1. Holographic Entanglement Entropy from Cond-mat to Emergent Spacetime
Gravity Theories and Their Avatars @CCTP, Crete　July 13-19, 2012
Holographic Entanglement Entropy from Cond-mat to Emergent Spacetime
Tadashi Takayanagi
Yukawa Institute for Theoretical Physics (YITP),
Kyoto University
Based on arXiv (JHEP 2012(125)1)
with N. Ogawa (RIKEN) and T. Ugajin (IPMU/YITP)
+ on going work with M.Nozaki (YITP) and S.Ryu (Ilinois,Urbana-S.)

2. Contents ① Introduction ② Entanglement Entropy and Fermi surfaces
③ Holographic Entanglement Entropy (HEE)
④ Fermi Surfaces and HEE
⑤　Emergent Metric via Quantum Entanglement
⑥ Conclusions

3. ① Introduction– from cond-mat viewpoint
AdS/CFT is a very powerful method to understand strongly coupled condensed matter systems.
Especially, the calculations become most tractable
in the strong coupling and large N limit of gauge theories.
In this limit, the AdS side is given by a classical gravity and
we can naturally expect universal behaviors such as the
no hair theorem in GR, η/s=1/4π, etc.
So, we concentrate on this limit for a while.

4. against perturbations by the Coulomb forces.
We would like to consider what is a universal properties for metallic condensed matter systems via AdS/CFT.
The metals are usually described by the Landau’s Fermi liquids. It is well-known that Fermi liquid states are stable
against perturbations by the Coulomb forces.
However, in strongly correlated
electron systems such as the
strange metal phase of high Tc
superconductors or heavy fermion
systems etc., we encounter
so called non-Fermi liquids.
Strange Metal
Fermi Liquid
Pseudo Gap
Mott Insulator SC
Figs taken from Sachdev

5. Can we obtain Landau’s Fermi liquids in the classical gravity limit ?
So, one of the main purposes of this talk is to answer the question:
Can we obtain Landau’s Fermi liquids in the classical gravity limit ?
Note: Several interesting setups of (non-)Fermi liquids have already been found.
(i) Probe fermions in SUGRA b.g.
(ii) Electron stars (Lifshitz metric in the IR)
⇒　∃Fermi surfaces, but, not in the leading order O(N2)
of the large N limit.
[Rey 07, Faulkner-Liu-McGreevy-Vegh 09, Cubrovic-Zaanen-Schalm 09,
DeWolfe-Gubser-Rosen 11, 12]
[Hartnoll-Tavanfar 2010]

6. Systems with Fermi surfaces ⇔ Fermi liquids or non-Fermi liquids
So, we concentrate on systems with Fermi surfaces.
To make the presentation simpler, we will work for 2+1
dim. systems with Fermi surfaces. But our analysis can be
generalized to higher dimensions, straightforwardly.
Fermi surface
Fermi sea

7. How to characterize the Fermi surfaces ?
Metals ⇒ Conductivity ?
But it seems difficult to find universal results for conductivity in
the gravity dual. This is because it is related to the propagation
of U(1) gauge fields in AdS, whose behavior largely depends on
the precise Lagrangian of gauge fields e.g. f(φ)F2 .
So we want to find a quantity whose gravity dual is closely
related to the metric (i.e. gravity field).
　　　 ⇒ We should look at a thermodynamical quantity !

8. One traditional candidate is the specific heat C.
For (Landau’s) fermi liquids, we always have the behavior
This linear specific heat can be understood if we note that we
can approximate the excitations of Fermi liquids by an infinite
copies of 2 dim. CFTs.
Fermi surface
Fermi sea
Fermi surface

9. In 2d CFT, we know In this way, we can estimate the specific heat of the Fermi liquids However, the linear specific heat is not true for non-Fermi liquids. This is because they have anomalous dynamical exponents z. (~infinite copies of 2d Lifshitz theory: )

10. To characterize the existence of Fermi surfaces, we need to look at a property which is common to both the FL and non-FL. ⇒ The entanglement entropy is a suitable quantity. After we concentrate on the systems with Fermi surfaces, we can distinguish between FL and non-FL by calculating the specific heat.

11. ② Entanglement Entropy and Fermi surfaces
Divide a quantum system
into two subsystems A and B:
We define the reduced density matrix for A by
taking trace over the Hilbert space of B .
(2-1) Definition and Properties of Entanglement Entropy
Example: Spin Chain A B

12. Now the entanglement entropy is defined by the
von-Neumann entropy
In QFTs, it is defined geometrically:

13. EE in QFTs includes UV divergences.
(2-2) Area law
EE in QFTs includes UV divergences.
In a (d+1 ) dim. QFT with a UV fixed point, the leading term of EE
is proportional to the area of the (d-1) dim. boundary :
where is a UV cutoff (i.e. lattice spacing).
Intuitively, this property is understood like:
Most strongly entangled
[Bombelli-Koul-Lee-Sorkin 86, Srednicki 93]
Area Law ∂A A

14. However, there are two known exceptions: (a) 1+1 dim. CFT
(b) ∃Fermi surfaces ( ) B A B
[Holzhey-Larsen-Wilczek 94, Calabrese-Cardy 04] A
[Wolf 05, Gioev-Klich 05]

15. (2-3) Fermi Surfaces and Entanglement Entropy
Why do Fermi Liquids violate the area law ? This can be understood if we remember that the Fermi liquids can be though of as infinite copies of 2d CFTs: We will mainly assume this choice of subsystem A below. B A B

16. Recently, there have been evidences that this logarithmic
behavior is true also for non-Fermi liquids (e.g.spin liquids).
[Swingle 09,10, Zhang-Grover-Vishwanath 11　etc.]
Intuitively, we can naturally expect this because the logarithmic
behavior does not change if we introduce the dynamical
exponent z in the 2d theory as
Therefore we find the characterization:
∃Fermi surface　⇔　Logarithmic behavior of EE

17. B A B To apply the AdS/CFT, we will embed the Fermi surface in a CFT.
In this case, the leading divergence still satisfies the area law.
But the subleading finite term has the logarithmic behavior:
if we assume
So, we will concentrate on the gravity dual whose entanglement entropy has this behavior in our arguments below. B A B

18. ③ Holographic Entanglement Entropy
Holographic Entanglement Entropy Formula [Ryu-TT 06]
is the minimal area surface
(codim.=2) such that
homologous

19. In spite of a heuristic argument [Fursaev, 06] , there has been no
complete proof. But, so many evidences and no counter examples.
[A Partial List of Evidences]
Area law follows straightforwardly [Ryu-TT 06]
Agreements with analytical 2d CFT results for AdS3 [Ryu-TT 06]
Holographic proof of strong subadditivity　[Headrick-TT 07]
Consistency of 2d CFT results for disconnected subsystems
[Calabrese-Cardy-Tonni 09] with our holographic formula [Headrick 10]
Agreement on the coefficient of log term in 4d CFT 　(~a+c)
[Ryu-TT 06, Solodukhin 08,10, Lohmayer-Neuberger-Schwimmer-Theisen 09,
Dowker 10, Casini-Huerta, 10, Myers-Sinha 10]
A direct proof when A = round ball [Casini-Hueta-Myers 11]
Holographic proof of Cadney-Linden-Winter inequality
[Hayden-Headrick-Maloney 11]

20. ④ Fermi Surfaces and HEE [Ogawa-Ugajin-TT 11]
(4-1) Setup of gravity dual For simplicity, we consider a general gravity dual of 2+1 dim. systems. The general metric can be written as follows (up to diff.) where f(z) and g(z) are arbitrary functions. We impose that it is asymptotically AdS4 i.e.

21. (4-2) Holographic EE Now we would like to calculate the HEE for this gravity dual. We choose the subsystem as the strip width l as before the minimal surface condition reads B A B
AdS Bdy

22. In the end, we obtain when the size of subsystem A is large
In the end, we obtain when the size of subsystem A is large . In this case, the minimal surface extends to the IR region deeply. ⇒ The logarithmic behavior of EE is realized just when n=1. We identify as a characteristic scale of the Fermi energy. Note: f(z) does not affect the HEE and is still arbitrary.

23. (4-3) Null Energy Condition To have a sensible holographic dual, a necessary condition is known as the null energy condition: In the IR region, the null energy condition argues

24. At finite temperature, we expect that the solution is given by a black brane extension of our background: The `non-extremal factors’ behave near the horizon From this, we can easily find the behavior of specific heat:

25. Combined with the null energy condition: , we obtain Notice that this excludes the Landau’s Fermi liquids (α=1). In summary, we find that classical gravity duals only allow non-fermi liquids. Comments: (i) This result might not be so unnatural as the non-Fermi liquids are expected in strongly correlated systems. (ii) Even in the presence of perturbative higher derivative corrections, the result does not seem to be changed.

26. (iii) Some miracle coincidences ?
AdS: No curvature singularity in the gravity dual
⇒ α=2/3 [11]
Shaghoulian
CMT: Spin fluctuations:
[Moriya, Hertz, Millis …. 70’-90’]
N Fermions + U(1) gauge:
⇒　α=2/3 (i.e. z=3)
[Lee 09, Metlitski, and S. Sachdev 10,
Mross-McFreevy-Liu-Senthil 10,
Lawler-Barci-Fernandez-Fradkin-Oxman 06]
Experiment: YbRh2(Si1-xGex)2
⇒　α=2/3
Examples of heavy fermions [Pepin 11, talk at KITP]

27. (iv) We can embed this background in an effective gravity theory:
[Earlier works, Gubser-Rocha 09,
Charmousis-Goutéraux-Kim-Kiritsis-Meyer 10,…]
if W and V behave in the large φ limit as follows [Ogawa-Ugajin-TT 11]

28. (v) This metric can also be regarded as a generalization of Lifshitz backgrounds so that it violates the hyperscaling. [Huijse-Sachdev-Swingle 11, Dong-Harrison-Kachru-Torroba-Wang 12]

29. q d-1 d `Landscape’ of (d+1) dim. Quantum Phases from HEE
Can we find examples
in realistic systems ?
Lower dimensional
structure
Power violation ?
(∃Hol. duals)
d-1 d q
Not allowed
Area Law
(QFTs)
Logarithmic
(Fermi surface)
Volume Law
(Non-local)

30. ⑤ Emergent Metric from Quantum Entanglement
(5-1) Basic Outline
In principle, we can obtain a metric from a CFT as follows:
a CFT state ⇒ Information (~EE) = Minimal Areas ⇒　metric
One candidate of such frameworks is so called the entanglement
renormalization (MERA) [Vidal 05 (for a review see )] as
pointed out by [Swingle 09].
[cf. Emergent gravity: Raamsdonk 09, Lee 09]

31. (5-2) Tensor Network (TN)
[See e.g. Vidal and references therein]
Recently, there have been remarkable progresses in numerical
algorithms for quantum lattice models, based on so called
tensor product states.
Basically, people try to find nice variational ansatzs for
the ground state wave functions for various spin systems.
⇒　An ansatz is good if it respects the quantum entanglement
of the true ground state.

32. Ex. Matrix Product State (MPS) [DMRG: White 92,…,
Rommer-Ostlund 95,..]
Spin chain

33. MPS and TTN are not good near quantum critical points (CFTs) because their entanglement entropies are too small: In general, A A

34. (5-3) AdS/CFT and (c)MERA
MERA (Multiscale Entanglement Renormalization Ansatz):
An efficient variational ansatz to find CFT ground states have been
developed recently. [Vidal 05 (for a review see )].
To respect its large entanglement in a CFT, we add (dis)entanglers.
Unitary transf.
between 2 spins

35. Calculations of EE in 1+1 dim. MERA
A= an interval (length L) A

36. A conjectued relation to AdS/CFT　[Swingle 09]

37. Continuous MERA (cMERA)
Now, to make the connection to AdS/CFT clearer, we would like
to consider the MERA for quantum field theories.
Continuous MERA (cMERA)
[Haegeman-Osborne-Verschelde-Verstraete 11]
K(s) : disentangler, L: scale transformation
Conjecture

38. (5-4) Emergent Metric from cMERA
[Nozaki-Ryu-TT, in preparation]
We conjecture that the metric in the extra direction is given by using the idea of the quantum metric (up to a constant c): Note: The quantum distance between two states is defined by

39. Comments
The denominator represents the total volume of phase space
at energy scale u.
The operation e^{iLdu} removes the coarse-graining procedure so that we can measure the strength of
unitary transformations induced by disentanglers (bonds).
　⇒　guu measures the density of bonds.
Consistent with the HEE: A B

40. (5-5) Emergent Metric in a (d+1) dim. Free Scalar Theory
Hamiltonian:
Ground state :
Moreover, we introduce the `IR state’ which has no real
space entanglement.

41. For a free scalar theory, the ground state corresponds to For the excited states, becomes time-dependent. One might be tempting to guess Indeed, the previous proposal for guu lead to
Density of bonds

42. Explicit metric Massless scalar (E=k) (ii) Lifshitz scalar (E=kν)
(iii) Massive scalar
Ｃａｐｐｅｄ　ｏｆｆ　ｉｎ　ｔｈｅ　ＩＲ

43. (5-6) Excited states after quantum quenches
m(t) m0 t
Note: There is an phase factor ambiguity of (A,B).
⇒　Different choices of time slices ?

44. t z Time dependent metric from the Quantum Quench Light cone
looks like a propagation of
gravitational wave. t z
We can also confirm the linear growth of EE: SA∝t.
This is consistent with the known CFT (2d)
[Calabrese-Cardy 05]
and with the HEE results (any dim.).
[Arrastia-Aparicio-Lopez 10 , Albash-Johnson 10, Balasubramanian-Bernamonti-de Boer-Copland-Craps- Keski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10, 11,….]

45. (5-7) Towards Holographic Dual of Flat Space If we consider the (almost) flat metric the corresponding dispersion relation reads This leads to the highly non-local Hamiltonian: [cf. Li-TT 10]

46. ⑤ Conclusions The entanglement entropy (EE) is a useful bridge between
gravity (string theory) and cond-mat physics.
Gravity Entanglement Cond-mat.
systems
Classical gravity duals + Null energy condition
⇒ a constraint on specific heat
　　⇒ Non-fermi liquids ! 　　　　　　　　　　　　　
Questions: String theory embeddings of the NFL b.g. ?
　　　　　　 　 [See also solutions in Singh 10, Narayan 12]

47. We studied the conjectured relation between AdS/CFT and
MERA. We especially employed the cMERA and proposed
a definition of metric in the extra dimension. This metric passes
several qualitative tests.
Many future problems: how to determine gtt ?
ambiguity of choices time slices ?
large N and strongly coupling limit ?
higher spin holography ? :