Quantum Geometry of the Laughlin State and Flat-Band models

• “classical locality” means the existence of anorthogonal basis of coordinate eigenstates[ˆx a , ˆx b ]=0hx|x 0 i =0, x 6= x 0ˆx a |x 0 i = x 0a |x 0 i• This structure is required for the equivalenceof the Schrödinger and Heisenbergformulations of quantum mechanics:(x) =hx| i• This equivalence fails for non-commuting coordinates•Projection into a “flat band” leads to non-commutingcoordinates

• band structure plus gauge-invariantinteractions:H = H 0 + V ({n i })n i = X c † i c iH 0 = Xi,j, 0t i ,j 0c † i c j 0= X nk" n (k)c † nk c nk• assume one or more “flat bands”are present near the Fermi energy• Projection into these bands: P 0EFemptylarge gaplarge gapfilled

• basis with “classical locality” |x i, i = c † i |0ihx i , |x j ,0 i = ij 0• projected (and renormalized) basisP 0 |x i , i = i | 0 (x i , )i h 0 (x i , )| 0 (x i , )i =1projected into the “flat band” renormalized to unit weight• “quantum geometry” is encoded in the overlaph 0 (x i , )| 0 (x j ,0 )i = S i ,j 0• renormalized states have a local gauge ambiguity under| 0 (x i , i)i 7! e i'(x i, ) | 0 (x i , i)i

• “quantum geometry” is encoded in the overlaph 0 (x i , )| 0 (x j ,0 )i = S i ,j 0• gauge-invariant “quantum distance”d 12 =(1 |S 12 |) 1 2obeys triangleinequalityThis is the “trivial distance” for orthonormal orbitals,where d = 0 (same), or d = 1 (different)• gauge-invariant “triangle Berry phases”S 12 S 23 S 31 = |S 12 S 23 S 31 |e i 123213

• after projection into the flat band, one endsup with a model of the formH = U({n i })a modified gauge-invariant interactionpotential, because of the renormalizationsn i = X c † ic ic i c † j 0 ± c † j 0 c i = S i ,j 0• The Hermitian matrix S is positive indefinite(has null modes) because there are moreorbitals per unit cell than “flat bands”

• Spin-orbit coupling makes the “quantumgeometry” richer, but for simplicity, I willnow drop the spin-degree of freedom• Consider the (spinless) boson “flat band”Hubbard model on the “fuzzy lattice”H = 1 2XiU i c † i c† i c ic i[c i ,c † j ]=S ijc † i ⌘ c† R,↵ U i = U ↵d ij = d ↵↵ 0(R R 0 )unit cellof Bravaislatticebasis oforbitals withinunit cell“fuzzy lattice” quantum distancesand triangle Berry phases areinvariant under lattice translationsnon-null eigenstates of S ij are Bloch bands whichencode the flat-band topological invariants

• A Landau level is a “flat band” with acontinuous translation symmetry, and the ν= 1/2 Laughlin state is the maximumdensityzero-energy ground state ofH = 1 2 U Zd 2 x2⇡`2B[c(x),c † (x 0 )] = S(x, x 0 )z = ! ax ap 2`Ba complexunit vectorc † (x)c † (x)c(x)c(x)= e z⇤ z 0 e 1 2 z⇤z e 1 2 z0⇤ z 0! ⇤ a! b = 1 2 (g ab + i✏ ab )2d metric with Euclideansignature, detg = 1.The metric g ab is the fundamental parameterof model wavefunctions based on Euclidean cft!2d antisymmetricsymbol

• If the integral over x is discretized in anyperiodic pattern of points {x i } that has oneflux quantum per unit cell, the Chernnumber of the single non-null band of Blocheigenstates of S ij is 1.• There are two independent two-body statesc † (x)c † (x)|0> per flux quantum, so at leasttwo discretization points per unit cell areneeded to reproduce the count of non-nulltwo-body states.• This discretization will reproduce the zeroenergystate count of the continuum model.

• Thus the continuum 1/2 Laughlin state willalso be the zero-energy ground state of a“Fuzzy-Lattice” flat-band honeycomb netboson Hubbard model with Chern number1, if the projected orbitals on each site aretaken to be Landau-level guiding-centerGaussian coherent states.H = 1 2 U X xc † (x)c † (x)c(x)c(x)Two orbitals, one fluxquantum, per unit cell• A “real” fuzzy-lattice Hubbard model obtainedfrom a projected flat-band structure willexhibit a Laughlin-like ground state if itsoverlap function is “sufficiently close” to theGaussian one

some related “mysteries”• Why are model wavefunctions related to(Euclidean) 2+0 d cft good models for theFQHE?• If the Laughlin state is a “lowest Landau levelSchroedinger wavefunction” why does itoccur in the second Landau level?• Why is it “holomorphic”?•What aspects of 1+1d cft apply to edgestates?

• The conformal group is the group ofcoordinate transformations that preservethe unimodular part of a metricds 2 = e 2'(x,t) v 1 dx 2 vdt 2(1+1)d“universal speed of massless particles”ds 2 = e 2'(x)g ab dx a dx b(2+0)d“Euclidean metric, det g = 1”

• after Landau quantization, residual guiding center degrees offreedom are non-commutativer = R + ˜R[R a ,R b ]= i`2B✏ abeliminatedby Landauquantizationantisymmetricsymbol• isomorphic to phase space, theyobey an uncertainty principleclassicalcoordinateOrRLandau orbitradius vectore⇥˜Rguiding centercoordinateguiding centerscannot be localizedwithin an area lessthan 2⇡`2B

• The metric defines the shape of the coherent state at the centerof the “symmetric gauge” basis of guiding-center statesdifferent choices of metric:(“squeezed” relative to each other)centralcoherentstate| 0 (g)i• Guiding-center “spin” (rotationoperator) is defined by the metricL(g) = g ab2`2BR a R b[L(g),a † (g)] = a † (g)a(g)| 0 (g)i =0| m (g)i = (a† (g)) mp m!| 0 (g)i

• Model cft-based states such as the Laughlin statehave a constant (flat, rigidly-fixed) metric• In real FQH states of electrons contained in anon-uniform background potential, the metricvaries locally and dynamically to allow theincompressible fluid to adjust to non-uniformflow induced by the background.• The metricg ab (r,t) then becomes an emergentdynamical collective degree of freedom of theFQH state.FDMH, Phys. Rev. Lett. 107, 116801 (2011)

holomorphicity:• coherent state basisā|¯zi =¯z|¯zi |¯zi = e¯zā†¯z ⇤ā|0iS(¯z, ¯z ⇤ ;¯z 0 , ¯z ⇤ )=h¯z|¯z 0 i = e¯z⇤ ¯z 01 2 (¯z0⇤ ¯z 0 +¯z ⇤ ¯z)• non-null eigenstates of the overlap define anorthonormal basisZ d¯z 0 d¯z 0⇤2⇡ S(¯z, ¯z⇤ ;¯z 0 , ¯z 0⇤ ) (¯z 0 , ¯z 0⇤ )= (¯z, ¯z ⇤ )• non-null eigenstates are degenerate with λ = 1(¯z, ¯z ⇤ )=f(¯z ⇤ )e 1 2 ¯z⇤ ¯zholomorphic!“accidentally” coincidewith lowest-Landau levelwavefunctions if ¯z = z ⇤ !!!

• This is the true origin of holomorphicfunctions in the theory of the FQHE• NOTHING to do with lowest Landau levelstates, derives from overlaps between statesin a non-orthogonal overcomplete basis!• Has obvious parallels in theory of flat-bandChern insulators, where the projected latticesitebasis is non-orthogonal and overcomplete|qL i = Y iZ d¯z⇤i d¯z i2⇡“Laughlinwavefunction”Yi

• It is a common misconception that the Laughlinstate is fundamentally “a lowest Landau-levelwavefunction” of Galileian-invariant Landau levels• The similarity to a lowest-LL wavefunction is entirelyaccidental, as should have been clear when it wasalso found in the second LL. The recent discoverythat Laughlin-like states occur in “flat band” Cherninsulators now makes this entirely clear!• The holomorphic character of the Laughlin state isentirely a property of the “quantum geometry” of theflat band (Landau level) encoded in s(r 1 ,r 2 ), which inturn depends on the choice of metric g ab .

• Origin of FQHE incompressibility is analogous toorigin of Mott-Hubbard gap in lattice systems.• There is an energy gap for putting an extra particlein a quantized region that is already occupied• On the lattice the “quantizedregion” is an atomic orbital witha fixed shape• In the FQHE only the area ofthe “quantized region” is fixed.The shape must adjust tominimize the correlation energy.e - energy gap preventsadditional electronsfrom entering theregion covered by thecomposite boson

• The metric (shape of the composite boson) has apreferred shape that minimizes the correlationenergy, but fluctuates around that shape• The zero-point fluctuations of the metric are seenas the O(q 4 ) behavior of the “guiding-centerstructure factor” (Girvin et al, (GMP), 1985)E / (distortion) 2• long-wavelength limit of GMP collective mode isfluctuations of (spatial) metric (analog of “graviton”)FDMH, Phys. Rev. Lett. 107, 116801 (2011)

1/3 Laughlin state If the central orbital is filled,the next two are emptyeThe composite bosonhas inversion symmetryabout its centerIt has a “spin”the electron excludes other particles froma region containing 3 flux quanta, creating apotential well in which it is bound−121 0 01332135213..........L = 1 2− L = 3 2s = 1

• crucial new physics:composite bosons couple to the combinationpeB(r)charge of compositeboson* gauge field isrelated toWen and Zee 1992~sK(r)guiding-center“spin” of bosonpeA µ (r)Gaussian curvatureof metric~s⌦ µ (r)analog of spin-connection+ Chern-Simons

• metric deforms (preserving det g =1)inpresence of non-uniform electric fieldfluidcompressedby Gaussiancurvature!potentialnear edgeproduces a dipole momemt

• multicomponent quantum Hall edge statesdo not have a universal speed, so are notLorentz and conformally invariant.• components of the cft energy momentumtensor:Momentum density is independent of v:T 0 x = T¯TEnergy density and stress are proportional to vT 0 0 = T x x = v(T + ¯T )Tracelessness (in flat space-time) is independent of vT 0 0 + T x x =0Energy current density is proportional to v 2T x 0 = v 2 (T ¯T )

The only speed-independent properties are• The (signed) Virasoro algebra of the Fouriercomponents of the momentum density (with thetopologically-conserved chiral central charge˜c = cThis is a fundamental quantity that has nothing todo with conformal invariance (and in fact mustvanish in a “true” (modular-invariant) 1+1d cft )It controls a “Casimir momentum” 124 ~˜c/L• Tracelessness of the energy-momentum tensor (1dpressure = energy density), which is true for linearlydispersingmodes, independent of their speed.¯c

• Tracelessness of the 2D stress tensor in theEuclidean field theory is the absence ofhydrostatic 2D pressureP = 1 2 T x x + T y y =0• Incompressible 2D quantum fluids at T = 0 donot transmit pressure through their bulkbecause of their energy gap (no gapless soundmodes) - they only transmit pressure aroundtheir edges via gapless edge excitations.• The traceless stress tensor of Euclidean 2dconformal field theory (and its dependence ona metric) may explain its applicability to FQHE

One final result• In the “trivial” non-topologically-orderedinteger QHE (due to the Pauli principle)˜c = ⌫ =Chernnumber˜c ⌫ =0• the (guiding-center) “orbital entanglementspectrum” of Li and Haldane is insensitive tofilled (or empty) Landau levels or bands, andallows direct determination of non-zero ˜cprevious methods used the onerous calculation ofthe“real-space” entanglement spectrum to find ˜c⌫

140"laughlin3_n_13.cyl16.14" using 2:4ORBITAL CUT120100806040200• Hall viscosity gives “thermally excited”momentum density on entanglement cut,relative to “vacuum”, at von Neumanntemperature T = 10 5 10 15 20 25 30 35 40signed conformalanomaly (chiral stressenergyanomaly)“CASIMIR MOMENTUM” term(NOT “real-space cut” which requiresthe Landau orbit degrees of freedom and theirform factor to be includedchiralanomalyvirasoro levelof sector

Yeje Park, Z Papic, N Regnault124 (˜c ⌫) = 1 24 1 1= 1336mean Virasoro level - expected0.10.080.060.040.020-0.02-0.04-0.06L_A = 148, plevel = 12, +1/6L_A = 149, plevel = 12L_A = 150, plevel = 12, +1/6L_A = 149, plevel = 111/361360 100 200 300 400 500 600 700 800 900L^2Matrix-product state calculation on cylinder with circumference L(“plevel” is Virasoro level at which the auxialliary space is truncated)