illustration by いらすとや

Nature abhors
a vacuum
A simple rigorous example ofthermalization in an isolated
macroscopic quantum system

Naoto Shiraishi and Hal Tasaki
webinar @ YouTube / 2023

a typical process of thermalization
equilibrium state withtemperature T and pressure pvacuum with zero pressure

we prove that this process
takes place in a dilute ideal
gas of fermions on a chain
evolving only by quantummechanical time-evolution(we treat the case T = ∞)nonequilibrium
state

thermalization
equilibriumstatemotivation
main result
frequently asked questions
essence of the proof

what is the origin of thermalization?
approach to thermal equilibrium
foundation of equilibrium statistical mechanics

question: does an isolated macroscopic quantum system
thermalize only by means of quantum mechanical timeiĤt
|
(t)i
=
e
| (0)ievolution?
<latexit sha1_base64="/l7d9Q20VrqMLD+BnxqQrwP2WAc=">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</latexit>

YES! supported by numerous theoretical arguments,
numerical simulations, and experiments in cold atoms
BUT, there were no concrete (and “realistic”) examples inwhich the presence of thermalization was establishedwithout relying on any unproven assumptions
we prove the presence of thermalization (in a restricted sense)
for low-density non-interacting fermions on a chainmotivation
main result
frequently asked questions
essence of the proof

model and initial state
N non-interacting fermions on the chain {1,…, L}
L large prime, N large positive integer, density ρ = N/L<latexit sha1_base64="bCqacHgvrpcZRr67V0R6v6ZYYVY=">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</latexit>

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

e

i✓

PL

i✓ †
i✓ †
ĉx+1 ĉx }x=1 {e ĉx ĉx+1 + ei✓<latexit sha1_base64="QTa92IjxRPM4459xqzIGsB/dBsg=">AAAB/3icbVDJSgNBEK2JWxy3qEcvg0HwYpgRXC5i0IvHCGaBZAw9nZ6kSc9Cd40Qxhz8FUU8KOLVL/DuRfwbO8tBEx8UvH6viq56Xiy4Qtv+NjIzs3PzC9lFc2l5ZXUtt75RUVEiKSvTSESy5hHFBA9ZGTkKVoslI4EnWNXrng/86g2TikfhFfZi5gakHXKfU4JaEsDgGlLYAw6PgNDRbwQC/WYubxfsIaxp4oxJ/vTDPIkfvsxSM/fZaEU0CViIVBCl6o4do5sSiZwK1jcbiWIxoV3SZnVNQxIw5abD/fvWjlZalh9JXSFaQ/X3REoCpXqBpzsDgh016Q3E/7x6gv6xm/IwTpCFdPSRnwgLI2sQhtXiklEUPU0IlVzvatEOkYSijszUITiTJ0+Tyn7BOSwcXDr54hmMkIUt2IZdcOAIinABJSgDhVu4h2d4Me6MJ+PVeBu1ZozxzCb8gfH+A8jZlcI=</latexit>e✓ 2 [0, 2⇡)<latexit sha1_base64="1K0UJZ5OY5ALJaJq5xJ6CeBmI3k=">AAACC3icbVDLSgMxFL1TX7W+prp0EyyCgpSZ4mtZdOOygn1AO5RMmmlDM5khyQhl6N6Nn+HWjQtF3PoD7vwb03YW2nog4dxz7iW5x485U9pxvq3c0vLK6lp+vbCxubW9Yxd3GypKJKF1EvFItnysKGeC1jXTnLZiSXHoc9r0h9cTv3lPpWKRuNOjmHoh7gsWMIK1kSK7CE+gYQDU3NhwBgLa4MAJVEwVm/q4a5ecsjMFWiRuRkqQoda1vzq9iCQhFZpwrFTbdWLtpVhqRjgdFzqJojEmQ9ynbUMFDqny0ukuY3RolB4KImmO0Giq/p5IcajUKPRNZ4j1QM17E/E/r53o4NJLmYgTTQWZPRQkHOkITYJBPSYp0XxkCCaSmb8iMsASE23iK5gQ3PmVF0mjUnbPy2e3p6XqVRZHHvbhAI7AhQuowg3UoA4EHuAZXuHNerRerHfrY9aas7KZPfgD6/MHL02Uzw==</latexit>

Lemma: all the energy eigenvalues of Ĥ are non-degenerate for most θ we choose such θ, e.g, θ = (4N)−(L−1)/2

initial statepick a normalized state | Φ(0)⟩ at random (with uniform
probability) from the Hilbert space where all particles
L−1
are in the left half-chain {1,…, 2 }fiequilibrium at T = ∞ con ned in the half-chain

time-evolution and thermalization<latexit sha1_base64="/l7d9Q20VrqMLD+BnxqQrwP2WAc=">AAACQ3icbVDLSgMxFL3j2/oadelmaBEUaZkRfGyEohuXFawW2loyadqGZjJDkhHK2L/wF/wPN/6Au/6AGxeKuBW8fSxs64WQk/MgyfEjwbVx3Z41Mzs3v7C4tJxaWV1b37A3t250GCvKijQUoSr5RDPBJSsabgQrRYqRwBfs1m9f9PXbe6Y0D+W16USsGpCm5A1OiUEqtLPwAE9QgBZw2AMD+3hSQEBCEwQwOMN1BwlkUX9CF0FPApfQxb07lnUnszU74+bcwTjTwBuBTD5dOXjs5TuFmv1aqYc0Dpg0VBCty54bmWpClOFUsG6qEmsWEdomTVZGKEnAdDUZdNB1dpGpO41Q4ZLGGbB/EwkJtO4EPjoDYlp6UuuT/2nl2DROqwmXUWyYpMOLGrFwTOj0C3XqXDFqRAcBoYrjWx3aIopQg7WnsARv8svT4OYw5x3njq68TP4chrMEO5DGWj04gTwWXoAiUHiGN/iAT+vFere+rO+hdcYaZbZhbKyfXznIpVI=</latexit>

time-evolved state | (t)i = e iĤt | (0)iN̂left the number of particles
L−1
in the left half-chain {1,…, 2 }<latexit sha1_base64="AyQHVO93uEm520GWADGBK/jzj1M=">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</latexit>N̂left| (0)i = | (0)iN

for the choice of | Φ(0)⟩−(ρ/3)Nfi

fi

Theorem: the following is true with prob. ≥ 1 − ethere exist suf ciently large T > 0 and a set G ∈ [0,T]−(ρ/4)Nwith | G | /T ≥ 1 − e
for any t ∈ G, the measurement result of N̂left satis es
Nleft 1−(ρ/4)N−≤ ϵ0(ρ) with prob. ≥ 1 − eN
2quantum mechanicalwith ϵ0(ρ) = 3ρ/2
probability

time-evolution and thermalization<latexit sha1_base64="/l7d9Q20VrqMLD+BnxqQrwP2WAc=">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</latexit>

time-evolved state | (t)i = e iĤt | (0)iN̂left the number of particles
L−1
in the left half-chain {1,…, 2 }<latexit sha1_base64="AyQHVO93uEm520GWADGBK/jzj1M=">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</latexit>N̂left| (0)i = | (0)iN

Theorem: it almost certainly happens that, for suf ciently
large and typical time t, the measurement result of N̂left
Nleft 1almost certainly satis es
≃
N
2
Nleft
since
= 1 at t = 0, we see thermalization!!
N
but the precision is ϵ (ρ) = 3ρ/2
0

fi

fi

the result is meaningful only for low enough density ρmotivation
main result
frequently asked questions
essence of the proof

does thermalization take place only in thespecial free fermion chain with a prime L ?

of course, the physics should not change when L is not aprime, but this is (so far) the only example where we canprove thermalization without relying on any unproved
assumptions
we indeed prove a general thermalization theorem undert wo assumptions
Assumption 1: energy eigenvalues are non-degenerate
Assumption 2: any energy eigenstate | Ψj⟩ satis es<latexit sha1_base64="oaW33qn7dnu9zwLOSxYbXKYqPi8=">AAACV3icbVHJTsMwEJ2ErZStwJGLxSJxoUqQWI4ILpwQSBSQSqkc12lNHSeyHaQq5HP4BX6AP0Bc+BA4cIFJisQ6ku037814eQ4SKYz1vGfHHRkdG5+oTFanpmdm52rzC2cmTjXjDRbLWF8E1HApFG9YYSW/SDSnUSD5edA/KPTzG66NiNWpHSS8FdGuEqFg1CIV13bgDiRQUNDFlWN2DAYEtOEabjHroWYhQzZHLkNGQwSkrA1Rycuqr55C/75bMW/CFXZuwBHk7dqKV/fKIH+B/wlW9lZf7h9upl6P27XHy07M0ogryyQ1pul7iW1lVFvBJM+rl6nhCWV92uVNhIpG3LSy0pecrCHTIWGscShLSvZ7R0YjYwZRgJURtT3zWyvI/7RmasPdViZUklqu2PCgMJXExqQwmXSE5szKAQLKtMC7EtajmjKLX1FFE/zfT/4Lzjbr/nZ96wTd2IdhVGAJlmEdfNiBPThE2xvA4AnenBFn1Hl23t1xtzIsdZ3PnkX4Ee78B3iVp88=</latexit>hj |P̂left |

N
i

2
j

fi

fi

assumption 1 is very plausible in a generic quantum
many-body systems
we have examples of interacting model whereassumption 2 is veri ed assuming assumption 1

why isolated systems? there can't be acompletely isolated system!
re

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quantum system
of interest

weak interaction with the surrounding
environment (bigger system)
g
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o u r sequantum system
of interest
perfectly isolated from the outside worldwhy isolated systems? there can't be acompletely isolated system!
re

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Landau and Lifshitz, “Statistical Mechanics” p.6

quantum system
of interest

weak interaction with the surrounding
environment (bigger system)
g
n
i
t
t
o u r sequantum system
of interest
perfectly isolated from the outside worldwhy isolated systems? there can't be acompletely isolated system!
re

g
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quantum system
of interest
surrounding environment (bigger system)
fashionable answer
modern experiments in cold atoms!
unfashionable answer
we wish to learn what isolated systems can do(e.g., whether they can thermalize)
after that, we may study the effect played by theenvironmentmotivation
main result
frequently asked questions
essence of the proof

basic concepts and strategies
two main concepts
ETH (energy eigenstate thermalization hypothesis)
all energy eigenstates | Ψj⟩ with Ej ≃ E are similar
von Neumann 1929, Deutsch 1991, Srednicki 1994

<latexit sha1_base64="MsoDdMBfwKxsumgamBGBWZMkAzs=">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</latexit>

P

4

1

effective dimension De↵ =j h j | (0)iDe the effective number of energy eigenstates that
constitute the initial state | Φ(0)⟩Tasaki 1998, Reimann 2008, Linden, Popescu, Short, Winter 2009

ff

ff

ff

essential conditions that guarantee the presence ofthermalization
von Neumann 1929
1) a strong version of ETH Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi 2010
Tasaki 1998, Reimann 20082) a version of ETH and large De Linden, Popescu, Short, Winter 20093) very large De Goldstein, Hara, Tasaki 2014, Tasaki 2016

basic believedconcepts
to be valid andin most strategies
suf ciently

quantum many-body systems
two maincomplex
concepts
ETH (energy eigenstate thermalization hypothesis)
all energy eigenstates | Ψj⟩ with Ej ≃ E are similar
von Neumann 1929, Deutsch 1991, Srednicki 1994

P<latexit sha1_base64="MsoDdMBfwKxsumgamBGBWZMkAzs=">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</latexit>

41effective dimension De↵ =j h j | (0)iDe the effective number of energy eigenstates that
constitute the initial state | Φ(0)⟩Tasaki 1998, Reimann 2008, Linden, Popescu, Short, Winter 2009

fi

fi

ff

ff

ff

believed
to be verylarge
for realisticthe
nonequilibrium
essential
conditions
thatguarantee
presence statesofin suf ciently complex quantum many-body systems
thermalization
von Neumann 1929
we use thisstrategy
Tumulka,and Zanghiin
20101) a strong version of ETH Goldstein, Lebowitz, Mastrodonato,
present
work
Tasaki 1998, theReimann
20082) a version of ETH and large De Linden, Popescu, Short, Winter 20093) very large De Goldstein, Hara, Tasaki 2014, Tasaki 2016

why does large De lead to thermalization
initial state | (0)i =<latexit sha1_base64="TrAGit+3h4gJ8dF6vWrAV/w+zS8=">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</latexit><latexit sha1_base64="U3p1x9NigMu3w1KqSmH/AJsdTOg=">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</latexit>Pj ↵j |

ji

iĤt
|
(t)i
=
e
| (0)i =time-evolved state

expectation value of an observable<latexit sha1_base64="wenw0PIS1iIpxdKm7U5geCb2ihM=">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</latexit>

P

P

iEj t
↵
e
|
j j

ji

⇤
i(Ej Ek )t↵↵eh j |Ô| k i
j,k j knon-degeneracy ( Ej ≠ Ek if j ≠ k )long-time average
R
P
1 T
2
limT "1 T 0 dth (t)|Ô| (t)i = j |↵j | h j |Ô| j i
P
1
2
4
1
if De↵ = ( j |↵j | ) ⇠ Dtot then |↵j | ⇠ Dtoth (t)|Ô| (t)i =<latexit sha1_base64="C6ykF7V/lnvFfNpMQIO1Ag20DXk=">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</latexit><latexit sha1_base64="jxHGNK7DN/DE8v26lr8gx3fJVXw=">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</latexit><latexit sha1_base64="Tai2Hdjo26iiX8dBsE0GOtwmYoI=">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</latexit>

dimension of the whole Hilbert space
RP
T
1and limT "1 T1 0 dth (t)|Ô| (t)i ⇠ Dtot j h j |Ô| j i<latexit sha1_base64="xfNIIZEa2+tmLyDCptl/YVb6OPk=">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</latexit>= hÔicanonical
T =1(there also is a version for nite T )fiff<latexit sha1_base64="vMIP3BayxzrxQ8VWGZKruP7Doi4=">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</latexit>structure of the proofℋleft : Hilbert space in which all particles are
L−1
in the half-chain {1,…, 2 }̂ : projection onto ℋleft
Pleft
general theory
we prove that a low-density lattice gas exhibits
thermalization under the t wo (plausible) assumptions
Assumption 1: energy eigenvalues are non-degenerate
Assumption 2: any energy eigenstate | Ψj⟩ satis es<latexit sha1_base64="oaW33qn7dnu9zwLOSxYbXKYqPi8=">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</latexit>hj |P̂left |

N
i

2
j

fi

analysis of the free fermion chain
Assumption 2 can be proved from the exact solution
Assumption 1 can be proved by using results from thenumber theorygeneral theory

proof that De is large

Assumption 2: any energy eigenstate | Ψj⟩ satis es<latexit sha1_base64="oaW33qn7dnu9zwLOSxYbXKYqPi8=">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</latexit>hj |P̂left |

N
i

2
j

Dleft : dimension of ℋleft| Φ(0)⟩ : a random normalized state from ℋleft
standard formula for a random state2|h (0)| j i|4 = |h (0)|P̂left | j i|4 =kP̂left | j ik4
Dleft (Dleft + 1)
2
2=
h j |P̂j | j i
Dleft (Dleft + 1)
Tr[P̂left ] = Dleft
N
P
P
2
⇥
2
1
De↵ = j |h (0)| j i|4 j h j |P̂left | j i
Dleft (Dleft + 1)
N2⇥2
1 (2/3)⇢N−1=
 Dtot e
D
ρ
close
to
if
is
small
tot
Dleft + 1
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ff<latexit sha1_base64="kKR70ErYarSZQRK0BFOxCJ3aCcE=">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</latexit>

analysis of the free fermion chain

proof of the absence of degeneracy

PL
i✓ †
i✓ †
= be
{e ĉxinĉx+1
e ĉx+1ĉx }Hamiltonian Ĥ muststudiedthe+numbertheory!
x=1<latexit sha1_base64="bCqacHgvrpcZRr67V0R6v6ZYYVY=">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</latexit>

energy eigenvalues<latexit sha1_base64="HXtRrCycm6kp9VKsw3YNfxwZxHE=">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</latexit>

En =

PL 1

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j=0 j
L

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⇣=e

PL 1

i 2⇡
L

j
n
⇣
]
j=0 j
PL
occupation numbers nj = 0, 1,
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number-theoretic facts
assume L is an odd prime, N ≤ (L − 1)/4, m1, …, mL−1 ∈ ℤ
L−1j
Lemma: if mj ≠ 0 for some j, then ∑j=1 mj ζ ≠ 0

no degeneracy for most θ
L−1
L−1
j
−(L−3)/2
Lemma: we have | ∑j=1 mj ζ | ≥ ( ∑j=1 | mj | )no degeneracy if θ ≠ 0 and | θ | ≤ (4N)−(L−1)/2

L−!1analysis of the free fermion chainµ=1

mµ ζ µ #= 0.

(3.16)

proof of the absence of degeneracy

The lemma is a straightforward
consequence of the classical result
by Gauss, known as theirreducibility of the cyclotomic polynomials of prime index. See, e.g., Chapter 12, Section 2of [47] or Chapter 13, Section 2 of[48].
The following lemma5 provides an
exL
plicit lower bound for | "L−1 m µ†i✓ †µ ζ |. i✓µ=1
x+1
Lemma 3.4 For any m , . . . , mx+1 xx=1 "L−1x1
L−1 ∈ Z such thatµ=1 |mµ | = M > 0, one has
# 1
#
!
#L−
#
1
µ#
#
m
ζ
≥
µ
# M
L 1 #µ=1L) 1(L−3)/2 .
2⇡i✓ (3.17

P= be studied{e ĉ inĉ the+e ĉ theory!
ĉ }Hamiltonian Ĥ must
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⇣=e

i 2⇡
L

j
n
⇣
]
j
j=0
Proof : The lemma is proved byusing standard facts about the fie
ld norm and algebraic P
integers. See, e.g., [49]. Let α = "Lm ζ ∈ Z[ζ] ⊂ Q[ζ] andoccupation numbers nj = 0, 1,
j=1 nj = N

En =j=0 nj cos( L j + ✓) = <[e<latexit sha1_base64="m/ULVwRUeU/nWtOrvjPUqPE+8Rs=">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</latexit>

L−1
µ=1

µ

σj (α) =

µ

L−
!1

mµ e

i2π jµ/L,(3.18)

uss
a
G
h
c
i
r
d
e
i
r
F
Carl

number-theoretic
facts
be its conjugate, where j = 1, . .
. , L − 1. Note that σ (α) = α, σ(α) = {σ|σ (α)| ≤ M . Let N : Q[ζ] → Q de(α)} , andnote the field norm of Q[ζ]. By deition, we haveassume L is an odd prime, N ≤ (L −fin1)/4, m1, …, mL−1 ∈ ℤ
µ=1

1

j

L−
$1

(L−1)/2

$

j

L−j

∗

L−1
j
(3.19
j=
1
Lemma: if mj ≠ 0 for somej=1 j, then ∑j=1 mj ζ )≠ 0
Since Lemma 3.3 guarantees σ (αN (α) =σj (α) =|σj (α)|2 .

) #= 0 for all j, we see that N (α)
j> 0. Note that α is analgebraic integer, and hence so are its conjugates σj (α) and the norm
N (α). It is known thatan algebraic integer that is rationa
l must be an integer. Since N (α)∈ Q, we see N (α) ∈ Z
and hence N (α) ≥ 1. This bound,with (3.19), implies

no degeneracy for most θ

L−1
L−1
j
−(L−3)/2
Lemma: we have | ∑j=1 mj ζ | 1≥ ( ∑j=1 | mj | )L−3 .
(3.20)%(L−1)/2
&
$ #
#2 −1
2
#σj (α)#
|α| ≥
≥
j=2

M

−(L−1)/2||nodegeneracy
if
and
θ
≠
0
θ
≤
(4N)
We are now ready to prove our physics theorems.

summaryfi

ff

we focused on the problem of thermalization(approach to thermal equilibrium) in isolated
macroscopic quantum systems
without relying on any unproved assumptions,
we proved that a free fermion chain exhibits
thermalization (in some weak sense)
the key observations were that a random
nonequilibrium initial state has a large De and thatthe absence of degeneracy can be proved by using somenumber-theoretic results
it is desirable to have examples of non-integrable
systems in which our (plausible) assumptions for thegeneral theory of thermalization can be justi ed