Renormalized entanglement entropy and the number of degrees of freedom

Transcription

1 Renormalize entanglement entropy an the number of egrees of freeom Hong Liu MIT Base on arxiv: with Mark Mezei

2 Goal For any renormalizable quantum fiel theory, construct an observable which coul: probe an characterize quantum entanglement at a given scale. track the number of egrees of freeom of the system at a given scale.

3 Quantum entanglement Quantum entanglement 1 2 ( ) Spooky non-locality Quantum gravity Quantum information Conense matter Bi-partite entanglement: entanglement entropy Σ A = Trρ A log ρ A (will focus on vacuum)

4 Entanglement entropy R Expect it to epen on physics at length scales ranging from size R all the way to short-istance cutoff δ. ominate by short-istance physics unpleasant features: A Σ + δ 2 δ : Short-istance cutoff (Bombelli et al, Srenicki) ill-efine in the continuum limit: Divergent for a renormalizable QFT Long range correlations har to extract. Even in the large R limit, still sensitive to all the shorter-istance.o.f., not clear it will reuce to the behavior of the IR fixe point.

5 Common practice: subtract the UV ivergent parts by han, often ambiguous (e,g. typically not invariant uner reparametrizations of the cutoff) Even after the subtraction, coul still epen on physics at scales much smaller than the size of the entangle region.

6 Free massive fiels For a free massive scalar fiel for a spherical region in the regime mr >> 1 in =3: Herzberg an Wilczek, Heurta S scalar (mr) =# R δ π 6 mr π mr + The finite part iverges linearly in R an oes not have a well efine limit in the large R limit. At long istances, the system contains nothing. Ieally, we woul have like to have the EE to go to zero.

7 Entanglement entropy contains too much information (junk): e.g. in the infinite R limit, it oes not reuce to physics at the IR fixe point, an still epens on physics at much shorter length scales. Woul like to be able to irectly probe entanglement relations at a given scale. Here we make a simple proposal.

8 Renormalize entanglement entropy For any entangling (smooth) surface Σ with a scalable size R: (R) = 1 ( 2)!! 1 ( 2)!! R R R R 1 R R 3 R R ( 2) (R) o R R 2 R R ( 2) (R) even. =2: S 2 (R) =R S R =4: =3: 3 (R) =R S(Σ) R 4 (R) = 1 2 R R(R R 2 )= 1 2 S(Σ) R 2 2 R 2 R S(Σ) R

9 Renormalize entanglement entropy Will show: UV finite, well-efine in the continuum limit R-inepenent for a scale invariant system For a general quantum fiel theory (R) (R) =s (Σ) s (Σ,UV) R 0 s (Σ,IR) R. It is most sensitive to egrees of freeom at scale R. For a sphere: s (sphere) = central charge (CFT) Monotonicity of S sphere (R) woul then lea to a c-theorem.

10 UV finiteness The ivergent part of EE shoul only epen on local physics at the cutoff scale near the entangling surface, iv = 2 σ hf (K ab,h ab ) Σ h: inuce metric, K: extrinsic curvature F: sum of all possible geometric invariants Grover, Turner, Vishwanath iv = S( Σ) iv For a scalable smooth surface Function F must be even in K. h ab R 2, K ab R, D a R 0 F 1+K 2 + K R R 4 + iv = a 1R 2 + a 2 R 4 +

11 CFT: iv = General QFT: a 1 1 R 2 δ 2 0 R 2 δ 2 0 UV finiteness (II) iv = a 1R 2 + a 2 R 4 + δ 2 0 a 1 = 1 δ0 2 h 1 (µδ 0 ), a 2 1 δ0 4, + + R δ 0 o + + R2 δ log R δ 0 even µ : some mass scale h 1 (µδ 0 )=c 0 + c 2 (µδ 0 ) 2α + c 3 (µδ 0 ) 3α + δ 0 : bare UV cutoff previously known from holographic calculations (Ryu, Takayanagi) Terms with negative powers of R are all UV finite. µδ 0 1 In a renormalizable theory: cannot contain any singular epenence on µ in the µ 0 limit.

12 UV finiteness (III) Given: iv = a 1R 2 + a 2 R 4 + (R) = 1 ( 2)!! 1 ( 2)!! R R R R 1 R R 3 R R ( 2) (R) o R R 2 R R ( 2) (R) even. will then get ri of all UV ivergent terms for any QFT. The ifferential operator also gets ri of finite terms of the same R-epenence. Such terms can be moifie by reefining the cutoff, thus not well efine in the continuum limit ( contaminate ). (R) is thus UV finite, an unambiguous (inepenent of reparametrizations of the cutoff).

13 = CFT For a scale invariant system, we must have: (R) =s (Σ) Converting it back to the EE itself, we then have R R δ 0 +( 1) 1 2 s (Σ) δ 2 0 R 2 δ R2 δ 2 0 o +( 1) 2 2 s (Σ) log R δ 0 + const even This agrees with what was previously foun from holographic calculations. (Ryu, Takayanagi) s (Σ) is the universal part of the entanglement entropy.

14 General QFTs In the small R limit: (R) S (Σ,UV), R 0 (R) s (Σ,UV), R 0 In the large R limit: R (R) epens on scales from δ 0 to R incluing all intermeiate scales of the system, µ 1,µ 2, Nevertheless (R) s (Σ,IR), R

15 Introucing a floating cutoff : General QFTs (II) 1 µ 1, δ δ 1, δ R µ 2 So that between an R there is no other physical length scales an the system can be well approximate by the IR fixe point, (δ 0,R,µ 1, )= (δ,r) S (Σ,IR) (δ,r) (R) s (Σ,IR), R In other wors, in the large R limit, the contributions to EE from various mass parameters must have the form: c 1 R 2 + c 2 R O(R)+O((µR) # ) (o )

16 General QFTs (III) Similarly, for any length L << R, L δ R (R) shoul not be sensitive to contributions from.o.f. at scale L. (Their contributions shoul be suppresse by positive powers of L/R.) (R) can be consiere to irectly probe an characterize entanglement at scale R. The R-epenence can be interprete as escribing the RG flow of entanglement entropy with istance scale.

17 (R) = 1 ( 2)!! 1 ( 2)!! R R Summary UV finite, well-efine in the continuum limit R-inepenent for a scale invariant system For a general quantum fiel theory R R 1 R R 3 R R ( 2) (R) o R R 2 R R ( 2) (R) even. (R) (R) =s (Σ) s (Σ,UV) R 0 s (Σ,IR) R. most sensitive to egrees of freeom at scale R. can be consiere as escribing the RG flow of entanglement entropy Note: efinition not unique, simplest

18 Gappe systems For a free massive scalar fiel for a spherical region in the regime mr >> 1 in =3: S scalar (mr) =# R δ π 6 mr S scalar (mr) =+ π 120 π mr mr + In o, for generic gappe systems, we expect: (e.g. =3) 3 (R) γ, R γ : In even : Topological entanglement entropy (Kitaev, Preskill; Levin, Wen) 2n (R) 0, R, n =1, 2,

19 An application: non-fermi liquis For a system with a Fermi surface, expect at large R: (R) A FS k 2 F (R) k 2 F R 2 A FS A Σ, R S Σ (R) k 2 F R 2 log(k F R) A FS A Σ log(a FS A Σ ) Similarly for higher co-imensional Fermi surfaces: (R) (k F R) n S Σ (R) Wolf; Gioev, Klich Swingle, Swingle, Senthil (inepenent whether the system has quasiparticles or not) (k F R) n log(k F R) n even (k F R) n no

20 Renyi entropy R n (A) = 1 1 n log Trρn A One can similarly efine renormalize Renyi entropies, an exactly parallel iscussion shows that for all n, Renyi entropies have the same structure as the entanglement entropy for a CFT. The results for the (non)-fermi liquis also apply to Renyi entropies.

21 EE an the number of.o.f. (R) characterizes entanglement at scale R. escribes the RG flow of entanglement entropy Coul vary R? (R) track the flow of the number of.o.f. as we Since RG flow leas to a loss of short-istance.o.f. R S(Σ) (R) R < 0? (much stronger conition) s (Σ,UV) >s (Σ,IR) i.e. a c-theorem.

22 =2 S 2 (R) =R S R For a CFT S 2 = c 3 Holzhey, Larsen, Wilczek For Lorentz-invariant, unitary QFTs S 2 (R) monotonic Casini an Huerta alternative proof of Zamolochikov s c-theorem Proof uses Lorentz symmetry an strong sub-aitivity conition S(A)+S(B) S(A B)+S(A B)

23 Higher imensions now epens on the shape of. (R) Σ =4: for a CFT s (Σ) 4 =2a 4 Will all shapes work? Σ 2 σ he 2 + c 4 a 4,c 4 : coefficients of trace anomaly I 2 vanishes for sphere, Σ s (sphere) 4 =4a 4 2 σ hi 2 For a general shape, will be a combination of a an c. Thus only for a sphere, o we always have s (Σ,UV) 4 >s (Σ,IR) 4 Soloukhin

24 Higher imensions (II) For all even spacetime imensions: s (sphere) 2n For all o imension: s (sphere) =4a 2n = (log Z) finite Myers, Sinha Casini, Myers, Heurta Casini, Myers, Heurta (log Z) finite : finite part of the Eucliean partition for the CFT on S There are supports that these quantities coul satisfy s (sphere,uv) >s (sphere,ir) Cary, Myers, Sinha Jefferis, Klebanov, Pufu an Safi even imension: coefficient of ivergent term, local expression

25 Thus now focus on a sphere S (R) if monotonic lea to the conjecture c-theorem in all imensions give a scale-epenent measure of the number of.o.f. for a general QFT.

26 =3 S 3 (R) =R S R S Free massive scalar an various holographic examples: Conjecture: S 3 (R) monotonically ecreasing with R an non-negative for all Lorentz invariant, unitary QFTs Monotonicity S (R) < 0 Casini an Huerta have given a proof shortly after ( ). But their proof oes not appear to give non-negativeness.

27 Free massive scalar fiel For a free massive scalar fiel for a spherical region in the regime mr >> 1 in =3: S scalar (mr) =+ π mr + S 3 R monotonic for all R mr

28 Importance of unitarity fz 5 S 3 RK z R Null energy conition monotonicity of f(z) monotonicity of S 3 (R)

29 S 4 (R) = 1 2 Various holographic examples: =4 R 2 2 S R 2 R S R S 4 (R) Nevertheless neither monotonic nor non-negative S 4 (R 0) > S 4 (R ) from a-theorem

30 Some examples R 2S S 3 RK 4 2S 4 RK See also Albash, Johnson IR: CFT R c R R R =4: Leigh-Strassler flow A toy flow R 2S 4 RK 2S 4 RK n orer transition R c R IR: gappe phases R 0.5 GPPZ flow (Girarello, Petrini, Porrati,Zaffaroni) Coulomb branch flow (Freeman, Gubser, Pilch, Warner)

31 =4 S 4 (R) neither positive-efinite nor always monotonic the function form shoul be moifie S 4 Monotonicity of or its improvement woul imply an inequality for S with least three erivatives. R 3 3 RS + R 2 2 RS<R R S Not clear it coul arise from the strong subaitivity conition. Despite this: S 4 characterizes scale-epenent entanglement Its non-monotonicity an non-positive-efiniteness likely reflects some interesting unerlying physics

32 >4 When two fixe points are close, i.e. s UV s IR s UV 1 S (R) is always monotonically ecreasing in holographic systems

33 Behavior near a UV fixe point In all holographic theories: For small R S (R) =s (UV) A(α)(µR) 2α + α = (source flow) α = (vev flow) S = s (UV) O(g 2 ) g: least relevant coupling Free massive fiel in 2+1 imension: A(α) > 0 See also Klebanov, Nishioka, Pufu, Safi m 2 RS 3 m 2 R 2 =0 =0 Klebanov, Nishioka, Pufu, Safi

34 Behavior near an IR fixe point For large R For α = < α 1 2 S (R) =s (IR) + B( α) + ( µr) 2 α S (R) =s (IR) + o 1 even # R + # s (IR) + O(g 2 ) outsie the above range: S = s (IR) + Cg γ γ = HL, Mezei, to appear Dimension of leaing irrelevant operator o B( α) > 0 R + even 2 α 1 o 2 α 1 even < 2 C: nonlocal Sign of C not efinite in =4

35 Phase transitions We also observe that in holographic systems, the entanglement entropy has phase transitions in the Lorentz-invariant vacuum as a function of size: can be first orer or secon orer Involving topology change or no topology change See also Klebanov, Nishioka, Pufu,Safi Not yet clear what these phase transitions tell us. Nishioka an Takayanagi Klebanov, Kutasov, Murugan Pakman, Parnachev Hearick Albash an Johnson.

36 Some examples 4.8 R R S 3 RK S 3 RK n orer transition =3 S 3 RK st orer transition 0.40 R R R R c c R R 2S 4 RK 2S 4 RK 2S 4 RK n orer transition R R c 0.5 R R c R = R 0.5 GPPZ flow (Girarello, Petrini, Porrati,Zaffaroni) Coulomb branch flow (Freeman, Gubser, Pilch, Warner)

37 2 n orer phase transitions involving topology change z R 1 <R c <R 2 R 1 R c R 2 ρ z =0

38 Summary For any renormalizable quantum fiel theory (not necessarily Lorentz-invariant), renormalize entanglement entropy: probe an characterize quantum entanglement at a given scale. for =2,3, C-function, caniate for a measure of the number of egrees of freeom of the system at a given scale (with Lorentz symmetry) Non-relativistic? an intrinsically finite efinition (mutual information)?