Quantum Dimers and Spin-Charge Separation Charge fractionalization can occur in two dimensions. We have proven in a simple but important model that “holons” are deconfined. 1 Outline: 1. Spin-charge separation 2. Quantum dimers and the Rokhsar-Kivelson Hamiltonian 3. The triangular lattice 4. Deconfinement of holons 5. Supersymmetry work with R. Moessner and S. Sondhi slightly related work with K. Schoutens, B. Nienhuis and J. de Boer 2 Important principle: All of condensed-matter and particle (and much of AMO) physics is many-body physics. In particular, the ground state of a quantum system is not a “vacuum”. In condensed-matter physics, we have the Fermi sea. In particle physics, the ground state is full of virtual particles, which can pop in and out. As a result, many counterintuitive effects occur in nature. 3 excitation = quasiparticle = particle These words all mean an eigenstate of the quantum Hamiltonian with localized momentum and energy relative to the ground state One might expect that the quasiparticles over a Fermi sea have quantum numbers (charge, spin) of an electron. Not necessarily true! 4 “RVB” phase in 2d Electrons (spin 1/2, charge ) on a square lattice, one per site. Say the interactions favor a spin singlet on neighboring sites. The ground state will be a superposition of “dimer” states like £ ¢ ¡ Each dimer has spin ¡ and charge , so the ground state has spin and charge for £ sites. This is called a resonating valence bond state. 5 To get excited state, break one bond If there is no confinement, you make two quasiparticles. This state has the same charge as the ground state, but different spin. ¤ ¢ Thus these particles have charge ¡ and spin !! ¥ They are called spinons. Anderson; Kivelson, Rokhsar and Sethna 6 Say you remove two electrons, and create a defect: ¦ ¢ This two-hole state has no spin, and charge relative to the ground state. Each ¦ quasiparticle has charge and spin ¡ !! This is like oblique confinement in QCD at §©¨ : quarks are free, but have no color. The electron has “split” into a spinon and a holon! This is called spin-charge separation. 7 We understand spin-charge separation in one dimension. It occurs in many one-dimensional models, such as the Hubbard model; one can often compute the spectrum exactly by using the Bethe ansatz. Charge fractionalization has been observed experimentally in 1+1 dimensions by studying the shot noise for tunneling between fractional quantum Hall edges. 8 We are only beginning to understand spin-charge separation in two dimensions. Many theorists believe that the ideas of spin-charge separation and fractional charge are crucial to understanding high-temperature superconductivity. In particular, Anderson proposed that superconductivity evolves out of a liquid of these valence bonds. A liquid is phase with a gap, but without order. The RVB superconductivity is argued to occur when the holons condense. 9 To show that this idea has merit, we need to find a concrete model in 2d which has 1. A ground state made of a superposition of valence-bond states (i.e. a liquid, not an ordered solid) 2. Gapped excitations 3. Deconfined holons The quantum dimer model on the triangular lattice exhibits this behavior. 1. Rokhsar and Kivelson 2. Moessner and Sondhi 3. Fendley, Moessner and Sondhi 10 The quantum dimer model is a slight simplification of the full electron model: the only states we consider are the dimers (valence bonds). We require that the dimers have hard cores and that they are close-packed, meaning that every site is touched by precisely one dimer. We create holons by removing sites from the lattice. A typical configuration on the square lattice is We need a quantum Hamiltionian to act on this two-dimensional Hilbert space. 11 Most simple Hamiltonians one might write down (e.g. a hopping term for dimers) violate the hard-core/close-packing constraints. The simplest way f or to act locally is to act on flippable plaquettes and One can flip a plaquette, i.e. ¦ or ¦ without affecting any other plaquettes. 12 The RK Hamiltonian is then ¦ ¨ £ where £ counts the number of flippable plaquettes. Rokhsar and Kivelson, 1988 It turns out that in the continuum limit, this Hamiltonian is not Lorentz invariant in 2+1 ¨ ¢ dimensions (the dynamical exponent ). Henley One can show that this Hamiltonian arises from a large- £ limit of certain antiferromagnetic models. Read and Sachdev Quantum dimer models are equivalent to lattice gauge theories. Moessner, Sondhi and Fradkin; Senthil and Fisher. They also are useful for describing geometrically-frustrated magnets, observed in neutron-scattering experiments. 13 ¨ It is simple to show that if , the RK model on the square lattice has property 1): the ground state is the equal-amplitude sum of all the allowed states. To see this, consider the operator , acting on a single plaquette . On non-flippable plaquettes, ¡ . On the two kinds of flippable plaquette, ¨ ¦ ¦ ¦ ¦ 14 ¨ Then when , ¨ ¡ !" Thus the eigenvalues of obey . The equal-amplitude state obeys ¨ ¡ " ! # so it is the ground state. It’s an RVB state! 15 Remarkably, one can compute all correlators in the ground state, e.g. the dimer-dimer correlator ¨ $ % %'& * " ! !" , ,.- # ( / )+* ) * % where , creates a dimer on link . & $ % % The computation of is a classical 2d computation, because " ! is the sum of # all classical 2d dimer configurations. This problem was solved long ago by Kasteleyn and by Fisher by inventing the Pfaffian technique. For example, the number of ways of putting 32 dominos on a chessboard is... 16 12,988,816. 1 The Pfaffian of a matrix is e.g. for a 0 0 matrix ¦ ¨ 2 2 2 2 2 2 2 3 5 3 5 5 3 Pf 4 4 4 Every index appears exactly once, just like our close-packing constraint. To solve the dimer model on any planar lattice, draw arrows on every link of the lattice so that there are an odd number of arrows around each plaquette. Define the matrix 2 so ¦ ¨ ¨ 2 ¤ 2 ¤ 8 8 6 that 6 if an arrow points from to , if an arrow points from to , 9 9 7 7 ¨ : 2 ¡ and 6 otherwise. Then the number of dimer configurations on this lattice is 7 3 = ¨ ¨ 2 : Pf 2 det < ; In modern language, one can write this as a Grassmanian path integral F A A D C C A ¨ B CED $ : @? > ; < 17 In the continuum limit on the square lattice, this turns into a massless free fermion. Unfortunately, this means that the correlators decay algebraically. For example, ¤ & $ % % # sq HG ¦ % %I& ) ) In dimer language, a holon is a monomer (a site without a dimer touching it). The & holon-holon correlator % % is # ¤ % %J& # sq HG 3 ¦ = % %K& ) ) Fisher and Stephenson, 1963 So the quantum dimer model on the square lattice does not satisfy criterion 2): when ¨ , the excitations are gapless. This coupling is a critical point separating two ordered phases. For , the dimers ML form columns; for they form staggered columns. MN 18 This line of approach to the problem was abandoned for a decade. Progress came when it was realized that the RK Hamiltonian on the triangular lattice has very different properties. The flip operator acts as e.g. ¦ O O : O O 19 The remarkable observation of Moessner and Sondhi is that on the triangular lattice, N P there is a whole RVB phase Q where the excitations are gapped. ¨ At the RK point , one can use the Pfaffian technique to compute correlators in the ground state. The dimer-dimer correlator falls off as T , ,K- X V VW = $ % %J& UT # tri SR where 3 5 = ¤ ¢ _ T ^ 3 ¨ ¨ a ¤ 0 0 a a a \ _ b Y 3 5 ¦ = 0[Z ¢ _ ^ ] ` The correlation length is only about one lattice spacing! 20 integ The where An The e dimer Green’ xact r al r c l l l e s i l s l xpression \ is s d e function bilinear u t t f u u t 8 s u ¥ n l n ¤ f in or g t ¢ the ¨ l $ w mvu t der t tr wJ w t f er i y mon w y ( y l can j k iv mions mon l y ed t w mon w r h t be b w' t y mon y z y m . x y m ¨ der {}| s Legendre s z ¡ x y z x z y | { iv x y | { | { / ed ¢ z y x z f z | { x y x y or | { 21 {}| (using ¢ _ tw ¥ in o ¥v ^ 1812!), t t t f p q p q p q p q er p q an p q p q p q p p p p mions n l p p p p u p p p u~ n identity p p p u p p n p p p n r p p r n p b l l p p n n ut p s u p u n umer in n p p s n nr n p u n it’ p n ~ p n the n f s u~ n s or p ~ n u u ical s l v n s u ~ same the er n s n ~ l p n v y p u p p r u ~ n alue s n complete ugly u l n s ~ nr r u n ~ n p s ~ ~ ro n p s r u n n . n w r n s s ~ r s n ~ is nr s n l s n u n l elliptic s s One can interpolate between the square and triangular lattices. Draw the triangular lattice as ¨ Just assign a variable weight for dimers on the diagonal bonds; ¡ is the square ¨ ¤ lattice and the triangular.