Fermi liquid & Non- Fermi liquids. Sung- Sik Lee McMaster University Perimeter Ins>tute

Transcription

1 Fermi liquid & Non- Fermi liquids 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 [r, p] =i < Ψ O Ψ >

4 But it is extremely difficult to solve 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 ψ> is complicated and sample dependent, but certain proper>es are simple and universal Specific heat is propor>onal to T, not T 0.99 or T 1.02 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 to one part in a billion, despite the presence of many defects Universality class (phase of maber) Different systems in one universality class (phase) share same proper>es at low temperatures although they are different microscopically

6 Universality class metal Quantum hall state New phase of maber yet to be discovered

7 One of the main goals in condensed maber physics is to classify different phases of maber 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 Low energy effec>ve theory book- keeping tool for the low energy manifold of many- body states Keep far fewer than the full degrees of freedom Emergent low energy degrees of freedom can be very different from microscopic degrees of freedom and obey different symmetry

9 Different phases of maber 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 maber Trivial insulator Topological insulator / IQHE Topologically ordered states / FQHE Specific heat exponen>ally small L d- 1 T # L d- 1 T # Entanglement entropy l d- 1 l d- 1 l d- 1 + topological term Rela>vis>c QFT L d T d term l d- 1 + universal ( log l for d=1 ) Fermi liquid L d T l d- 1 log l L l Non- Fermi liquid L d T a (a < 1 ) l d- 1 log l

11 Different phases of maber entanglement Fermi Liquid Non- Fermi Liquid FQHE; Topologically Ordered Phases Rela>vis>c QFT Trivial Insulator IQHE; Topological Insulator # of Gapless mode

12 Fermions at finite density

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

14 Interac>ng Fermions p y k+q k - q p y k k p x p n k1,σ 1,n k2,σ 2,...> is no longer an eigenstate due to quantum fluctua>ons caused by interac>ons

15 Energy level Perturba>on theory? Typically, the strength of interac>ons is much larger than the energy spacing between non- interac>ng states E n V V>> E 1 L E 0 We expect that there are strong mixings between non- interac>ng eigenstates, leading to new types of excita>ons What do new eigenstates look like? Do they behave as point- like par>cles, or something else?

16 Landau Fermi Liquid When the interac>on between electrons are short- ranged (e.g. screened Coulomb repulsion), 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 The total energy has non- linear terms : H = k,σ k n k,σ + 1 V n k1,σ 1,n k2,σ 2,...> Why does this descrip>on work so well? F σ,σ (k, k )n k,σ n k,σ k,k,σ,σ

17 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 scaberings is small : only forward scaberings are important par>cles created near FS have long life >me Low energy eigenstates are s>ll labeled by occupa>on numbers of par>cle

18 Non- Fermi liquid

19 A route to non- Fermi liquid : long- range force ky K+q k - q kx Fermi surface + gapless boson k V(q)~1/q a k Non- forward scaberings are enhanced by long- range interac>ons (singular in momentum space) mediated by gapless boson : 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 What replaces the par>cle descrip>on? Can one find a new degree of freedom which captures low energy excita>ons?

20 A route to non- Fermi liquid : long- range interac>on mediated by gapless boson ky kx Fermi surface + gapless boson Gapless boson describes sou collec>ve fluctua>ons of many- body systems : Quantum fluctua>ons of order parameters in metals Sta>s>cal fluctua>ons in the frac>onal quantum Hall state Collec>ve modes in spin liquid

21 Different roads to non- Fermi liquids Cri4cal mode Boson momentum Fermion- boson coupling Cri>cal current cri>cality (emergent) Gauge field 0 a µ j µ Ferromagne>c cri>cality Ferromagne>c order 0 φ c i,α σ αβc i,β Charge density wave cri>cality Charge density order Q 0 φ e i Q r i c i,α c i,α Spin density wave cri>cality Spin density order Q 0 φ e i Q r i c i,α σ αβc β

22 Interac>ng fermions K+q q k Fermions at different angles are mixed with each other through singular small angle scatterings.

23 Locality in the angular direc>on At low energies, fermions with angle θ couples only with the bosonic modes whose momenta are tangential to the Fermi surface at that angle. As a result, modes at different angles are decoupled except for the modes at the anti-pole.

24 Minimal theory : two- patch theory A closed Fermi surface consist of con>nuously many decoupled theory, each of which describes low energy excita>ons at angle θ and θ+π

25 Chiral Fermi surface B For chiral Fermi surface, one- patch theory is the minimal theory A stack of quantum hall layers creates a two- dimensional chiral Fermi surface [Balents and Fisher]

26 2+1 dimension is special Stronger quantum fluctua>ons D More gapless modes D=2+1 is unique : below the upper cri>cal dimension (D=3+1) with an extended Fermi surface Abundant low energy modes + strong quantum fluctua>ons = novel physics

27 Non- Fermi liquids in 2+1D Coupling between fermion and boson become strong even though bare coupling is weak (characteris>c of low dimensionality) It is useful to find a limit where one can solve the theory exactly : generalize N=2(spin flavor) to a large N It has been believed that quantum fluctua>ons are unimportant in the large N limit (due to self averaging effect), which turns out to be incorrect

28 Feynman graph : organizing & visualizing quantum fluctua>ons i=1,2,,n As boson propagates, boson drags cloud of par>cles and holes around it For large N, boson gets dressed heavily with fermion clouds, and fluctua>ons of boson are suppressed 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 ~ 1/N Energy of fermion is modified by an amount propor>onal to 1/N Fermions on the Fermi surface has small energy ~ 1/N

29 Bare vertex vs. dressed vertex (a) (b) Naively, one expect (b) is smaller than (a) by a factor of N - 4 (- 4 = 1 [one addi>onal fermion loop] 5 [5 addi>onal boson excita>ons] ) This naïve expecta>on is wrong for the following reasons : Although bosons are very weakly fluctua>ng, slow fermions are very vulnerable to small perturba>on (b) include more sou channels of fluctua>ons than (a)

30 Small perturba>on (weak coupling ~ 1/N) + large suscep>bility (small energy ~ 1/N) = O(1) response F = 1 N f 1 2N x2 x =1

31 Strong quantum fluctua>ons near FS In diagram (b), there are 4 more channels via fermions can stay right on the Fermi surface Whenever fermion is on the Fermi surface, the effect of interac>on is magnified by a factor of N because fermion on the FS has only energy ~ 1/N which comes from interac>on enhanced by N 4 same magnitude as (a)

32 There are infinitely many graphs (paths) that are important even in the large N limit Feynman graphs are organized by their topology, not by the number of ver>ces All graphs that can be drawn on the plane without a crossing (planar diagrams) are important (genus expansion) A sign of new kind of emergent degrees of freedom [SL (2009)]

33 Emergent string. g=0 g=1 sphere torus 2d surface made of Feynman graphs = world sheet of string Genus expansion : weak coupling expansion of string [ t Hoou] Precise nature of the theory is yet to be understood

34 Recent developments In non- chiral (two- patch) theory, non- planar diagrams are also important : possible break- down of genus expansion [Metlitski and Sachdev (10)] è not clear how Feynman diagrams are organized even in the large N limit Perturba>ve expansion is possible in a double limit : large N + small ε [Mross, McGreevy, Liu, Senthil (10)] : non- renormaliza>on of boson kine>c energy due to non- locality introduced for generic value of ε In chiral (one- patch) theory, non- perturba>ve effects are expected to be significantly suppressed

35 Summary Non- Fermi liquid is the souest, most entangled and the least understood phase of maber Conven>onal paradigms fail : new insight needed