Maximal quantum chaos of the

critical Fermi surface

Probing Complex Quantum Dynamics through Out-of-time-ordered Correlators

MPIPKS, Dresden October 12, 2021

Subir Sachdev

HARVARDTalk online: sachdev.physics.harvard.edu

Aavishkar Patel Berkeley

Maria Tikhanovskaya Harvard

1. Critical Fermi surfaces: large N theory

2. Gu-Kitaev theory of OTOCs with spatio-temporal chaos

3. Maximal quantum chaos of the critical Fermi surface

Fermi surface coupled to a gauge field

kx

Occupied states

Empty states

ky

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A metal with a Fermi surfaceminimally coupled to a gauge field A

L = c†k

✓@

@⌧+ "(�ir� gA)� µ

◆ck +

1

2(r⇥A)2

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"(k) < 0

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"(k) > 0

Fermi surface coupled to a gauge field

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• Gauge fluctuation at wavevector q couples most e�ciently tofermions near ±k0.

• Expand fermion kinetic energy at wavevectors about ±k0. InLandau gauge A = (a, 0).

Fermi surface coupled to a gauge field

M. A. Metlitski and S. Sachdev,Phys. Rev. B 82, 075127 (2010)

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L[ ±, a] =

†+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

�g a⇣ †+ + � †

� �

⌘+

1

2(@ya)

2

Large N theory of a critical Fermi surface

Ilya Esterlis, Haoyu Guo, Aavishkar Patel, S.S. arXiv: 2103.08615

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Main idea:IntroduceN flavors of fermions and bosons,and examine an ensemble of theories withdi↵erent Yukawa couplings. In the largeN limit, every member of the ensemble isexpected to have the same critical proper-ties, and so it is easier to study the averagetheory.

Large N theory of a critical Fermi surface

Ilya Esterlis, Haoyu Guo, Aavishkar Patel, S.S. arXiv: 2103.08615

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N flavors of fermions ±↵,

M flavors of a boson a↵, anda “Yukawa coupling” g↵�� which is a random function in

flavor space. Note: there is no spatial randomness. Take

the large N limit with M/N fixed.

L = †+↵

�@⌧ � i@x � @2y

� +↵ + †

�↵

�@⌧ + i@x � @2y

� �↵

� g↵��N

a↵⇣⌘+↵

†+� +� + ⌘�↵

†�� ��

⌘+

1

2(@ya↵)

2

⌘±↵ = ±1 depending upon nature of a↵: gauge field, Higgs

field, order parameter . . . .

g↵�� = 0 , |g↵�� |2 = g2

Large N theory of a critical Fermi surface

Ilya Esterlis, Haoyu Guo, Aavishkar Patel, S.S. arXiv: 2103.08615

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We can now proceed just as in the SYK model: we obtain a theory for Green’sfunctions which are bilocal in both space and time. Using the spacetime co-ordinate X ⌘ (⌧, x, y), we can write the averaged partition function

Z � =

ZDG(X1, X2)D⌃(X1, X2)DD(X1, X2)

⇥D⇧(X1, X2) exp [�NI(G,⌃, D,⇧)] .

The G-⌃-D-⇧ action is now

I(G,⌃, D,⇧) =g2

2Tr (G · [GD])� Tr(G · ⌃) + 1

2Tr(D ·⇧)

� ln det⇥�@⌧1 � i@x1 � @2

y1

��(X1 �X2) + ⌃(X1, X2)

⇤

+1

2ln det

⇥��K@2

y1

��(X1 �X2)�⇧(X1, X2)

⇤.

where we have introduced notation

Tr (f · g) ⌘Z

dX1dX2 f(X2, X1)g(X1, X2) .

Large N theory of a critical Fermi surface

Ilya Esterlis, Haoyu Guo, Aavishkar Patel, S.S. arXiv: 2103.08615

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Saddle-point equations

G(k, i!) =1

i! � kx � k2y � ⌃(k, i!), D(k, i!) =

1

k2y �⇧(k, i!)

⌃(r, ⌧) = g2D(r, ⌧)G(r, ⌧), ⇧(r, ⌧) = �g

2G(�r,�⌧)G(r, ⌧)

Exact solution at low energies:

⌃(k,!) = g4/3

T2/3�

⇣!

T

⌘,

where �(z) is a universal scaling function, obtained by analytical continu-ation from imaginary Matsubara frequencies !n = (2n� 1)⇡T

�

✓i!n

T

◆= �i sgn(!n)

25/3

3p3H1/3

✓|!n|� ⇡T

2⇡T

◆

Hr(n 2 Z+) =nX

j=1

1

jr

1. Critical Fermi surfaces: large N theory

2. Gu-Kitaev theory of OTOCs with spatio-temporal chaos

3. Maximal quantum chaos of the critical Fermi surface

Slides(mostly)

byMaria

Out-of-time-order correlator (OTOC)

• Definition

Assume the leading contribution comes from ladders

Eigenvalue equation

(connected part)

<latexit sha1_base64="wELZ6HtWr4GHd83LKtAiH/vRzMM=">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</latexit>

Here � ⌘ 1/T = 2⇡.

Structure of OTOC – early time regime

Yingfei Gu, Alexei Kitaev 2018

Exponential growth

Consider a single mode ansatz for early times

Josephine Suh, Alexei Kitaev 2017

Yingfei Gu, Alexei Kitaev 2018

Exponential growth

Consider a single mode ansatz for early times

Vertex functions

Josephine Suh, Alexei Kitaev 2017

Structure of OTOC – early time regime

Yingfei Gu, Alexei Kitaev 2018

Exponential growth

Vertex functionsConsider a single mode ansatz for early times

Josephine Suh, Alexei Kitaev 2017

Structure of OTOC – early time regime

Yingfei Gu, Alexei Kitaev 2018

Exponential growth

“Scramblon”Consider a single mode ansatz for early times

Josephine Suh, Alexei Kitaev 2017

Structure of OTOC – early time regime

Yingfei Gu, Alexei Kitaev 2018

Exponential growth

Function to be found

Consider a single mode ansatz for early times

Josephine Suh, Alexei Kitaev 2017

Structure of OTOC – early time regime

• What is C(p)? • A way to find it is to use the ladder identity from [Gu, Kitaev 2018] 1) Write down the self-consistency condition:

Yingfei Gu, Alexei Kitaev 2018

Structure of OTOC – ladder identity

• What is C(p)? • A way to find it is to use the ladder identity from [Gu, Kitaev 2018] 2) Apply the single-mode ansatz:

Yingfei Gu, Alexei Kitaev 2018

Structure of OTOC – ladder identity

• What is C(p)? • A way to find it is to use the ladder identity from [Gu, Kitaev 2018] 3) Find C(p):

Important for us:

Yingfei Gu, Alexei Kitaev 2018

Structure of OTOC – ladder identity

OTOC - early time behavior:

<latexit sha1_base64="KERxedpUkjM5aQcYORSO2rtUJgA=">AAACH3icbVDLSgMxFM3UV62vUZdugkVoN2WmlOqy2I0rqWAf0Cklk2ba0MxMSO4IZeifuPFX3LhQRNz1b0wfC209EDiccy439/hScA2OM7MyW9s7u3vZ/dzB4dHxiX161tJxoihr0ljEquMTzQSPWBM4CNaRipHQF6ztj+tzv/3ElOZx9AgTyXohGUY84JSAkfp2tV6QRexpHuJ77NFYY0+wAApeoAhNvTGRkuBFRPJpWp56ig9HUOzbeafkLIA3ibsiebRCo29/e4OYJiGLgAqiddd1JPRSooBTwaY5L9FMEjomQ9Y1NCIh0710cd8UXxllgINYmRcBXqi/J1ISaj0JfZMMCYz0ujcX//O6CQQ3vZRHMgEW0eWiIBEYYjwvCw+4YhTExBBCFTd/xXRETDNgKs2ZEtz1kzdJq1xyq6XKQyVfu13VkUUX6BIVkIuuUQ3doQZqIoqe0St6Rx/Wi/VmfVpfy2jGWs2coz+wZj94CqFz</latexit>

C(p) ⇠ N cos

✓(p)⇡

2

◆

Structure of OTOC: saddle point vs pole Fourier transform:

Early time:

Estimate the integral:

1) Saddle point solution:

2) Pole contribution:

Yingfei Gu, Alexei Kitaev 2018

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Note: p1 and psare pure imaginary: i.e.p1 = i|p1| and ps = i|ps|.

1) Saddle point solution:

2) Pole contribution:

Yingfei Gu, Alexei Kitaev 2018

Saddle point contribution – weak quantum chaos

Pole contribution – maximal chaos

Structure of OTOC: saddle point vs pole

<latexit sha1_base64="BbHdm0KkJqLGH+rTzphPq984sI0=">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</latexit>

Maximal chaosrequires a largebutterfly velocity,or (0) not too far

from 1.

<latexit sha1_base64="kqC9sZlf0ULHsZ3+FyQo0VqVaYQ=">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</latexit>

Note: p1 and psare pure imaginary: i.e.p1 = i|p1| and ps = i|ps|.

1. Critical Fermi surfaces: large N theory

2. Gu-Kitaev theory of OTOCs with spatio-temporal chaos

3. Maximal quantum chaos of the critical Fermi surface

Slides(mostly)

byMaria

Eigenvalue equationInvariance under adding a ladder

Gauge field

As in [Patel, Sachdev 2016]

<latexit sha1_base64="/ZHG3S5tz/2sLzHs8Kfg2qHEA5E=">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</latexit>

We find maximal chaos with �L = 2⇡T , and butterfly velocity v1 = 9.67g�4/3T 1/3

<latexit sha1_base64="/kOuV78BjdNGyFnjSmEBtGAXi2c=">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</latexit>

Note: p1 and pxare pure imaginary: i.e.p1 = i|p1| and px = i|px|.

Outlook<latexit sha1_base64="5wmwqAiFMWdBPzPmwiZ+y0wlN5I=">AAAFLXicfVTBbhxFEJ1kF0gMAQducCnhRTjSetldO7GlCClyJIgUKwrCjiN5HVPTXTPTck/30N2z62XkA1/DEfgYDkiIKx/BherZSRSQSZ9qqmq6Xr1X1WmllQ/j8e/Xrvf6b739zo2ba+++d+v9D9Zvf/jM29oJOhJWW/c8RU9aGToKKmh6XjnCMtV0nJ4/jPHjOTmvrDkMy4pOS8yNypTAwK6z272PZynlyjQqUKl+oMu1WbTgAF1OMHgygFCQdUvIrANjzdb3NXpVoQtKaAK5NFgq4cFmgCCcYjdq+IpcqcDXLkNBI/gWpeTkyioTALW2Cw9+6bkOoxAgbFnVoQUU78l0LUK9+hTWORLR9ENAI6FC5ZTJW1uS8SostxY4J1DGB0yVZgTkR9C10cxaippU13QJD6/EBwV6KPFClRwRBVoPCxUKGMw08yjx7AC+hCnMKgWHgxWKtA6BXKaXMCdtBYPgbK9KOHzRTL7YvhyMXvJ4TFpvScpYHgmOmS4p9kgXlTVkguKSubMLLsfeVXGBhuMsPdCcDDfWcdXB8oV1YcuhyWPTDANX/LTAEDx3ocHUZUou3ikpd0StQBkb0pbAvBIKvokRQuXsXEkGxzpf1ZfyoNtZIGPrvBjB5tf1EB6rgDQfwnQ82bsPj4kHsEQzhCeKhrBPLo+h6fjO1To8Wkpnu9EByrKoMEv42qR954OzJv8ceNxMqMuOmg6sB9eOB/JQxexFwThDhIp6gctXag7fICRs7ms8Z7QHqm7BTu5E0cjIV6twtr4xHt29N93em8J4NG5PNHZ3trd3YNJ5NpLuPD1b/3smrahLFlZo9P5kMq7CadMtC99ee6pQnGNOJ2xy/+SHcq4q35qnTUvSJXzGQdl2llnemNb7+s8Nlt4vy5QzeYMK/99YdF4VO6lDtnfaKMPrRkasCmW1hmAhPg4gVdw2ll8qXC1zJD5OGD8hazNP/L6YPBTNLNBFWCjJdZqd0e5dZSJbLymB/zeeTUeTe6Odb6YbD/Y73m4knySfJpvJJNlNHiSPkqfJUSJ6P/Z+6v3S+7X/c/+3/h/9P1ep1691/3yU/Ov0//oH4Ny7yw==</latexit>

• Large N theory for non-quasiparticle dynamics of a critical Fermi surface.

Saddle point allows systematic computation of fluctuation corrections,

and pairing and density-wave instabilities.

• Critical Fermi surface has maximal chaos with �L = 2⇡T , and butterfly

velocity ⇠ T 1/3.

• Well-defined regime of exponential growth of chaos can exist even in sys-

tems with short-range interactions and a small number of degrees of free-

dom on each site provided the butterfly velocity is large enough. (Gu,

Kitaev, 2018; Keselman, Nie, Berg, 2020)

• Hydrodynamic e↵ective theory for ‘strong’ quantum chaos provides ratio-

nale for why it is always maximal, with �L = 2⇡T (Blake, Liu, 2021).

Outlook<latexit sha1_base64="5wmwqAiFMWdBPzPmwiZ+y0wlN5I=">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</latexit>

• Large N theory for non-quasiparticle dynamics of a critical Fermi surface.

Saddle point allows systematic computation of fluctuation corrections,

and pairing and density-wave instabilities.

• Critical Fermi surface has maximal chaos with �L = 2⇡T , and butterfly

velocity ⇠ T 1/3.

• Well-defined regime of exponential growth of chaos can exist even in sys-

tems with short-range interactions and a small number of degrees of free-

dom on each site provided the butterfly velocity is large enough. (Gu,

Kitaev, 2018; Keselman, Nie, Berg, 2020)

• Hydrodynamic e↵ective theory for ‘strong’ quantum chaos provides ratio-

nale for why it is always maximal, with �L = 2⇡T (Blake, Liu, 2021).

Outlook<latexit sha1_base64="5wmwqAiFMWdBPzPmwiZ+y0wlN5I=">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</latexit>

• Large N theory for non-quasiparticle dynamics of a critical Fermi surface.

Saddle point allows systematic computation of fluctuation corrections,

and pairing and density-wave instabilities.

• Critical Fermi surface has maximal chaos with �L = 2⇡T , and butterfly

velocity ⇠ T 1/3.

• Well-defined regime of exponential growth of chaos can exist even in sys-

tems with short-range interactions and a small number of degrees of free-

dom on each site provided the butterfly velocity is large enough. (Gu,

Kitaev, 2018; Keselman, Nie, Berg, 2020)

• Hydrodynamic e↵ective theory for ‘strong’ quantum chaos provides ratio-

nale for why it is always maximal, with �L = 2⇡T (Blake, Liu, 2021).

Outlook<latexit sha1_base64="5wmwqAiFMWdBPzPmwiZ+y0wlN5I=">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</latexit>

• Large N theory for non-quasiparticle dynamics of a critical Fermi surface.

Saddle point allows systematic computation of fluctuation corrections,

and pairing and density-wave instabilities.

• Critical Fermi surface has maximal chaos with �L = 2⇡T , and butterfly

velocity ⇠ T 1/3.

• Well-defined regime of exponential growth of chaos can exist even in sys-

tems with short-range interactions and a small number of degrees of free-

dom on each site provided the butterfly velocity is large enough. (Gu,

Kitaev, 2018; Keselman, Nie, Berg, 2020)

• Hydrodynamic e↵ective theory for ‘strong’ quantum chaos provides ratio-

nale for why it is always maximal, with �L = 2⇡T (Blake, Liu, 2021).