Laughlin spin liquids states on lattices

1. Laughlin spin liquids states on lattices A. Nielsen, I. Cirac and G. Sierra MPQ-Garching, IFT-Madrid Workshop on Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations Institut d'Etude Scientifique de Cargese (IESC), France, 24 June- 6 July 2013

2. Aim: To investigate interesting phenomena in many-body systems. Examples: Topological phases,superconductivity, quantum Hall effect, magnetism, edge states, ... Method: Construct simple models with particular physical properties. Wave function and/or Hamiltonian Why?: Fundamental understanding of how the phenomena can arise. Experimental simulations under well-controlled conditions. Practical applications.

3. Wave functions from CFT Why?

4. Wave functions from CFT Analytical wave function Why? Entanglement spectra Mathematical properties Parent Hamiltonians Excited states Correlation functions

5. Wave functions from CFT 1D and 2D Topological properties Physical properties Analytical wave function Why? Critical systems Entanglement spectra Mathematical properties Parent Hamiltonians Excited states Correlation functions

6. Wave functions from CFT G. Moore and N. Read (1991) 1D and 2D Topological properties Physical properties Analytical wave function Why? Critical systems Entanglement spectra Mathematical properties Parent Hamiltonians Excited states Correlation functions

7. Wave functions from CFT G. Moore and N. Read (1991) 1D and 2D Topological properties Physical properties Analytical wave function Why? Critical systems Entanglement spectra Mathematical properties Generalization of matrix product states Parent Hamiltonians Excited states Correlation functions

8. Based on • ”Infinite matrix product states, Conformal Field Theory and • the Haldane-Shastry model” J. I. Cirac and G. Sierra, Phys. • Rev. B 81, 104431 (2010); arXiv:0911.3029. • ”Quantum spin Hamiltonians for the SU(2)k WZW model”, • A. E. B. Nielsen, J. I. Cirac, G. Sierra, J. Stat. Mech. P11014 • (2011), arXiv:1109.5470. • - “Laughlin Spin-Liquid States on Lattices Obtained • from Conformal Field Theory". • Phys. Rev. Lett. 108, 257206 (2012); arXiv:1201.3096 • - ”Fractional quantum Hall states in lattices: Local models • and physical implementation”, A. E. B. Nielsen, G. • Sierra, J. I. Cirac; arXiv:1304.0717.

9. Plan of the talk • Infinite MPS and CFT • c=1 gaussian model applied to the XXZ model • WZW model SU(2)@1 and the s=1/2 Haldane-Shastry model • SU(2)@k -> Parent Hamiltonians from null vectors • Spin 1 HS model • Kalmeyer-Laughlin wave function and some properties • Truncated spin Hamiltonian and Fermi-Hubbard model for KL

10. Matrix Product States (see P. Corboz, B.Estienne,F. Verstraete talks) Consider a 1D spin 1/2 system with N sites and Hamiltonian The GS wave function is given in a local spin basis by The MPS is an ansatz of the form Where are dimensional matrices and vectors Bond dimension / Schmidt number / m of the DMRG

11. The entanglement entropy in a bipartition A U B scales as (1D area law) In a critical system described by a CFT (periodic BCs) c = central charge hence one needs very large matrices to describe critical systems Correlation length Another alternative is to choose infinite dimensional matrices: Infinite dimensional MPS = iMPS

12. CFT -> iMPS Consider a chiral massless boson with two point correlator Chiral vertex operators are the normal order exponentials Two point correlator is given by (scaling dimension) The vertex operators act on the Fock spaces generated by all the oscillators of the boson field. This allows for an infinite dimensional version of the MPS:

13. vacuum of CFT The iMPS is given by Using the vertex correlators one gets The delta function implies are variational parameters obtained by minimization of the GS energy and the symmetries of the Hamiltonian

14. Graphical representation of iMPS physical degrees Conformal map auxiliary space

15. Excitations in iMPS In CFT : excited states <-> conformal fields (primary and descendants)

16. Quenching ?

17. Example: anisotropic spin 1/2 Heisenberg model Phases of the model Translational invariance -> Marshall sign rule -> Minimize the energy ->

18. Overlap of exact and the CFT wave functions (N=20) The CFT wf is exact in two cases isotropic ferromagnetic chain XX chain At the isotropic AFH model Haldane-Shastry chain

19. Renyi entropy and spin correlators Spin Critical regime corresponds to a c=1 CFT Related to the Luttinger parameter in Lauchli talk

20. The iMPS and the Haldane-Shastry model (1988) HS wave function in the hard core boson variables positions of the bosons Map: This is the GS of the Haldane-Shastry Hamiltonian HS state: 1D Laughlin state of bosons at filling fraction 1/2

21. Relation between HS and iMPS Take Using the map CFT interpretation

22. The WZW model CFT with c =1 and two primary fiels Fusion rule: Unique conformal block (N even) The Haldane-Shastry state:

23. Generalization: Take any CFT iMPS = chiral correlators of CFT Examples: SU(2)@k WZW model with k=1,2,… The number of wave functions is given by the fusion rules

24. Parent Hamiltonians from null vectors SU(2)@k Kac-Moody algebra / current algebra Primary fields Null vector (Gepner-Witten) highest weight vector of a multiplet with spin

25. The whole multiplet of null fields can be written as K’s are Clebsch-Gordan coefficients Impose the decoupling of null fields in a correlator of primary fields Using the Ward identity

26. one finds that the conformal blocks satisfy a set of algebraic equations where

27. Define the operators Satisfy: Parent Hamiltonian (frustration free) GS is (are) their number is determined by Fusion Rules

28. SU(2)@k=1 Here Parent Hamiltonian standard HS Hamiltonian

29. Equations for spin correlators From the decoupling eqs. One gets a linear system of equations for spin-spin correlators In the uniform case we recover the Gebhard-Vollhardt result But we also find an exact formula for finite N

30. Four point spin correlator

31. SU(2)@k=2 Primary fields: Fusion rule: The spin 1 field –> one conformal block (like for s=1/2) Parent Hamiltonian (See also M. Greiter for a s=1 Hamiltonian)

32. Spectrum in the uniform case There are not accidental degeneracies-> No Yangian symmetry Not like in s=1/2 HS model which has the Yangian

33. Spin 1 wave function SU(2)@k=2 = Boson + Ising (c= 3/2 = 1+ 1/2) Primary spin 1 fields (h=1/2) Majorana fermion In the uniform case we expect the low energy spectrum of this model to be described by SU(2)@k=2 model Look at-> Renyi entropy and spin-spin correlator

34. Renyi entropy

35. Spin-spin correlator CFT prediction Suggest existence of log corrections (Narajan and Shastry)

36. CFT -> 2D wave funtions

37. Kalmeyer-Laughlin wave function (1987) Bosonic Laughlin wave function at filling fraction 1/2 • - Hard-core bosons located on a square lattice • -> spin singlet state • KL state describes a spin liquid: gap in the bulk, gapless edge • excitations, topological order, degeneracy depending on the • on the topology of the space, abelian anyons, etc.

38. Relation with SU(2)@k=1 model Take the z’s in the square lattice One can show that for a large lattices including the gaussian factor No need to add this factor by hand

39. Parent Hamiltonian Sum of two body and three body terms Breaks time reversal g(z), h(z): generic

40. If z’s are one the circle this reduces to the HS Hamiltonian 1D Haldane-Shastry state CFT model 2D Kalmeyer-Laughlin state • For Hamiltonians on a square lattice and a torus in the infinite • size limit see: • D. F. Schroeter, E. Kapit, R. Thomale, M. Greiter (2007). • - E. Kapit and E. Mueller (2010).

41. Spin-spin correlation function 200 spins on the sphere

42. Area law Topological entanglement entropy

43. Truncated Hamiltonian for the KL state Overlap

44. Spectrum on the torus In the limit N >>1 one expects 2 degenerate GS’s and a gap to the third excited state Many body Chern number =1 for two degenerate GS Agrees with the expected value for Laughlin state at ½ filling

45. Derivation from a Fermi-Hubbard Hamiltonian Couplings: U, t , t’

46. U>>t, t’ -> Mott insulating regime: each site occupied by single fermion Get an effective spin Hamiltonian to third order in t/U is Chern number for the effective model is well inside the topological region

47. Conclusions • CFT is a tool to propose wave functions for spin lattice systems in D=1,2. Provides a realization of infinite dimensional MPS. • Unification of the Haldane-Shastry and Kalmeyer-Laughlin models, obtaining generalizations to non-uniform models and higher spins. • Proposed a truncated Hamiltonian in the same universality class as the parent Hamiltonian for KL state and its realization with fermions trapped in optical lattices. • Spin version of the Moore-Read wave function with potential • applications to Topological Quantum Computation.

48. Thank You Merci beaucoup