Solvable models of quantum matter without quasiparticles Subir - - PowerPoint PPT Presentation

Solvable models of quantum matter without quasiparticles

HARVARD

Subir Sachdev July 23, 2018 XIX International Congress on Mathematical Physics, Centre Mont-Royal, Montreal, Canada

Fermi liquid theory of metals, insulators, semiconductors Theory of superconductivity (pairing of quasiparticles) Theory of disordered metals and insulators (diffusion and localization of quasiparticles) Theory of metals in one dimension (collective modes as quasiparticles) Theory of the fractional quantum Hall effect (quasiparticles which are `fractions’ of an electron)

Quasiparticles are ubiquitous:

• Note: The electron liquid in one dimension and the fractional

quantum Hall state both have quasiparticles; however, the quasi- particles do not have the same quantum numbers as an electron.

What are quasiparticles ?

• Quasiparticles are additive excitations:

The low-lying excitations of the many-body system can be identified as a set {nα} of quasiparticles with energy εα E = P

α nαεα + P α,β Fαβnαnβ + . . .

In a lattice system of N sites, this parameterizes the energy

• f ∼ eαN states in terms of poly(N) numbers.

• Quasiparticles eventually collide with each other. Such

collisions eventually leads to thermal equilibration in a chaotic quantum state, but the equilibration takes a long

• time. In a Fermi liquid, this time diverges as

τeq ∼ ~EF (kBT)2 , as T → 0, where EF is the Fermi energy.

What are quasiparticles ?

• Quasiparticles eventually collide with each other. Such

collisions eventually leads to thermal equilibration in a chaotic quantum state, but the equilibration takes a long

• time. In a Fermi liquid, this time diverges as

τeq ∼ ~EF (kBT)2 , as T → 0, where EF is the Fermi energy.

What are quasiparticles ?

• This time is much longer than the ‘Planckian time’ ~/(kBT),

which we will find in systems without quasiparticle ex- citations. τeq ~ kBT , as T ! 0.

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

A simple model of a metal with quasiparticles

Pick a set of random positions

Place electrons randomly on some sites

A simple model of a metal with quasiparticles

Place electrons randomly on some sites

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Electrons move one-by-one randomly

A simple model of a metal with quasiparticles

Fermions occupying the eigenstates of a N x N random matrix

tij are independent random variables with tij = 0 and |tij|2 = t2

A simple model of a metal with quasiparticles

H = 1 (N)1/2

N

X

i,j=1

tijc†

icj − µ

X

i

c†

ici

cicj + cjci = 0 , cic†

j + c† jci = δij

1 N X

i

c†

ici = Q

A simple model of a metal with quasiparticles

ρ(ω) = − 1

πIm G(ω)

−2t − µ

2t − µ

ω

Feynman graph expansion in tij.., and graph-by-graph average, yields exact equations in the large N limit: G(τ) ≡ −Tτ D ci(τ)c†

i(0)

E G(iω) = 1 iω + µ − Σ(iω) , Σ(τ) = t2G(τ) G(τ = 0−) = Q. G(ω) can be determined by solving a quadratic equation.

A simple model of a metal with quasiparticles

εα level spacing ∼ 1/N

ρ(ω) = − 1

πIm G(ω)

Let εα be the eigenvalues of the matrix tij/ √ N. The fermions will occupy the lowest NQ eigen- values, upto the Fermi energy EF . The single- particle density of states is ρ(ω) = (1/N) P

α δ(ω − εα), and ρ0 ≡ ρ(ω = 0).

ρ0

−2t − µ

2t − µ

ω

A simple model of a metal with quasiparticles

Quasiparticle excitations with spacing ∼ 1/N

There are 2N many body levels with energy E =

N

X

α=1

nαεα, where nα = 0, 1. Shown are all values of E for a single cluster of size N = 12. The εα have a level spacing ∼ 1/N. Many-body level spacing ∼ 2−N

A simple model of a metal with quasiparticles

The grand potential Ω(T) at low T is (from the Sommerfeld expansion) Ω(T) E0 = N ✓ π2 6 ρ0T 2 + O(T 4) ◆ + . . . where ρ0 ⌘ ρ(0) is the single particle density of states at the Fermi level. We can also define the many body density of states, D(E), via Z = e−Ω(T )/T = Z ∞

−∞

D(E)e−E/T The inversion from Ω(T) to D(E) has to performed with care (it need not commute with the 1/N expansion), and we obtain D(E) ⇠ exp π r 2Nρ0(E E0) 3 ! , E > E0 , 1 N ⌧ ρ0(E E0) ⌧ N and D(E) = 0 for E < E0. This is related to the asymptotic growth of the partitions

• f an integer, p(n) ⇠ exp(π

p 2n/3). Near the lower bound, there are large sample- to-sample fluctuations due to variations in the lowest quasiparticle energies.

Fermi liquid state: Two-body interactions lead to a scattering time

• f quasiparticle excitations from in (random) single-particle eigen-

states which diverges as ∼ T −2 at the Fermi level.

A simple model of a metal with quasiparticles

Now add weak interactions H = 1 (N)1/2

N

X

i,j=1

tijc†

icj − µ

X

i

c†

ici +

1 (2N)3/2

N

X

i,j,k,`=1

Uij;k` c†

ic† jckc`

Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U 2. We compute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the free particle Hamitonian with energy "↵. By Fermi’s Golden rule, for "↵ at the Fermi energy 1 ⌧↵ = ⇡U 2⇢2 Z d"d"d"f(")(1 − f("))(1 − f("))("↵ + " − " − ") = ⇡3U 2⇢2 4 T 2 where ⇢0 is the density of states at the Fermi energy, and f(✏) = 1/(e✏/T + 1) is the Fermi function.

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

The Sachdev-Ye-Kitaev (SYK) model

Pick a set of random positions

Place electrons randomly on some sites

The SYK model

Place electrons randomly on some sites

The SYK model

Place electrons randomly on some sites

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

Entangle electrons pairwise randomly

The SYK model

The SYK model

This describes both a strange metal and a black hole!

• A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)
• S. Sachdev and J.

Ye, PRL 70, 3339 (1993)

(See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit; T.A. Brody, J. Flores, J.B. French, P .A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981))

The SYK model

H = 1 (2N)3/2

N

X

i,j,k,`=1

Uij;k` c†

ic† jckc` − µ

X

i

c†

ici

cicj + cjci = 0 , cic†

j + c† jci = δij

Q = 1 N X

i

c†

ici

Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U 2 N → ∞ yields critical strange metal.

The SYK model

U U

Σ =

G

G

G

• S. Sachdev and J.

Ye, Phys. Rev. Lett. 70, 3339 (1993)

Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex- act equations in the large N limit: G(iω) = 1 iω + µ − Σ(iω) , Σ(τ) = −U 2G2(τ)G(−τ) G(τ = 0−) = Q.

• S. Sachdev and J.

Ye, Phys. Rev. Lett. 70, 3339 (1993)

The SYK model

X

• A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex- act equations in the large N limit: G(iω) = 1 iω + µ − Σ(iω) , Σ(τ) = −U 2G2(τ)G(−τ) G(τ = 0−) = Q. Low frequency analysis shows that the solutions must be gapless and obey Σ(z) = µ − ei(⇡/4+✓) A √z + . . . , G(z) = Ae−i(⇡/4+✓) √z where A = (π/U 2 cos(2θ))1/4. The value of θ is universally related to Q by a Luttinger-Ward functional analysis similar to that used to establish the Luttinger theorem of Fermi liquid theory: Q = 1 2 − θ π − sin(2θ) 4

The SYK model

At T > 0, we obtain a solution with a conformal structure G(τ) = −A e−2πET τ √ 1 + e−4πE ✓ T sin(πTτ) ◆1/2 , 0 < τ < 1/T , where the ‘particle-hole asymmetry’ is determined by E e2πE = sin(π/4 + θ) sin(π/4 − θ) .

• S. Sachdev, PRX 5, 041025 (2015)
• A. Georges and O. Parcollet PRB 59, 5341 (1999)

Feynman graph expansion in Uijk`, and graph-by-graph average, yields ex- act equations in the large N limit: G(iω) = 1 iω + µ − Σ(iω) , Σ(τ) = −U 2G2(τ)G(−τ) G(τ = 0−) = Q.

X

GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, 134406 (2001)

Many-body level spacing ∼ 2−N = e−N ln 2

• W. Fu and S. Sachdev, PRB 94, 035135 (2016)

Non-quasiparticle excitations with spacing ∼ e−Ns0

The SYK model

There are 2N many body levels with energy E. Shown are all values of E for a single cluster of size N = 12. The T → 0 state has an entropy SGP S = Ns0, where s0 < ln 2 is determined by integrating ds0 dQ = 2πE . At Q = 1/2, s0 = G π + ln(2) 4 = 0.464848 . . . where G is Catalan’s constant.

The SYK model

• J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P

. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, arXiv:1611.04650;

• R. Davison, Wenbo Fu, A. Georges,

Yingfei Gu, K. Jensen, S. Sachdev, arXiv.1612.00849 ; A.M. Garcia-Garcia and J.J.M. Verbaarschot, arXiv:1701.06593; D. Bagrets, A. Altland, and A. Kamenev, arXiv:1702.08902;

• D. Stanford and E. Witten, arXiv:1703.04612; A. Kitaev and S.J. Suh, arXiv:1711.08467;

Yingfei Gu and S. Sachdev, unpublished.

Ω(T) E0 = N  s0T 1 2(γ + 4π2E2K)T 2 + O(T 3)

• + 2T ln

✓U T ◆ . . . is the grand potential, where K = dQ/dµ ⇠ 1/U is the compressibility/N, γ ⇠ 1/U will appear later in the co-efficient of the Schwarzian, and the N 0 term arises from fluctuations about the large N theory described by the Schwarzian. The inversion from Ω(T) to the many-body density of states, D(E), requires terms in Ω(T) which are exponentially small in N (not shown above) from the Schwarzian action, yielding terms which are not small in D(E). We obtain D(E) =

∞

X

p=−∞

e2πpE d ✓ E p2 2NK ◆ where NQ + p is the integer fermion number, d(E) = 0 for E < E0, and d(E) ⇠ exp (Ns0) sinh ⇣p 2Nγ(E E0) ⌘ , E > E0 , e−cN ⌧ γ(E E0) ⌧ N There are exponentially more low energy states than for the quasiparticle case, and D(E) self-averages down to energies exponentially small in N.

A simple model of a metal with quasiparticles

The grand potential Ω(T) at low T is (from the Sommerfeld expansion) Ω(T) E0 = N ✓ π2 6 ρ0T 2 + O(T 4) ◆ + . . . where ρ0 ⌘ ρ(0) is the single particle density of states at the Fermi level. We can also define the many body density of states, D(E), via Z = e−Ω(T )/T = Z ∞

−∞

D(E)e−E/T The inversion from Ω(T) to D(E) has to performed with care (it need not commute with the 1/N expansion), and we obtain D(E) ⇠ exp π r 2Nρ0(E E0) 3 ! , E > E0 , 1 N ⌧ ρ0(E E0) ⌧ N and D(E) = 0 for E < E0. This is related to the asymptotic growth of the partitions

• f an integer, p(n) ⇠ exp(π

p 2n/3). Near the lower bound, there are large sample- to-sample fluctuations due to variations in the lowest quasiparticle energies.

The SYK model

• J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P

. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, arXiv:1611.04650;

• R. Davison, Wenbo Fu, A. Georges,

Yingfei Gu, K. Jensen, S. Sachdev, arXiv.1612.00849 ; A.M. Garcia-Garcia and J.J.M. Verbaarschot, arXiv:1701.06593; D. Bagrets, A. Altland, and A. Kamenev, arXiv:1702.08902;

• D. Stanford and E. Witten, arXiv:1703.04612; A. Kitaev and S.J. Suh, arXiv:1711.08467;

Yingfei Gu and S. Sachdev, unpublished.

Ω(T) E0 = N  s0T 1 2(γ + 4π2E2K)T 2 + O(T 3)

• + 2T ln

✓U T ◆ . . . is the grand potential, where K = dQ/dµ ⇠ 1/U is the compressibility/N, γ ⇠ 1/U will appear later in the co-efficient of the Schwarzian, and the N 0 term arises from fluctuations about the large N theory described by the Schwarzian. The inversion from Ω(T) to the many-body density of states, D(E), requires terms in Ω(T) which are exponentially small in N (not shown above) from the Schwarzian action, yielding terms which are not small in D(E). We obtain D(E) =

∞

X

p=−∞

e2πpE d ✓ E p2 2NK ◆ where NQ + p is the integer fermion number, d(E) = 0 for E < E0, and d(E) ⇠ exp (Ns0) sinh ⇣p 2Nγ(E E0) ⌘ , E > E0 , e−cN ⌧ γ(E E0) ⌧ N There are exponentially more low energy states than for the quasiparticle case, and D(E) self-averages down to energies exponentially small in N.

No quasiparticles The SYK model

• A. Eberlein,
• V. Kasper, S. Sachdev, and
• J. Steinberg, PRB 96, 205123 (2017)
• A. Georges and O. Parcollet

PRB 59, 5341 (1999)

• Rapid local thermal equilibration (of fermion correla-

tors) in a ‘Planckian’ time τeq ∼ ~ kBT , as T → 0. Established by solution of Schwinger-Keldysh equations for a quench.

No quasiparticles The SYK model

• A. Eberlein,
• V. Kasper, S. Sachdev, and
• J. Steinberg, PRB 96, 205123 (2017)
• S. Sachdev, Quantum Phase

Transitions, Cambridge (1999)

• A. Georges and O. Parcollet

PRB 59, 5341 (1999)

• Rapid local thermal equilibration (of fermion correla-

tors) in a ‘Planckian’ time τeq ∼ ~ kBT , as T → 0. Established by solution of Schwinger-Keldysh equations for a quench.

• Presence of quasiparticles should slow down

thermalization, so all quantum systems obey τeq > C ~ kBT , as T → 0. Absence of quasiparticles ⇔ Fastest possible thermalization

Other quantum models without quasiparticles

Absence of quasiparticles ⇔ Fastest possible thermalization

• Rapid local thermal equilibration in a ‘Planckian’ time

τeq ∼ ~ kBT , as T → 0.

• Ising model in a transverse field in 2 dimensions at its quan-

tum critical point, g = gc. Described by the Wilson-Fisher fixed point of φ4 quantum field theory in 2+1 dimensions H = − X

hiji

σz

i σz j − g

X

i

σx

i

σx,z

i

are the Pauli operators on site i.

• Other strongly-coupled conformal field theories.

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

• A. Kitaev, 2015
• S. Sachdev, PRX 5, 041025 (2015)

X X X

At frequencies ⌧ U, the iω + µ can be dropped, and without it equations are invariant under the reparametrization and gauge transformations. The singular part of the self-energy and the Green’s function obey Z β dτ2 Σsing(τ1, τ2)G(τ2, τ3) = δ(τ1 τ3) Σsing(τ1, τ2) = U 2G2(τ1, τ2)G(τ2, τ1)

G(iω) = 1 iω + µ − Σ(iω) , Σ(τ) = −U 2G2(τ)G(−τ) Σ(z) = µ − 1 A √z + . . . , G(z) = A √z

• A. Kitaev, 2015
• S. Sachdev, PRX 5, 041025 (2015)

Z β dτ2 Σ(τ1, τ2)G(τ2, τ3) = −δ(τ1 − τ3) Σ(τ1, τ2) = −U 2G2(τ1, τ2)G(τ2, τ1) These equations are invariant under τ = f(σ) G(τ1, τ2) = [f 0(σ1)f 0(σ2)]1/4 g(σ1) g(σ2) e G(σ1, σ2) Σ(τ1, τ2) = [f 0(σ1)f 0(σ2)]3/4 g(σ1) g(σ2) e Σ(σ1, σ2) where f(σ) and g(σ) are arbitrary functions. By using f(σ) = tan(πTσ)/(πT) we can now obtain the T > 0 solution from the T = 0 solution.

• A. Kitaev

Let us write the large N saddle point solutions of S as Gs(τ1 − τ2) ∼ (τ1 − τ2)−1/2 Σs(τ1 − τ2) ∼ (τ1 − τ2)−3/2. The saddle point will be invariant under a reperamateri- zation f(τ) when choosing G(τ1, τ2) = Gs(τ1 − τ2) leads to a transformed e G(σ1, σ2) = Gs(σ1 − σ2) (and similarly for Σ). It turns out this is true only for the SL(2, R) transformations under which f(τ) = aτ + b cτ + d , ad − bc = 1. So the (approximate) reparametrization symmetry is spon- taneously broken down to SL(2, R) by the saddle point.

After introducing replicas a = 1 . . . n, and integrating out the dis-

• rder, the partition function can be written as

Z = Z Dcia(τ) exp "

• X

ia

Z β dτ c†

ia

✓ ∂ ∂τ µ ◆ cia U 2 4N 3 X

ab

Z β dτdτ 0

• X

i

c†

ia(τ)cib(τ 0)

• 43

5 . For simplicity, we neglect the replica indices, and introduce the identity 1 = Z DΣ(τ1, τ2) exp " N Z β dτ1dτ2Σ(τ1, τ2) G(τ2, τ1) + 1 N X

i

ci(τ2)c†

i(τ1)

!# .

The Schwarzian theory of the SYK model

• A. Georges and O. Parcollet

PRB 59, 5341 (1999)

• A. Kitaev, 2015
• S. Sachdev, PRX 5, 041025 (2015)

X Then the partition function can be written as a path integral with an action S analogous to a Luttinger-Ward functional Z = Z DG(τ1, τ2)DΣ(τ1, τ2) exp(NS) S = ln det [δ(τ1 τ2)(∂τ1 + µ) Σ(τ1, τ2)] + Z dτ1dτ2Σ(τ1, τ2) ⇥ G(τ2, τ1) + (U 2/2)G2(τ2, τ1)G2(τ1, τ2) ⇤ At frequencies ⌧ U, the time derivative in the determinant is less important, and without it the path integral is invariant under the reparametrization and gauge transformations τ = f(σ) G(τ1, τ2) = [f 0(σ1)f 0(σ2)]1/4 g(σ1) g(σ2) G(σ1, σ2) Σ(τ1, τ2) = [f 0(σ1)f 0(σ2)]3/4 g(σ1) g(σ2) Σ(σ1, σ2) where f(σ) and g(σ) are arbitrary functions.

The Schwarzian theory of the SYK model

Reparametrization and phase zero modes We can write the path integral for the SYK model as Z = Z DG(⌧1, ⌧1)DΣ(⌧1, ⌧2)eNS[G,Σ] for a known action S[G, Σ]. We find the saddle point, Gs, Σs, and only focus on the “Nambu-Goldstone” modes associated with breaking reparameterization and U(1) gauge symmetries by writing G(⌧1, ⌧2) = [f 0(⌧1)f 0(⌧2)]1/4Gs(f(⌧1) − f(⌧2))eiφ(τ1)iφ(τ2) (and similarly for Σ). Then the path integral is approximated by Z = Z Df(⌧)D(⌧)eNSeff[f,φ].

The Schwarzian theory of the SYK model

• J. Maldacena and D. Stanford, arXiv:1604.07818;
• R. Davison, Wenbo Fu, A. Georges,

Yingfei Gu, K. Jensen, S. Sachdev, arXiv.1612.00849;

• A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia, arXiv:1802.07746;

Yingfei Gu and S. Sachdev, unpublished

The Schwarzian theory of the SYK model

Symmetry arguments, and explicit computations, show that the effective action is Seff[f, φ] = K 2 Z 1/T dτ(∂τφ + i(2πET)∂τf)2 − γ 4π2 Z 1/T dτ {tan(πTf(τ)), τ}, where f(τ) is a monotonic map from [0, 1/T] to [0, 1/T], the couplings K, γ, and E can be related to thermodynamic derivatives and we have used the Schwarzian: {g, τ} ≡ g000 g0 − 3 2 ✓g00 g0 ◆2 . Specifically, an argument constraining the effective at T = 0 is Seff  f(τ) = aτ + b cτ + d, φ(τ) = 0

• = 0,

and this is origin of the Schwarzian.

• J. Maldacena and D. Stanford, arXiv:1604.07818;
• R. Davison, Wenbo Fu, A. Georges,

Yingfei Gu, K. Jensen, S. Sachdev, arXiv.1612.00849;

• A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia, arXiv:1802.07746;

Yingfei Gu and S. Sachdev, unpublished

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

Black holes

• Black holes have an entropy proportional to their sur-

face area, and a temperature, TH = ~c3/(8πGMkB).

• Black holes relax to thermal equilibrium in a

‘Planckian’ time ∼ ~/(kBTH) = 8πGM/c3.

• Black holes in d + 1 spatial dimensions are similar to

a quantum system without quasiparticles in d spatial dimensions.

SYK models and black holes

ζ

~ x

ζ = ∞

charge density Q Holographic Metals and the Fractionalized Fermi Liquid

Subir Sachdev

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 23 June 2010; published 4 October 2010) We show that there is a close correspondence between the physical properties of holographic metals near charged black holes in anti–de Sitter (AdS) space, and the fractionalized Fermi liquid phase of the lattice Anderson model. The latter phase has a ‘‘small’’ Fermi surface of conduction electrons, along with a spin liquid of local moments. This correspondence implies that certain mean-field gapless spin liquids are states of matter at nonzero density realizing the near-horizon, AdS2 R2 physics of Reissner- Nordstro ¨m black holes.

151602 (2010) P H Y S I C A L R E V I E W L E T T E R S

week 8 OCTO

105, 151602 (2010) Black hole horizon

SYK models and black holes

S2

AdS2 × S2 ds2 = (d⇣2 − dt2)/⇣2 + d~ x2 Gauge field: A = (E/⇣)dt

Bekenstein-Hawking entropy of AdS2 horizon at T = 0 ⇔ Ns0 entropy of SYK model

Connections of SYK to gravity and AdS2 horizons

• Reparameterization and gauge

invariance are the ‘symmetries’ of the Einstein-Maxwell theory of gravity and electromagnetism

• SL(2,R) is the isometry group of AdS2.

ds2 = (dτ 2 + dζ2)/ζ2 is invariant under τ 0 + iζ0 = a(τ + iζ) + b c(τ + iζ) + d with ad − bc = 1.

SYK models and black holes

ζ

~ x

ζ = ∞

charge density Q

Einstein-Maxwell-theory

AdS2 × S2 ds2 = (d⇣2 − dt2)/⇣2 + d~ x2 Gauge field: A = (E/⇣)dt

S2

• S. Sachdev, PRX 5, 041025 (2015)

S4D = Z d4x p −ˆ g ✓ ˆ R + 6/L2 − 1 4 ˆ Fµν ˆ F µν ◆ ,

• Has Reissner-N¨
• rdstrom-AdS charged black hole solution, with charge den-

sity Q, a near-horizon AdS2 × S2 geometry, and surface electric field E.

• The Bekenstein-Hawking black hole entropy S4D obeys the same relation

as the SYK model ∂S4D ∂Q = 2πE , where E is identified from the spectral asymmetry of probe particle Green’s functions in both cases.

ζ

~ x

ζ = ∞

charge density Q

Einstein-Maxwell-theory

AdS2 × S2 ds2 = (d⇣2 − dt2)/⇣2 + d~ x2 Gauge field: A = (E/⇣)dt

S2

P . Nayak, A. Shukla, R.M. Soni, S.P . Trivedi, and V. Vishal, arXiv:1802.09547;

• A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia,

arXiv:1802.07746

S4D = Z d4x p −ˆ g ✓ ˆ R + 6/L2 − 1 4 ˆ Fµν ˆ F µν ◆ ,

In the small black hole size limit, T ⌧ 1/R, where R is the radius of the black hole, the theory dimensionally reduces to an Einstein-Maxwell-dilaton theory in two dimensions (the Jackiw-Teitelbaum model), along with Maxwell term S2D = Ns0 + Z d2xpg ✓ Φ(R Λ) Z(Φ) 4 FabF ab ◆ . The dilaton Φ represents the radial oscillations of the small black hole.

ζ

~ x

ζ = ∞

charge density Q

Einstein-Maxwell-theory

AdS2 × S2 ds2 = (d⇣2 − dt2)/⇣2 + d~ x2 Gauge field: A = (E/⇣)dt

S2

There are no bulk quantum fluctuations of the metric in two-dimensional gravity, and there a further dimensional reduction to a 0+1 dimensional theory representing fluctu- ations of the AdS2 boundary: this 0 + 1 dimensional turns

• ut to be precisely the Schwarzian theory obtained for the

SYK model.

• J. Maldacena, D. Stanford, and Zhenbin

Yang, arXiv:1606.01857;

• K. Jensen, arXiv:1605.06098;
• J. Engelsoy, T.G. Mertens, and H.

Verlinde, arXiv:1606.03438 P . Nayak, A. Shukla, R.M. Soni, S.P . Trivedi, and V. Vishal, arXiv:1802.09547;

• A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia,

arXiv:1802.07746

S2D = Ns0 + Z d2x√−g ✓ Φ(R − Λ) − Z(Φ) 4 FabF ab ◆ .

• 1. Random matrix quasiparticle model

q=2, complex SYK

• 2. Matter without quasiparticles

q=4, complex SYK

• 3. The Schwarzian theory

4.Connections to black holes with AdS2 horizons

• 5. Connections to strange metals

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu O

Described by a Hubbard model

• n the Cu sites

SM FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

SM FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

Strange Metal

TSDW Tc T0

2.0

α"

1.0

SDW

Superconductivity

BaFe2(As1-xPx)2

Resistivity ∼ ρ0 + AT α

• S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,
• H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and
• Y. Matsuda,

Physical Review B 81, 184519 (2010)

AF +nematic

Strange Metal

H = X

x

X

i<j,k<l

Uijkl,xc†

ixc† jxckxclx

SYK quantum islands of electrons with random hopping between them.

Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, 216601 (2017)

Coupled SYK Islands

|Uijkl|2 = 2U 2 N 3

Gaussian distribution and |ti j,x,x0|2 = t2

0/N.

formalism, one studies

lx +

X

hxx0i

X

i,j

tij,xx0c†

i,xcj,x0

distribution

t U

See also A. Georges and O. Parcollet PRB 59, 5341 (1999)

H = X

x

X

i<j,k<l

Uijkl,xc†

ixc† jxckxclx

Can also use non-random t, and the same U on all “islands”.

lx +

X

hxx0i

X

i,j

tij,xx0c†

i,xcj,x0

distribution

t U

See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Pengfei Zhang, PRB 96, 205138 (2017) Debanjan Chowdhury, Yochai Werman, Erez Berg, T. Senthil, arXiv:1801.06178

Coupled SYK Islands

Low ‘coherence’ scale Ec ∼ t2 U

Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, 216601 (2017) See also A. Georges and O. Parcollet PRB 59, 5341 (1999)

Coupled SYK Islands

Low ‘coherence’ scale Ec ∼ t2 U

For T < Ec, the resistivity, ρ, and entropy density, s, are ρ = h e2 " c1 + c2 ✓ T Ec ◆2# s ∼ s0 ✓ T Ec ◆

Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, 216601 (2017) See also A. Georges and O. Parcollet PRB 59, 5341 (1999)

Coupled SYK Islands

For Ec < T < U, the resistivity, ρ, and entropy density, s, are ρ ∼ h e2 ✓ T Ec ◆ , s = s0

Low ‘coherence’ scale Ec ∼ t2 U

Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, 216601 (2017) See also A. Georges and O. Parcollet PRB 59, 5341 (1999)

Coupled SYK Islands

Quantum matter without quasiparticles

• S. Sachdev, Quantum Phase

Transitions, Cambridge (1999)

• Rapid local thermal equilibration (of fermion correla-

tors) in a ‘Planckian’ time τeq ∼ ~ kBT , as T → 0.

• Presence of quasiparticles should slow down

thermalization, so all quantum systems obey τeq > C ~ kBT , as T → 0. Absence of quasiparticles ⇔ Fastest possible thermalization

Quantum matter without quasiparticles

• Planckian dynamics is realized in the ‘solvable’ SYK

models

• Black holes thermalize in a time ∼ ~/(kBTH), where TH

is the Hawking temperature.

• A Schwarzian theory of a time reparameterization mode,

with SL(2,R) symmetry, describes the quantum dynam- ics of – the SYK models – black holes with near-extremal AdS2 horizons