1. Gauge invariance and topological order in quantum many-particle systems 오시가와 마사기 (Masaki Oshikawa) 동경공대 (Tokyo Institute of Technology) 2005 년 10 월 28 일 @ 한국고등과학원

2. Commensurability and Luttinger’s theorem implications of (fractional) particle density (“old” stuffs) Ground-state degeneracy and topological order what is the topological order and when do we find it? (more recent developments)  

3. Quantum phases and transitions (at T=0) gap Phase I Phase II critical point (gapless) Typical example: Ising model with a transverse field in d-dim. (equivalent to classical Ising in (d+1)-dim.) ordered phasedisordered phase

4. Renormalization Group Critical point = gapless RG fixed point There is always a relevant perturbation! We have to fine-tune the coupling to achieve the criticality

5. However ……. there are many gapless systems in cond-mat physics, without any apparent fine-tuning! solids, metals, etc. …… Why is the gapless phase “protected”? Nambu-Goldstone theorem: gapless excitations exist if a continuous symmetry is spontaneously broken explains gapless phonons in solids but what about metals?? Let’s seek a new mechanism……

6. Magnetization process of an antiferromagnet (at T=0 ) classical picture H

7. magnetization curve H m saturation

8. Magnetization process in quantum antiferromagnets Long history of study Exact magnetization curve for S=1/2 Heisenberg antiferromagnetic chain (Bethe Ansatz exact solution) Quantitave difference from classical case No qualitative difference??

9. New feature in the quantum case Shiramura et al. (1998) [H. Tanaka group, Tokyo Inst. Tech.]

10. H m magnetization plateau difficult to understand in classical picture! Quantization condition for a plateau: n : # of spins per unit cell of the groundstate S : spin quantum number (M.O.-Yamanaka-Affleck ’97) T=0 10

11. Understanding the quantum magnetization process At T=0, the system should be in the ground state magnetization curve = magnetization of the ground state for the Hamiltonian (which depends on the magnetic field)

12. Hamiltonian: Magnetic field (Zeeman term) Exchange interaction (typical example) Let us assume that the interaction is invariant under the rotation about z-axis (direction of the applied field)

13. We can choose simultaneous eigenstates of and They are also always eigenstates of no change in the eigenstates even if the magnetic field is changed! how does the ground-state magnetization increase by the magnetic field?

14. E H lowest energy state with lowest energy state with  g.s. magnetization = MM+1 plateau of width  gap

15. For any (finite size) quantum magnet (with the axial symmetry) the magnetization curve at T=0 consists of plateaus and steps! In the thermodynamic limit (infinite system size) “gapless” (  ! 0 above the ground state) : smooth magnetization curve “gapful” (  remains finite above the g.s.): plateau

16. H T=0 gapful! gapless m when can the quantum magnet be gapful? gapful phases are rather “special”!

17. Quantum magnet as a many-particle system e.g. consider S=1/2 “down” “up” empty site occupied by a particle particle hopping interaction particle creation op. annihilation op.

18. When can the quantum many-particle system on a lattice be gapful? usually, particles can move around, giving gapless (arbitrary low-energy) excitations A finite excitation gap may appear if the particles are “locked” by the lattice to form a stable ground state. ( particles are then mobilized only by giving an energy larger than the gap.)

19. To have the particles “locked”, the density of the particles must be commensurate with the lattice. 1 particle/ unit cell (= 2 sites) add extra particles (“doping”) mobile carriers

20. commensurate density particle density (# of particle/site) # of sites/ unit cell of the g.s. # of particles/unit cell of the g.s. particles may be “locked” to form an insulator, with a finite gap incommensurate density particles are mobile, forming a conductor with gapless excitations 20 (possibly with SSB of translation symmetry ---- will come back on this later)

21. Finite-temperature transition near the plateau magnetization//H vs. T MFT T m

22. Magnon BEC picture Tsuneto-Murao 1971...........Nikuni et al. 2000 singlet on dimer (lowest) triplet on dimer vacuum magnon (boson) magnetic fieldchemical potential ordering transition magnon BEC Dispersion:(near the bottom)

23. Consequences of the BEC picture condensed magnons Quantum spin system in a field = “particles” with a tunable chemical potential Nikuni, MO, Oosawa, Tanaka 2000

24. Back to the quantization….. e.g. consider S=1/2 “down” “up” empty site occupied by a particle commensurability condition

25. Is it really true? physical properties of the system (such as magnetization curve): generally depends on Hamiltonian ground state in strongly interacting system: very complicated! why would the commensurability condition be valid in strongly interacting systems??

26. d=1 A generalization of Lieb-Schultz-Mattis argument (1961) shows There are q degenerate groundstates if = p/q and if the system has a gap (M.O.-Yamanaka-Affleck, 1997) d ¸ 2 Topological argument (with assumptions) Relation to Drude/Kohn argument Rigorous proofs (M.O. 2000) (M.O. 2003) (Hastings 2004, 2005)

27. Insulator vs. conductor Linear response theory Drude weight D=0 : insulator D>0 : conductor (Kohn, 1963)

28. Real-time formulation of D initial condition: ground state at t=0 taking t! 1, T ! 1 (as long as the linear response theory is valid)

29.  circumference: E uniform electric field cf. Laughlin (1981)

30. energy gain 30

31. (unit flux quantum) choose and take the limit Hamiltonian at t=T with the unit flux quantum is equivalent to that at t=0 with  =0 Does the groundstate go back to the groundstate? If so, the energy gain =0 thus the system is an insulator (no Aharonov-Bohm effect)

32. No change in the momentum?! As long as we choose constant-A gauge, Hamiltonian is translational invariant. Momentum is gauge-dependent!! large gauge transf.

33. To compare the momentum, we compare and lattice translation operator cross section Total momentum change (after large gauge tr.) and has same momentum (Lieb-Schultz-Mattis, 1961)

34. Momentum P x is defined modulo 2  The final state must be different from the initial state (g.s.) if  Z (for appropriate C) In order to have an insulator for an incommensurate particle density  Z, one must have low-energy state with the extra momentum 1 dim. 2dim.: 3 dim. and higher: no constraint from D=0 (M.O. 2003)

35. Application to gapless system non-interacting electrons = free Fermi gas Fermi sea Consider a system of electrons (fermions)

36. Landau’s Fermi liquid theory Interacting electrons: what happens?? elementary excitation: “quasiparticles” collective excitation in terms of electrons but behaves like free fermions “Fermi sea” of quasiparticles What is the volue of the “Fermi sea”? Luttinger’s theorem : V F is not renormalized by interactions

37. In some cases, the original proof by Luttinger does not apply, or is questionable…. eg. one dimensional systems systems involving localized spins (Kondo lattice) non-Fermi liquids Alternative approach?

38.  E cf. Laughlin (1981) adiabatically insert unit flux quantum (again!)

39. calculate the momentum change due to the flux insertion (i) by Fermi liquid theory (or any effective theory) (ii) using the large gauge transformation

40. Applications electrons coupled to localized spins (Kondo lattice) localized spins do contribute to Fermi sea volume! (if low-energy excitations are exhausted by Fermi liquid) “Fractionalized Fermi liquid” a phase that has similar low-energy excitations as the Fermi liquid but violates Luttinger’s theorem (with fractionalized spin exc.) (Senthil-Sachdev-Vojta, 2003)

41. M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000). [From http://sachdev.physics.harvard.edu/]

42. Effect of flux-piercing on a topologically ordered quantum paramagnet N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002). 1 2 3 L x -1 L x -2 LxLx vison LyLy [From http://sachdev.physics.harvard.edu/]

43. Flux piercing argument in Kondo lattice Shift in momentum is carried by n T electrons, where n T = n f + n c In topologically ordered, state, momentum associated with n f =1 electron is absorbed by creation of vison. The remaining momentum is absorbed by Fermi surface quasiparticles, which enclose a volume associated with n c electrons. A Fractionalized Fermi liquid. cond-mat/0209144 [From http://sachdev.physics.harvard.edu/]

44. “Bose volume” The present argument actually applies to system of boson as well. The momentum change due to applied electric field is “quantized”! The corresponding “Luttinger’s theorem” gives a quantization of magnus force in lattice bose systems at T=0 (Vishwanath and Paramekanti, 2004)

45. Summary Quantum many-particle systems on a periodic lattice  : # of particles / unit cell Topological restrictions: If the system is gapless the “Fermi(Bose) volume” is quantized -- “Luttinger’s theorem” If the system is gapful for  Z there must be q-fold groundstate degeneracy

46. conductor or insulator? (Kohn, 1963) magnetization plateau Luttinger’s theorem (1960) topological quantization gauge invariance and QHE (Laughlin, 1981) Lieb-Schultz-Mattis theorem (1961) Haldane conjecture (1983)

47. Topological restrictions: If the system is gapless the “Fermi(Bose) volume” is quantized -- “Luttinger’s theorem” If the system is gapful there must be groundstate degeneracy what does this mean? “Usually” it is a consequence of Spontaneous Symmetry Breaking characterized by a local order parameter e.g. Neel order

48. Topological degeneracy There is also an “unusual” possibility that the groundstate degeneracy is due to a “topological order” Characteristics of the topological degeneracy (i) Degeneracy (# of g.s.) depending on the topology of the system (sphere, torus….) well known for Fractional Quantum Hall Liquids [ cannot be understood with the ordinary SSB] (ii) Absence of the local order parameter

49. Topological degeneracy degenerate g.s.: indistinguishable by any local operator ground-state degeneracy N depends on topology of the system g=0 g=1 g=2 not a consequence of a ordinary SSB….. a signature of a topological order!

50. Quantum many-particle systems on a periodic lattice  : # of particles / unit cell Topological restrictions: If the system is gapful for  Z there must be some kind of order, either the standard SSB with a local order parameter or a topological order

51. Systematic determination of order parameter S. Furukawa, G. Misguich and M.O., cond-mat/0508469 How to find the order parameter without a prior knowledge? measure all the correlation functions? is there a better way? In a quantum system, ground-state (GS) degeneracy signals some kind of order! can be found without knowing the order parameter!

52. Suppose there are two-fold (quasi-) degenerate GSs below the gap, in a system of finite size L (sufficiently large) Energy gap and usually the degeneracy is a consequence of SSB Symmetry-Breaking GSs and (linear combinations of and ) Order parameter: an observable which can distinguish these GSs

53. : observable defined on area  is an order parameter, if “Difference” of the two GSs w.r.t.  for any normalized Information on the expectation value of arbitrary observable on  is encoded in the reduced density matrices

54. whereare eigenvalues of if “diff” is non-zero on an area   there is an order parameter defined on  Properties of “diff” Maximum is achieved with the “optimal order parameter” If  µ 

55. Simple examples Neel ordered state diff = 2 already for  = 1 spin  Spontaneously dimerized state  diff = 0 for single spin (no order parameter) diff = 3/2 for two spins

56. S=1/2 ladder with 4-spin exchange studied by many people gap [schematic phase diagram] 0.070.1476 0.39 2-fold degenerate GSs in the both phases --- what are the order parameters for them? Phase I Phase II 

57. Symmetry-breaking GSs: two possibilities Finite-size (quasi-) GSs and : real (“time-reversal” invariant) : real in S z -basis (“time-reversal” invariant) “time-reversal” invariant “time-reversal” breaking so calculate both “diff1” and “diff2” separately We can’t know a priori which is the case;

58. Phase IPhase II Numerical result on 14x2 system (with periodic BC) 0* : exactly zero due to symmetries, even in a finite system Optimal order parameters on minimal area Phase I (leg) dimer order Phase II scalar chiral order (broken “time reversal”) reproduced known results!

59. crossing point of diff1 and diff 2: agrees very well with the exact

60. Quantum Dimer Model on Kagome Solvable Hamiltonian h: hexagon in the Kagome  loop involving only one hexagon h Misguich-Serban-Pasquier 2002 Zheng-Elser dimer shift along the loop

61. Exact solution GS(s): “Rokhsar-Kivelson” type RVB state Finite gap above the GS(s) GS degeneracy depends on the topology of the system cylinder: 2-fold, torus: 4-fold ……… Exact realization of “Z 2 spin liquid” What is the order parameter? “topological degeneracy” (uniform superposition of “short-ranged” valence bond states)

62. Order parameter of Kagome QDM We can show that between any (linear combinations of ) and for any local area  absence of local order parameter! stability of qubit against decoherence Expected property for the topological degeneracy, but is here shown explicitly and rigorously (cf. Ioffe-Feigel’man 2002)  system

63. Non-local order parameter For the “diff” to be non-zero,  must extend over the system non-local order parameter necessary to detect the “topological order” 

64. QDM on triangular lattice consider Rokhsar-Kivelson wavefunctions (in topologically distinct sectors) Is there a local order parameter? – apparently NO (not exactly solvable!)

65. Possible developments can we identify a “new” order parameter? combination with QMC/DMRG etc. relation to DMRG, (quantum) information theory degeneracy > 2 : optimization also on systematic evaluation of the stability of many-body “topological” qubits

66. How to detect the topological order Vishwanath-Paramekanti 2004 Even*Odd system: Momenta of the GSs: (0, 0) & ( ,0) whether the system has the SSB of translation symmetry or the topological order Even*Even system: Gauge argument Momenta of the GSs: (0,0) & ( ,0) SSB (0,0) & (0,0)topological Flux insertion = vison insertion

67. What is “order”? What is “phase”? We are just beginning to understand…. 감사 합니다