consensus.It will polarize a population into two opposingideological factions when high certainty of be-lief on one central statement implies high cer-tainties of belief on all other statements, andlow certainty of belief on that central statementimplies low certainties of belief on all other state-ments. One faction accepts the premise of thecentral statement and thus accepts all the otherstatements as true; the other rejects the premiseof the central statement and thus rejects allthe other statements as false. How can we bet-

ter understand the dynamics of belief systemsin which individuals’ certainties of belief are mod-ified by network mechanisms of interpersonalinfluence toward a consensus on a set of inter-dependent beliefs?An analyzable problem on belief system dy-

namics can be posed as follows. Let us startfrom a state of heterogeneity in a population ofindividuals (i) with various levels of certainty ofbelief on the truth values of two or more truthstatements and (ii) with a common set of logicalconstraints that associate these statements. Inthis population, levels of certainty of belief aboutone statement are associated with levels of cer-tainty of belief about another statement and,more generally, an individual’s level of certaintyof belief about one statement is some mixtureof that individual’s certainty of beliefs aboutother statements. Let each individual’s certaintyabout each statement be subject to disturbance.Cognitive consistency theory posits that the dis-turbance will cause a within-individual changethat recalibrates their certainties of beliefs toachieve consistency. Let each individual in thispopulation be embedded in a social networkthat allows interpersonal influences on individ-uals’ beliefs. With such a network, cognitive con-sistency effects are now competing with effectsof other individuals’ displayed beliefs.In our model (Fig. 1), individual nodes have

different certainties of belief on multiple truthstatements, which may be changed throughtheir interactions with others. The nodes mayvary in their levels of closure-openness to influ-ence. Each node’s integration of their own andothers’ displayed certainties of belief may besubject to logical interdependencies among state-ments. These interdependencies can be expressedas a matrix of logic constraints.The dynamics of this n-individual belief sys-

tem on m truth statements is defined by thetensor matrix equation (18)

Xðkþ 1Þ ¼ AWXðkÞCT þ ðI − AÞXð0Þ

where k ¼ 0; 1;…. The Xð0Þ is a n�m matrixof n individuals and m truth statements withtruth values (true or false) on which individ-uals have heterogeneous certainties of beliefin the ½0; 1� interval, such that xij ¼ 0:50 cor-responds to an i with maximum uncertaintyon the truth value of statement j of the mstatements; xij ¼ 1 corresponds to an i with

SCIENCE sciencemag.org 21 OCTOBER 2016 • VOL 354 ISSUE 6310 321

1Center for Control, Dynamical-Systems and Computationand Department of Sociology, University of California, SantaBarbara, CA, USA. 2Engineering and Technology Institute(ENTEG), University of Groningen, Netherlands. 3TheInstitute of Problems of Mechanical Engineering of RussianAcademy of Sciences (IPME RAS) and ITMO University, SaintPetersburg, Russia. 4National Research Council (CNR-IEIIT),Torino, Italy. 5V.A. Trapeznikov Institute of Control Sciencesof the Russian Academy of Sciences (ICS RAS), Moscow,Russia.*Corresponding author. Email: friedkin@soc.ucsb.edu

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Observation of a nematic quantum Hall liquid on the surface of bismuth

YazdaniBenjamin E. Feldman, Mallika T. Randeria, András Gyenis, Fengcheng Wu, Huiwen Ji, R. J. Cava, Allan H. MacDonald and Ali

DOI: 10.1126/science.aag1715 (6310), 316-321.354Science 

, this issue p. 316Sciencereflected in the orientations of the electrons' elliptical orbits.placed in an external magnetic field. The exchange interactions in the material caused a loss of symmetry, which was

used scanning tunneling microscopy to image the wave functions of electrons on the surface of bismuthet al.Feldman number of exotic materials. However, establishing a direct connection between the interactions and nematicity is tricky.crystal lattice. This loss of symmetry, caused by interactions and dubbed electronic nematicity, has been observed in a

The electronic system in a strongly correlated material can sometimes be less symmetrical than the underlyingRelating interactions and nematicity

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REFERENCES

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