￼

Washington University in St. Louis
Physics Colloquium on March 3, 2021

Unexpected turns ofa humble theorem

Masaki Oshikawa
ISSP, UTokyo￼

Quantum Many-Body Systems
From Kozuma Group (Tokyo Tech)
web page(spinless) Hubbard model<latexit sha1_base64="q3lDUvy5ettyz7Ti5YmrUHTyUiI=">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</latexit>

H=

Xh

hj,ki

t

⇣

†
†
cj ck + ck cj
2⌘+ V n j nki￼It’s a hard problem!
Fermion: each site is either empty (0) or occupied (1)Λ sites: Hamiltonian is 2Λ✕2Λ matrix
huge even for moderately large size Λ
cf. quantum computing
Numerical algorithms: great advancements(Quantum Monte Carlo, Density-Matrix Renormalization
Group, Tensor Network, … ) but still challenging
Exact solution: available only for the standard model in 1D(no longer exactly solvable in 2D and higher,
or by inclusion of next-nearest-neighbor coupling etc.)3Quantum fluctuations can drive the system at T=0
into different quantum phases, and cause quantum phasetransitions between quantum phases
gapped (off-critical) But there areEcontinuum of
excited states

gapless systems
E
without any
apparent ne-tuning…
(phonons, metals…)

gapless (critical)

gap
ground state

ground state

4
fi

fi

￼

Quantum Many-Body Systems

g. by Subir Sachdev

￼General Principles?
Symmetries of the model<latexit sha1_base64="q3lDUvy5ettyz7Ti5YmrUHTyUiI=">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</latexit>

H=

Xh

hj,ki

t

⇣

†
†
cj ck + ck cj

cj !e cj

U(1) symmetry c† !e i✓ c†j+ V n j nkiparticle numberi✓<latexit sha1_base64="fegnrYLo4fp/yRxwcagZSc7zBMk=">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</latexit>⌘<latexit sha1_base64="cTWugXywhp5Nyg/2ddkGDcm+a88=">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</latexit>

j

N⌘

†
nj ⌘ cj cj !nj

X

nj

j

conserved

Noether’s theorem
Can we say something about the energy spectrum?
5￼

Nambu-Goldstone Theorem
e.g. spin waves

Spontaneous breaking of a continuous symmetry (e.g. U(1))“slow twist”
Gapless excitations
There are many gapless systems
without a SSB (metals, etc.), however.
Any other mechanism for gaplessness?
yes, if there is also alattice translation invariance6gapless (critical)E￼

Nambu-Goldstone Theorem
e.g. spin waves

Spontaneous breaking of a continuous symmetry (e.g. U(1))“slow twist”
Gapless excitations
There are many gapless systems
without a SSB (metals, etc.), however.
Any other mechanism for gaplessness?
yes, if there is also alattice translation invariance6gapless (critical)
E

Lieb-Schultz-Mattis 1961, M.O.-Yamanaka-Af eck 1997,…
Number of particles: conserved ← U(1) symmetry
Lattice translation symmetry +spatial inversion or time reversal symmetrye.g. 1D spinless Hubbard model with periodic b.c. cL ⌘ c0<latexit sha1_base64="2SRDTvNlA4x/iWvtAfRoa96D12U=">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</latexit><latexit sha1_base64="YecP+iZzquvCvbTwkBvj0CTAu9Q=">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</latexit>

L-1

H=

t

L
X1 ⇣

†
†
cj+1 cj + cj cj+1

j=0⌘

Lattice translation T<latexit sha1_base64="DdnFCT/ic8h2enDyE5pn3UakmUo=">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</latexit>

Translation inv.

L
X1

+V

nj nj+1

j=0<latexit sha1_base64="SJ5p4fSgPmHK+yQWy7Rs3RL+lKk=">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</latexit>

T cj T[T , H] = 0<latexit sha1_base64="s4zslnFUSJkuGu9EifV+NCBVom4=">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</latexit>

7
fl

￼

Lieb-Schultz-Mattis Theorem in 1D1= cj+1￼

LSM Variational Argument
H| 0 i = E0 | 0 i
Ground state(very complicated — we don’t need to know it exactly
its EXISTENCE is enough!)<latexit sha1_base64="dOjsxssre936ngu6eKENxY7/46g=">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</latexit><latexit sha1_base64="JpJ3xGeEC/v3AOUsRLwgxYyZfSM=">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</latexit>

e

i✓N

=e

i✓

P

j nj

global U(1) transformation<latexit sha1_base64="YuHEh/LWg30EXIw7sMsrzRdAoNc=">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</latexit>“Slow twist” (NOT symmetry)<latexit sha1_base64="GAXfN1HgtBViNQdtbpF9XSRZadM=">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</latexit>†i✓<latexit sha1_base64="QHTr63bGOH1KpWcCzXoJ6eEClNE=">AAADAXichVI9TxRBGH5Y/MDzg1MbEpsNlzNWl/eIAWNF1MLG8HlAAnjZXQcY2a/szl0CG2zs/AMWVBAtiJEQW0sa/4AFP4FYYmJj4bN7mxAlyEx253mfeZ+ZeeYdN/Z1akSO+qz+S5evXB24Vrl+4+atwertO3Np1Ek81fIiP0oWXCdVvg5Vy2jjq4U4UU7g+mreXX+az893VZLqKJw1G7FaDpzVUK9ozzGk2tUhr/3aXjKRrV5mmmBNGWfLJtmu1qQhRbPPgmYJaijbZFQ9xhJeIYKHDgIohDDEPhyk7ItoQhCTW0ZGLiHSxbzCFirUdpilmOGQXed/ldFiyYaM8zXTQu1xF59fQqWNunyXPTmRb/JJjuX3uWtlxRr5WTY4uj2tituD74Zmfl2oCjgarJ2qqKifq3HZgwt8GazgUeFH019cMLlTr7dHd/P9yczj6Xp2X3blBz3uyJEc0mXY/el9nFLT2/858+nuzxj3qpAQvShdTFChyORRfo8ZM2eL+8zKsZJ3PoLmvyU/C+ZGGs3Rhkw9rI0/KZ/DAO5hGA9Y8zGM4zkm0eIp3uADPmPfemvtWQfWl16q1Vdq7uKvZn39A4NVrgs=</latexit>

U cj U = exp

✓

◆

2⇡ij
cj
L

cj ! e cj

01
X 2⇡ijU ⌘ exp @nj ALjconsistent with PBC<latexit sha1_base64="BJbVYro75ejum5dJBJDzs1OHmNY=">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</latexit>

8

◆

2⇡i0
U c0 U = exp
c0 = c0
L
✓
◆
2⇡iL
†
U cL U = exp
cL = cL
L†<latexit sha1_base64="iclmT3d62Psx0MxCUZ6irVSBJNk=">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</latexit>✓cL ⌘ c0<latexit sha1_base64="2SRDTvNlA4x/iWvtAfRoa96D12U=">AAAC9XichVLLShxBFD3TvsaJj9FshGwGhxFXwx0JUVwNmkUWiuNjVPAxdJelFvbL7p4RbfID+YEI4kKJgrjwI9z4Ay78BMnSQDYuvN3TIImoVXTXvafuubdO3TJcU/kB0V1Ka2lta+9Id2Y+dHX39Gb7+hd9p+4JWRWO6XjLhu5LU9myGqjAlMuuJ3XLMOWSsTMZ7S81pOcrx14I9l25ZulbttpUQg8YWhe1qdyq3K2rRk7UqJbNU5HikXtplBIjj2RUnOw9VrEBBwJ1WJCwEbBtQofPcwUlEFzG1hAy5rGl4n2J78gwt85RkiN0Rnf4v8XeSoLa7Ec5/ZgtuIrJn8fMHAp0Sxf0QDd0Sff0+GquMM4RnWWfV6PJlW6t98fA/N93WRavAbafWcwovMoxeFrv6AqwibFYj2J9boxESkWzRuPg58P8+FwhHKJT+s0aT+iOrlml3fgjzmbl3NEbZ36u/pX9Zhc8tqYTFTPMkIxEXnSPIUcuxPcZJmsmmvwISv+3/KWxOFIsfSnS7Od8eSJ5Dml8wiCGueejKOMbKqjyKTwc4gSn2p52rP3SzpuhWirhfMQ/Q7t6AlsaqY8=</latexit>￼

LSM Variational Argument<latexit sha1_base64="+vTm4n+kA3z2ZPpFKmXiraZBOJM=">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</latexit>†

U HU =t<latexit sha1_base64="YecP+iZzquvCvbTwkBvj0CTAu9Q=">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</latexit>

H=<latexit sha1_base64="Ym9H5zWqqq8nB6tCl2KCyLbD4xM=">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</latexit>U † HU<latexit sha1_base64="gcCguMsweCGP+3yRqkFPom6NhLE=">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</latexit>

h

0|

†

U HU

t

L
X1 ⇣

e

j=0
L
X1 ⇣

2⇡i/L †
2⇡i/L †
cj+1 cj + e
cj cj+1††
cj+1 cj + cj cj+1

j=0

⌘

L
X1

+V

⌘

+V

L
X1

nj nj+1

j=0

nj nj+1

j=0⌘
expectation
2⇡i X ⇣ ††H =t
cj+1 cj cj cj+1
value vanishes
L j✓ ◆2 X ⇣
⌘
2⇡
1
†
†
+t
cj+1 cj + cj cj+1 + O( 2 )LLj1
H | 0 i = O( )L<latexit sha1_base64="zkZxGos/zkBbplyFikYJcbmxKf4=">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</latexit>

9

isalow-energy
U| 0 istatecould be (almost) identical to | 0 i<latexit sha1_base64="UISLxds+xY09y20hgyxWiSGW2sE=">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</latexit><latexit sha1_base64="zkZxGos/zkBbplyFikYJcbmxKf4=">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</latexit>

U|

0i

Are they different?
<latexit sha1_base64="2JSev1zuEkNAJrNjK25KxSFUF8s=">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</latexit>

T|

0i = eiP0|“ lling factor”(particle # / site)

0i<latexit sha1_base64="aYNXt3z7ZuWvc1GlkWDNla2QJfk=">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</latexit><latexit sha1_base64="qTvBjLx9jX/XcawF/naZqgo52No=">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</latexit>†<latexit sha1_base64="YuHEh/LWg30EXIw7sMsrzRdAoNc=">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</latexit>

U TU =e2⇡iP

j nj /L01
X 2⇡ijU ⌘ exp @nj ALj<latexit sha1_base64="+HrLHGtEakMaFR7u0sFM7tpNidg=">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</latexit>

T (U|

T =e

iP0 +2⇡⌫i
i)
=
e
(U|
0

2⇡⌫i

0 i)

⌫=

T

P

j nj

L

N
=
L

is a low-energy state
different from | 0 i
if ν is NOT integer!
<latexit sha1_base64="zkZxGos/zkBbplyFikYJcbmxKf4=">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</latexit>

10
fi

￼

Does it mean anything?

U|0i<latexit sha1_base64="UISLxds+xY09y20hgyxWiSGW2sE=">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</latexit>

Quantum Many-Body System (in 1D) with- global U(1) symmetryAND- lattice translation symmetry
WITH a fractional (non-integer) lling factor ν- gapless excitations above the ground stateOR- multiple, degenerate ground states below gap- unique ground state below gap“featureless (trivial) insulator”
11
fi

￼

Statement of LSM theorem￼Intuitive picture for the LSM theorem:
gapped phase needs the particles to be“locked”, and the density of the particles
must be commensurate with the lattice.
1 particle/
unit cell(= 2 sites)￼

Intuitive picture for the LSM theorem:
gapped phase needs the particles to be“locked”, and the density of the particles
must be commensurate with the lattice.
1 particle/
unit cell(= 2 sites)
add extra
particles(“doping”)

￼Intuitive picture for the LSM theorem:
gapped phase needs the particles to be“locked”, and the density of the particles
must be commensurate with the lattice.
1 particle/
unit cell(= 2 sites)
add extra
particles(“doping”)
mobile carriers

￼

Original LSM paper in 1961
not about the famous theorem!?

13

Main Result:
Exact solution of
S=1/2 XY chain
by mapping tofree fermions(Jordan-Wignertransformation)￼

14Apparently, the authors themselves did notthink the theorem was too important.
They proved the theorem just for S=1/2 chain atzero magnetic eld…
Perhaps the theorem had drawn little attention
for more than 20 years after its birth in 1961
15
fi

￼

Where was the LSM theorem??

Appendix….

Apparently, the authors themselves did notthink the theorem was too important.
They proved the theorem just for S=1/2 chain atzero magnetic eld…
Perhaps the theorem had drawn little attention
for more than 20 years after its birth in 1961
15
fi

￼

Where was the LSM theorem??

￼

Haldane “Conjecture” in 1981
S=1/2, 3/2, 5/2, …..
“Quantum Critical”✓ ◆⌘1~i · S~j i /hSr

Gapless

S=1, 2, 3…..
Non-vanishing excitation gap (“Haldane gap”)
✓
◆
r
~
~
hSi · Sj i / exp⇠
Against the “common sense” at the time
16

“conjecture”

Generalization of the original LSM theorem for S=1/2
to arbitrary spin quantum number S
S: half-odd-integer→ gapless or
2-fold g.s. degeneracy

integer S : no constraint from LSM→ may have a unique gapped ground state
consistent with Haldane conjecture!
17

fl

￼

Af eck-Lieb 1986

M.O.-Yamanaka-Af eck
1997
Identify, say, Sz = -S state as “vacuum”
increase Sz by 1
add a particle (magnon)
Spin S: Sz = -S, -S+1, … , S-1, Sz
Sj =S + nj<latexit sha1_base64="dWkaEbar5MpxiOpnz8PmJLQosU8=">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</latexit>

magnetization per sitezm = hSj i =<latexit sha1_base64="AXz9is3xWdGFTcQZKLBPXBP76mI=">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</latexit>

S + hnj i =S+⌫zero magnetization (ground state of antiferromagnet)
fractional
lling
if
and
⌫=S
m=0
only if S is half-odd-int<latexit sha1_base64="ECDoPG3Bjdc8BXNz3uJyNElxx1o=">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</latexit><latexit sha1_base64="O/yDahZjhK4sTeV635WPkmwRxso=">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</latexit>

fl

18

fi

￼

Spin System as Many Particles

Standard(?) view:
topological term of the O(3) non-linear sigma model
present only for half-odd-integer spin S
Intuitive(?) view:half-odd-integer spin S: fractional (1/2+integer) lling
integer spin S: integer lling → can be “trivial” insulator
naturally obtained by generalizing the LSM theorem tomany particle systems [Yamanaka-MO-Af eck 1997]z
m = hSj i =<latexit sha1_base64="AXz9is3xWdGFTcQZKLBPXBP76mI=">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</latexit>

S + hnj i =S+⌫zero magnetization (ground state of antiferromagnet)⌫=S<latexit sha1_base64="ECDoPG3Bjdc8BXNz3uJyNElxx1o=">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</latexit>

m=0<latexit sha1_base64="O/yDahZjhK4sTeV635WPkmwRxso=">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</latexit>

fi

fl

19
fi

￼

Why Haldane Gap?

￼

Why not in LSM?

?!

20

￼

Why not in LSM?

?!
Perhaps refers to this paper20￼

Why not in LSM?

?!
Perhaps refers to this paper

But no mention is actually made on
the generalization of LSM theorem?!
20

LSM in 1961 thought they can generalize their theorem
to general S,
but then realized the proof “fails” for integer S
So they scrapped the generalization(until Af eck-Lieb paper in 1985,
but after Haldane conjecture)…. perhaps they just missed the clue ofthe “Haldane gap”??

21
fl

￼

Maybe….

￼Higher Dimensions?
LSM twist in x direction<latexit sha1_base64="K2lEDdgaWGRB/U90AmUxuU/GN54=">AAADPXichVI7bxQxEJ4sr2R55IAmUpoVp0OhOfkiBAgJKQIKCiLyuEsixdHKa3x3VrwPeX2nC9b+Af4ARaogUUT5BzQUNPwBinQ0FAeIJkhpKJjdW+kEUYitXc98nm/Gn8dBomRqCDmccM6dv3Dx0uSUe/nK1WvTles31tK4p7lo8VjFeiNgqVAyEi0jjRIbiRYsDJRYD7af5PvrfaFTGUdNs5OIrZB1ItmWnBmE/EqThsx0OVO2lfkD75FHxSCxVIm2mXNpWzNu52kiPZnZ5/4g82jaC31L+4JbnWXewIvGnku17HTNncyvVEmdFMM7aTRKowrlWIorQ6DwEmLg0IMQBERg0FbAIMW5CQ0gkCC2BRYxjZYs9gVk4CK3h1ECIxii2/jvoLdZohH6ec60YHOsovDTyPSgRj6TfXJEPpEDMiS/T81lixz5WXZwDUZckfjTr2dWj89khbga6I5ZyKidyglwhmfoMtCGB4UeifqSAsmV8lGN/qs3R6sPV2r2NnlLvqPGPXJIPqLKqP+Lv1sWK7v/OfO4+lP0R13QaC2WKl4gQyCSe/k9WoxsFvdpy9XNJz6Cxr8tP2mszdcb9+pk+W514XH5HCZhFm7BHPb8PizAM1iCFp7iPXyBIXxzPjhfnR/Oz1GoM1FybsJfwzn+AwU5x7w=</latexit>

2⇡i X
xn~r
Lx

Ux = exp

Ly<latexit sha1_base64="7LGXtar24LkOrEKU9w8W9OmO1Vw=">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</latexit>~r!

~r = (x, y) 2 Z<latexit sha1_base64="LFFCps7AgnRymDSspqITsVoFMew=">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</latexit>2

Lx<latexit sha1_base64="k+VZPhHLcVKifI7Q/G4MzU0E6Yw=">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</latexit>1
Ly
O( 2 ) ⇥ Lx Ly = O( )Lx
Lx<latexit sha1_base64="CG1gREPbsHsjQSWUPeW5tIt+bxA=">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</latexit>

Energy gain due to the twist
Not small….?!
22￼

Anisotropic Limit
LSM variational argument works, if

Ly /Lx ! 0<latexit sha1_base64="QTQ5dNd8fZRaedPNdd6zuopvyUk=">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</latexit>

while Lx , Ly ! 1 , as already pointed out in LSM(1961)<latexit sha1_base64="QHqFugiWnBH8D7CPyM9xe1uQTd8=">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</latexit>

But is this really
2D limit?
Can we show LSM
for isotropic 2D
limit?

23

For example, consider a many-particle system on thesquare lattice of Lx × Ly with periodic boundary conditions
assume particle number conservation (U(1) symmetry)Φ

assume that the system is gapped, and consider the
adiabatic insertion of unit ux quantum through the “hole”
M. O. 2000
24
fl

￼

Many Particles on Periodic Lattice

(i) Increase Aharonov-Bohm ux Φ adiabatically from
0
|
i
!
|
0 to Φ0(=2π)
00iHamiltonian for the nal state isΦdifferent from the original one, but
we can(ii) eliminate the unit ux quantum by
the large gauge transformation
Ux H(= 2⇡)Ux
Ux = exp

1

= H(

2⇡i X
xn~r
Lx
~
r

0
0i

= 0)

!

variational argument replacedby adiabatic process
25
fl0i ! |

fl|0
0 i ! Ux |fi￼Adiabatic Flux Insertion￼

Large Gauge Transformation
Initial Groundstate |Tx |0i = eiPx(0)|

Final State

0i
0i

Tx |

|

0
0 i = Fx |0
iPx(0)|
0i = e00igroundstate of H(2⇡)

groundstate of H(0)Large gauge transformation0˜| 0 i ⌘ Ux |Ux = exp

0
0i

2⇡i X
xn~r
Lx
~
r

!

must be a groundstate of H(0)i(0˜
Tx | i = e
0

Ux

1

2⇡i X
n~r
Lx

Tx Ux = Tx exp

~
r

P
(0)
2⇡
Px + Lx
r
~
r n~
26

)| ˜ 0 i
0

!

0i

X
2⇡
Px(0) ! Px(0) +
n~r
Lx
~
r

total number of particles(conserved)

We are usually interested in the thermodynamic limit
for a xed particle density (particle # / unit cell) νpSuppose ⌫ =and choose Ly to be a coprime with q
q
2⇡
p
Px =
Lx Ly ⌫ = 2⇡Ly
Lxq

Lattice momentum is de ned modulo 2π
momentum shifted if q ≠ 1 (fractional lling)
The nal state is different from the initial ground state
ground-state degeneracy!
fi

fi

fi

27
fi

￼

Momentum ShiftLSM 1961, Af eck-Lieb 1985,M.O.-Yamanaka-Af eck 1997,
M. O. 2000, Hastings 2004,…
Periodic (translation invariant) lattice
U(1) symmetry

unit cellconserved particle numberν : number of particle per unit cell ( lling fraction)
ν = p/q“ingappability”- system is gaplessmust be in a nontrivial phase!
OR- gapped with q-fold degenerate ground states
gapped with unique ground state
fl

fi

28
fl

￼

LSM in arbitrary dimensions

￼

Recent Developments

Non-symmorphic lattice with “glide symmetry”:“effective unit cell” is half of the unit cell
⌫
⌫e↵ =
2
LSMOH-type restriction
even when ⌫ 2 Z
29￼

Crystallographic Symmetries

30

Proofs/arguments for the original LSM do not work0<latexit sha1_base64="wHcN6ZEs4P3gIv6OPGwaMakEo2w=">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</latexit>

1D LSM

1

X
2⇡i
zA
@
U = exp
jSj
L j

Lieb-Schultz-Mattis
1961

variational low-energy state
2D and higher
Adiabatic insertion of magnetic ux(U(1) gauge eld)
M.O. 2000
Hastings 2004〜
fl

31
fi

￼

LSM for Discrete Symmetry?

Proofs/arguments for the original LSM do not work0<latexit sha1_base64="wHcN6ZEs4P3gIv6OPGwaMakEo2w=">AAADIXichVJNaxRBEH0ZPxI3alZzEbw0LivxstQGJUEQQvTgQTHJZpNAJi4zk95NJ/NFT+9iMuwf8A94EAQVD+LP0IMeDCJ4CP6C4DGCFw/WzA4EDTHdzFTV63pV/brbjX2VGKK9IevU6TNnh0fOlUbPX7g4Vr50eSmJutqTTS/yI73iOon0VSibRhlfrsRaOoHry2V36262vtyTOlFRuGi2Y7kWOJ1QtZXnGIZa5emmuCNs+SRObV+2zYSw29rx0kk7VkL10wd9YSfdoLUpNkXj8Q5bW6vOhrnRb5UrVKN8iKNOvXAqKMZcVN6HjXVE8NBFAIkQhn0fDhKeq6iDEDO2hpQxzZ7K1yX6KDG3y1mSMxxGt/jf4Wi1QEOOs5pJzva4i8+fZqZAlb7RWzqgj/SO9un3sbXSvEa2l2227oAr49bY0yuNXyeyArYGG4csZlSP5bg8gxN0GbQxnetRrC/OkUypN+jR23l20Li9UE2v0yv6wRpf0h69Z5Vh76f3Zl4uPP/Png+73+N4cAuavYeFikfMkIxkUXaOKWcu5ueZFraUTX4E9X+v/KizNFmr36rR/M3KzGzxHEZwFdcwwXc+hRncxxyavIsX+IDP2LVeW5+sL9bXQao1VHDG8dewvv8BNIW6fg==</latexit>

1D LSM

1

X
2⇡i
zA
@
U = exp
jSj
L j

Lieb-Schultz-Mattis
1961

variational low-energy state
2D and higher
Adiabatic insertion of magnetic ux(U(1) gauge eld)
M.O. 2000
Hastings 2004〜
fl

31
fi

￼

LSM for Discrete Symmetry?

[1D]MPS-based “proof” Chen-Gu-Wen 2011
Field-theory argument Fuji 2014
Mathematical proof Ogata-Tachikawa-Tasaki 2020
[2D and higher]
Many statements for space group symmetries etc.
Po-Watanabe-Jian-Zalatel 2017, Else-Thorngren 2020
But the argument is either at abstract level, or relying on
Schmidt decomposition (OK for xed width but…)or “trivial” degeneracy of odd-site sytems
Watanabe-Po-Vishwanath-Zalatel 2015
So we will try to give a convincing physics argument…
32
fi

￼

LSM for Discrete Symmetry

￼

Rigorous Proof for 1D in 202033

￼

Example: XYZ model
Yuan Yao (ISSP→RIKEN) & M.O.
arXiv:2010.09244“XYZ” spin model on the square lattice of size L1 ✕ L2
<latexit sha1_base64="hIPcZBN69rFL6qHIrXfp1+Q1v4k=">AAADmHichVLLbhMxFL3TgVLMowGEhGAzogotD0UOEgKBkCIoooAQbdK0gU4ZzbhOMqrnIY8TkYzmB/gBFqxAYoFY8BFs+AEWXbFgVbEsEhsW3HmgNq1Kbc343HPPsX1tO6FwI0XphjamHzo8fmTiKDl2/MTJydKp00tR0JOMN1kgAtly7IgL1+dN5SrBW6HktucIvuys30/zy30uIzfwF9Ug5Kue3fHdtstshZRV+mx6tuoyW8RziXHXMKOeZ8WmsP2O4IbZ5yyWybV/YNowZZZJiCl4W82Qx1bLaLx8hZZckYxE04lxFSXPkRyMSAa7JS+QHI5IhjskxJRup6suE0Ks0hSt0KwZe0G1AFO1J+z7jyvja/NBaRNMWIMAGPTAAw4+KMQCbIiwr0AVKITIrUKMnETkZnkOCRD09lDFUWEju47/DkYrBetjnM4ZZW6Gqwj8JDoNKNNv9CPdol/pJ7pJ/+w7V5zNke5lgKOTe3loTb4+1/h9oMvDUUF324WO8r4eB7t3QF0K2nArq8fF+sKMSStl+Rr94Zutxu16Ob5E39OfWOM7ukG/YJV+/xf7sMDrb/+z5+3VZzHOb0EielpU8QwdHJk0Ss8xRuVidp5xMZK04yOo7r7yvWDpeqV6o0IX8DXcg7xNwAW4CDN45zehBnMwD01g2lntjjarPdDP6zX9of4ol45phecMjDS9/hcH8+T3</latexit>

H=

X

y y
x x
JX S~r S~r0 + JY S~r S~r0 + JZ S~rz S~rz0

h~
r ,~
r0 i

On-site discrete symmetry of Z2 ✕ Z2 (dihedral sym.)(π-rotation of spins about x, y, and z axes)
Lattice translation symmetry T1, T2
We can “twist” the boundary condition along x-directionby π-rotation about z-axis
34

￼Twisted Boundary Condition
X<latexit sha1_base64="4vnOu7CqUxaJ3ovGAj+jkB1w/dI=">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</latexit>

H

twist

=

y y
x x
z z
JX S~r S~r0 + JY S~r S~r0 + JZ S~r S~r0h~r ,~r 0 i2seam
/

X

+

y y
z z
JY S~r S~r0 + JZ S~r S~r0x x
JX S~r S~r0h~r ,~r 0 i2seam<latexit sha1_base64="6ZlVd3auyvR6lav073D+wJRLEzg=">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</latexit><latexit sha1_base64="8OdZmEXhWSpV6HbSKPbR2VoH0Wc=">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</latexit>

L2<latexit sha1_base64="hIzhOqsSMdRqVdaJVQ62QPj6qDI=">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</latexit>[T1 , Htwist

<latexit sha1_base64="4+8X7sz6o9LNzvexCvUgbABAcUw=">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</latexit>T̃1 ⌘ T1

L1
<latexit sha1_base64="jUfJMerpZHG3PWquwEkTsyNzMQs=">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</latexit>but] 6= 0Ye[T̃1 , H

twist

]=0

translation+ discrete gauge tr.

z
i⇡S~
r

~
r 2seamGlobal π-rotation of spins about x-axis<latexit sha1_base64="odrTIKEgfIY8jHYTl2Z0y2vkw10=">AAADH3ichVI7b9RAEP5iXuF45IAGicbidIjqNBfxElUEFDSI5C6XRIpzlu1Mwip+sfZZCdb9Af4ABTRBokD8CyhAAiEaivsJEWWQaCgY+yydIArZlb0z3843s9/OurGvkpRoNGUcO37i5Knp07UzZ8+dn6lfuLiURAPtcc+L/EivuE7Cvgq5l6rU55VYsxO4Pi+7W/eL/eWMdaKicDHdiXktcDZDtaE8JxXIrt/q9K1Y2dumxU8HKjOtWEfrdm5l7OV6ODS5nysBldntb0/goV1vUIvKYR402pXRQDXmo/oeLKwjgocBAjBCpGL7cJDIXEUbhFiwNeSCabFUuc8YoibcgUSxRDiCbsl/U7zVCg3FL3ImJduTKr58WpgmmvSd3tI+faJ3tEe/D82VlzmKs+zI6o65HNszzy93fx3JCmRN8WTCEkbzUI4rMzhCV4oN3Cn1KNEXl0ih1BvXyJ692O/e7TTza/SafojGXRrRB1EZZj+9NwvcefmfM0+qPxB/3AUt1qNKxWNhsCCFV9xjLpGL5X3m1VorpjyC9r8tP2gszbbaN1u0cKMxd696DtO4gqu4Lj2/jTk8xDx6copXeI/P+GLsGh+Nr8a3cagxVXEu4a9hjP4AMbK7tA==</latexit>

Rx⇡ ⌘

Y
~
r

e

x
i⇡S~
r

⇡
twist
[Rx , H]=0<latexit sha1_base64="N1X9icZTVaROoackOS21fnLKujU=">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</latexit>

35

￼

Two Symmetries under the Twisted BC<latexit sha1_base64="4+8X7sz6o9LNzvexCvUgbABAcUw=">AAADN3ichVK7btRAFL0xjyTLIws0kWgsVouoVtcRiIgqAgoayGN3k0hxYtnemzCKX4xnLZKRf4AfoEgVpBQobRpqGn6AIjQUIEREhYJEQ8G11yKCKGRG9px75p47c2bGSwKRKsT9IePM2XPnh0dGaxcuXro8Vr9ydT6N+9Knrh8HsVz03JQCEVFXCRXQYiLJDb2AFrz1B8X8QkYyFXHUURsJLYfuWiRWhe8qppz6E1uJoEe6kzuWadOzvsjMTgETGfccbWfka5mbtohMW9FzpVMunucmrWjBScJsr2z+Sctzp97AFpbNPA6sCjSgajNx/QBs6EEMPvQhBIIIFOMAXEi5L4EFCAlzy6CZk4xEOU+QQ421fc4iznCZXef/GkdLFRtxXNRMS7XPqwT8SVaa0MT3+BoP8R3u4gH+OrGWLmsUe9ng0RtoKXHGXoy3f56qCnlU8PRIxYrmiRqPe3iKLwWrMFn6EewvKZnCqT9YI9t8edi+N9fUN/EVfmOP27iPb9lllP3wd2Zpbus/ez5a/SHHg1uQjB5XLqZZQcwUUXGOmjM75XnqaqwVnR+B9e+VHwfzEy3rTgtnbzem7lfPYQSuww24xXd+F6bgEcxAl3exBx/gE3w23hgfjS/G10GqMVRprsFfzfj+GyTAxWo=</latexit>T̃1 ⌘ T1

Y

e

z
i⇡S~
r

<latexit sha1_base64="odrTIKEgfIY8jHYTl2Z0y2vkw10=">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</latexit>

⇡
Rx ⌘

~
r 2seam<latexit sha1_base64="t/w6wU+ERycwgeqChJhnH+4dXoo=">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</latexit>⇡⇡T̃1 Rx = Rx T̃1 (

1)

2SL2

<latexit sha1_base64="HCokr7y0qnyge3OVuEzLbuuLlRs=">AAADUHichVJNTxRBEH3M+oHjByteTLxM3CzBg5saIsGQkBDlwMUILAskLLuZGZu1w3xlZnYDdPYP+Ac8eILEg/FneOHCxcQDP4F4XBM9YKRmdpJVcaU7M1X1ul51v662Q1fGCdHJiFa4cvXa9dEb+s1bt++MFe+Or8VBO3JEzQncINqwrVi40he1RCau2AgjYXm2K9btnefp+npHRLEM/NVkLxRbntXy5bZ0rIShZrElGkrWQ2lUG/tNVe8IR0XdbtcYwLu/wfqcMfnYfNRQU9VhKca/CzaLJapQNoyLjpk7JeRjKSieoo5XCOCgDQ8CPhL2XViIeW7CBCFkbAuKsYg9ma0LdKEzt81ZgjMsRnf43+JoM0d9jtOaccZ2eBeXv4iZBsr0hT5Qj47oI53S2dBaKquRnmWPrd3nirA59uZ+9fulLI9tgtcDFjPKQzk2T+8SXQm28TTTI1lfmCGpUqe/R2f/ba86u1JWE3RIX1njAZ3QJ1bpd74575fFyrv/nHmw+wLH/S5E7L3IVbxkhmAkjdJ7VJy5mt2nyq2eTn4E5t8tv+isTVXM6QotPynNP8ufwyge4CEmueczmMcillDjUxyjhzP81D5rP7RfhZF+qpZb3MMfo6CfA6Vbzb4=</latexit>

e

z
x
i⇡S~
i⇡S
r
~
r

e

Y

e

x
i⇡S~
r

~
r

x
z
2S i⇡S~i⇡Sr~r= ( 1) ee

The two symmetry operators anticommute
if S is half-odd-integer and L2 is chosen to be odd
Ground states (and all the eigenstates)of the Hamiltonian under the twisted b.c.
are exactly two-fold degenerate!
Hirano-Katsura-Hatsugai 2008
for U(1) symmetric systems36

￼What Does This Mean?
This argument does not apply directly to periodic b.c.
If the degeneracy is only an artifact of the twisted b.c.
it would not mean much.
But we argue that the degeneracy is “robust”
and present also for the periodic b.c.

37

￼

Physical (Hand-Waving) Argument
If the ground state is gapped and unique under
the periodic b.c., the system should not have any order(conventional or topological). The absence of the order
implies that the system should be insensitive to the twistof the b.c. (in a large enough system)
Therefore, the exact ground-state degeneracy under thetwisted b.c. does imply some order (conventional or
topological), and the (quasi) ground-state degeneracy
under the periodic b.c.
38￼More Formal (Less Hand-Waving?) Approach
imaginary time
“Quantum Transfer Matrix”
along spatial direction
Betsuyaku 1984<latexit sha1_base64="SGitzLEXHGz5s6Bf9eYSOqxAj5k=">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</latexit>H̃<latexit sha1_base64="hTMu4jWqJ6Lu579pzy/UBObEuVw=">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</latexit><latexit sha1_base64="AKaoSCWzSM4DTZRPiS0eoBimrgg=">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</latexit>

T =e

Z = Tre
H

H̃
L1 H̃

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Trotter-Suzuki decomposition→ path integral formulation<latexit sha1_base64="0ncPFyoWOXeR7DxIKuFLMx+HtbE=">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</latexit>

space

Z = Tre39

H⇣⇠ Tr e

HA /N

e

HB /N

⌘N

￼

Twisted BC and QTM<latexit sha1_base64="p01vrAhrLcHxhgzAjTwW8vgOzqs=">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</latexit>

Z<latexit sha1_base64="SGitzLEXHGz5s6Bf9eYSOqxAj5k=">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</latexit>

H̃

twist

= Tr

⇣

⇡
L1 H̃
R̃z e

⌘

globalsymmetry
operator⇡R̃z (π-rotation about z)
unitary<latexit sha1_base64="lTJ5mzW3RylVdPAD8hf+3XlOvj4=">AAAC+XichVLNTttAEP4wtIX0BxcuSFwiolQ9RZOqVaueUOHABQGBBCRCI9tZ6Ar/yd5YAosX4AU4wAUkEC1HHoFLX6AHHiHimEpcOHTsWIoA0ezK3plv55vZb2dN35ahIroe0AaHnj1/MTySe/nq9ZtR/e1YLfRagSWqlmd7wZpphMKWrqgqqWyx5gfCcExbrJrbM8n+aiSCUHruitrxxYZjbLlyU1qGYqih63Ul7aaIK3vf675s7OYbeoFKlI78Y6OcGQVkY9HT26ijCQ8WWnAg4EKxbcNAyHMdZRB8xjYQMxawJdN9gT3kmNviKMERBqPb/N9ibz1DXfaTnGHKtriKzV/AzDyK9Id+Uod+0wW16e7JXHGaIznLDq9mlyv8xuj+xPJtX5bDq8KPHosZxSc5Jk+njy6FTXxJ9UjW56dIotTq1oh2DzrLXyvF+B2d0A1rPKZrumKVbvTXOl0SlcP/nLlXfZb9bhcCtuYzFQvMEIwkXnKPMUeupPcZZ2sumfwIyg9b/tiofSiVP5Vo6WNh+lv2HIYxiSm8555/xjTmsIgqnyLCEU5xpsXasXau/eqGagMZZxz3hnb5DxAZq1E=</latexit>cf.) “topological defect line” in CFT
40

￼

Proof* by Contradiction
Suppose that the ground-state of the original Hamiltonianis gapped and unique under the periodic b.c.
→ the “ground state” | ˜ 0 i of the QTM Hamiltonian H̃ mustbe also unique(cf. zero-temperature entropy in the thermodynamic limit)
Symmetry of H̃⇡→ the “ground state” is also an eigenstate of R̃z<latexit sha1_base64="SGitzLEXHGz5s6Bf9eYSOqxAj5k=">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</latexit><latexit sha1_base64="xzEvllF7VcwWGNr8lmYMHeLFu1o=">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</latexit><latexit sha1_base64="SGitzLEXHGz5s6Bf9eYSOqxAj5k=">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</latexit><latexit sha1_base64="lTJ5mzW3RylVdPAD8hf+3XlOvj4=">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</latexit>⇡ ˜R̃z | 0 i = ⇣0 | ˜ 0 i<latexit sha1_base64="sJGGqxnZ4zx9iY8ym8WqKVlNtVU=">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</latexit>|⇣0 | = 1
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Z

twist

⇡
L1 H̃ ˜PBC˜
⇠ h 0 |R̃z e| 0 i = ⇣0 Z

up to exponentially small corrections
41

Z

twist

2R

+

⇣0 = 1<latexit sha1_base64="0+qRHExMOUVdGXj4vWMUPVETZg8=">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</latexit>￼

Proof* by Contradiction<latexit sha1_base64="LG68FDf8g4ZPG001BvQ6xct3WcI=">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</latexit>Z⇡
L1 H̃ ˜PBC˜
⇠ h 0 |R̃z e| 0 i = ⇣0 Z

twist

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Z

twist

⇠Z

⇣0 = 1<latexit sha1_base64="0+qRHExMOUVdGXj4vWMUPVETZg8=">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</latexit>

PBC

up to exponentially small corrections
Then the zero-temperature entropy in the thermodynamic
limit must be zero under the twisted b.c.
→ Htwist must have a unique ground state
Contradiction with the exact ground-state degeneracyof Htwist→ assumption (unique gapped g.s. of H ) was wrong→ H with the periodic b.c. must have degenerate groundstates!
*: not really rigorous
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Ruelle “Statistical Mechanics: rigorous results” 1988

Free energy density in the thermodynamic limit
at a xed temperature
43
fi

￼

Thermodynamic LimitBut we needzero-temperature entropy
given by g.s. degeneracy dlog Z( , L) ⇠ log d + ✏0 L + O(exponentially small)<latexit sha1_base64="TTGQ/98IcINv2rOk5jbDm1cg8TI=">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</latexit>

the O(1) quantity log d to be well-de nedin the limit β〜L → ∞cf.) “topological entanglement entropy”
Kitaev-Preskill / Levin-Wen

44

Furukawa-Misguich 2007fi

￼

Thermodynamic Limit￼

What We Have Shown
As a simple example, consider “XYZ” spin modelon the square lattice of the size L1 ✕ L2
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H=

X

y y
x x
z z
JX S~r S~r0 + JY S~r S~r0 + JZ S~r S~r0

h~
r ,~
r0 i

On-site discrete symmetry of Z2 ✕ Z2
(π-rotation of spins about x, y, and z axes)
Lattice translation symmetry T1, T2
for half-odd-integer spin (and L2 is odd),
if the system is gapped, the ground-state must be
degenerate under the periodic b.c.
implying (conventional or topological) order45

￼Generalizations and Limitations
Similar constraint if the “spins” within the unit celltransforms a projective representation of the symmetry
SU(N) symmetry etc.
We can often obtain the ground state degeneracy > 2
for the twisted b.c., but at present we cannot deduce
the number of ground states for the periodic b.c.
(other than it must be > 1). Maybe the number of thedegenerate ground states under the twisted b.c. is also
robust??
46

- Lieb-Schultz-Mattis theorem is one of the few very-

general yet powerful constraints on quantum
many-body systems
Started as a humble result in an Appendix and had
been overlooked for many years, its generality has
been gradually appreciatedActive topic of research for generalization, rigorous
proof, etc. (related to anomaly in eld theory,
topological phases,….) also in recent years
47
fi

￼

Summary