Late time physics of holographic quantum chaos
Julian Sonner

East Asian String Webinar
7 August 2020

work with

Alexander Altland (Cologne)[1603.04856], [1707.08013], [1903.00478],[1903.03143], [1907.10061], [2008.02271]

Contents
1. Introduction
context and motivation
2. The effective field theory of quantum chaoscausal symmetry breaking and sigma model
3. Bulk picture: minimal string (field) theory
Open/closed duality with causal symmetry breaking

Quantum chaos and holography
AdS/CFT relates gravity (often in AdS) to unitary field theory (often CFT)
Goal: use the quantum dynamics of field theory to shed light on theparadoxical behaviour of black holes.

Puzzle: are you saying black holes are just lumps of coal? Where does
all the intrigue hide? Where does the universality of black-hole physics
come from?
Resolution: the long-time dynamics of complex Hamiltonians ischaotic and at the same time universal. This replaces the (naive)
semiclassical late time physics of bulk gravity.

of [57–59], but far from equilibrium.

2The
The
Collapse Thermalization
State➛ BH formation (& evaporation)
premise:boundary:
Quantum quench

bulk:
(zk )

(z1 )
(z2 )

Black-hole collapse(z3 )[T. Anous, T. Hartman, A. Rovai, J. Sonner]

Figure 2.1: A shell made up of individual null dust particles collapses to form a BTZ black hole.
We have labelled the particles by their dual operator insertion on the boundary in anticipation of our
CFT construction in this paper.

The strategy: Study quantum thermalisation at all relevant timescales

2.1➛ Quantum chaos, quantum ergodicity
Motivation from the gravity side

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the original signal.
“spectral form factor”
log K(⌧ )Compared to (4.7) which deals with early times,
regenesis is concerned with the time scales of orderthe scrambling time of a system.
For maximally chaotic systems with a gravity
dual, the phenomenon has a beautiful geometric interpretation. The thermal field double state corresponds to an extended Schwarzschild geometry,
which has a non-traversable wormhole, the so-called
log ⌧Einstein-Rosen bridge. Turning on the coupling potential V at t = 0 injects a pulse of negative energy
RMT universality
which makes the Einstein-Rosen bridge traversable:[from
et al.]
the signal
sent Bohigas
much earlier
in the right system can
FIG. 9. Universal behavior
K(⌧ as) fora chaotic sysnow simply propagate through the wormhole to show
2D Blackofholes
RMT
1, whereistem [152]. ⌧ is measured in units of
up in the left system. See Fig. 8.
average level spacing. The universal RMT behavior corConnecting
quantum
chaos
on
the
field
theoryside
Renaissance in recent years: 2D BHs
areto exactly
described
by bothresponds
the linear ramp
followed by a plateau,
and interesting gravitational phenomena in this way,in dotted lines in the figure.
gives
a new and excitingensembles
perspective on potential
random-matrix[Kitaev,indicatedStanford
et al., Maldacena et al.]laboratory realizations of quantum many-body systems with holographic duals. Evidently as the relWeird feature of lower dimensions portantor important“observable”generalto probe thelesson?
level distribution
evant technology is at an early stage, one must iniis the spectral form factor K(⌧ ), which is defined as
tially target the simplest known realizations, namely<latexit

The Big Picture

Bohigas-Giannoni-Schmidt: (paraphrased)

A classically chaotic system behaves* quantum mechanically likea random matrix ensemble (at late times)
15

Our Result
Boundary theories in holographic duality* exhibit late-time RMTbehaviour in accordance with BGS.

The main technical insight:
RMT behaviour is universal because it follows from a simple
symmetry breaking pattern and the associated Goldstone EFT!

Causal symmetry
breaking

The EFT ofquantum chaos

In particular, this is true for individual quantum systems!

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sha1_base64="8p6/5pxMaWu83kY9cigl68U8X9I=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgiCWRAQ9Fj3UYwX7AW1aNttNu3SzCbsTpYRe/CtePCji1Z/hzX/jts1BWx8MPN6bYWaeHwuuwXG+rdzS8srqWn69sLG5tb1j7+7VdZQoymo0EpFq+kQzwSWrAQfBmrFiJPQFa/jDm4nfeGBK80jewyhmXkj6kgecEjBS1z6odE5xW7AAFO8PgCgVPeJK56xrF52SMwVeJG5GiihDtWt/tXsRTUImgQqidct1YvBSooBTwcaFdqJZTOiQ9FnLUElCpr10+sAYHxulh4NImZKAp+rviZSEWo9C33SGBAZ63puI/3mtBIIrL+UyToBJOlsUJAJDhCdp4B5XjIIYGUKo4uZWTAdEEQoms4IJwZ1/eZHUz0uuU3LvLorl6yyOPDpER+gEuegSldEtqqIaomiMntErerOerBfr3fqYteasbGYf/YH1+QOcYpXD</latexit><latexit

If opposite causality propagators are equal, clearly 𝞀 = 0
there is a “causal symmetry” G+ $ G (actually a GL(n|m) symmetry)<latexit

1

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R2 (!) =<sha1_base64="7q/7QC/lriJH6mAyr9GiFmP0Hl8=">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</latexit>

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Causal symmetry and its breaking: an aperçu
Key observable: spectral correlations

⌘
⇣
D ⇣
!
2
⇢ E
⇢ E+
2

:= ⇢(E) = G (E + i✏)+

Goldstones of causal symmetry fix R2⌘E!
2

Define density via the discontinuity

G (E
i✏)

Causal GL(n|m)

RMT Universality

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ramp and plateau behavior. Left panel:

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sha1_base64="eR6a7Q+eMW2jWAQKjXzjwfKWoqs=">AAACEnicbVA9SwNBEN3zM8avqKXNYhC0MNyJoGXQxjJ+RAO5GPY2k2Rx9/bYnVPCkd9g41+xsVDE1srOf+MmRlDjg4HHezPMzIsSKSz6/oc3MTk1PTObm8vPLywuLRdWVi+sTg2HKtdSm1rELEgRQxUFSqglBpiKJFxG10cD//IGjBU6PsdeAg3FOrFoC87QSc3CNtLQCkXh6oyGp6LTRWaMvqWhVtBh3162c9ZvFop+yR+CjpNgRIpkhEqz8B62NE8VxMgls7Ye+Ak2MmZQcAn9fJhaSBi/Zh2oOxozBbaRDV/q002ntGhbG1cx0qH6cyJjytqeilynYti1f72B+J9XT7F90MhEnKQIMf9a1E4lRU0H+dCWMMBR9hxh3Ah3K+VdZhhHl2LehRD8fXmcXOyWAr8UnOwVy4ejOHJknWyQLRKQfVImx6RCqoSTO/JAnsizd+89ei/e61frhDeaWSO/4L19AuqrnP4=</latexit><latexit<latexitt⇠e )!⇠eFigure 4.1: Spectral form factor at half filling ⌫ = 12 for vary⇣ S⌘
di↵erent temperatures at N
e = 14. In both cases we see the ch
Need to control effects of O e
in bulk & boundary!
<latexit(Doubly) non-perturbative spectroscopy

SYK spectral
Form factor[Cotler et al.; JS, Vielma]SS= 1. One sees sev

Contents
1. Introduction
context and motivation
2. The effective field theory of quantum chaoscausal symmetry breaking and sigma model
3. Bulk picture: minimal string (field) theory
Open/closed duality with causal symmetry breaking

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sha1_base64="h24la5TNbnw23ROBcpxqD2kyR0s=">AAACAHicbVA9SwNBEN3zM8avUwsLm8UgWIW7IGgZtLGMYD4gOY69zV6yZG/32J1TkpDGv2JjoYitP8POf+MmuUITHww83pthZl6UCm7A876dldW19Y3NwlZxe2d3b989OGwYlWnK6lQJpVsRMUxwyerAQbBWqhlJIsGa0eBm6jcfmDZcyXsYpixISE/ymFMCVgrd41Ho445gMWje6wPRWj3iUVgJ3ZJX9mbAy8TPSQnlqIXuV6eraJYwCVQQY9q+l0IwJho4FWxS7GSGpYQOSI+1LZUkYSYYzx6Y4DOrdHGstC0JeKb+nhiTxJhhEtnOhEDfLHpT8T+vnUF8FYy5TDNgks4XxZnAoPA0DdzlmlEQQ0sI1dzeimmfaELBZla0IfiLLy+TRqXse2X/7qJUvc7jKKATdIrOkY8uURXdohqqI4om6Bm9ojfnyXlx3p2PeeuKk88coT9wPn8AUBOWNg==</latexit><latexitz$zRemark: therea “Weyl ansymmetry”:
possesses
interesting
Weylsymmetry
under the exchan
pectral determinantZ(4) is1
2⌦↵
1
Im @z2 function
Z(2) (ẑ) unchanged.. As we
Zz(2)2 , (ẑ)⇢(z) = ⌥correlation(2.9)
which leaves ,the spectral
will discuss⇡Lz1 =z2 =z
z1 =z2 =E±i0
discrete symmetry is key to the understanding of the non-perturbative structure<latexit

tslight
is written
natural way
in terms
times
normalized
by theinverseabuseinofa notation,in eachcaseofthe
matrixẑ is taken
to be
the diagonal m
discussedthesecharacteristic
probes,we 1now
introducea case.
class of
aux,
z
)
in
the
first
The
dimension oderedenergyarguments,
e.g.
ẑ
=
diag(z2
Spectral determinants
hx willoutbe
toclearbe themost
objects
on which
we base ourwill turnalwaysfromtheconvenient
context, andthe utilityof arranging
energies in mewill
theemerge
form ofinratiosof determinants
and we are
principally
our subsequentdevelopments.
Let
us briefly interested
pause andinnote that cof departure:
ratios of determinants,e.g.
tion needs to bePoint
placedon the infinitesimal
imaginary parts
given to the energy argum
e denominators, where they determine the pole structure of the spectral determindet(z1 H)
det(z1 H)det(z2 H)
Z(4) (ẑ) = Z(2) allows us to extract.
(2.8) density v)fically,=
the spectral determinant
the spectral
det(z2 H)
det(z3 H)det(z4 H)
ent, viz.

notation, in each case the matrix ẑ is taken to be the diagonal
matrix⌦↵1
Im @dimension
Z(2)diag(z(ẑ) 1 , z2 ) in ,⇢(z) case.
= ⌥ The.
W (z)z2 Z(2) (ẑ)of thisthe first
uments,e.g.= @ẑz2 =⇡L correlation:
For example, zwe=z recover the spectral
z1 =z2 =E±i0
1 =z2canclear from the context, and the utility of arranging energies in matrix
Retardedhe subsequent
other hand, developments.
the spectral twoLet
point
functionrequirestheuseof theratio Z(4) :ur
us briefly
pause
and
note
that
carefulAdvanced
laced on the infinitesimal imaginary parts given to the energy arguments,
E
D
2
Re
@
+
where they determine
the= pole structureZof (zthe, zspectral
determinants.
,
z
,
z
)
.
h⇢(E)⇢(E 0 )i
1
2
(4)
3
4
2
2
+
3 @z4
z1 =zvia
ral determinant Z(2) allows ⇡usL to@zextract
the spectral density
3 =E theFFz2 =z4 =E 0

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a 4L dimensional
vectorof dimension
with 2L 4L
Grassmanniancompo
is now agradedgraded object
and theaction invariant
under
GL(2L|2L) integrals produce the determ
er components.
Here, theGrassman, while the c-number integrals produce those in the denominator.
<latexit

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symmetry
We introduced 2L Grassmann numbers ⌘<latexit

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First we exponentiate

We can trivially write a determinant as a (functional) integral

det (EH) =Zd(¯⌘ , ⌘)e⌘¯(E H)⌘y to rewrite the ratios (2.8) is by means of a graded Gaussian inˆi ¯(ẑ H)d( , ¯) ,Z(4) (ẑ) = e(Inverse determinants are introduced by boson integrals)

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e energy arguments have been S
split into E14⇥4 , and the matrix of energy di↵eren
P system
ysics
of
the
are
modes
drawn
from isthe⇠ flavore loss) GoldstoneIR
theoryisofgiven
by Goldstones
on coset
we assume i !i (i.e.
= 0!
without
generality.
The integral
above
setmanif
up su
egration over the commuting (anti-commuting) variables produces the determina
M =entering
GL(2|2)/⇥ GL(1|1)). that the integrat
enominator (numerator)
the (GL(1|1)
spectral determinant.
Note= { µa } and ¯ = { ¯µa } live in a tensor product space, HC ⌦ HF , where HC is t
<latexit

Construction of the theory
Then we integrate
ined in section 2.5 above, our starting point is the Gaussian integral representati1), of the An
spectraldeterminant
Eq.us:(2.8),
now
integrated over the invariant ensemb
exampleto motivate
actualRMTˆˆ
V (H)+i ¯(ẑ H)¯
hZ(ẑ)i = dH d( , ) e.
(3
T ¯rite = Due( + ,to integral, ⌘ + , ⌘ )only, H-singlets= ( ¯+ , ¯can, ⌘¯+survive:, ⌘¯ ), soGL(2L|2L)
that ± denotec-number integ➛ GL(2|2)
ables, ¯ their conjugates (note the minus sign in front of ¯ required by convergenc
Results in an action over only a (2|2) graded matrix with causal
re independent Grassmann variables, and we have made the H-average explicit. W
symmetry GL(2|2)!
the energies as a matrix,

The presence of a non-zero density of states 𝜌 spontaneously breaksẑ = E + !
ˆ +GL(2|2)
i ⌧3 , ➛!
ˆGL(1|1)= diag(!
1 , !2 , !3 , !4 ),x GL(1|1)(3

conjugation respects the residual U(1|1)⇥U(1|1) symmetry. More generally, the Goldstone
modes Q(x) are fluctuating degrees of freedom, where x parameterizes the base space ofthe e↵ective theory. In this case, the Goldstone modes are constructed by writing T as
Constructing
a product of an elementin the full group G thetimes EFT
an element of the preserved subgroup
physics of the system are flavor Goldstone modes drawn from the manifold3
H. This is achieved by letting T (x) ! T (x)H(x), H(x) 2 U(1|1) ⇥ U(1|1) so that H
Used RMT
as amotivating
butT0mechanism
is general:largeweakly
becomes
an exact
local
symmetry,example,
and T (x) !
T (x), T0 2 U(2|2)
a global one,M=
GL(2|2)/(GL(1|1)⇥
GL(1|1)).
(2.13)
brokenbythedi↵erencesoftheenergyarguments
entering
theargumentẑ.
GL(2L|2L) projects on singlets GL(2|2) ⟹ broken to causal sectors

4. E↵ective action: Historically, the present form of nonlinear sigma models systems was
ne modes:thephysics isthe light
modes,
thee↵ecCansinceusewell-known
coset
toby
construct
EFT
pioneeredin theirapplicationtoe↵ectively
themethod
physics dominatedof[CCWZ]
disordered
metals.
There,
x is a real
space
ory willbe given
integralthe soft manifold. The convergence of thiscoordinate,
and bythe anactionassumesoverthe form
requires a reduction to theˆcoset space U(2|2)/U(1|1)⇥ U(1|1), where the bosonic
ˆ
2
i
S[Q]
=
iv
str
(ẑQ)
dV
+
F
str
r
Qr
Q dV
+ ··· ,
(2.15)nic) sector of U (2|2) is the (pseudo)unitary group iin two
dimensions.
Geometri-e bosonic sector of the coset is a two-dimensional hyperboloid, and the fermionicwhere ‘str’ is the matrix trace generalized to ‘supermatrices’containing
commuting and[Wegner,
Efetov]wo-dimensional sphere. A convenient
way to represent this manifold
is in terms of
4
2
anti-commuting elements. The dimensionful quantities v and F can be identified (seerix field
Where
Q is analogous
to states
the pionin thechiralsection3) withthe density of
per field
volume,⇢(z),
and Lagrangian:the di↵usion constant D
tf

respectively. Note the structural similarity to the chiral Lagrangian of QCD [18] (albeit
1
Q = T ⌧3 T ,pattern)
T which
2 U(2|2)(2.14)
with a di↵erent symmetry-breaking
shows that the di↵usion term entersin the same way into the e↵ective field theory of chaos as the Pion decay constant enterssymmetrybreaking
vacuum
0
0
5
ss
s
ss
chirale↵ective. However,Lagrangian,S[Q]ofdoes
the thePaulimatrixin Lagrangiancausal space(⌧3 ) unlike= ( the) QCD, andthe action
T by
3 is intofsha1_base64="dapTOiy76w5Z/lgnP4xUhMpLdEw=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ItQ9OKxQtsU2lg220m7dLMJuxuhlP4ILx4U8erv8ea/cdvmoK0PBh7vzTAzL0wF18Z1v53C2vrG5lZxu7Szu7d/UD48aukkUwybLBGJaodUo+ASm4Ybge1UIY1DgX44upv5/hMqzRPZMOMUg5gOJI84o8ZKfoPcEHz0e+WKW3XnIKvEy0kFctR75a9uP2FZjNIwQbXueG5qgglVhjOB01I305hSNqID7FgqaYw6mMzPnZIzq/RJlChb0pC5+ntiQmOtx3FoO2NqhnrZm4n/eZ3MRNfBhMs0MyjZYlGUCWISMvud9LlCZsTYEsoUt7cSNqSKMmMTKtkQvOWXV0nrouq5Ve/hslK7zeMowgmcwjl4cAU1uIc6NIHBCJ7hFd6c1Hlx3p2PRWvByWeO4Q+czx/7PI6r</latexit>

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3Well, oftechnicallythepioninfieldreallyistheory,
W, wheretion
theresidual
U(1|1)⇥U(1|1)
symmetry.
Moretheconjugation,
Goldstone
Therespects
presenceanti-unitary
symmetries
the microscopic
such
as timegenerally,
reversal, charger chiral symmetries, restricts the set of symmetry-compatible
continuous transformations and changes the
W
Q(x)
are
fluctuating
degreesoffreedom,wherexparameterizes
the
base
space
of
T
=
e
Goldstone mode manifold. However, for simplicity, we focus on the simplest setting, where H is just hermitian.

Comments
Showed “single-particle version”, more relevant is many-bodyversion, which lives instead on many-body Hilbert space (harder)
We used RMT average to motivate singlet projection. Generalquantum system: take late time limit (+ perhaps ‘mild’ average)“Altland-Zirnbauer” classification of 10 RMT ensembles becomes
Cartan classification of 10 symmetric spaces (Weyl symmetry)

Generalises easily to ratios of more spectral determinants

We used graded (‘susy’) approach. Also common to use replicas.

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gimes,
boththeactiondescribing
singleparticle
systems Eq. how(2.15),
ving
into
the
bulk
story
in
section4,
letus
explainqualitatively
theWait long enough so that all non-ergodic modes have decayedystems,Eq. Thouless(2.16)
collapse
zero
mode action
lly produces
correlations
thethetype
associatedwith Euclidean wormhole(definetime) of tofrom the study of low-energy QCD⇡ — or indeed any other context where
= —
i the
str(Qẑ),bosons dominate theS(Q)physics
physical manifestation of the coset(
2
modes (aka pions) associated to the generators of the broken symmetry.
Remember: wanted to evaluate spectral determinants. This is now=llyQwriten = Q is the homogeneous zero mode and the level spacing define=B
he systems are physicallyequivalentmodels (which
T = exp(W),
W to= random matrix,
B̃
ibed by the action (2.17) as we will demonstrate in section 3):
2 easily done in terms of the “pions”
v n 1 defined through the e↵ectiveof Hilbert space✓ dimension◆⇡

P<latexit

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is morethan
one causal
symmetrybreaking
saddle B and B̃sthematrixpion
fields,expandedinthebrokengeneratorse random-matrix physics: The partition sum describing a chaotic quan
yergodicare (1|1)
supermatrices.
To form
leading order in these generators, the acti⇣
⌘ theregimeassumes“standard” saddle SSS = 0
2 B̃ , where for the time being we expand around the ‘[B, B̃]ˆ
M== 2isH2 str⇥ SB
S(Q)e have written the exponent
toleading(quadratic)order only, which is jushZ(ẑ)i = dQ e,(
S
gic and which we will correct systematically.
Expandingarounde quadra“Andreev-Altshuler”
saddle S
AA ⇠ this<latexit

ation to the SYK model), or a ‘many body localized’ phase with st
fluctuationsof Qn (seeRef. [21] forive contributions:wormholes
andreview.)
baby universes

The universal content of RMT

F

tf

f

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Coefficient of diagram is fixed!
12
where the marked (red) points are the insertionsFT(sof the projectors,
which
appearonthebound) ⇠ t, ⇢(z) = ⌥Im @z2(z) = @z2 Z(2) (ẑ)
gives linearW
ramp<latexit

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ator
insertionsaresimilarly
up (topology
fromB,cylinderB̃ matrices.
Specifically,in omatrixwill
bebuilt
clear
fromthethecontext,
andthe utility
Simplest
diagram:“wagonwheel”
of/ annulus)
weightEq. (2.21)
canalways
be represented
asn of the two-point function at leading order we are instructed to compute theform will
emergef in
oursubsequentdevelopments.
Let
us
brieflf
f
E P = { ,f 0 ,f } project onto thematrices
ue hstr(B B̃P )str(B̃BP )i,D where the
f
f
)str(
B̃BP
) infinitesimal imaginary parts
str(B
B̃P
attention
needs
to
be
placedon
theor. The matrices B and B̃ each have one index acting in the retarded sector
in inthethedenominators,
wherethey ofdetermine
thein polestructurecting
advanced sector, andthe order
multiplication
the precedingue Specifically,e↵ectively projects
on the⇠advanced (red) andthe once
spectralallowsus retarded
to extracs 2 on the(2) once=determinant Z⇠
s required by the prescription in (2.8). We may then represent the matrices
resolvent, viz.
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framework.
These
take
the
form
of ratiosof determinants
and
(1|1)
supermatrices.
To
leading
order
in
these
generators,
theactiontakes⇣⌘
namely] =two2iscases,
str B B̃, where for the time being we expand around the ‘standard
Topological expansion of EFTe written the exponent to leading (quadratic) order only, which is justified by
det(z
det(z
H)
H
1
1
nd which
we exhibit
will correct
Expanding aroundthisquadratic
limit,Let ussomeoftheleading
contributions
toZsystematically.
Z
(ẑ)
=
(ẑ)
=
(2)
(4)
det(z
H)
det(z3 H
nsider matrix integral averages of the
type2
ˆ
⇣
⌘
!
) (. .each
By a slightof B̃)
notation,
the matrixẑ is ta⇠ eS (2.21)s :=h. . . iabuse⌘ d(B,
e 2is str(B B̃in. ) , case⇡ble line notation, the Wick contraction of these matrices with the Gaussian(2.22)

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set of arrows that can be drawn on the two closed loops of the ribbon graph above, so thateach each is always anti-parallel to its neighboring edge (again, we refer to Appendix A fora more microscopically resolved discussion of this point.) Correspondingly, the bulk surface
HigherTopologies
is non-orientable which is achieved by the crosscap insertion, as realized in bulk languagein [25]. Finally, by expanding to yet higher order in the B, B̃ matrices, we can find highergenus contributions, the first non-trivial one coming from expanding to order str((B B̃)4 ). ThisFirst correction: “Sieber-Richter pair” (topology of cylinder with one handle)
adds one handle as inED
str(B B̃P f )str(B̃BP f )str((B B̃)4 )(2.24)In the The
unitary
class, it actuallythat a further
genus-one classcontraction exists, which
coefficientof this turns
termout
depends
on symmetrycancels precisely against the one shown here, but in other symmetry classes diagrams like theone above give non-vanishing contributions. Since the translation between ribbon graph andis zero. one
In fact
all corrections
vanishBy(cansurfaceInbyunitaryeye can classbecomecoefficient
a little complicated,
can proceed
more abstractly.
tracingshow
thisusingone
Duistermaat-Heckmann)
the loopsof thisdiagramsees that it is indeed orientable and that it has F = 2 faces, V = 2
vertices and P = 4 propagators, meaning that it has a single handle H = 1. This follows directlyfrom the classic formula in toplogy F P + V = 2 2H. The marked points correspondingsha1_base64="wF6pf9c0Tg7PmKYITILqNsZcbX4=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBC8GHYloMegF48RzAOSNcxOZpMx81hmZoWw5B+8eFDEq//jzb9xkuxBEwsaiqpuuruihDNjff/bW1ldW9/YLGwVt3d29/ZLB4dNo1JNaIMornQ7woZyJmnDMstpO9EUi4jTVjS6mfqtJ6oNU/LejhMaCjyQLGYEWyc1zUN2Xp30SmW/4s+AlkmQkzLkqPdKX92+Iqmg0hKOjekEfmLDDGvLCKeTYjc1NMFkhAe046jEgpowm107QadO6aNYaVfSopn6eyLDwpixiFynwHZoFr2p+J/XSW18FWZMJqmlkswXxSlHVqHp66jPNCWWjx3BRDN3KyJDrDGxLqCiCyFYfHmZNC8qgV8J7qrl2nUeRwGO4QTOIIBLqMEt1KEBBB7hGV7hzVPei/fufcxbV7x85gj+wPv8ATZ7juA=</latexit>

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4s,(2.25)
simplicity we included orientable diagrams/surfaces only. In the semiclassicallyexact<latexit

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unitary class, the expansion truncates after the first contributions, and we obtain the spectral
wo point function as1
sin2 s2sins2isR2 (s) = Re 2 (1 e)=.
(2.26)22s (see also Figure s1 for a description in terms
bove graphical representation suggests

R2 (s) =
Re

2is
(1
e
)=
2
2s
1
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s 2
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+···

+

s 4

s2

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+

+···

.

⌘

,
+···

“standard” saddle,⌘
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2

latexithe presence of the Altshuler-Andreev saddle, QAA , defined in the beginning of the section
uffices for the present purposes. Referring the Reader to section 3 for the technical fine print,
So far only diagrams around “standard saddle”. Full picture:0
a repetition
expansion,
now around+ · · · QAA gives rise to the full structure+= thees perturbative
R (s) of⌘<s 4

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2is s
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latexit2<

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nto
of these computations, a stationary phase analysis extended for
pacethe(seetechnicalities
Fig. 1 for an illustration.)
nceTechnically,of the Altshuler-Andreev
saddle,in theof theAA , defined
the Goldstone mode
matrixQ
integrals
Eq. (2.18)
are beginning
simple enoughto besection
doablethe present
purposes.
Referring
the Reader
tofor
section
3 for purposes
the technicalfine
print,
nr closed
form forall symmetry
classes [29].
However,the present
it is notnecessaryThingsget(doubly)
non-perturbativeo delve
into
the technicalitiesof thesenow
computations,a stationary
analysisextended foronof theperturbative
expansion,around QAA
gives risephaseto thefull structureass, the expansion truncates after the first contributions, and we obtain the spectralThis
resultswe
inincluded
the famous
sinediagrams/surfaces
kernel [F. Dysononly.
(1970)]
whereforsimplicity
orientable
In the semiclassically exact
function as⌘,(2.25)(2.26)

To summarise

next: the terms in the topological expansion are arranged intosame
topologies
as theexpansionof e↵ective
the JT matrixine 1: The thegeneric
late-time
behavior
described
by thetheorymodelof quantum
chaos.
Euclidean
wormholes
andbehavior
baby universes.
is not a– the spectral form
op line shows
the ramp
and plateau
of a simpleThis
observablecoincidence…
r K(⌧ ) – andthe bottom line shows how the di↵erent qualitative regimes are obtained in
gma model. The ramp behavior is initially well described by small perturbative corrections
d the standard saddle (the north pole of the sphere in the sigma model geometry H2 ⇥S 2 ).

Contents
1. Introduction
context and motivation
2. The effective field theory of quantum chaoscausal symmetry breaking and sigma model
3. Bulk picture: minimal string (field) theory
Open/closed duality with causal symmetry breaking

gredient to develop a bulk understanding is, once again, the determinant operator
det(E

±

H) = eTr log(E ± H)

Causal symmetry breaking
in
the
bulk
A bulk picture(4.1)

his object should be thought of as inserting a D-brane in the bulk at position E (see for example
ill now describe the emergence of the sigma-model from bulk considerations. The main
5, 36]), andLet
we distinguishherebetween an advanced
a retarded
brane, depending
us write thedeterminant
in terms braneof theandloop
operatordient
to develop a bulk understanding is, once again, the determinant operator
the sign of the (infinitesimal) imaginary part of the energy, as indicated. The Hermitian
atrix H – the Hamiltonian from the±point of viewof ourdiscussion of quantum chaos above± H)
Tr log(Edet(EH) = e(4.1)
corresponds to the presence of an additional stack of D-branes, one for each dimension of the
any-body Hilbert
space.
One way corresponds
to see this is byto
notingthat in
thestring
determinantoperator canSuchadeterminantabranetheory.
object should be thought of as inserting a D-brane in the bulk at position E (see for example
obtained as
the exponentialof the loop operator
Eachinsertionof6]), and we distinguish here between an advanced brane and a retarded brane, depending
e sign of the (infinitesimal) imaginary
part
energy,as indicated. The Hermitian
W(E) = Tr
logof(EtheH),(4.2)
274
8 Toroidal
compactification
and T -dualityx H – the Hamiltonian from the pointof viewof ourdiscussionof quantum chaos aboveadds
a world
sheetboundarylabelledby thewith
fixedE.
esponds
thepresenceof
an inadditional
ofwordsheet
D-branes,one
forenergy
each dimension
of
the
hich
acts toinserta boundaryan open stackstringboundary
conditionsuchatthe string
ends Oneon the‘E-brane’.
Upon
to formthe determinant,-body
Hilbertnow
space.
way
to see thisis byexponentiating
noting that thedeterminantoperatorthecancending
powersof the Tr logofE theHloop
insertioncorrespond to open string world sheets with antained as
the exponential
operator
Textbook
example with
with the right combinatorics to count all possible ways thecreasing numberof boundaries
σtwoonboundary
insertions
ring can end
the D-brane[35].
Tosummarisethen,H
an L ⇥ L matrix corresponding
W(E) = Tr log (E H) , is(4.2)
σ
the presence of a (large) stack of L D-branes, while E corresponds to the position of aacts
to insert“probe”
a boundary
in anopenthatstringwordsheetboundary
conditionngle
additional
D-brane.
Given
these
additionalwith
branesare addedin ordersuch
tohe string
nowofends
on the‘E-brane’. Upon
exponentiating
to form
nstruct
ratiosspectraldeterminants
as in (2.8),
we sometimes
refertheto determinant,these objects theas[source: Polchinski Volume I]
pectralbranes”.
WeTrwill
returnmore concrete
microscopic
such objects
ding powersof thelog E
H toinsertioncorrespondto open realizations
string worldofsheetswith anFig. 8.5. Exchange of a closed string between two D-branes. Equivalently, a
low innumber
the context
of minimal withstringvacuum
theory,we withcanonetake
L branes
to be of ways
ZZ typeloop ofwhereancombinatorics
open string
end oneach
D-brane.
asingof boundaries
the
right
tothe
count
all possible
the
2

1

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should be thought of as inserting a D-brane in the bulk at positiond we We
distinguishhere andbetweenan advancedbrane
andbya theretarded
have “advanced”“retarded”
spectral branes,
definedanalytic
continuation inimaginaryenergyof the(infinitesimal)
part of the energy, as indicatedthe TheHamiltonian
from
the
point
of
view
of
our
discussionofqua
auxiliary objects
now are physical: they are open strings
ds tostretching
the presence
anbranes
additional
stackbranesof D-branes, one for eachbetweenofsea
and spectral
Hilbert space. One way to see this is by noting that the determiad as the exponential of the loopµoperator<latexit

usal symmetry breaking in the bulk
physicalofrole
the psi’s from bulk consider
w describe the A
emergence
theforsigma-modelo develop
a bulkunderstanding
again,bythe
determinantIn this way
the spectral
determinantsis,areonce
represented“spectralbranes” o
ft

tf

f

det(E

±

O(1) number of “spectral
branes”
Tr log(E ± H)
H) = e “sea stack” of L >> 1 branes(think of L D3 branes e.g.)

W(E) = Tr log (E
H) ,

a ¯b
ab
⇧
=
with
µ µ and we may e↵ectively concentrate
metry breaking mechanism applies
on the light secto
3
odes contained in ⇧, in µ=1
other words the Goldstone sector Q.
about over the ‘sea’ degrees of freedom associated with the stack of L
Goldstones
mesonic
fluctuations
aroundthis
this vev(c.f.group,
holographic
QCD)
ndex
a. This areprojects
ontosinglets under‘color’leaving
nsform in with their a, b indices in the adjoint of the ‘flavor’ group.
pectral correlations, for example of pair correlations (2.6), the causal
This bulk interpretation of the chaotic sigma model is illustrated in Figure 2.
LX30
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ed to exponentiate the D-brane operator as open string modes. These
Paton factors µa , µ = 1, 2 . . . L, a = 1, 2 . . . n which allow them to end
Bulk
breakings making up the‘sea’causal
and/orsymmetry
on any of then probe branes. Noteh is supersymmetric by design, with a supergroup acting upon the
Causalis the[39]non-abelian
symmetryof coincident
spectralbranesedom(seesymmetryfor example
for a generaldescriptionof suchobjects).
USY perspective reveal that the bulk manifestation of the Goldstone
Brokenarespontaneously
by a condensateof string bound-statesa modele↵ective boundstates of strings

Making this fully concrete: MMST
This picture can be made fully explicit in minimal string theory:

Take Liouville (as a world sheet theory) and couple to (q,p) minimalmodel matter CFT
Minimal string theory has D-branes of the kind we need:Sea brane
Spectral brane

ZZ braneFZZT brane

There exists a direct map between matrix model and worldsheet
formulation ➛ can explicitly prove every assertion we made abovewhich is a variant of the Kontsevich model [45]. Equivalent derivations of this mdouble-scaled matrix theories have appeared before in [38, 46], although not in
associated mean field spectral density
supersymmetric formalism as we have utilized here. In the absence of sources, ⇣ˆMatrix model picture1
action is invariant under graded
generallineartransformations
a
!
T
aT
,T 2G
S0 p
3/4
e
2
L
S0
⇢(⇣)
=
⇣
,
e
⌘ 3/2 . of the partition sum Z(⇣)(4.
the Efetov-Wegner theorem securesthenormalization=⇡g
To connect bulk (= worldsheet) and boundary ( = matrix
model)
1 ⇢In the presence of sources, a parameterization of a =followed by integ
i2
focus again on determinant operatorsbtain thea two-dimensionalflavor ⇢,matrix
model
Grassmann variables
leads
to describing the fine structure of edge of singspacings by expansion aroundthe symmetry
unbrokensolution right◆✓ˆ⇣ 3at the⌘ edge, Ā(0)p
In the double-scaled
limit (at the
spectral
edge)
these
are computed
S0 str
g
ˆ
@
@
d
d
e
+
⇣
1
2
3
ˆ
Defining A = Ā(0) + 21/4 a where
factor upfronte the fluctuation
matrix a, (2,1)=the scaling cubic
by a graded variantZ(of⇣)Kontsevich’s
matrix
model
(q,p)
=
@⇣2 order in
2⇡a and ⇣, we obtain
duced for convenience, and expanding @⇣to 1leading⇣⌘ commuting variables,
where the diagonal matrix ⌘ ˆa|⇢= =0 econtains
the1 , 2 an
S0 str a3 +⇣aˆ
3hZ(ẑ)i'daeof integration contours safeguarding convergence is, implicit to the definition of(4.thFinally the factor (2⇡) 1 ensures the proper normalization of the bosonic measure, w
h is a notvariantof thespecify
Kontsevichmodel(4.4)[45].above.
Equivalent
derivationsof thismodel
fro
explicitlyin Equation
In orderto obtain the
spectraldensitle-scaledmatrixhave
appeared[38,
althoughin termsofwithz example:! ⇣,theories
a further⇣2 derivative
to beintaken,
beforesetting
the energy
argum
For
spectraldensity has
isbeforegiven
by
Airy46],function,
withnot
rsymmetricformalism
as results
we have
In oftheabsence of sources, ⇣ˆ = ⇣1, th
asymptotic
behavior
to eachother. Thisin utilized
the Airyhere.
densitysates
n is invariant under graded general linear transformations a ! T aT 1 , T 2 GL(2|2), an0
2S0 /32
2
2S0 /3
⇢(⇣)
=
e
Ai(x)
,
x
=
e
+
xAi
(x)
Efetov-Wegner theorem secures the normalization of the partition sum Z(⇣) = ⇣1.
e S0 pn the presence of sources, a parameterizationof a = 1 i ⇢2 followed by integration ov⇠⇣,
Grassmann variables ⇢, leads to ⇡◆ ˆused the standard⇣ 3asymptotics⌘
where in the second ✓line
we
haveof the Airy fu
S
ˆ
@
@
d d
0

Open / closed duality
Kontsevich’s matrix model is interpreted as the effective theory of
the spectral brane, having integrated out sea branes

Individual contributions to the determinant operators can becalculated directly in the world-sheet approach. For example:[4, 48],h (⇣1 ) (⇣2 )i ' eDisk(⇣1 )+Disk(⇣2 )+ann(⇣1 ,⇣2 )+ 12 (ann(⇣1 ,⇣1 )+ann(⇣2 ,⇣2 ))te that the individual factors contributing to this answer are precisely the ingre
world sheet disk amplitude with one FZZT boundary
oduced Disk(𝝵above,1):namely
disk diagrams with a boundary on either of the spectral b
diagrams
withsheetbothcylinder
boundaries
on the withsametwobrane,ann(𝝵amplitudeFZZTann(⇣boundaries
1,𝝵2): world
1,2 ) annulus
1,2 , ⇣1,2 ), and fi
ulus diagram with one boundary on each spectral brane, ann(⇣1 , ⇣2 ).
the next section, we will analyse the connected annular contribution ann(⇣1 , ⇣2
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4.2.1

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latexit<+1

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2⇡ 2

+
1

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Leading⇣ ) =singularity
Re W (⇣ , ⇣ )+ reg.
R (⇣1

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2

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2<<latexit

2
1 the worldsheet setting
ant to compute the bulk annulus diagram,
which
in
=
=
(4.23)
2
Some years ago Martinec computed
c 2s<2 1. annular partition2⇡ 2 (⇣1 ⇣the2)
ding contributions from worldsheet ghosts Zghost , the 2d Liouville partition funcfunction
in full
generality.
We
only
The terms
we leftout are regularin thelimitneed:⇣1 ! ⇣2 , as indicated in the first line. Takwell as the (2,p)
minimal
model
CFT,
Z
,
leading
to
the
final
result,mattering the real part is equivalent to fixing a prescription to analytically continue the function
W (⇣1 , ⇣2 )annulus , whichˆ 1leads to the singular⌧behavior we indicated. This is the bulk counterpart) expansioncos(4⇡sP )EFT= of chaoscos(4⇡s1of the cylinder contribution in
the topologicalof1 thegiven in Eq.
2
1 P Tr⇠ s(2.22)
Tr 2=,(⇣it1 ,is⇣reassuring(4.18)
Z(2,p)andH
HP ,coincides with that prediction.
2) =thatdP
the singular part of theindeedP diagram2⇡P sinh 2⇡ cosh 2⇡ b and retarded (blue) boundaries
0
In this way it becomes
clear how to map the badvanced (red)<latexit

o computing the contribution of the Goldstone modes to the leading singularity
The spectral correlation function is then obtained via
ndard saddle in the bulk, i.e. the exact bulk equivalent of the EFT diagram2

2

annulusannulus2
In order to further illustrate the universality of the EFT,
let(⇣usW(2,p)
Z(2,p) (⇣1 , ⇣2how) , the leading
1 , ⇣briefly
2 ) = @⇣ ,⇣highlight
1

1

2

+
1

annulus

2

(2,p)

2

in that representation to the properties of the FZZT boundary state given by the boundary±
Figure3: Thetwo-point function. Its universal part gives
cosmological constants ⇣1,2
(see
Figure3).annulus contribution to the spectral2
the correct
EFT singular contributionof 1/2s. The diagram is identical to (2.22), althoughleadingWhile
singulardiagramcontributing
to
the
ratio
(4)
higher-genus
contributions,suchas(2.24)
insymmetryclasses asdi↵erent
from withthe two branes,
here it represents a minimal-string +worldsheet interpreted
a 2D spacetime
also
as theinLaplacetransform
a Euclidean
wormhole. theThe bulk
2D gravity
1 ofunitary one might be possible
toknowncomputeprinciple(butdifficultin practice),
in- amplitude
gives the same leading singularity including the numerical coefficient as (2.22), coming fromterpretation of the Althshuler-Andreev
saddleremains
elusive, although it seems promising to
the Goldstone
sectorofcausalsymmetry.
⌧
annulus2
attempt to constructthese contributions
fromlookalike construc1
1
2 applying the Riemann-Siegel
s
⇠
Tr +
Tr
=
1, 2
tion to contributions
like(4.22).
This
would
amountto giving a bulk semiclassical description
H
H
36
1
2
(2,p)
which theis theAltshuler-Andreev(2, p) version of what wecalled ann(⇣) above,and where
1 , ⇣2weof the leading singularity aroundsaddle,which
discussedfrom therr
EFT perspective in (2.25). The same is true forpthe physics
ofspectralcorrelations
inµ the far2
2 1,2 ,
⇣1,2 = 2⇡ cosh 2⇡bs
,
=
b=
.
(4.19)
2p⇡b2
infrared, where the level rigidity symptomatic for systems with hard quantum
chaossinreflectsin
restoration
causal symmetry
andintegration
overthefull flavor
manifold.
Since
Figure 3: aTheannulusofcontributiontospectraltwo-pointfunction.
Itsmatrix
universal
part
gives
Thetheconstantµ is thebulk
cosmologicalconstantof 2D gravity
and wehave thechosen the nor2
2
1supersymmetricof theenergyvariables ⇣1,2
as toreproduce
the leading
density of states (4.8). A
the correct
EFT singular contributionof EFT
1/2s
The accessible
diagramisinsoidenticalto (2.22),
although
non-perturbative
structure ofmalization
theis. most
theformulation,Withfactors:
EFT
prediction
R2 (⇣1to study⇣worldsheet) the= extension=
exposition
ofoftherelevant
minimal-string
calculations
can
beone
consulted
in [43, 44, 49].
2detailedhere it represents
a minimal-stringinterpretedasaspacetimewith two
it would bedesirable
thestringworldsheet
tobranes,that is222⇡ (⇣1 ⇣2 ) 2D
2s2 formalism
also known
transform
a Euclidean
wormhole. The 2D gravity amplitude
dualastothe
theLaplacesupersymmetric(i.e.ofgraded)flavor framework.
2Z

4.3 also
A three-dimensional
example:T ⇥ Ifunction,
wormholes
Remark: cando this in 3D,
thankstopartition
wormholeof respect
CotlerJensenating theFZZT annulus(4.18), twice withto theandenergy arguments
gives the same leading singularity including the numerical coefficient as (2.22), coming fromThe first non-trivial contribution to the spectral correlation function is obtained by di↵erentithe Goldstone sector of causal symmetry.
2
The ramp(4.20)

a shift of S0 by 32 log(↵).28

Wormholes and baby universes
Example: Z0,2

gh it is a somewhat special case from the perspective of the above formulas,
Taking Liouville as our theory of 2D gravity, the annulus is a Euclidean
corresponds
ing to workout theformula twofor Z
0,2 explicitly.
wormhole
connecting
boundaries(JT This“doubletrumpet”) to the case w
asymptotic boundaries are connected by a “double trumpet”
Boundary 1

Boundary 2tly, in thiscase one
Working
veryhashard one might be able to compute the same process!
!
Zwith the exchange
universe inb2the intermediate
stateb2
1 of a baby 1/2
1/2
2 1
2 2
bdb
e
e
Z0,2 ( 1 , 2 ) =
1/2
1/2
1/2
1/2
(2⇡)
(2⇡)
0
1
2
p
Boundary 11 2
Boundary 2.
=
2⇡( 1 + 2 )

(1

(1

(1

theitKontsevich
generates
allthese,
as well
as growand 2 ! model
it,easily
and taket, we
get thelinearly
ontinue However,1 ! 1 +
2
arbitrarilymany
multi boundaries.
contribution
to thespectralform factor It is thus a “Universe Field Theory”

1tConclusions

Conclusions
Chaos in the RMT sense is ubiquitous. The reason is that its physicalcontent follows from a universal principle, the breaking of causal
symmetry
In this way we understand why and how generic quantum chaotic
systems at late times look like random matrix theories
Bulk picture of causal symmetry breaking in terms of spectral banes
Gives meaning to bulk Euclidean wormholes and baby universes evenin theories which are not defined via ensembles (e.g. in higher
dimensions)Outlook

Connection to replica wormholes and Page curve?
Refactorization? Erratic fluctuations?
Explicit bulk picture in higher dimensions? Maldacena-Maoz
wormholes…

Plug: results on operator ergodicity coming up with Nayak andVielma…

thank you for your attention