Quantum Computing And Quantum Topology

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Modeling and Classification of Quantum Hall StatesA story of electrons under extreme conditions or of translation invariant symmetric polynomialsZhenghan WangMicrosoft Station Q & UC Santa BarbaraTexas, March 23, 2015

Classical Hall Effect---Maxwell’s MistakeWhen electrons discovered?

Birth of Integer Quantum Hall EffectNew Method for High-Accuracy Determination ofthe Fine-Structure Constant Based on QuantizedHall Resistance,K. v. Klitzing, G. Dorda and M. PepperPhys. Rev. Lett. 45, 494 (1980).These experimental data, available to the public 3 yearsbefore the discovery of the quantum Hall effect, containalready all information of this new quantum effect so thateveryone had the chance to make a discovery that led to theNobel Prize in Physics 1985. The unexpected finding in thenight of 4./5.2.1980 was the fact, that the plateau values inthe Hall resistance x-y are not influenced by the amount oflocalized electrons and can be expressed with high precisionℎby the equation 𝑅𝐻 2 𝑒

Fractional Quantum Hall Effect (FQHE)D. Tsui enclosed the distance between B 0 and theposition of the last IQHE between two fingers ofone hand and measured the position of the newfeature in this unit. He determined it to be threeand exclaimed, “quarks!”H. StormerThe FQHE is fascinating for a long list of reasons,but it is important, in my view, primarily for one: Itestablished experimentally that both particlescarrying an exact fraction of the electron charge eand powerful gauge forces between these particles,two central postulates of the standard model ofelementary particles, can arise spontaneously asemergent phenomena.R. LaughlinIn 1998, Laughlin, Stormer, and Tsuiare awarded the Nobel Prize“ for their discovery of a new formof quantum fluid with fractionallycharged excitations.”D. C. Tsui, H. L. Stormer, and A. C. GossardPhys. Rev. Lett. 48, 1559 (1982)

How Many Fractions Have Been Observed?𝑁 𝑁 𝑒 100filling factor or fraction𝑁𝑒 # of electrons𝑁 # of flux quantaHow to model the quantumstate(s) at a filling fraction?What are the electrons doingat a plateau?1/32/34/35/37/38/31/5 1/7 1/9 2/11 2/13 2/152/5 2/7 2/9 3/11 3/13 4/153/5 3/7 4/9 4/11 4/13 7/154/5 4/7 5/9 5/11 5/13 8/156/5 5/7 7/9 6/11 6/13 11/157/5 9/7 11/9 7/11 7/13 22/158/5 10/7 13/9 8/11 10/13 23/1511/5 12/7 25/9 16/11 20/1312/5 16/717/1119/7m/5, m 14,16, 192/173/174/175/176/178/179/173/19 5/21 6/23 6/254/19 10/215/199/1910/195/27/219/8Pan et al (2008)

Three Answers1. Art: electrons find “partners” and dance2. Physics: patterns of long range entanglement3. Math: (2 1)-TQFT or modular tensor category in nature

State and Energy At each moment, a physical system is insome state-a point in 𝑅 𝑛 or a vector ( 0)in some Hilbert space (a wave function). Each state has an energy. States of the lowest energy win: stableHow to find the lowest energy states(ground states) and understand theirproperties (excitations responses)?

Classical Electrons on2𝑆Thomson’s Problem:Stable configurations of N-electrons on 𝑆 2minimizing total Coulomb potential energy𝐸𝑖𝑗 1𝑖 𝑗 𝑑𝑖𝑗, 𝑑𝑖𝑗 distance between 𝑖, 𝑗What happens if 𝑁 ?

Mathematical Quantum Systems A triple Q (L, H, c), where L is a Hilbert space with apreferred basis c, and H an Hermitian matrix.Physically, H is the Hamiltonian. A non-zero vector in Lis a quantum state wave function. Given a quantum system, find the ground state manifold:the eigenspace of H with the smallest eigenvalue:L Vi,where Vi is the eigenspace of H with eigenvalue energy𝜆𝑖 , i 0,1, , in an increasing order.V0 is the ground state manifold with energy 𝜆0and others are excited states. A linear algebra problem that needs a quantum computer.

Quantum Hall SystemsN electrons in a plane bound to the interface between twosemiconductors immersed in a perpendicular magnetic fieldPhases are equivalence classes of groundstate wave functions that have similarproperties or no phase transitions as N (N 1011 𝑐𝑚 2)Fundamental Hamiltonian:1H 1 𝑁 2𝑚 [ 𝑗 q A(𝑧𝑗 )] 2 𝑉𝑏𝑔 (𝑧𝑗 )} 𝑗 𝑘 V(𝑧𝑗 -𝑧𝑘 )Model Hamiltonian:1H 1 𝑁 2𝑚 [ 𝑗 q A(𝑧𝑗 )] 2 } ?, e.g. 𝑗 𝑘 (𝑧𝑗 -𝑧𝑘 ), 𝑧𝑗position of j-th electron

Electrons in Plane Technology made 2D possibleCoulomb potential is translation invariantPauli exclusion principle: spin degeneracySpin deg. resolved by magnetic fieldLorentz force:𝐹 𝑞𝑣 𝐵Quantum phases of matter at T 0.

Many Electrons in a Magnetic Field Landau solution: electron at position 𝑧,single electron wave function 𝜓𝑚 many electrons 𝑝(𝑧)𝑒1 𝑧241 4 𝑧 2𝑚𝑧 𝑒,,𝒑 𝒛 polynomial---describe how electronsorganize themselves under extreme conditions 𝜈 1, 𝑝 𝑧1 , 𝑧2 , , 𝑧𝑁 𝑖 𝑗 (zi zj ). p(z) for 𝜈 1?3

Laughlin State for 1/3Laughlin 1983, Nobel 1998N electrons at zi in ground stateGaussian 𝟏/𝟑 i j(zi-zj)32/4- z e i i

Laughlin Right?Physical Predictions:1. Elementary excitations have charge e/3 (Laughlin 83, Nobel 98)2. Elementary excitations are abelian anyons (Arovas-Schrieffer-Wilczek 84)Experiments:Laughlin wave function is a good model

Enigma of 5/2 FQHER. Willett et al discovered 5/2 in1987 Moore-Read State, Wen 1991Greiter-Wilczek-Wen 1991Nayak-Wilczek 1996Morf 1998 MR (maybe some variation) is a good trial state for 5/2Bonderson, Gurarie, Nayak 2011,Willett et al, PRL 59 1987A landmark (physical) proof for the MR state“Now we eagerly await the next great step: experimentalconfirmation.”---WilczekExperimental confirmation of 5/2:charge e/4 , but non-abelian anyons ?

Pfaffian StateG. Moore, N. Read 1991Pfaffian state (MR w/ charge sector) 𝑷𝒇 Pf(1/(zi-zj)) i j(zi-zj)22/4- z iiePfaffian of a 2n 2n anti-symmetric matrix M (𝑎𝑖𝑗 ) is 𝑛 n! Pf (M) d𝑥 1 d𝑥 2 d𝑥 2𝑛 if 𝑖 𝑗 𝑎𝑖𝑗 d𝑥 𝑖 d𝑥 𝑗Physical Theorem:Elementary excitations are non-abelian anyons, called Ising anyon Read 09

A Mathematical Classificationjoint work with X.-G. Wen (MIT and PI) How to label UNIQUELY a fractional quantumHall (FQH) state?A collection of model wave functions {Ψ𝑘 }--classification of FQH states. How to calculate topological properties of FQHstates from wave functions?E.g. Statistics of anyons unitary representationsof the braid groups.

Wave Function of Bosonic FQH State Chirality:p(z1, ,zN) is a polynomial (Ignore Gaussian) Statistics:symmetric anti-symmetric divided by 𝑖 𝑗 (zi-zj) Translation invariant:p(z1 c, ,zN c) p(z1, ,zN) for any c ℂ Filling fraction: lim𝑁𝑁 , where 𝑁 is max degree of any zi

Polynomial of Infinite Variables A sequence of translation invariant symmetricpolynomials {Pk 𝑝(𝑧1 , , 𝑧𝑁𝑘 )} is called a 𝜈polynomial of infinite variables if there is a𝑁𝑘positive 𝜈 ℚ such that 𝑙𝑖𝑚𝑘 𝜈, where𝑑𝑘𝑑𝑘 maximum degree of 𝑧1 in P𝑘 𝑝(𝑧1 , , 𝑧𝑁𝑘 ) is a model wave function of 𝑁𝑘electrons in a magnetic field. When a 𝜇-polynomial of infinite variablesrepresents a FQH state?

ExamplesLaughlin: 1/q, 𝑁𝑘 𝑘, q even𝑷𝒌,𝟏/𝒒 i j(zi-zj)qPfaffian: 1/𝑞, 𝑁𝑘 2𝑘, q odd𝑷𝒌,𝑷𝒇 Pf(1/(zi-zj)) i j(zi-zj)q

General ConstructionsGiven 𝑓(𝑧1 , , 𝑧𝑁 ) polynomial Symmetrization:𝑆 𝑓 𝑧1 , . , 𝑧𝑁 𝜎 𝑆𝑁 𝑓(𝑧𝜎 1 , , 𝑧𝜎𝑁)is symmetric. Note 𝑆 𝑧1 𝑧2 0. Center-of-mass substitution:(𝑐)(𝑐)T 𝑓 𝑧1 , . , 𝑧𝑁 𝑓(𝑧1 , , 𝑧𝑁 ),z1 zN(𝑐)𝑧𝑖 zi . Note T 𝑧1 𝑧2N 0.

Pauli Exclusion Principle Ψ𝑘 2 is probability The probability for two electrons at the sameposition is zero, so 𝑝(𝑧1 , , 𝑧𝑁𝑘 ) 0 whenever𝑧𝑖 𝑧𝑗 for some 𝑖 𝑗. But this is encoded in theVandermonde factor (𝑧𝑖 𝑧𝑗 ). How about more than 𝑎 2 electrons in thesame position?Poly. 𝑝(𝑧1 , , 𝑧𝑁𝑘 ) “vanish” at certain powers{Sa} when a particles are brought together

Pattern of Zeros QuantifiedGeneralized Pauli Exclusion PrincipleGiven poly. 𝑝(𝑧1 , , 𝑧𝑁𝑘 ) :𝑝 𝑧1 , , 𝑧𝑁𝑘 Sa,k min{𝑖𝑖𝑖𝐼 , I i , , i , 𝑧𝐼 𝑧 1 𝑧 2 𝑧 𝑛𝑐𝑧1n𝐼 𝐼𝑛1 2𝑎𝑗 1 𝑖𝑗 }---minimaltotal degrees of a variables.If 𝑆𝑎,𝑘 𝑆𝑎 for all 𝑘 such that Nk 𝑎, then the sequence{Sa} of integers is called the pattern of zeros (POZ) ofthe polynomial of infinite variables.Morally, {Sa} model wave function and encode manytopological properties of the FQH state.

CFT ExamplesLaughlin: Sa qa(a-1)/2, 1/q, 𝑁𝑘 𝑘𝑷𝟏/𝒒 i j(zi-zj)qPfaffian: Sa a(a-1)/2-[a/2], 1, 𝑁𝑘 2𝑘𝑷𝟏/𝟐 Pf(1/(zi-zj)) i j(zi-zj)In a CFT, if Ve is chosen as the electron operator and aconformal block as a W.F.If Va (Ve)a has scaling dimension ha, thenSa ha-a h1

Quantum Hall State A quantum Hall state at filling fraction 𝜈 isa 𝜈-polynomial of infinite variables whichsatisfies the UFC and nCF conditionsand whose POZ has even 𝚫𝟑 . Classification:1) find all possible POZs of FQH states,2) realize them with polynomials,3) when POZs are FQH states?

Fuse a-electronsGiven a-electrons at {𝑧𝑖 , 𝑖 1, , 𝑎}set 𝑧𝑖 𝑧1𝑎 𝜆𝜉𝑖 ,where 𝑧1𝑎 ( 𝑖 𝑧𝑖 )/𝑎 , and 𝜉𝑖 2 1.Imagine 𝑧𝑖 as vertices of a simplex, then 𝑧1𝑎is the barycenter of the simplex. As 𝜆 0,𝑧𝑖 𝑧1𝑎 keeping the same shape.Sphere S2a-3 of {𝜉𝑖 } parameterizes the shapeof the a-electrons.

Unique Fusion ConditionTake a-variables zi fusing them to z1(a)The resulting polynomials (coefficients of 𝜆𝑘 )pk z1a , 𝜉1 , , 𝜉𝑎 ; 𝑧𝑎 1 , , 𝑧𝑛depend on the shape {𝜉𝑖 } 𝑆 2𝑎 3 of {zi}.If the resulting polynomials of z1(a),za 1, ,znfor each k of pk (𝜉𝑖 ) span 1-dim vectorspace, the poly. satisfies UFC.

Derived PolynomialsGiven 𝑝(𝑧1 , 𝑧𝑁 ), if all variables are fusedto new variables zi(a) and UFC is satisfied,𝑎then the resulting new polynomial 𝑝(𝑧𝑖 ) iswell-defined, and called the derivedpolynomial.Derived polynomials for Laughlin states:𝒂𝒃 𝒒𝒂𝒃𝒂𝒂 𝒒𝒂𝟐 𝒂 𝒃 𝒊,𝒋 (𝒛𝒊 𝒛𝒋 ) 𝒂 𝒊 𝒋 (𝒛𝒊 𝒛𝒋 )

n-cluster FormIf there exists an n 0 such that for any 𝑘, 𝑛 𝑁𝑘 ,then the derived polynomial of n-clusters is𝒏 𝑸𝒏 𝒂 𝒃 (𝒛𝒂 𝒛𝒃 )The poly. has the n-cluster form (nCF)nCF reduces pattern of zeros to a finite problem:Sa kn Sa kSn kma k(k-1)mn/2, where 𝜈 n/m.

Pattern of Zeros ClassificationTheorem (Wen-W.)If a 𝝂-polynomial of infinite variable {p(zi)} satisfy UFC and nCF for n,set m Sn 1-Sn. Then1) mn even, and n/m2) Sa b-Sa-Sb 03) Sa b c-Sa b-Sb c-Sc a Sa Sb Sc 04) S2a even5) 2Sn 0 mod n6) Sa kn Sa kSn kma k(k-1)mn/2POZ is not complete data for FQH state.Puzzle: Need 𝚫𝟑 Sa b c-Sa b-Sb c-Sc a Sa Sb Sc to be EVEN!

What Are Polys of Infinite Variables? Represent topological phases of matter. Universal properties of topological phasesof matter are encoded by TQFTs/modulartensor categories/CFTs, so polynomials ofinfinite variables are TQFTs/MTCs/CFTs.How does this connection manifest, e.g.how to derive MTCs/CFTs from POZs?

(2 1)-TQFTModular Tensor Category (MTC)Topological Phase of MatterTopological Quantum ComputationTopological phases of matter are TQFTs in Nature and hardwarefor hypothetical topological quantum computers.

Laughlin wave function is a good model. Enigma of 5/2 FQHE R. Willett et al discovered 5/2 in1987 Moore-Read State, Wen 1991 Greiter-Wilczek-Wen 1991 Nayak-Wilczek 1996

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