F. Duncan M. Haldane Geometry of "Flux Attachment"
CESC Cargèse, Corsica, France July 2017
Geometry of "flux attachment" in the fractional quantum Hall effect.
F. Duncan M. Haldane Princeton University
• Flux attachment as emergent gauge-field with geometry • Geometry and energy of the “flux attachment” that forms “composite particles”
Supported by DOE DE-SC-0002140 and W. M. Keck Foundation
Thursday, August 3, 17 • landau level geometry • laughlin state, unsolved problems • why is QHE (gapped 2+1) well represented by 2d cft? • status of projector models with conformal block ground states
Thursday, August 3, 17 • The quantum Hall fluid in a Landau level on the flat plane
H = "(pi)+ V (xi xj) ˆ n i i
Thursday, August 3, 17 H = "(p )+ V (x x ) i i j i i a b a a [x ,x ]=0 [x ,pb]=i~ b [pa,pb]=i~eB✏ab Kronecker symbol (not metric) • symmetries: spatial translation symmetry • spatial inversion symmetry plays a fundamental role in the QHE: we will impose "(p)="( p) • full O(2) rotational symmetry defined by a metric is not a fundamental symmetry of the QHE, and will not be required. Thursday, August 3, 17 H = "(pi)+ V (xi xj) "(p)="( p) i i a b a a [x ,x ]=0 [x ,pb]=i~ b [pa,pb]=i~eB✏ab spatial translation and xi xi + c • p 7! ±p inversion symmetries i 7! ± i • decomposition of the 2D Heisenberg algebra Landau orbit radius vector ¯a ¯b ab 2 2 ⇡` 2 is the [R , R ]=i✏ `B B ¯a 1 ab area per R =(eB) ✏ pb [Ra,Rb]= i✏ab`2 B quantum of Landau orbit guiding center [Ra, R¯b]=0 magnetic flux Ra = xa R¯a 0 = h/e Thursday, August 3, 17 H = "(pi)+ V (xi xj) "(p)="( p) i i a a ¯a ¯a ¯b ab 2 xi = Ri + Ri [R , R ]=i✏ `B a b ab 2 b [R ,R ]= i✏ ` pai = (eB)✏abR¯ B i [Ra, R¯b]=0 • will also specify that �(p) = �(p1,p2) is an entire function of each component of p (e.g. a polynomial) and • has a unique minimum at p = 0 with no other stationary points (this ensures a simple Landau-level structure) Thursday, August 3, 17 • semiclassical Landau quantization • like phase space [px,py]=i~eB py There is no reason that Landau orbits with different px index n should be congruent (have the same shape) semiclassical Landau levels are localised on contours of constant "(p) 1 that enclose momentum-space area 2⇡~eB(n + 2 ) (analog of Bohr-Sommerfeld quantization) Thursday, August 3, 17 • Quantum treatment: [px,py]=i~eB 1 2 2 usual model h = (px) +(py) (harmonic oscillator, separable) 2m generic model h = "(px,py) (bivariate non-separable function of non-commuting coordinates) • To be well-defined, �(p) must have an absolutely convergent expansion in p - p0 for all (c-number) p0 , so must be an entire function of both px and py, e.g. a polynomial, Thursday, August 3, 17 • macroscopic degeneracy of Landau levels [Ra, "(p)] = 0 "(p) = E | n↵i n| n↵i basis within degenerate Landau level • one independent state for each quantum of magnetic flux through the plane Thursday, August 3, 17 • Coherent states of a Landau orbit are defined by a guiding-center metric • The coherent state centered at the origin is defined by "(p) n(0, g˜) = En n(0, g˜) g˜ RaRb| (0, g˜)i = p(det| g ˜)(` )i2 (0, g˜) ab | n i B | n i • we can always choose the metric to be unimodular (determinant = 1), as it just defines a complex structure z(x,y) coherent states minimize the uncertainty of the guiding center Thursday, August 3, 17 • factorization of a metric to define a complex structure 1 2 (˜gab + i✏ab)=ea⇤eb detg ˜ =1 a complex vector example a zg˜(x, y)=eax /`B g˜ab = ab 1 a unimodular metric defines a e = p2 (1,i) (dimensionless) complex structure (up to a U(1) ambiguity) (Euclidean metric) z ei z 7! Thursday, August 3, 17 • The Schrödinger wavefunction of a Landau level coherent state defined by a metric has the form 1 z⇤z n(x, y)=fn(z⇤;˜g)e 2 a holomorphic function of z* that depends of the choice of guiding-center metric to define the coherent state • so far, we have complete freedom of choice to choose this metric: is there a “natural choice”? Thursday, August 3, 17 • Yes, the natural choice of metric is provided by the “Hall viscosity tensor” of the Landau orbit: 1 (n) n↵ pa,pb n↵ =2⇡~⌘ h | 2 { }| i ab • viscosity is the linear relation between stress and flow-velocity gradient. Stress is a mixed-index tensor (momentum current density) which is traceless in gapped incompressible quantum fluids a ac d ac ae bf H ac ca = ⌘ @ v ⌘bd = ✏ ✏ ⌘ be df ⌘bd = ⌘db b bd c { }{ } stress velocity gradient Traceless condition dissipationless H 1 ⌘ ab cd = 2 (⌘ac✏bd + ⌘ad✏bc + ⌘bc✏ad + ⌘bd✏ac) { }{ } rank-2 symmetric Hall- rank-4 tensor viscosity tensor odd under time- odd under time- even under time-reversal reversal reversal Thursday, August 3, 17 • every Landau level has a “natural metric” proportional to its Hall viscosity tensor, which characterises its shape • It also has a natural effective mass tensor defined by its reponse to polarization: (v)=v p "(p) Lagrangian operator L · (v) (v) = L (v) (v) L | n↵ i n | n↵ i L (v)= E + 1 m(n)vavb + O(v4) n n 2 ab • finally, it also has a topological spin Thursday, August 3, 17 • special case, rotationally invariant Landau levels: ab ["(p), g˜ papb]=0 1 n z⇤z general n, coherent state n(x, y) (z⇤) e 2 / 1 z⇤z n=0, coherent state (x, y) e 2 0 / 1 z⇤z (x, y)=f(z)e 2 n=0, general state This structure is non-generic, only applies to case where �(p) has rotational symmetry. Thursday, August 3, 17 • Filled LLL (rotational symmetry) 1 z⇤z = (z z ) e 2 i i i j i inspiration for Laughlin 1 (m) m z⇤z = (z z ) e 2 i i L i j i Thursday, August 3, 17 • quartic polynomial case Figure 2.7: Contour plots in the momentum space of a typical member in each class 2 2 2 2 2 2 2 of the quartic term. The quartic terms are I) 5p1 + p2,p1 +5p2 , II) p1,p1 + p2 , III) p2,p2 and IV) p4. Notice that only the contours{ for class I} are closed.{ } { 1 2} 1 Thursday, August 3, 17 33 2 Figure 3.5: Contour plots of ln 10(z,z⇤) where 10(z,z⇤)isthecoherentstatein the 10th Landau level for class| I quartic| term (2.45)withc 1=4> 2(top)and c 1=1< 2 (bottom). Both plots show piece-wise contours with the four spikes and the line charges as branch cuts. Another common feature is the existence of four maxima along the directions of the central cross and four saddle points along those of the spikes. Despite the similarities, the two plots also show qualitatively di↵erent Thursday, August 3, 17 shapes of the semiclassical orbits. We see that c 1 > 2correspondstoconcave shapes while c 1 < 2 to convex ones. 44 Figure 3.6: A “ridge” (dashed) defined as the boundary between two regions where the gradient field flows to the origin (inside) or infinity (outside). The roots are those of the coherent state in the 10th Landau level for class I quartic term with c 1=2. Contours are also shown. Notice that the local maxima and the saddle points are all right on the ridge. The area enclosed by the ridge is ⇡n, corresponding to an area of 2⇡~eB on the momentum plane, which is consistent with (shifted) semiclassical quantization. turbation that breaks rotational invariance, the new zeros will only appear where the original wavefunction has a small amplitude, namely, around the origin or along the Gaussian tail separated by the peak in between which contains most of the weight of the wave function. In Fig.(3.7)weshowthatthepeakregioncanencompassasmuch Thursday, August 3, 17 as 90 percent of the total weight. The existence of a clean separation between the central zeros and the rest of the pattern by an annulus-like region, or more rigorously a region with Euler characteristic + =0,definedby n V,V R , is critical for our definition of a topological spin, | | 2 45 • The effective continuum Hamiltonian is H = "(p )+ V (x x ) i i j i i Thursday, August 3, 17 Generic model with translation and inversion symmetry only, no rotational symmetry H = "(p )+ V (x x ) "(p)="( p) i i j i i shape of Landau orbit around equipotentials around point charge guiding center (from 3D dielectric tensor) Thursday, August 3, 17 • The “holomorphic lowest Landau level wavefunction” is a property of a SO(2) rotationally-invariant system: x = R + R˜ [Ra,Rb]= i`2 ✏ab B ˜a ˜b 2 ab e [R , R ]=i`B✏ R˜ x [Ra, R˜b]=0 x angular momentum R ~ a b ˜a ˜b O L = 2 ab(R R R R ) 2`B 1 = ~ a†a b†b 2 Tw o sets of ladder operators: guiding center Landau level Thursday, August 3, 17 • Now write the Laughlin state as a Heisenberg state, not a Schrödinger wavefunction: x y m R + iR L (a† a†) 0 a 0 =0 a = | i/ i j | i i † i b 0 =0 lowest Landau level condition i| i In the Heisenberg form, we see that the LLL condition is quite incidental to the Laughlin state, which involves guiding-center correlations Thursday, August 3, 17 • The fundamental form of the Laughlin state does not reference the details of the Landau level in any way: a !aR m a† = L(˜g) (ai† aj†) 0 ai 0 =0 | i/ | i | i p2`B i Thursday, August 3, 17 • Topological states of matter have been a major theme in the recent developments in understanding novel quantum effects. • key questions are: why do they occur,what features of materials favor such states, and how can we understand the energetics that drives their emergence. • I will principally discuss the fractional quantum Hall effect, but this is a general question Thursday, August 3, 17 • thirty years after its experimental discovery and theoretical description in terms of the Laughlin state, the fractional quantum Hall effect remains a rich source of new ideas in condensed matter physics. • The key concept is “flux attachment” that forms “composite particles” and leads to topological order. • Recently, it has been realized that flux attachment also has interesting geometric properties Thursday, August 3, 17 • I will talk about what has been interesting me for the last few years: • What is an “incompressible quantum fluid” such as the one described by Laughlins wavefunction. • What “fluid dynamics” describes it? Thursday, August 3, 17 2 d q iq (x x ) H = "(p )+ V˜ (q) e · i j i 2⇡ i i Thursday, August 3, 17 • Girvin-Macdonald-Platzman Lie algebra N iq R ⇢(q)= e · i i=1 X 1 2 [⇢(q), ⇢(q0)] = 2i sin( q q0` )⇢(q + q0) 2 ⇥ B • q = 0 generator = N, is in kernel. d2q`2 H = V˜ (q)( 1 ⇢(q)⇢( q)) 2⇡ n 2 Z rapidly-decreasing (Gaussian) function at large q (Fourier transform of an entire function) Thursday, August 3, 17 • “compactification” on the torus L mL + nL L L =2⇡N `2 2 { 1 2} 1 ⇥ 2 B Bravais Lattice of periodic translations • reciprocal lattice (discrete set of allowed wavevectors) iq L e =1 a ea eb = ab · L = L ea · Euclidean metric b ✏abL 2 N qa`B iq q0` 2 N ` e ⇥ B =1 ⇢ B ⇣ ⌘ • reciprocal vectors q1 and q2 are equivalent if q q = N q 1 2 2 • There are ( N ) distinct reciprocal vectors in a “Brillouin zone” q =0 1 q • a reciprocal vector q is even if 2 is also an allowed reciprocal vector Thursday, August 3, 17 distinct reciprocal vectors in a • the inversion-symmetric pbc is iq R N N e · i = ⌘(q) | i | i ⌘(q)=1if q is even, 1 if not. • why is there a Brillouin zone? • The pbc means the allowed translations compatible with the pbc are L/N • In fact, we can with full generality work only on this lattice. naive formula based in the dz dz⇤ z⇤z idea that these are 1 2 = 2^⇡i (f1(z))⇤f2(z)e h | i Schrödinger wavefunctions 1 0 z⇤z = R (f (z))⇤f (z)e N 1 2 x X sum over x = L/N in the unit cell Thursday, August 3, 17 ¯ ¯ gcd(p, q)=1 • for N = p N, N = qN q N q iq R pq iQ q`2 e · i =(⌘(q)) e ⇥ B | ↵i | ↵i i=1 ! Y ⇣ ⌘ many-body translation quantum Q number takes ( N ¯ ) 2 distinct values If N ¯ is even (odd), one (four) of these have inversion symmetry ↵ is a q-fold exact topological degeneracy Thursday, August 3, 17 many-body Q lives in a “Brillouin zone” of N¯ = gcd(N,N ) points ¯ N¯ even N odd • quantum Hall states always have inversion- symmetric Q. • If they occur only at Q =0, the elementary droplet has p particles with flux attachment q • If they occur on the zone boundary, p and q must be doubled and N ¯ halved (e. g., Moore- Read state) p/q =1/2 2/4 The physical FQH (as opposed to algebraic)! p, q obey gcd(p, q) 2 Thursday, August 3, 17 0.9 laughlin 10/30 0.8 ތࠓ goes into continuum 0.7 0.6 (2 quasiparticle 0.5 bosonic “roton” E 0.5 + 2 quasiholes) 0.4 fermionic 0.3 “roton” “roton” 0.2 2 Moore-Read “kF ” ⌫ = 0.1 4 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2k k lB gap incompressibility B Collective mode with short-range V1 Collective mode with short-range three-body pseudopotential, 1/3 filling (Laughlin state is pseudopotential, 1/2 filling (Moore-Read state is exact ground state in that case) exact ground state in that case) momentum ħk of a quasiparticle-quasihole pair is • b proportional to its electric dipole moment pe ~ka = abBpe gap for electric dipole excitations is a MUCH stronger condition than charge gap: doesn’t transmit pressure! (origin of Virasoro algebra in FQHE ?) Thursday, August 3, 17 ⇢(q) = N⌘(q)N P expectation with a h i q,0 translationally- invariant = ⌫N ⌘(q)N P inversion-symmetric density q,0 matrix 1 P 0 iq (q1 q2) = e ⇥ q1,q2 (N )2 = 1 is q1 and q2 are equivalent, q = 0 if not X ⇢(q)=⇢(q) N⌘(q)N P q,0 • Guiding-center structure function (2-point function) ⇢(q) ⇢( q) = N Sgc(q) S = ⌫(1 + ⇠⌫) h i 1 1 for fermions/bosons ± • for an uncorrelated state P Sgc(q) S =0 ( q,0 = 0) 1 Thursday, August 3, 17 • for an completely uncorrelated (mixed) state P Sgc(q) S =0 ( q,0 = 0) 1 • for a correlated pure state S gc ( q ) S is a rapidly-decreasing function away from 1the center of the (geometric) Brillouin zone (defined by a metric) lim 0 Sgc( q)=0 ! • for gapped FQH state (topologically degenerate multiplet) 4 ab , cd 2 lim 0 Sgc( q) {{ } { }}qaqbqcqd`B ! ! First discovered by GMP ab , cd {{ } { }} is positive, [ ab , cd ] satisfies a bound involving the Hall viscosity tensor H{ } { } This bound seems to be saturated in conformal-block models Thursday, August 3, 17 • For a (commutative) 2D liquid with density n0 1 iq (r r ) S(q)= e · i j N h i=j X6 2 1 d q iq r g(r) 1= e · (S(q) 1) 2⇡n 2⇡ 0 Z pair correlation standard structure function factor For the guiding-center liquid, there is a self-duality • 2 2 2 d q0`B iq q0` Sgc(q) S = ⇠ e ⇥ B (Sgc(q0) S ) 1 2⇡ 1 Z 1 0 iq (Ri Rj ) P = e · ij N q 1 (fermion/boson) X Thursday, August 3, 17 ± • rotationally-invariant states have a global metric, are eigenstates of 1 L = g˜ RaRb 2`2 ab i i B i X 2 ab • Sgc(q) is a function of q =˜g qaqb analytic on real-q2 axis • conjecture: for conformal block states Sgc(q) is an entire function of q2 Thursday, August 3, 17 Sgc(q) S 1 O( q 4) | | q | | Thursday, August 3, 17 1 H = 1 2ln( + 1 z 2 2 0 z z | 2 | i| 1 i Thursday, August 3, 17 Becomes a • electron in 2D Landau orbit “fuzzy object” (bound to 2D surface) after kinetic energy is quantized cyclotron e- magnetic flux orbit density B normal x to 2D surface = x guiding center 1 ~ 2 R `B = eB ✓| |◆ [Rx,Ry]= i`2 B non-commutative geometry Thursday, August 3, 17 1 3 z⇤z = (z z ) e 2 i i Laughlin 1983 i j i 30 years later: my answer: unanswered question: hidden geometry we know it works, but why? Thursday, August 3, 17 some widespread misconceptions about the Laughlin state No Landau level was specified: all • “it describes particles in thespecifics of the Landau level are lowest Landau level” hidden in the form of U(r12) “It is a Schrödinger Non-commutative geometry has no • Schrödinger representation (this wavefunction” requires classical locality); it only has • “Its shape is determined by a Heisenberg representation. the shape of the Landau The interaction potential U(r12) orbit” determines its geometry (shape) • “It has no continuously- tunable variational Its geometry is a continuously- parameter” variable variational parameter Thursday, August 3, 17 gab = 1 (Ra Ra), (Rb Rb) inverse 2 h |{ i j i j }| i metric Fundamental symmetries of the incompressible quantum fluid H = U(R R ) [Ra,Rb]=i✏ab`2 i j i i B i Thursday, August 3, 17 •Anatomy of Laughlin state electron with “flux attachment” Chiral edge mode with chiral anomaly to form a “composite and Virasoro anomaly boson” - geometric e edge dipole moment - determined by Hall e Text viscosity fractionally-charged Topological and geometric bulk properties e/3 quasiholes obeying revealed by entanglement spectrum of cut (Abelian) fractional statistics Thursday, August 3, 17 • the essential unit of the 1/3 Laughlin state is the electron bound to a correlation hole corresponding to “units of flux”, or three of the available single- particle states which are exclusively occupied by the particle to which they are “attached” • In general, the elementary unit of the FQHE fluid is a “composite boson” of p particles with q “attached flux quanta” • This is the analog of a unit cell in a solid.... Thursday, August 3, 17 • The Laughlin state is parametrized by a unimodular metric: what is its physical meaning? correlation holes in two states with different metrics • In the ν = 1/3 Laughlin state, each electron sits in a correlation hole with an area containing 3 flux quanta. The metric controls the shape of the correlation hole. • In the ν = 1 filled LL Slater-determinant state, there is no correlation hole (just an exchange hole), and this state does not depend on a metric Thursday, August 3, 17 • quantum solid • unit cell is correlation hole • defines geometry • repulsion of other particles make an attractive potential well strong enough to bind particle solid melts if well is not strong enough to contain zero-point motion (Helium liquids) Thursday, August 3, 17 but no broken symmetry • similar story in FQHE: • “flux attachment” creates correlation hole • defines an emergent e- geometry • potential well must be strong enough to bind electron • continuum model, but similar physics to Hubbard • new physics: Hall viscosity, model geometry...... Thursday, August 3, 17 3 3 composite boson: if the central = a† a† 0 • | Li i j | i orbital of a basis of eigenstates of i Thursday, August 3, 17 • Origin of FQHE incompressibility is analogous to origin of Mott-Hubbard gap in lattice systems. • There is an energy gap for putting an extra particle in a quantized region that is already occupied • On the lattice the “quantized region” is an atomic orbital with a - fixed shape e • In the FQHE only the area of the “quantized region” is fixed. energy gap prevents additional electrons The shape must adjust to from entering the minimize the correlation energy. region covered by the composite boson Thursday, August 3, 17 1/3 Laughlin state If the central orbital is filled, the next two are empty The composite boson has inversion symmetry e about its center It has a “spin” 1 3 5 2 2 2 1 1 0 0 ..... L = 2 1 1 1 L = 3 − 3 3 3 ..... − 2 s = 1 the electron excludes other particles from a region containing 3 flux quanta, creating a potential well in which it is bound Thursday, August 3, 17 2/5 state 1 3 5 2 2 2 1 1 0 0 0 ..... L =2 e e 2 2 2 2 2 ..... − L =5 − 5 5 5 5 5 s = 3 g L = ab RaRb 2`2 i i B i X ab 2 a b 2 ab Q = d rr r ⇢(r)=s`Bg Z second moment of neutral composite boson charge distribution Thursday, August 3, 17 hopping of a “composite fermion” (electron + 2 flux quanta) e e e e e e 2/5 boson is quasiparticle of 1/3 state 1/3 boson is quasihole of 2/5 state Jain’s “pseudo Landau levels” 2/5 1/3 2/5 1/3 2/5 Thursday, August 3, 17 • The composite boson behaves as a neutral particle because the Berry phase (from the disturbance of the the other particles as its “exclusion zone” moves with it) cancels the Bohm-Aharonov phase • It behaves as a boson provided its statistical spin cancels the particle exchange factor when two composite bosons are exchanged p particles ( 1)pq =( 1)p fermions q orbitals ( 1)pq =1 bosons Thursday, August 3, 17 • The metric (shape of the composite boson) has a preferred shape that minimizes the correlation energy, but fluctuates around that shape E (distortion)2 • The zero-point fluctuations of the metric are seen as / the O(q4) behavior of the “guiding-center structure factor” (Girvin et al, (GMP), 1985) • long-wavelength limit of GMP collective mode is fluctuations of (spatial) metric (analog of “graviton”) FDMH, Phys. Rev. Lett. 107, 116801 (2011) Thursday, August 3, 17 • An “intrinsic metric” measures lengths in dimensionless units, like unit cells in a solid • to describe the “intrinsic metric tensor” we need a coordinate system • It could be the Euclidean Laboratory frame, but doesn’t have to be! Thursday, August 3, 17 Laughlin’s model wavefunction has provided the inspiration for the modern understanding of the fractional quantum Hall effect 1 2 m z⇤z /` (z z ) e 4 i i B / i j i Thursday, August 3, 17 I will give a somewhat heretical reinterpretation of the Laughlin state • Despite what Laughlin told us, its holomorphic structure has nothing to do with the electrons being in the “Lowest Landau Level” • It should not be regarded as a “wavefunction”, but as a Heisenberg state of guiding centers, which obey a “quantum geometry” • It was proposed as a “trial wavefunction” with no apparent variational parameter: it does in fact have such a parameter: its metric. Thursday, August 3, 17 • Perhaps one of the most surprising (and very fruitful) aspects of the Laughlin state is its connection to conformal field theory. • Its “conformal block” property was noticed as an empirical observation, but has never really been explained. • Incompressible (bulk) FQHE states are essentially unlike gapless cft’s (the conformal group here is the “(2+0)d” conformal orthogonal group, not the “(1+1)d” Lorentz variant) Thursday, August 3, 17 • The conformal orthogonal group CO(2) is a profound local extension of the global SO(2) rotation group (that can be regarded as “the rotation group on steroids” !) • non-generic “Toy models” with CFT properties are particularly simple to treat, because the CFT makes their generic topological properties easy to expose, but the topological properties do not require conformal invariance • I will argue that SO(2) rotational invariance is a “toy model” feature that should not be part of a fundamental theory of the FQHE, just as the SO(3) and Galilean invariance of the free electron gas should not be part of the theory of metals. Thursday, August 3, 17 • The “standard model” for the QHE is usually taken to be the Galileian-invariant @ Newtonian-dynamics model pa = i~ eAa(x) @xa 1 e2 1 H = abp p + 2m ia ib 4⇡✏ ✏ d(x , x ) i i Euclidean metric of 2D plane (derived from the spatial metric of an inertial frame in which the plane on which the electrons move non-relativistically is embedded) a Cartesian coordinates x = x ea e e = a · b ab Thursday, August 3, 17 • However, the continuous translational symmetry “plane” on which the electrons move is an emergent symmetry of a low-density of electrons moving on a crystal lattice plane, and generically does NOT have the rotational invariance of Newtonian dynamics • The only generic point symmetry of a crystal plane is 2D inversion (180o rotation in plane) 2D plane of epitaxial quantum well embedded in 3D crystal Thursday, August 3, 17 • The effective continuum Hamiltonian is H = "(p )+ V (x x ) i i j i i Thursday, August 3, 17 Generic model with translation and inversion symmetry only, no rotational symmetry H = "(p )+ V (x x ) "(p)="( p) i i j i i shape of Landau orbit around equipotentials around point charge guiding center (from 3D dielectric tensor) Thursday, August 3, 17 • The “holomorphic lowest Landau level wavefunction” is a property of a SO(2) rotationally-invariant system: x = R + R˜ [Ra,Rb]= i`2 ✏ab B ˜a ˜b 2 ab e [R , R ]=i`B✏ R˜ x [Ra, R˜b]=0 x angular momentum R ~ a b ˜a ˜b O L = 2 ab(R R R R ) 2`B 1 = ~ a†a b†b 2 Tw o sets of ladder operators: guiding center Landau level Thursday, August 3, 17 • Now write the Laughlin state as a Heisenberg state, not a Schrödinger wavefunction: x y m R + iR L (a† a†) 0 a 0 =0 a = | i/ i j | i i † i b 0 =0 lowest Landau level condition i| i In the Heisenberg form, we see that the LLL condition is quite incidental to the Laughlin state, which involves guiding-center correlations Thursday, August 3, 17 • The fundamental form of the Laughlin state does not reference the details of the Landau level in any way: a !aR m a† = L(˜g) (ai† aj†) 0 ai 0 =0 | i/ | i | i p2`B i Thursday, August 3, 17 • The original form of the Laughlin state is a finite-size droplet of N particles on the infinite plane. • Somewhat confusingly, in this droplet state the metric parameter fixes both the shape of the droplet state and the shape of the correlation hole around each particle formed by “flux attachment”: e correlation hole edge of droplet Thursday, August 3, 17 • to remove the edge, compactify on the torus with NΦ flux quanta: • An unnormalized holomorphic single- particle state has the form N N = (a† w ) 0 , w =0 | i i | i i i=1 i=1 Y X Weierstrass sigma function (z)=z (1 z )exp(z + 1 ( z )2) L L 2 L L=0 Y6 Filled Landau level N = N = ( a†) (a† a†) 0 | filledLLi i i i j | i i Thursday, August 3, 17 • Laughlin state on torus (⌫ =1/m, m > 1) m m m (˜g) ( a† w ) (a† a†) 0 | L i/0 i i j 1 i j | i j=1 i Thursday, August 3, 17 • The Laughlin state is indeed a variational trial state, we must choose its metric to minimize the correlation energy ˜ 2 1 V (q) fn(q) iq (R R ) H = | | e · i j N 2⇡`2 q B ! i Thursday, August 3, 17 • The Laughlin states are also the exact zero- energy ground states of the metric- dependent “pseudopotential” interaction 1 2 2 1 q2`2 iq (R R ) H(˜g)= V L (q ` )e 2 B e · i j N m0 m0 B q ! i Thursday, August 3, 17 Degenerate (flat) Landau levels E • H = U(Ri Rj) empty i effective Coulomb repulsion is analytic at origin because of smoothing by Landau- orbit form factor Thursday, August 3, 17 This is the entire problem: nothing other than this matters! H = U(R R ) • H has translation and i j i [(Rx Rx), (Ry Ry)] = 2i`2 gap 1 2 1 2 B • relative coordinate of a pair of particles behaves like a single particle two-particle energy levels Thursday, August 3, 17 Solvable model! (“short-range pseudopotential”) E2 symmetric 1 • 2 (A + B) 1 (r )2 antisymmetric B 2 12 2 (r12) 2`2 U(r12)= A + B 2 e B rest all 0 `B 0 ⇣ ⇣ ⌘⌘ Laughlin state • • m=2: (bosons): all pairs m avoid the symmetric state m E2 = ½(A+B) = a† a† 0 | L i i j | i i Thursday, August 3, 17 • Furthermore, the local electric charge density of the fluid with ν = p/q is determined by a combination of the magnetic flux density and the Gaussian curvature of the metric e peB J 0(x)= sK (x) e 2⇡q g ✓ ~ ◆ Topologically quantized “guiding center spin” Gaussian curvature of the metric Thursday, August 3, 17 • In fact, it is locally determined, if there is an inhomogeneous slowly-varying substrate potential H = v (R )+ V (R R ) n i n i j i i deformation y near edge vn(x) x Thursday, August 3, 17 • “skyrmion”-like “cone”-like structure moves charge away from quasihole by introducing negative Gaussian curvature fluid density distance from center in an effective theory, core of quasihole may collapse into a cone singularity of the metric. Thursday, August 3, 17 One final result • In the “trivial” non-topologically-ordered integer QHE (due to the Pauli principle) c˜ = ⌫ =Chernnumber c˜ ⌫ =0 • the (guiding-center) “orbital entanglement spectrum” of Li and Haldane is insensitive to filled (or empty) Landau levels or bands, and allows direct determination of non-zero c˜ ⌫ previous methods used the onerous calculation of the“real-space” entanglement spectrum to find c˜ Thursday, August 3, 17 "laughlin3_n_13.cyl16.14" using 2:4 140 ORBITAL CUT 120 100 80 signed conformal 60 anomaly (chiral stress- chiral energy anomaly) anomaly 40 “CASIMIR MOMENTUM” term (NOT “real-space cut” which requires 20 the Landau orbit degrees of freedom and their form factor to be included virasoro level of sector 0 0 5 10 15 20 25 30 35 40 • Hall viscosity gives “thermally excited” momentum density on entanglement cut, relative to “vacuum”, at von Neumann temperature T = 1 Thursday, August 3, 17 Yeje Park, Z Papic, N Regnault 1 1 1 1 (˜c ⌫)= 1 3 = 24 0.1 24 36 L_A = 148, plevel = 12, +1/6 0.08 L_A = 149, plevel = 12 L_A = 150, plevel = 12, +1/6 0.06 L_A = 149, plevel = 11 1/36 0.04 1 0.02 0 36 -0.02 -0.04 mean Virasoro level - expected -0.06 0 100 200 300 400 500 600 700 800 900 L^2 Matrix-product state calculation on cylinder with circumference L (“plevel” is Virasoro level at which the auxialliary space is truncated) Thursday, August 3, 17 • In a 2D Landau level, we apparently start from a Schrödinger picture, but end with a “quantum geometry” which is no longer correctly described by Schrödinger wavefunctions in real space because of “quantum fuzziness” (non-locality) • It remains correctly described by the Heisenberg formalism in Hilbert space. Thursday, August 3, 17 • the essential unit of the 1/3 Laughlin state is the electron bound to a correlation hole corresponding to “units of flux”, or three of the available single- particle states which are exclusively occupied by the particle to which they are “attached” • In general, the elementary unit of the FQHE fluid is a “composite boson” of p particles with q “attached flux quanta” • This is the analog of a unit cell in a solid.... Thursday, August 3, 17 • The Laughlin state is parametrized by a unimodular metric: what is its physical meaning? correlation holes in two states with different metrics • In the ν = 1/3 Laughlin state, each electron sits in a correlation hole with an area containing 3 flux quanta. The metric controls the shape of the correlation hole. • In the ν = 1 filled LL Slater-determinant state, there is no correlation hole (just an exchange hole), and this state does not depend on a metric Thursday, August 3, 17 • quantum solid • unit cell is correlation hole • defines geometry • repulsion of other particles make an attractive potential well strong enough to bind particle solid melts if well is not strong enough to contain zero-point motion (Helium liquids) Thursday, August 3, 17 but no broken symmetry • similar story in FQHE: • “flux attachment” creates correlation hole • defines an emergent e- geometry • potential well must be strong enough to bind electron • continuum model, but similar physics to Hubbard • new physics: Hall viscosity, model geometry...... Thursday, August 3, 17 • The composite boson fluid covers the plane, and provide an intrinsic dimensionless spatial distance measure on the plane, analogous to measuring distances in lattice units in the solid. • The effective field theory should only involve a connection compatible with the intrinsic spatial metric, not the connection compatible with the Euclidean metric. Thursday, August 3, 17 • space-time connection compatible with a time- dependent intrinsic spatial metric gab(x,t) f = @ f b f rµ a µ a µa b a = 1 gac @ g + d (@ g @ g ) µb 2 µ bc µ b cd c bd • unusual feature, connection 1-form carries only a a µ spatial indices b = µbdx • Geometric Chern-Simons 3-form is analog of gravitational CS form, but trace is over spatial indices a d b + 2 a b c =2! d! b a 3 b c a ^ ^ ^ ^ spin connection Thursday, August 3, 17 • conserved Gaussian curvature current of intrinsic metric: gab = pgg˜ab unimodular part J µ = 1 ( µ@ µ@ ) @ g˜ab +˜gab@ ln pg g 2 a t 0 a b b 1 µ⌫ ab cd + 8 ✏ ✏acg˜bd @⌫ g˜ @ g˜ (Brioschi formula) µ @µJg =0 • any non-singular time-dependent symmetric spatial tensor field can define a conserved Gaussian curvature current Thursday, August 3, 17 a a • three dynamical ingredients gab, v ,P : • a “dynamic emergent 2D spatial metric” gab(x,t) with g ≡ det g, and Gaussian µ µ⌫ curvature current Jg = ✏ @⌫ ! (x,t) • a flow velocity field va(x,t) • an electric polarization field Pa(x,t) a composite boson current • µ µ a µ Jb = pg(x,t)( 0 + v (x,t) a ) here a is a 2D spatial index, and µ is a (2+1D) space-time index. The fluid motion is non-relativistic relative to the preferred inertial rest frame of the crystal background Thursday, August 3, 17 (pe)2 effective bulk action: H = • 2⇡~K U(1) Chern-Simons field S = d2xdt U(1) condensate field L0 H Z “spin connection” ~ 1 = ✏µ⌫ (K b @ b + ! @ ! ) of the metric L0 4⇡ µ ⌫ µ ⌫ µ +J (~(@µ' bµ S!µ)+peAµ) b = pg "(v,B) U(g, B, P) (E + ✏ vbB)P a H a ab kinetic energy metric-dependent of flow correlation energy Thursday, August 3, 17 • shape of correlation hole (flux attachment) fluctuates, adapts to environment (electric field gradients) S new property: - “spin” couples e to curvature e- geometric distortion (preserving inversion symmetry) shape=metric creates “curvature” of metric • polarizable, B x electric dipole = momentum, origin of “inertial mass” - x e electric polarizability Thursday, August 3, 17 1/3 Laughlin state If the central orbital is filled, the next two are empty The composite boson has inversion symmetry e about its center It has a “spin” 1 3 5 2 2 2 1 1 0 0 ..... L = 2 1 1 1 L = 3 − 3 3 3 ..... − 2 s = 1 the electron excludes other particles from a region containing 3 flux quanta, creating a potential well in which it is bound Thursday, August 3, 17 2/5 state 1 3 5 2 2 2 1 1 0 0 0 ..... L =2 e e 2 2 2 2 2 ..... − L =5 − 5 5 5 5 5 s = 3 g L = ab RaRb 2`2 i i B i X ab 2 a b 2 ab Q = d rr r ⇢(r)=s`Bg Z second moment of neutral composite boson charge distribution Thursday, August 3, 17 • Furthermore, the local electric charge density of the fluid with ν = p/q is determined by a combination of the magnetic flux density and the Gaussian curvature of the metric e peB J 0(x)= sK (x) e 2⇡q g ✓ ~ ◆ Topologically quantized “guiding center spin” Gaussian curvature of the metric Thursday, August 3, 17 • In fact, it is locally determined, if there is an inhomogeneous slowly-varying substrate potential H = v (R )+ V (R R ) n i n i j i i deformation y near edge vn(x) x Thursday, August 3, 17 • “skyrmion”-like “cone”-like structure moves charge away from quasihole by introducing negative Gaussian curvature fluid density distance from center in an effective theory, core of quasihole may collapse into a cone singularity of the metric. Thursday, August 3, 17 One final result • In the “trivial” non-topologically-ordered integer QHE (due to the Pauli principle) c˜ = ⌫ =Chernnumber c˜ ⌫ =0 • the (guiding-center) “orbital entanglement spectrum” of Li and Haldane is insensitive to filled (or empty) Landau levels or bands, and allows direct determination of non-zero c˜ ⌫ previous methods used the onerous calculation of the“real-space” entanglement spectrum to find c˜ Thursday, August 3, 17 "laughlin3_n_13.cyl16.14" using 2:4 140 ORBITAL CUT 120 100 80 signed conformal 60 anomaly (chiral stress- chiral energy anomaly) anomaly 40 “CASIMIR MOMENTUM” term (NOT “real-space cut” which requires 20 the Landau orbit degrees of freedom and their form factor to be included virasoro level of sector 0 0 5 10 15 20 25 30 35 40 • Hall viscosity gives “thermally excited” momentum density on entanglement cut, relative to “vacuum”, at von Neumann temperature T = 1 Thursday, August 3, 17 Yeje Park, Z Papic, N Regnault 1 1 1 1 (˜c ⌫)= 1 3 = 24 0.1 24 36 L_A = 148, plevel = 12, +1/6 0.08 L_A = 149, plevel = 12 L_A = 150, plevel = 12, +1/6 0.06 L_A = 149, plevel = 11 1/36 0.04 1 0.02 0 36 -0.02 -0.04 mean Virasoro level - expected -0.06 0 100 200 300 400 500 600 700 800 900 L^2 Matrix-product state calculation on cylinder with circumference L (“plevel” is Virasoro level at which the auxialliary space is truncated) Thursday, August 3, 17 • In a 2D Landau level, we apparently start from a Schrödinger picture, but end with a “quantum geometry” which is no longer correctly described by Schrödinger wavefunctions in real space because of “quantum fuzziness” (non-locality) • It remains correctly described by the Heisenberg formalism in Hilbert space. Thursday, August 3, 17 quarks crystalline (rigid) continuum description atomic nuclei leptons condensed matter on lengthscales Higgs electrons larger than atoms photons Broken Lorentz, Galileian,and continuous rotational invariances • A very effective approach for understanding the essential physics is to remove all unnecessary non-generic ingredients from the description. • Only translation and (possibly) inversion symmetries are generic in a continuum description of phenomena in homogeneous crystalline condensed matter on a larger-than- atomic scale Thursday, August 3, 17 • The essential property of the uniform incompressible FQHE fluids is unbroken spatial inversion symmetry and a gap for excitations that carry an electric dipole moment (= momentum) • The momentum gap means that these fluids do not transmit forces through their bulk, unlike “classical incompressible fluids” Thursday, August 3, 17 • Top-level model (Schrödinger): pi = i~ r eA(r) r r A(r)=B H = "(pi)+ V0(ri rj) r ⇥ i i bare Coulomb interaction not necessarily quadratic controlled by (possibly anisotropic) (no Galilean invariance dielectric tensor of medium should be assumed) (no rotational invariance should be assumed) "(p)="( p) • model has inversion symmetry if , but even this need not be assumed Thursday, August 3, 17 a r = r e ea eb = ab pa = ea p a · · orthonormal basis Euclidean metric dynamical momentum displacement of tangent vectors of 2D plane (covariant index) (contravariant index) of 2D plane: a = 1,2 antisymmetric (2D Levi-Civita) symbol • Two independent Heisenberg algebras: ¯a ¯b 2 ab [pa,pb]=i~eB✏ab [R , R ]=i`B✏ a a organize as [Ra, R¯b]=0 [r ,pb]=i~ b a b [Ra,Rb]= i`2 ✏ab [r ,r ]=0 B ¯a 1 ab 2 a a ¯a 2 2⇡ R = ~ ✏ pb` R = r R 2⇡` = ~ > 0 B B eB Landau orbit Landau orbit guiding- quantum area radius vector center displacement (per h/e flux quantum) • Note: origin of guiding-center displacement has a A(r) A(r) gauge ambiguity under 7! + constant Thursday, August 3, 17 "(p) = E • Landau quantization | ni n| ni discrete spectrum of macroscopically- degenerate Landau levels • Project residual interaction in a single partially occupied “active” Landau level, all other dynamics is frozen by Pauli principle when gap between Landau levels dominates interaction potential a b 2 ab [R ,R ]= i`B✏ residual problem is non- commutative quantum H = V (R R ) n i j geometry! i Thursday, August 3, 17 original V0(x) [Ra,Rb]= i`2 ✏ab (not smooth) B H = V (R R ) n i j i • The potential Vn(x) is a very smooth (in fact entire) function that depends on the form- factor of the partially- occupied Landau level • The essential clean-limit symmetries are translation and inversion: R a R i 7! ± i Thursday, August 3, 17 [Ra,Rb]= i`2 ✏ab Vn(x) B H = V (R R ) n i j i • The quadratic expansion of this even function around the origin defines a natural “interaction metric” • The problem is often simplified by giving it a continuous rotation symmetry that respects this metric, but this is non-generic, and not necessary. • This metric and a rotation symmetry are important in model FQH wavefunctions based on cft, which have a stronger conformal invariance property. Thursday, August 3, 17 • It is straightforward to solve the two-body Hamiltonian: R12 = R1 R2 Vn(x) a b 2 ab E1 [R12,R12]=2i`B✏ E2 equivalent to a one- E3 E etc. 4 particle problem H = Vn(R12) • If there is a rotational symmetry, the energy levels (called “pseudopotentials”) completely characterize the interaction potential. • a large gap between energy levels favors flux attachment with a shape close to that of the “interaction metric” Thursday, August 3, 17 • The non-commutative quantum geometry does not have a Schrödinger representation because there is no orthogonal local basis within its Hilbert space • We can create an unfaithful Schrödinger representation by adding back a now-unphysical copy of the Landau orbit-degrees of freedom: physical unphysical local basis Projection into physical basis of holomorphic states Thursday, August 3, 17 • This looks like just mapping the quantum geometry back into a “lowest-Landau-level problem” • But important new features appear when the problem is “compactified on the torus” by imposing (quasi)periodic boundary conditions Thursday, August 3, 17 • Guiding-center translation operator • Periodic Boundary Conditions on a primitive region with flux NΦ in units h/e : smallest translation compatible with pbc Bravais lattice of periodic translations NOTE: choice of a basis L1, L2 is a “modular choice” Thursday, August 3, 17 • On the Torus (pbc) there are two distinct type of bases: (a). Geometry-independent, orthonormal bases that depend on a modular choice: This is a standard “Landau basis” Thursday, August 3, 17 • The other type of basis is a non-orthogonal basis of modular-invariant geometry- dependent holomorphic states: zeroes Weierstrass sigma function “almost holomorphic modular invariant” (modular invariant) Thursday, August 3, 17 • Holomorphic Laughlin states • Also get very useable forms for other states (composite fermion Fermi liquid states) • A surprise: only need to evaluate holomorphic states on the lattice of points in the primitive region of the torus! Thursday, August 3, 17 • On the torus, the Heisenberg algebra is compactified to the unitaries (discrete set of reciprocal vectors q) • To construct the unfaithful Schrödinger representation, we now use compactified dual variables: • We get a modular-invariant lattice-based Schrödinger representation 2 on (NΦ) sites (square of one-particle Hilbert space dimension) Thursday, August 3, 17 • A corollary: based on the naive lowest-Landau-level interpretation, one expects that the overlap between two holomorphic states must be calculated as primitive region of torus • In fact primitive lattice sum This allows a lattice-based Monte Carlo evaluation of model state properties on the torus, which would not have been guessed in the LLL picture Thursday, August 3, 17 • Furthermore, the projections of the lattice sites into the physical (holomorphic) space is essentially a coherent state representation, and all operators have a diagonal representation on the lattice. • A very economical Monte-Carlo method for model wavefunction properties (with huge speed-ups compared to previous continuum methods) is obtained and has been tested! (with Ed Rezayi, Scott Geraedts, Jie Wang) Thursday, August 3, 17 summary • While the “lowest Landau level wavefunction” (LLLWF) interpretation of model FQH and CFL states is entrenched in the “common wisdom” it is misleading. The system projected into a Landau level is a “quantum geometry” with no faithful Schrödinger representation. • modular-invariant holomorphic model states have an intrinsic metric that is the shape of “flux attachment”, fixed by the interaction in translationally-invariant systems • A new lattice-based approach to systems on the torus is obtained after discarding the LLLWF interpretation. Modular invariance is a key property. Thursday, August 3, 17 • Flux attachment is a gauge condensation that removes the gauge ambiguity of the guiding centers, giving each one a “natural” origin, so they define a physical electric dipole moment of the “composite particle” in which they are bound by the “attached flux”. • This is analogous to how the “the vector potential becomes an observable” (in a hand-waving way) in the London equations for a superconductor. R × (fuzzy) region from which particles other that those making up the “composite particle” are center of flux-attachment excluded Thursday, August 3, 17 • quantum solid • unit cell is correlation hole • defines geometry • repulsion of other particles make an attractive potential well strong enough to bind particle solid melts if well is not strong enough to contain zero-point motion (Helium liquids) Thursday, August 3, 17 • In Maxwell’s equations, the momentum density is j i ij i ⇡i = ✏ijkD Bk D = ✏0 Ej + P • The momentum of the condensed matter is p = d B ⇥ electric dipole moment • in 2D the guiding-center momentum then is b pa = eB✏ab R • The electrical polarization energy of the dielectric composite particle then gives its energy-momentum dispersion relation, with no involvement of any “Newtonian inertia” involving an effective mass Thursday, August 3, 17 • The Berry phase generated by motion of the “other particles” that “get out of the way” as the v × R vortex-like “flux-attachment” moves with the particle(s) it encloses can be formally- described as a Chern-Simons gauge field that cancels the Bohm-Aharonov phase, so that the composite object propagates like a neutral particle. Thursday, August 3, 17 • If the composite particle is a boson, it condenses into the zero-momentum (zero electric dipole-moment) inversion-symmetric state, giving an incompressible-fluid Fractional Quantum Hall state, with an energy gap for excitations that carry momentum or electric dipole moment (“quantum incompressibility”, no transmission of pressure through the bulk). • All FQH states have an elementary unit (analogous to the unit cell of a crystal) that is a composite boson under exchange. • It may be sometimes be useful to describe this boson as a a bound state of composite fermions (with their own preexisting flux attachment) bound by extra flux (Jain’s picture) Thursday, August 3, 17 0.9 laughlin 10/30 0.8 ތࠓ goes into continuum 0.7 0.6 (2 quasiparticle 0.5 bosonic “roton” E 0.5 + 2 quasiholes) 0.4 fermionic 0.3 “roton” “roton” 0.2 2 Moore-Read “kF ” ⌫ = 0.1 4 Topologically-degenerate FQH state 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2k k lB gap incompressibility B Collective mode with short-range V1 Collective mode with short-range three-body pseudopotential, 1/3 filling (Laughlin state is pseudopotential, 1/2 filling (Moore-Read state is exact ground state in that case) exact ground state in that case) momentum ħk of a quasiparticle-quasihole pair is • b proportional to its electric dipole moment pe ~ka = abBpe gap for electric dipole excitations is a MUCH stronger condition than charge gap: fluid does not transmit pressure through bulk! Thursday, August 3, 17 • The composite particle may also be a py fermion. Then one gets a Fermi surface in empty momentum-space = electric dipole space, and a gapless anomalous Hall effect which is filled quantized when the Berry phase cancels the px Bohm-aharonov phase. (HLR-type state) • There must be a distribution of dipole moments (or momentum) of the composite fermions, centered at the inversion-symmetric zero-moment state which has lowest energy. These are quantized by a pbc, and no two composite fermions can have the same diople moment. Thursday, August 3, 17 • Fermi surface quasiparticle formulas for anomalous Hall effect (FDMH 2006) • in 2D: e2 H = (n + ) 2⇡~ 2⇡ ei Berry phase for Integer determined moving a quasiparticle around Fermi surface (arc) at edge Thursday, August 3, 17 • holomorphic representations of guiding- center states a R a a = w a† + w a [a, a†]=1 p2`B 1 w = g wb (wa)⇤wa = 2 (gab + i✏ab) a ab det g =1 • This is the Girvin-Jach formalism, except they implicitly assumed the metric gab was the Euclidean metric of the plane. In fact, it is a free choice, not fixed by the any physics of the problem. Thursday, August 3, 17 • Then, once a metric (i.e., a complex structure) has been chosen, a one-particle state can be described as = f(a†) 0 a 0 =0 | i | i | i holomorphic • Both the “vacuum” |0⟩ and the function f(z) vary as the metric is changed (a Bogoliubov transformation) • Normalization/overlap: dz dz⇤ z⇤z = ^ f (z)⇤f (z)e h 1| 2i 2⇡i 1 2 Z Thursday, August 3, 17 • When compactified on the torus with flux NΦ , the modular-invariant formulation is N f(z) ˜(z w ) wi =0 / i i=1 i Y X Bravais lattice in complex plane 1 C ( L )z2 ˜(z L )=e 2 2 { } (z L ) |{ } |{ } “almost holomorphic modular Weierstrass sigma function invariant” (Eisenstein series) Thursday, August 3, 17 • In the Heisenberg-algebra reinterpretation N = (a† w ) 0 wi =0 one particle | i ˜ i i | i i=1 i N =1 Y X • The filled Landau level is filled Level = ˜ (ai† aj†) ˜( iai†) 0 | i 0 1 | i N = N i Thursday, August 3, 17 • A previously unknown (?) identity that I recently guessed and found was indeed true, and which dramatically transforms calculations on torus (e.g., orders of magnitude Monte-Carlo speedup) dz dz⇤ z⇤z = ^ f (z)⇤f (z)e h 1| 2i 2⇡i 1 2 Z⇤ mL1+nL2 1 0 z = N N 2 { } z 2 X (NΦ ) points replace integral over fundamental region by a modular-invariant finite sum Thursday, August 3, 17 • with Ed Rezayi, I found a remarkable clean composite Fermi liquid model state on the flat torus, inspired by a construction by Jain on the sphere. • On the torus, the state is precisely equivalent to the usual treatments of the Fermi gas with a pbc. • It is very accurate as compared to exact diagonalization of the Coulomb interaction, and amazingly “almost” (99.99%) particle-hole symmetric at ν = 1/2. Thursday, August 3, 17 1 Composite Fermi liquid (HLR-like) at ⌫ = • m gives Chern-Simons gives bf? / Z2 m (m 2) f( zi )= ˜(zi zj) det Mij ˜( zi wk) { } ij i i M ( z ; d )=edj⇤zi/m 0 ˜(z z d + d¯) ij { k} { k} i j i k=i Y6 L d (particle number, not flux) a set of dipole moments i 2 N Thursday, August 3, 17 • There are vastly more possible choices of 0.028677091503 0 1110101000 0.0286770915235 0 dipole “occupations” than independent 0001010111 0.0171543754946 0 1110110000 0.0172785391733 0 0001001111 states: The “good” ones0.00272205658268 are clusters that0 1111000001 0.00272205658096 0 0000111110 minimize ¯ 20.00741749061239 0 1111000010 i di d 0.00741749061624 0 0000111101 | | 0.0131254758865 0 1111000100 0.0131430302064 0 0000111011 Computing0.0172785391469 ph 0 symmetry 1111001000 0.017154375511 0 P 0000110111 0.0141743825022 0 1111010000 0.0141743825338 0 (with Scott 0000101111 Geraedts) 0.00547651410185 0 1111100000 model0.00547651412427 state is numerically 0 very close 0000011111 to p-h symmetry when k’s are clustered # Z_{COM} overlap with PH-conjugate in opposite charge sector 1- overlap 0 0.999998870263 1.1297367517e-06 1 0.999999369175 6.3082507884e-07 2 0.99999860296 1.39704033186e-06 3 0.99999860296 1.3970403312e-06 4 0.999999369175 6.30825078063e-07 5 0.999998870263 1.12973675237e-06 6 0.999999369175 6.30825079173e-07 7 0.99999860296 1.39704032942e-06 8 0.99999860296 1.39704032909e-06 9 0.999999369175 6.30825078507e-07 Thursday, August 3, 17 • particle-hole symmetry, and Kramers Z2 structure (Scott Geraedts and Jie Wang) Thursday, August 3, 17 A many-body ansatz for Berry phase we confirmed all paths are real, by Kramers These are ED results on exact Coulomb interaction states, with the exact particle hole symmetry, with occupation patterns obtained by finding the model states they have high overlap with Thursday, August 3, 17 • is there an analog of Dirac cone point ? State with the quantum numbers of +1 an inversion symmetric Fermi sea with a single hole at the center (has an even number of particles) 1 A state on the Torus with these quantum numbers is a parity doublet • as a hole is moved into the bulk, the ansatz must fail as if goes through the inversion- symmetric point! Thursday, August 3, 17 13 some region z -40 beyond the truncation point using the recurrence rela- quadratic + quartic dispersion (Yu Shen + FDMH) tion, but we found that the eigenvector of the truncated • tridiagonal matrix was accurate enough for finding the -40 -20 0 20 40 pattern13 of roots, when calculated with high precision. A typical layout (Class I) of the roots is presented in 10 some region z VIII. OTHER PROPERTIES OF THE LANDAU ORBIT FIG. 7. Black region accounting for 90 percent of the to- There are some other important characteristic physical tal weight (probability) of 20, the coherent state in the properties of the Landau orbit. The simplest is its dia- 2 2 4 4 20th Landau level of the Hamiltonian px + py +2px +3py + magnetic magnetic moment normal to the plane, given 2 2 3 4 p ,p + px,p . The annulus is bounded by contours at { x y} { y} by FIG. 1. Root distribution on the complex plane of thethe sameFIG. value 2. of Rootthe amplitude distribution 20 and on cleanly the complex separates plane of the the central zeroes from the rest of| the| structure. truncatedThursday, antiholomorphic August 3, 17 part of the coherent state in the truncated antiholomorphic part of the coherent state in the @En 10th Landau level f (z⇤) for class I quartic term (113) with µn = (122) 10th Landau level f10(z⇤) for class I quartic term (113) with 10 @B c 1 = 4 at a truncation of 500 fixed-parity states, i.e., the "(p) c 1=4atdi↵erent truncations N. Here N is not the de-tive di↵erence from class I is that the classical dispersion truncated f10(z⇤) is a polynomial of degree 998 with only even gree of the polynomial (truncated f10(z⇤)), but the number"(p) is consistently zero along one (for classes II, IV) or which in the Newtonian model with Galilean invariance n powers. Ten central zeroes, with two being degenerate at the of fixed-parity states (in this case (z⇤) with n being an even reduces to µn = ~eBsn/me. two (fororigin, class form III) a radial cross directions due to the onC thesymmetry momentum of (113). The number) we kept for diagonalization. For example, N =500plane. As a result, the classical orbits for4 these three The second property is the e↵ective mass that charac- 0 998 peripheral zeroes also have interesting features, such as four means we kept states from (z⇤) to (z⇤) .ForeachN,weclasses are not closed. Though the noncommutativity of terizes the kinetic energy of a guiding center that flows “spikes” pointing to the origin and line charges with constant used a su ciently high precision in diagonalization so thatp gives rise to e↵ective quadratic terms for classes II along a line of constant En(x), if there is slow adiabatic a density that bound 2D regions of quasi-uniform distributions. further increasing the precision would not do any better inand III, thus closing the quantum orbits, we expect that spatial variation of the Hamiltonian. This is given by the The arcs on the boundary are generated due to finite trunca- terms of the locations of the zeroes. As N increases, the re-a clean separation between the central zeroes and the linear response to a perturbation that breaks the inver- tion and should not be regarded as part of the zero pattern. gion covered by the zeroes expands. However, the existingouter structure by a set of contours cannot be achieved sion symmetry of the Landau orbit around its guiding pattern of zero distribution does not change, which justifiesfor any of the three classes, which is verified in Fig.4. center. The perturbed Landau levels are given by the validity of our results. We now combine quartic terms with quadratic terms. a The parametrizationbut the weaker is condition that the spectrum of H is stable(H v pa) n,↵(v) = En(v) n,↵(v) , (123) ab c d | i | i (i.e., bounded below) only requires that ✏ac✏bdA1 ui ui 2 2 4 4 where v is a parameter of the perturbed Hamiltonian. VII. A CASE STUDY: THE QUARTIC H0= inAp classesx + Bpy II,+ Cp III,x + IV.Dpy Then DISPERSION The class2 2 I and III quartic3 terms3 have a C symmetry + E px,py + F px,py + G px,py . (121)4 { } { ab } ab{ }ab n a b 4 with inverse metricg ˜0 .IfA1 g0 the full model also(v) H (v) = E + 1 m v v + O(v ), (124) / n↵ n↵ n 2 ab We will now present a detailed case study of the sim-The additionhas this of symmetry, quadratic and terms all does Landau not orbits introduce have theh same | | i qualitatively new features although the Cab4 symmetry ab plest model dispersion that exhibits the generic property metric given by the inverse ofg ˜0 . If, in addition,and g ˜1 genericallyab breaksab down. However, as we go to high- that there is no congruence between the shapes of di↵er- =˜g2 =˜g0 ,soc = 1, class I models have a continuous energy states, the quartic terms will eventually dominate. E (v)=E 1 mn vavb + O(v4), (125) SO(2) symmetry. n n 2 ab ent semiclassical orbits. Therefore we expect to see a quasi-C4 symmetry gradu- We first review the purely quadratic case, where the n This is the model with quadratic and quartic terms inally appear as the Landau index n increases, as shown in For the Newtonian model, mab = me ab. Though both quartic term is absent, Aabcd = 0. In this case n n the dispersion: Fig.5. Due to this asymptotic C4 symmetry,2 the generic mab and gab are proportional to the Euclidean metric features of the root structure and the contours still hold, in the Newtonian model, there is in general no propor- ab abcd 1 ab 1 H = E0g˜ ab,E0 > 0, (110) H = A1 papb + 4 A2 pa,pb pc,pd . (108)as illustrated in Fig 6. 2 tionality between them unless there is a discrete three-, { }{ } To make connection with the usual Galilean theory, four- or six-fold crystal rotational symmetry of the lat- ab Up to the addition of the quadratic Casimir, which iswe remarkwhere thatg ˜ asis we the reduce inverse the of weights a unimodular of the quartic metricticeg ˜ab planeand on which the electrons move (and no tangen- a constant, and does not a↵ect the semiclassical orbits,terms,there the central is a pseudo-Galilean zeroes will shrink invariance toward the with origin ab replacedtial component of the magnetic flux), in which case they and the outer zeroes will be pushed toward2 infinity1 (but both remain proportional to the Euclidean metric which there are four possible classes of quartic terms compat- byg ˜ab, and me replaced by (E0`B/~) . However, the ab generic patterns don’t change), consistent with the fact is invariant under the crystal symmetry. ible with strict monotonicity. Ifg ˜i are the inverses of resemblance to the Newtonian model is superficial, as unimodular metrics: that at(E the`2 limit) 1 ofand pureg ˜ quadraticcan be terms spatially-varyingfn(z⇤) becomes (andThe their generic lack of any relation between the viscos- a monomial0 B (z )n and allab the zeroes stay at the origin. ity tensor and the e↵ective mass tensor can be traced product⇤ can have Gaussian curvature) on a completely- ˜ab,cd ab cd ab cd 1 ab ab ab We conclude this section by revisiting the semiclassi- to the fact that the viscosity derives from perturbations A2,I =˜g1 g˜2 +˜g2 g˜1 , 2 (˜g1 + g2 )=cg˜0 , geometrically-flat plane with a Cartesian coordinate sys- ˜ab,cd ab c d a b cd tem. In this non-generic case, the Landau-orbit metric is A2,II =˜g1 u1u1 + u1u1g˜1 , g˜ independent of the Landau index, with all the zeroes ˜ab,cd a b c d a b c d a b a b ab ab A2,III = u1u1u2u2 + u2u2u1u1,u1u1 + u2u2 = g˜0 , of the Landau-orbit coherent state at its origin. Aab,cd = uaub uc ud. (109) The next-simplest case is the purely-quartic class I 2,IV 1 1 1 1 model, which also has the non-generic feature of a com- where > 0, and c 1. Strict monotonicity also requires mon metricg ˜ab for all Landau orbits, because of its hid- ab ab that the quadratic term A has the form 0g˜0 , 0 0, den C four-fold rotational symmetry. The generic model 1 4