Low energy effec,ve theories for metals. Sung-Sik Lee McMaster University Perimeter Ins,tute

Transcription

1 Low energy effec,ve theories for metals Sung-Sik Lee McMaster University Perimeter Ins,tute

2 Goal of many-body physics : to extract a small set of useful informa,on out of a large number of degrees of freedom C/T = A + B T 2 T 2 { (x i, p i ) i=1,,10 23 }

3 The rule is simple

4 Brute force method is not useful in most cases It is in general impossible to solve the Schrodinger equa,on for interac,ng par,cles Size of Hilbert space grows exponen,ally with the number of par,cles possibili,es

5 Universality class metal Topological phases New phase of ma]er yet to be discovered

6 ψ> is complicated and system dependent, but low energy proper,es can be simple and universal Specific heat is propor,onal to T for a large class of metals at low temperatures Hall conductance in quantum Hall states is quan,zed as integer or frac,onal mul,ple of e 2 /h Different systems in one universality class (phase) share common proper,es at low temperatures although they are different microscopically

7 One of the main goals in condensed ma]er physics is to classify different phases of ma]er and understand low temperature proper,es using effec,ve theories Energy level Energy landscape E n E 0 T Low energy degrees of freedom

8 Origin of simplicity at low energies At long wavelength, short distance informa,on is averaged out

9 Different phases of ma]er Gapped phases (no IR d.o.f.) (trivial) insulator Topological phases (sub-extensive IR d.o.f.) Quantum Hall liquids, topological insulator Gapless phases (extensive gapless modes) Rela,vis,c CFT (z=1; graphene, Ising cri,cal point ) Fermi surface (metal)

10 Different phases of ma]er entanglement Metals Rela,vis,c QFT Topological Phases Trivial Insulator # of Gapless mode

11 Metals

12 Fermi Gas Ground state p y H = X k, 0 k n k, p y Excited state k F p x p x Many-body eigenstates are labeled by a set of occupa,on numbers of singlepar,cle states n k1, 1,n k2, 2,...>

13 Interac,ng Fermions p y k+q k -q p y k k p x p Quantum fluctua,ons n k1, 1,n k2, 2,...> is no longer an eigenstate

14 Fermi Liquids In certain metals, the low temperature proper,es of interac,ng fermions are remarkably similar to those of the non-interac,ng Fermi gas Specific heat : C ~ T Magne,c suscep,bility : χ ~ const. Landau postulated that low energy eigenstates of the interac,ng fermions are s,ll labeled in the same way the non-interac,ng eigenstates are labeled n k1, 1,n k2, 2,...> 0 The total energy has non-linear terms : H = X k, kn k, + 1 V X k,k 0,, 0 F, 0 (k, k 0 )n k, n k 0, 0

15 Microscopic jus,fica,on of Landau Fermi Liquid theory p y [Shankar,Polchinski] k+q k -q k k k F k E k p x k k k k At low energies, the phase space for non-forward sca]erings is small : only forward sca]erings are important par,cles created near FS have long life,me Low energy eigenstates are s,ll labeled by occupa,on numbers of quasipr,cle

16 Non-Fermi liquids

17 ky Strongly Correlated Metals kx e - [Custers et al.(2003)] Son collec,ve modes in the system (such as order parameter fluctua,ons at quantum cri,cal point) can cause strong quantum fluctua,ons of FS

18 A route to non-fermi liquid : long-range force ky K+q k -q kx Fermi surface + gapless boson k V(q)~1/q 2 k Non-forward sca]erings are enhanced by long-range interac,ons mediated by collec,ve modes Bare fermion quickly decays into a complicated superposi,on of states Single par,cle is no longer a good basis to understand low energy proper,es

19 Examples of non-fermi liquids Collec&ve mode Momentum of collec&ve mode Cri,cal current cri,cality (Emergent) Gauge field 0 Nema,c cri,cality Nema,c order 0 Charge density wave cri,cality Charge density order Q 0 Spin density wave cri,cality Spin density order Q 0

20 In d=2 K+q Hot Fermi surface q Hot spot k Q=0 Nema,c, ferromagne,c QCP Spin liquids with emergent gauge boson Q 0 Spin & CDW QCP

21 Theore,cal Status of Non-Fermi liquids in 2+1D Coupling between fermion and boson become strong even though bare coupling is weak (characteris,c of low dimensionality) A small parameter is needed to study the system in a controlled way

22 Different routes to tame quantum fluctua,ons Pro Con Large N Easy to keep symmetry Not controlled Dynamical tuning Easy to keep symmetry Breaks locality Tune dimension Keeps locality UV/IR mixing Tune co-dimension Keep locality Break some symmetry

23 Large N i=1,2,,n For large N, collec,ve modes gets dressed heavily with fermion clouds This appears to suggest that effect of fluctua,ng boson on fermion is small therefore processes which involve excita,ons of mul,ple bosons are systema,cally suppressed for a large N However, quantum fluctua,ons are not tamed due to an extensive gapless modes

24 Dynamical tuning ky K+q k -q kx Fermi surface + gapless boson k V(q)~1/q a k Tune a from 2 to 1+ε, which makes the boson s,ffer E Breaks locality of the theory k

25 Tuning dimensions d (space dim) m (FS dim)

26 Tuning dimensions d (space dim) Size of FS enters as scale (UV/IR mixing) m (FS dim)

27 Tuning co-dimensions d (space dim) 3 2 k x k z m (FS dim)

28 Perturba,ve non-fermi liquid (Q=0) ψ + k K F K F k ψ Perturba,ve Non-Fermi liquid ε d = 2 d = 5/2 d = 3 Strongly interac,on Non-Fermi liquid Marginal Fermi liquid Fermi liquid with decoupled boson

29 Perturba,ve non-fermi liquid (Q 0) d = 2 ε d = 3 Cri,cal dim Perturba,ve window

30 Summary Effec,ve theory captures universal low energy physics Fermi liquid theory successfully describes conven,onal metals Effec,ve theories for non-fermi liquid metals should incorporate quantum fluctua,ons in a controlled way