Macroscopic Irreversibility
in Quantum Systems
ETH and Equilibration
in a Free Fermion Chain
Hal Tasaki
webinar @ YouTube / January 2024

we prove the presence of macroscopic
irreversible behavior (ballistic diffusion) in afree fermion chain initially in a non-random
state and evolving under quantum
mechanical unitary time-evolution
the proof is based on an accumulation of ideas andmethods (in particular, ETH = energy eigenstate
thermalization hypothesis) developed to understandthermalization in isolated macroscopic quantum
systems, as well as new results speci c to the freefermion chainfi

illustration of Humpty Dumpty from "Through the Looking Glass", by John Tenniel, 1871

introduction/motivation
model and the main theorem
ingredients of the proof

the emergence of macroscopic
irreversibility
a physical system governed by a deterministic reversible
time-evolution law may exhibit irreversible behavior

it is essential that the system has a large degree of
freedomeven an ideal gas may exhibit irreversible behavior

irreversible expansion in aclassical ideal gas

N free classical particles on the interval [0,L] withperiodic boundary conditions ( N ≫ 1 )t = 0 xj(0) = 0 for all j
vj drawn randomly and uniformly from [−v0, v0]t ≫ L / v0

xj(t) = vj t mod L are almost uniformly
distributed in the interval [0,L]

irreversible expansion (or “ballistic diffusion”)
the initial velocities must be chosen randomlyclassical irreversibility vsquantum irreversibilityc l a s s ic a l s ys te ms
macroscopic irreversibility can be proved as aprobabilistic statement that is valid for the majority of
random initial conditions
there are always exceptional initial conditions that does
not lead to irreversibility
s
m
e
t
s
y
s
m
u
t
n
qua
macroscopic irreversibility can be proved without
introducing randomness (either in the Hamiltonian or in
the initial state)
we provide a simple rigorous example

introduction/motivation
model and the main theorem
ingredients of the proof

model and non-degeneracy

N non-interacting fermions on the chain {1,…, L}
L a large prime
N a large positive integer

Hamiltonian Ĥ =<latexit sha1_base64="bCqacHgvrpcZRr67V0R6v6ZYYVY=">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</latexit>

PLei✓<latexit sha1_base64="vzG9JRGcT5WERkeG20JDhRepfBQ=">AAAB/XicbVDJSgNBEK2JWxy3qEcvg0HwFGYEl4sY9OIxglkgiaGn05M06ekZumuEOAR/RZAcFPHqJ3j3Iv6NneWg0QcFr9+roqueHwuu0XW/rMzc/MLiUnbZXlldW9/IbW5VdJQoyso0EpGq+UQzwSUrI0fBarFiJPQFq/q9i5FfvWVK80heYz9mzZB0JA84JWikHjC4gRQ4DAGha14IBAatXN4tuGM4f4k3Jfmzd/s0fvy0S63cR6Md0SRkEqkgWtc9N8ZmShRyKtjAbiSaxYT2SIfVDZUkZLqZjrcfOHtGaTtBpExJdMbqz4mUhFr3Q990hgS7etYbif959QSDk2bKZZwgk3TyUZAIByNnFIXT5opRFH1DCFXc7OrQLlGEognMNiF4syf/JZWDgndUOLzy8sVzmCALO7AL++DBMRThEkpQBgp38ABP8GzdW0PrxXqdtGas6cw2/IL19g02zZWC</latexit>

i✓ †
i✓ †
ĉx+1 ĉx }x=1 {e ĉx ĉx+1 + ei✓✓ 2 [0, 2⇡)<latexit sha1_base64="1K0UJZ5OY5ALJaJq5xJ6CeBmI3k=">AAACC3icbVDLSgMxFL1TX7W+prp0EyyCgpSZ4mtZdOOygn1AO5RMmmlDM5khyQhl6N6Nn+HWjQtF3PoD7vwb03YW2nog4dxz7iW5x485U9pxvq3c0vLK6lp+vbCxubW9Yxd3GypKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9cTv3lPpWKRuNOjmHoh7gsWMIK1kSK7CE+gYQDU3NhwBgLa4MAJVEwVm/q4a5ecsjMFWiRuRkqQoda1vzq9iCQhFZpwrFTbdWLtpVhqRjgdFzqJojEmQ9ynbUMFDqny0ukuY3RolB4KImmO0Giq/p5IcajUKPRNZ4j1QM17E/E/r53o4NJLmYgTTQWZPRQkHOkITYJBPSYp0XxkCCaSmb8iMsASE23iK5gQ3PmVF0mjUnbPy2e3p6XqVRZHHvbhAI7AhQuowg3UoA4EHuAZXuHNerRerHfrY9aas7KZPfgD6/MHL02Uzw==</latexit>

periodic b.c.

<latexit sha1_base64="QTa92IjxRPM4459xqzIGsB/dBsg=">AAAB/3icbVDJSgNBEK2JWxy3qEcvg0HwYpgRXC5i0IvHCGaBZAw9nZ6kSc9Cd40Qxhz8FUU8KOLVL/DuRfwbO8tBEx8UvH6viq56Xiy4Qtv+NjIzs3PzC9lFc2l5ZXUtt75RUVEiKSvTSESy5hHFBA9ZGTkKVoslI4EnWNXrng/86g2TikfhFfZi5gakHXKfU4JaEsDgGlLYAw6PgNDRbwQC/WYubxfsIaxp4oxJ/vTDPIkfvsxSM/fZaEU0CViIVBCl6o4do5sSiZwK1jcbiWIxoV3SZnVNQxIw5abD/fvWjlZalh9JXSFaQ/X3REoCpXqBpzsDgh016Q3E/7x6gv6xm/IwTpCFdPSRnwgLI2sQhtXiklEUPU0IlVzvatEOkYSijszUITiTJ0+Tyn7BOSwcXDr54hmMkIUt2IZdcOAIinABJSgDhVu4h2d4Me6MJ+PVeBu1ZozxzCb8gfH+A8jZlcI=</latexit>estandard free fermion Hamiltonian with nearestiθ
neighbor hopping except for the phase factor eLemma: all energy eigenvalues of Ĥ are non-degenerate
except for a nite number of θ, in particular, for any θ ≠ 0such that | θ | ≤ (4N + 2L)−(L−1)/2

Tasaki 2010, 2016, Shiraishi, Tasaki 2023−(L−1)/2fi

we choose such θ, e.g, θ = (4N + 2L)

S an arbitrary subset of {1,…, L} with | S | sites
P
N̂S = x2S n̂x the number of particles in S
<latexit sha1_base64="drSucDD3FZ+sInhiZ6WRsh29f0E=">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</latexit>

N̂S

|S|

μ = L the equilibrium value of N
δ > 0 an arbitrary (suf ciently small) precision<latexit sha1_base64="q2crWYxu5hmu9EC+MCrbM/ESUHM=">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</latexit>| (t)i = e

iĤt| (0)ithe total length ofTheorem: for any | Φ(0)⟩ and any (suf ciently large) T,
there exists a set ∈ [0,T] with ℓ( )/T ≤ e
δ2N̂S− 8μ(1 − μ) N̂⟨Φ(t) | P[−μ ≥ δ] | Φ(t)⟩ ≤ e
NA<latexit sha1_base64="p7yWkmsJjxtaqqJIeSDqDpDc0MM=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KolI9Vj14rGC/YA2lM120y7dbMLuRCihf8OLB0X06J/x5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGt1O/9ci1EbF6wHHC/YgOlAgFo2ilED6AAQUJBK57pbJbcWcgy8TLSRly1Hulr24/ZmnEFTJJjel4boJ+RjUKJvmk2E0NTygb0QHvWKpoxI2fzW6ekFOr9EkYa1sKyUz9PZHRyJhxFNjOiOLQLHpT8T+vk2J45WdCJSlyxeaLwlQSjMk0ANIXmjOUY0so08LeStiQasrQxlS0IXiLLy+T5nnFq1aq9xfl2k0eRwGO4QTOwINLqMEd1KFho03gCV7g1UmdZ+fNeZ+3rjj5zBH8gfP5A0Gojz0=</latexit>

fi

0

fi

𝒜

for any t ∈ [0,T]\
𝒜

𝒜
𝒜

main theorem

δ2− 8μ(1 − μ) N

s.t.

negligibly smallTtmain theorem (less formal)
S an arbitrary subset of {1,…, L} with | S | sites
P
N̂S = x2S n̂x the number of particles in S
<latexit sha1_base64="drSucDD3FZ+sInhiZ6WRsh29f0E=">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</latexit>

N̂S

|S|

μ = L the equilibrium value of N
δ > 0 an arbitrary (suf ciently small) precision<latexit sha1_base64="q2crWYxu5hmu9EC+MCrbM/ESUHM=">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</latexit>| (t)i = e

iĤt| (0)i

Theorem: let | Φ(0)⟩ be an arbitrary initial state.
for a suf ciently large and typical time t, the measurement
N̂S

result of N in | Φ(t)⟩ almost certainly equals theequilibrium value μ (within the precision δ).
A<latexit sha1_base64="p7yWkmsJjxtaqqJIeSDqDpDc0MM=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KolI9Vj14rGC/YA2lM120y7dbMLuRCihf8OLB0X06J/x5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGt1O/9ci1EbF6wHHC/YgOlAgFo2ilED6AAQUJBK57pbJbcWcgy8TLSRly1Hulr24/ZmnEFTJJjel4boJ+RjUKJvmk2E0NTygb0QHvWKpoxI2fzW6ekFOr9EkYa1sKyUz9PZHRyJhxFNjOiOLQLHpT8T+vk2J45WdCJSlyxeaLwlQSjMk0ANIXmjOUY0so08LeStiQasrQxlS0IXiLLy+T5nnFq1aq9xfl2k0eRwGO4QTOwINLqMEd1KFho03gCV7g1UmdZ+fNeZ+3rjj5zBH8gfP5A0Gojz0=</latexit>

fi

fi

0

T

t

irreversible expansion| Φ(0)⟩ any initial state where all particles are in S
<latexit sha1_base64="fzpmSiPorwhe0E9Y5hzZIMAbD+w=">AAACFXicbZA7SwNBEMfnfMb4igppbA6DYBXuLKKNEGJjJQmaByQh7G32kiV7D3bnhHDkS9j4CSztbSyUYCvY2fhZ3ORSaOLALr/9zwyz83dCwRVa1pextLyyurae2khvbm3v7Gb29msqiCRlVRqIQDYcopjgPqsiR8EaoWTEcwSrO4PLSb5+x6TigX+Lw5C1PdLzucspQS0FmSw8AoILEghQiPWrrwk1XcMIOnCj74QvwO5kclbemoa5CPYMcsVs5Zs/lcblTuaz1Q1o5DEfqSBKNW0rxHZMJHIq2CjdihQLCR2QHmtq9InHVDuebjUyj7XSNd1A6uOjOVV/d8TEU2roObrSI9hX87mJ+F+uGaF73o65H0bIfJoMciNhYmBOLDK7XDKKYqiBUMn1X03aJ5JQ1EamtQn2/MqLUDvN24V8oaLdKEESKTiEIzgBG86gCFdQhqq2/B6e4RXejAfjxRgb70npkjHrOYA/YXz8AIx8m0A=</latexit>

N̂S
N =1

| Φ(t)⟩ for suf ciently large and typical t<latexit sha1_base64="oR7cEu72IPXGi5jVIwVZe+Q5r4E=">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</latexit>

N̂S
N 'µ

“time s arrow” has emerged from the unitary
time evolution in an isolated macroscopic
quantum system!
we don t have to introduce randomness in theinitial state or the Hamiltonian
it is essential that we focus on a macroscopic
N̂S

fi

’

’

observable N

time-reversal “paradox”| Φ(0)⟩ any initial state where all particles are in S
<latexit sha1_base64="fzpmSiPorwhe0E9Y5hzZIMAbD+w=">AAACFXicbZA7SwNBEMfnfMb4igppbA6DYBXuLKKNEGJjJQmaByQh7G32kiV7D3bnhHDkS9j4CSztbSyUYCvY2fhZ3ORSaOLALr/9zwyz83dCwRVa1pextLyyurae2khvbm3v7Gb29msqiCRlVRqIQDYcopjgPqsiR8EaoWTEcwSrO4PLSb5+x6TigX+Lw5C1PdLzucspQS0FmSw8AoILEghQiPWrrwk1XcMIOnCj74QvwO5kclbemoa5CPYMcsVs5Zs/lcblTuaz1Q1o5DEfqSBKNW0rxHZMJHIq2CjdihQLCR2QHmtq9InHVDuebjUyj7XSNd1A6uOjOVV/d8TEU2roObrSI9hX87mJ+F+uGaF73o65H0bIfJoMciNhYmBOLDK7XDKKYqiBUMn1X03aJ5JQ1EamtQn2/MqLUDvN24V8oaLdKEESKTiEIzgBG86gCFdQhqq2/B6e4RXejAfjxRgb70npkjHrOYA/YXz8AIx8m0A=</latexit>

N̂S
N =1

| Φ(t)⟩ for suf ciently large and typical t<latexit sha1_base64="oR7cEu72IPXGi5jVIwVZe+Q5r4E=">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</latexit>

N̂S
N 'µ

new initial state | Ξ(0)⟩ = | Φ(T0)⟩* for a typical T0
?
!
x
o
d
p a ra<latexit sha1_base64="oR7cEu72IPXGi5jVIwVZe+Q5r4E=">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</latexit>

N̂S
N 'µ

time-evolved state | Ξ(T0)⟩ = e

̂ 0
−iHT

<latexit sha1_base64="fzpmSiPorwhe0E9Y5hzZIMAbD+w=">AAACFXicbZA7SwNBEMfnfMb4igppbA6DYBXuLKKNEGJjJQmaByQh7G32kiV7D3bnhHDkS9j4CSztbSyUYCvY2fhZ3ORSaOLALr/9zwyz83dCwRVa1pextLyyurae2khvbm3v7Gb29msqiCRlVRqIQDYcopjgPqsiR8EaoWTEcwSrO4PLSb5+x6TigX+Lw5C1PdLzucspQS0FmSw8AoILEghQiPWrrwk1XcMIOnCj74QvwO5kclbemoa5CPYMcsVs5Zs/lcblTuaz1Q1o5DEfqSBKNW0rxHZMJHIq2CjdihQLCR2QHmtq9InHVDuebjUyj7XSNd1A6uOjOVV/d8TEU2roObrSI9hX87mJ+F+uGaF73o65H0bIfJoMciNhYmBOLDK7XDKKYqiBUMn1X03aJ5JQ1EamtQn2/MqLUDvN24V8oaLdKEESKTiEIzgBG86gCFdQhqq2/B6e4RXejAfjxRgb70npkjHrOYA/YXz8AIx8m0A=</latexit>| Ξ(0)⟩ = | Φ(0)⟩*

N̂S
N =1

T0 is not typical with respect to | Ξ(0)⟩main theorem

for suf ciently large and typical t,
N̂S

fi

fi

it holds for | Ξ(t)⟩ that N ≃ μintroduction/motivation
model and the main theorem
ingredients of the proof

strong ETH boundPL

Hamiltonian Ĥ =energy eigenstates<latexit sha1_base64="bCqacHgvrpcZRr67V0R6v6ZYYVY=">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</latexit><latexit sha1_base64="tjIjmg9/SCVQr0G24Sz2LVG/iHY=">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</latexit>

|

i✓ †
i✓ †
ĉx+1 ĉx }x=1 {e ĉx ĉx+1 + e††
k i = âk1 · · · âkN |0i
PL
†
1/2
ikx †
âk = Lĉx
x=1 e<latexit sha1_base64="8youlETGf5I5JRu+AXA10e0tOWk=">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</latexit><latexit sha1_base64="LVjOkAiRccmkzzH3qidzSjUTor0=">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</latexit>

k = (k1 , . . . , kN )essential technical result in the present work

Lemma: for every energy eigenstate | Ψk⟩, we have<latexit sha1_base64="nuUPOMGsQn3m7lPbWpWeuL5GiRI=">AAADM3icjVLLjtMwFL0Jr1JeBZZsIkZIjARVMkgDyxFsWKEi6MxITaZyXKe16kdkO0gl9BvY8gNlzYYfYYGEWIAQW/6BG7dMywwLHCU59557fI8feSm4dXH8JQjPnD13/kLrYvvS5StXr3Wu39i3ujKU9akW2hzmxDLBFes77gQ7LA0jMhfsIJ8+afiDV8xYrtVLNytZJslY8YJT4jClOwtYgAACCsb4Zxj1wAKHIdSIc9CYHWFmBnIV1TCFOT5vIIUJKh1mehinyHM/y2ADN1UOCjBYSbFyrXmGmqbLCz/bMr6PvITqWG+8foy+UnTBcD6H6jWbIf//jhfexZ+Vpv7bhh1E9xAdYV3Tf9PruucR1jUuH6wc3oXk2O02Muh+2NmKu7Ef0WmQrMDW3vbb12Xr/bvesPMpHWlaSaYcFcTaQRKXLquJcZwKNm+nlWUloVMyZgOEikhms9qf+Ty6g5lRVGiDr3KRz24qaiKtnckcKyVxE3uSa5L/4gaVKx5lNVdl5Ziiy0ZFJSKno+YCRSNuGHVihoBQw9FrRCfEEOrwmrVxE5KTSz4N9ne6yW539znuxmNYjhbcgtt+Wx/CHjzFQ+0DDT4En4NvwffwY/g1/BH+XJaGwUpzE/4a4a/fVhfWQg==</latexit>

h

N̂S

k |P̂

⇥ N̂

S

N

µ

⇤

|

ki  2 e

2N
3µ(1 µ)μδ
almostcertainly
equals(withintheprecision)Nin every energy eigenstatestrong ETH (energy eigenstate thermalization hypothesis)in the form of large-deviation
von Neumann 1929, Deutsch 1991, Srednicki 1994, Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi 2010, Tasaki 2016

“ergodicity” theoremP| (0)i = k ↵k | k i
initial state
P
iEk t
|
(t)i
=
↵
e
| ki
time-evolved statek k
expectation valueP<latexit sha1_base64="SvaIkbQv50e7xkuin05hNuvFnz8=">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</latexit><latexit sha1_base64="CCdUZn7ySi1lKsEL6LFsAe0aMmE=">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</latexit><latexit sha1_base64="2Wuw3+6TPSSVuSfWNaauIEsZRAk=">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</latexit>⇤i(Ek Ek0 )t
0
↵
↵
e
h
k,k0 k kh (t)|P̂ | (t)i =k |P̂ |

k0 i

k |P̂ |

ki

Lemma: all energy eigenvalues of Ĥ are non-degenerate
long-time averageR<latexit sha1_base64="3kbArMmKbOQIxszay52gQ71wazg=">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</latexit>

limT "1 T

1

T
dt h
0

(t)|P̂ | (t)i =

P

2
|↵
|
k h
k

Lemma: for every energy eigenstate | Ψk⟩, we have<latexit sha1_base64="nuUPOMGsQn3m7lPbWpWeuL5GiRI=">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</latexit>

h

k |P̂

limT "1 TR
1 T<latexit sha1_base64="8XJIWpZARiXdJHBOzZAZu0/lSdA=">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</latexit>

⇥ N̂

S

N

µ

⇤

|

ki  2 e

2N
3µ(1 µ)Theorem: for an arbitrary initial state | Φ(0)⟩, we have
dt h (t)|P̂
0

⇥ N̂

S

N

µ

⇤

| (t)i  2 e

2N
3µ(1 µ)the main theorem is a simple corollarysummary

we proved the presence of macroscopic irreversible
behavior (ballistic diffusion) in a free fermion chain
initially in a non-random state and evolving underquantum mechanical unitary time-evolution
the proof is based on an accumulation of ideas
and methods (in particular, ETH) developed tounderstand thermalization in isolated
macroscopic quantum systems, as well as new
results speci c to the free fermion chain

remaining issues
fiextensions to other observables, time scale for
equilibration, treatment of non-integrable systems, …