Statistical Transmutations in Doped Quantum Dimers

Transcription

1 Statistical Transmutations in Doped Quantum Dimers Arnaud Ralko, Institut Néel, Grenoble. Annecy, 8 april 0.

2 Collaborators Didier Poilblanc (LPT - Toulouse / France) Pierre Pujol (LPT - Toulouse / France) Masaki Oshikawa (ISSP - Tokyo / Japan) Daniel Cabra (IFLP - La Plata / Argentina) Carlos Lamas (IFLP - La Plata / Argentina)

3 Scope Quantum Spin Liquids in Mott insulators Doped quantum dimer models Statistical transmutation Quantum phase transitions

4 Physical Motivations What is the general situation for the QAF on lattices? E Mott Insulators Doping Metals C > C Cooling J X i,j S i S j? Anderson (87) High Tc Superconductors electron, spin up Frustration: electron, spin down Colomb U >> J<0 J>0?

5 Heisenberg Model What are the possible scenarios? H = J S i.s j SU() + spatial symmetries: translations, point group Presence of a magnetic long range order: SU() broken D: up to Néel temperature D: only at T=0K Low energy excitations: Spin waves h S z i U(), spin rotation along z-axis S i.s j m q cos(q.(r i r j )) as r i r j

6 Other scenario: Spin liquids No magnetic long range order Shastry & Sutherland (8) (r i r j ) S i.s j e Even at T=0K E E s = (r i r j ) /a SU() is preserved Ground state made of Spin Singlets C S=0 states s C = p How to dope it? E i Removing electrons (holons) Magnetic field h (spinon) i = + 0i = p i = s h c 0i i h

7 Dimers in the nature: SrCu (BO ) Shastry & Sutherland (8), Kageyama et al. (005) No magnetic long range order (T ) T / e s/t Doping the singlet GS Increasing H Triplet Magnetization Plateau Static dimer background Bose-condensation of triplets Exotic phases: SF, SS

8 Dimers in the nature: SrCu (BO ) Shastry & Sutherland (8), Kageyama et al. (005) No magnetic long range order (T ) T / e s/t Doping the singlet GS Increasing H Triplet Magnetization Plateau Static dimer background Bose-condensation of triplets Exotic phases: SF, SS Holon quantum dynamics Statistics of Holons? Connection with High-Tc

9 Dimers in the nature: ZnCu(OH)6Cl PRL 04, 470 (00) P H Y S I C A L R E V I E W L E T T E R S Most frustrated spin-/ system Helton et al. PRL 00 G. (color online). (a) The in-phase component of the ac sceptibility, measured at 00 Hz with an oscillating field of Oe. (b) A scaled (T ) plot of theh/t ac susceptibility data measured at nzero applied field, plotted as 0 act with ¼ 0:66 on the y is and B H=k B T on the x axis. Inset: A scaled plot of the dc agnetization, showing MT 0:4 vs B H=k B T. Ideal spin-/ QAF Non magnetic order up to 50mK Mendels et al. PRL 007 Finite susceptibility (Z liquid?) GS at the proximity of a QCP and/or influenced by disorder FIG. (color online).

10 Impurities in the Kagomé Spin-/ QAF Real system: Zinc impurities ranging in 6% to 0% Hopping term added in the effective H Hao & Tcherbyshyov (00) t Possible comparison to NMR experiments What is the parent insulating GS? Strange critical behavior connected to holons

11 Doped Quantum Dimer models

12 Effective models derived from microscopic systems Heisenberg Spin-orbital Quantum Dimer Models Rokhsar & Kivelson (88), Moessner & Sondhi (0), AR et al. () Vernay, AR, Becca, Mila (06) Quantum Dimer Model: projection onto the singlet subspace = ( ) = O, = Id + 4 A + 6 B + Sutherland (88) H e, = h H i 'O,

13 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term

14 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term J: Kinetic term

15 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term J: Kinetic term t: Holon hopping term +t +t Dimers in competition Doping with holons

16 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term J: Kinetic term t: Holon hopping term +t +t

17 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term J: Kinetic term t: Holon hopping term +t +t

18 Rokhsar & Kivelson (88) Quantum Dimer Models Dimer background quantum dynamics H = v( + ) J( + ) v: Potential term J: Kinetic term t: Holon hopping term +t +t But what are the holon properties? H = H v + H J + H t Analitycal methods (Jordan-Wigner transformation) Exact diagonalizations, Quantum Monte-Carlo AR et al. PRL(0), PRB(0)

19 H = v( + ) J( + ) +t +t Triangular frustrated Square not frustrated Kagomé * role of the parent insulating state * importance of the statistics * effect of the frustration even more frustrated How to properly define the holon statistics? AR et al. PRL(0), PRB(0)

20 Complex structure of the Hamiltonian We start with the unfrustrated case J>0, t>0, bosons t t H a H a H b F B H = H v + H J + H t F B H c Four Hamiltonian classes t t H d J J H a H a 0 F B same spectrum Depending the sign of J, bosons transmutes in fermions

21 D Jordan-Wigner transformation Fradkin (88), Wang (9) From fermionic to fractional statistics: Fermion Anyons a i a + j + ei a + i a j = i,j a i = ei i c i i X l6=i c i c + j n l arg(z i z l ) + c+ i c j = i,j z l arg(z i z j ) = arg(z j z i ) ± z i Fermion Semion Boson 0 / Anyons are hard-core thanks to the Pauli principle of initial fermions Mean-field: a flux tube 0 attached on each electron Aharonov-Bohm phase when anyons are exchanged: e e ~ H ~A ~ dl = e i

22 Application to H = H v + H J + H t Rewrite dimers Impose constraint on site i Write down the Hamiltonian Dimer Kinetic h (J) i,j,k,l = b + i,j = p c + i, c+ j, c + i, c+ j, a + i a i + X b + i,j b i,j = jn.n. J(b+ i,j b+ k,l b j,kb l,i + h.c.) l k Dimer Potential h (v) i,j,k,l = v(b+ i,j b i,jb + k,l b k,l + b + j,k b j,kb + l,i b l,i) i j Perform the Jordan-Wigner h (J) i,j,k,l h( J) i,j,k,l = a i = e i i c i J( b + i,j b + k,l b j,k bl,i + h.c.) h (v) i,j,k,l h(ṽ) i,j,k,l =ṽ( b + i,j b i,j b+ k,l b k,l + b + j,k b j,k b+ l,i b l,i ) b i,j = e i( i+ j ) bi,j J = ṽ = J v (with fixed gauge) Equivalence proved J F B J

23 arxiv: v [cond the resonating VB state (RVB), a state with only expolattices lead to quite different insulating states. Indeed, nentially decaying correlations and no lattice symmetry an ordered plaquette phase appears on the square lattice breaking. A simple realization of RVB has been proimmediately away from the special RK point, whereas a posed by Rokhsar and Kivelson (RK) in the framework RVB liquid phase is present in the triangular lattice of an effective quantum dimer model (QDM) with only local processes and orthogonal dimer coverings []. Even though the v relevancejof these models t for the description staggered liquid columnar of SU() Heisenberg models is still debated, this approach is expected to capture the physics of systems that natrvb urally possess singlet ground states (GS). For instance, 5 v specific quantum dimer models have recently been dej RK rived from a spin-orbital model describing LiNiO [], or sites 88 sites antiferromagnet [4]. In a refrom the trimerized kagome.0 ofu doped (at T=0) generalizing : u : u cent work, a family : holon : : u QDMs : u 5 esf esf the so-called RK point of Ref.[] has been constructed SS.5 columnar plaquette staggered and investigated[5], taking advantage of a mapping to.0 classical dimer models [6] that extends the mapping of the RK model onto a classical model at infiniteesf temperps SF or Boseliquid? esf esf 4eSF SS esf 0.5 FIG. : (color online) Schematic phase diagrams for the triature, with evidence of phase separation at low doping. angular and the square lattice However, the soluble models of Ref.[5] are ad hoc con structions, and this call :for the investigation of similars t PSt esf : u : u : holon : u u : u In this Letter, we investigate in details Complex esf the properties esf PS SF or Boseliquid? esf esf 4eSF issue in the context of more realistic models. In that resites 88 sites.5 of model () on the square and triangular lattices at fi.0 spect, a natural minimal of : u : u : holon : u :model u : uto describe the motion 5 nite doping. Building esf on the differences between theesf two.0 charge.5 carriers in a sea of dimers is the two-dimensional SS lattices in the undoped case, we investigate to which exps Complex Fermiliquid esf esf 4eSF quantum hard-core PS esf Complex dimer-gas Hamiltonian: esf esf tent the properties of the doped system are governed by F of the insulating parent state. This investithebnature c c Bt F c c H = v Nc cc J 0.0 PS SF or Boseliquid? esf esf 4eSF SS esf esf gation is0.based on exact Diagonalisations and extensive ) c 0 (c,c (c,c ) Complex :Fermiliquid esf esf 4eSF Green s Function Monte-Carlo (GFMC) simulations [7] : u PS : u holon : u : u : u () essentially free of the usual finite-size limitations [8]..5.0 : u : u where the sum : holon : u runs : u over : u all configurations of the PS on (c) Phase separation: At small t, it is expected that esf holes esf Complex esf PS SF or Boseliquid? esf esf 4eSF t.0 t H=H +H +H Under doping : generic -e superconducting phase a PS esf Complex x a d b Hb Hc esf c Ha x b c square Zero doping at J>0, rich phase diagram Hd x FIG. 0. Phase diagrams of the four models (a), (b), (c) and Fermiliquid esf fromesf (d)esf at V / J PS = 0. Complex and t/ J = 0.5 derived Fig.6 4eSF and d Fig.8. All have both the e-superfluid (e-sf) and e-superfluid triangular Quantum phase transitions

24 TABLE V. Classification of the possible phases, including various superfluid (SF) phases, that may occur in doped QDM s on the triangular lattice. Such phases can be distinguished from the sign of the compressibility κ, the long-distance properties ( means short-range, LR means long-range) of various correlations, or the effective charge deduced from periodicity of the GS energy versus a magnetic flux inserted through a torus. sgnb and sgnf were defined in Ref. to analyze the node content of the GS wave function. Quantum phase transitions <0 >0 LR >0 LR >0 >0 88 sites >0 : holon : u : u : u >0 >0 ak al ai aj LR LR LR (weak) 5 ai Si,j aj a PS SS e-sf e-sf sites.0 Bose liquid : u : u Fermi liquid.5 Complex phase bi,j bi,j bk,l bk,l κ b Observables Phases LR SS sgnb sgnf Flux periodicity e e 5 e e e e esf e 0 0 < sgnf < 0 < sgnb < esf 0 < sgnf < 0 < sgnb <.0 SS esf operator in a complicated manner. Namely, (n) (n% ) S:k,i S al aisf or Boseliquid? : u holonak:)(a u j : u :l,j u PS: u esf esf PS SF or Boseliquid? esf esf 4eSF function is characteristic of order. The wave vector at 5 which the associated structure factor diverges defines the wave vector. t t PS esf Complex esf esf esf 4eSF c In principle, one can use the new correlations G i,j and sites 88 sites n n% : holon : u : u : u Pi,j,k,l to refine the previous phase diagrams (Ni,j,k,l was used : u : u 5 SS previous work to determine esf the and SS regions). esfto a (n) (n% ) in + ai S al )(aj Sk,j ak Complex Fermiliquid esfof theesf 4eSF of the numerical results doped QDM s, PS esf Complex l,i esf esf d easepsthe analysis n n% String a classification of the various possible phases based on simple B F B F + {loop terms}, (45) = Bi,j Bk,leSF isorprovided in Table V. PS SF Boseliquid? esf esf 4eSF esf b 0.0considerations SS We note that there is no phase where there is a charge-e Complex esf esf from 4eSF where first part of:fermiliquid the side is obtained a : u PS :the u holon right-hand : u : u : u condensation simultaneously with a dimer long-range order x closure relation involving all pairs of retraceable strings b b b and a Si,jeSF aj.) This is because ( LR for both b :% u= n, : ui.e., : holon : u : u : u i,j k,l i,j k,l i n PS esf Complex esf esf esf 4eSF c 0.0 PS 0. SF or Boseliquid? t t existence of the dimer long-range order leads to confinement (n) (n ) (n) (n ) % Sk,i Sl,j + Sl,i Sk,j = bij bkl, (46) of holons Ha x Hb Hc Hd e-sf phase FIG. 0. Phase diagrams of the four models (a), (b), (c) and Fermiliquid esf fromesf PS esf Complex esf (d)esf at V / J PS = 0. Complex and t/ J = 0.5 derived Fig.6 4eSF and d and the rest corresponds to pair hopping dressed with extra C. Numerical results Fig.8. All have both the e-superfluid (e-sf) and e-superfluid x n

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