￼

Protecting
Quantum Criticality— when is the system
conducting/insulating?
APCTP-IACS-SNBNC Workshop
on Computational Methods for
Emergent Quantum Matter 17-25 Nov 2022
at SNBNC/IACS Kolkata, India

Masaki Oshikawa
ISSP, UTokyo[Sendai Science Museum Video for Grade 3]

Conductor

Insulator
2fi

￼

Fundamental Classi cation

￼Free Electron Systems
single electronenergyBand
gap

Metal, conducting

Fermi
Level

insulator
3￼Interacting Many-Electrons
We can no longer describe physics in terms of
single electron
We cannot apply the “band picture” in general
We can still characterize the system in terms of
many-body excitation gap!
gapped≒insulator

gapless≒conductor

continuum of
excited states

Total energyof the system

continuum of
excited states

many-body gap
Δground state

4

ground state￼

Different Cultures…

not just different languages, butdifferent backgrounds, different points of view5￼

Solid State Physics
Electrons form conducting metal by default(e.g. in free space)
Some mechanism is needed for electrons toform an insulator by opening a gap(CDW instability, etc.)6Many-body gap Δ = inverse “correlation time”(in imaginary time formalism)<latexit sha1_base64="3cCJLJkeh3/OWLmTuT2PeKH/wGY=">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</latexit>hO(⌧ )O(0)i / e⌧

Correlation length is often proportional to 1/Δ
gapped (≒ insulator)

nite correlation length
off-criticalgapless (≒ conductor)∞ correlation length
critical
7fi

￼

Statistical Mechanics

fi￼Transverse-Field Ising Model<latexit sha1_base64="g9R3G73g2xZ5fougzMnZs8ubTYc=">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</latexit>

H=

X

z z
j k

g

X

x
k

j

hj,ki

gapped (off-critical) TFIM in d dimensions
Econtinuum of
excited states

gapless (critical)↕
E
Classical Ising Modelin d+1 dimensions

gap
ground state

ground state

8

g. by Subir Sachdev

Gapless, critical systems (≒ conductor)
are special, usually requires ne-tuning of parameter(s)However, there are many gapless, critical systems
in condensed matter physics which apparently
do NOT require any ne-tuning(Metals, phonons, …..)Why?
There must be some mechanism to protect
quantum criticality (gaplessness)….

fi

9
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￼

Stat Mech Point of View￼

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?
10￼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 invariance11gapless (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 invariance11gapless (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>

12
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

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P

j nj

global U(1) transformation<latexit sha1_base64="YuHEh/LWg30EXIw7sMsrzRdAoNc=">AAADMHichVJLaxRBEK6Mr7g+supF8NK4rMTL0htExVNQQQ+KeW0SyISlp1O720nPw56exdjMH/APePCk4EHEi3/Bi3cRDCjiMXqM4MWDNbMDQUNMNzNd9XV9Vf11dZBolVrOt8a8Q4ePHD02frx24uSp0xP1M2cX0zgzEjsy1rFZDkSKWkXYscpqXE4MijDQuBRs3Cr2l4ZoUhVHC3YzwdVQ9CPVU1JYgrr1O34o7EAK7To58/FhpoY1Hx8ljvkae3aS+WkWdteZ3zNCuik/UUyx9dzdy1lUwEb1B/Yyy7v1Bm/xcrC9RrsyGlCNmbi+DT6sQQwSMggBIQJLtgYBKc0VaAOHhLBVcIQZslS5j5BDjbgZRSFFCEI36N8nb6VCI/KLnGnJllRF02eIyaDJP/JXfIe/56/5Nv+9by5X5ijOsklrMOJi0p14cn7+14GskFYLg10WMZr7cgKa4QG6LPTgeqlHkb6kRAqlclRj+PjpzvyNuaa7xF/wH6TxOd/i70hlNPwpX87i3LP/nHm3+m3yR10wZN2vVDwgBhJSeMU9OopcKO/TVWutmPQI2v+2fK+xONVqX23x2SuN6ZvVcxiHC3ARJqnn12Aa7sIMdOgUb+ADfIYv3lvvk/fV+zYK9cYqzjn4a3jf/wDKu8D5</latexit>“Slow twist” (NOT symmetry)<latexit sha1_base64="GAXfN1HgtBViNQdtbpF9XSRZadM=">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</latexit>†i✓<latexit sha1_base64="QHTr63bGOH1KpWcCzXoJ6eEClNE=">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</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>

13

◆

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=">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</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>

14

is
a
low-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>

15
fi

￼

Does it mean anything?

U|0i<latexit sha1_base64="UISLxds+xY09y20hgyxWiSGW2sE=">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</latexit>If the lling factor ν=p/q is non-integer(p, q: coprimes)
there are q independent low-energy states including
the ground state
gapped (off-critical)

gapless (critical)
EE
continuum of
excited statesORgap
q-fold degenerate
ground state
16
fi

￼

Consequences

ground state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”
17
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

Yamanaka-M.O.-Af eck 1997
Typical model: Hubbard model at half- llingX †Xt
cj+1, cj, + U
nj," nj,#<latexit sha1_base64="wxJt16SyvIMjxE7j1Yj81qVeQOw=">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</latexit>

H=

j,

j

Total electron lling (# per site): ν=1 (integer)→ LSM cannot be applied??
Nevertheless, ν↑=ν↓=1/2 (non-integer)→ ground-state degeneracy with gap
OR gaplessBut we don’t know the nature of gapless excitations
fi

fl

19
fi

￼

LSM for Spinful Electrons

“Spin-charge separation”
If the gapless excitations are genuinely spin excitations,the system is (electrically) insulator but spin conductor
If the gapless excitations are genuinely charge excitations,the system is (electrically) conductor but spin insulator<latexit sha1_base64="wxJt16SyvIMjxE7j1Yj81qVeQOw=">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</latexit>

H=

X †
X
t
cj+1, cj, + U
nj," nj,#j,free electron metalat half llingj

Spin gap
gapless charge excitations
Luther-Emery Liquid

Charge gap
Mott insulator
gapless spin excitations
U

0
20

fi

￼

Charge Gap and Spin Gap

￼

Haldane “Conjecture” in 1981
<latexit sha1_base64="hMai9u9L20X59U+EsYDkye/g/W0=">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</latexit>

H=J

X
j

~j · S~j+1
S

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

Non-vanishing excitation gap (“Haldane gap”)✓◆
r~i · S~j i / exp
hS⇠
Against the “common sense” at the time
21

“conjecture”

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

22

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

23
fi

￼

Why Haldane Gap?

￼

Higher Dimensions?
LSM twist in x direction<latexit sha1_base64="K2lEDdgaWGRB/U90AmUxuU/GN54=">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</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….?!
24￼

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?

25

￼

Vector Potential: U(1) Gauge Field
Global U(1) symmetry in Quantum Mechanics
enhanced to U(1) gauge symmetry(~r) ! (~r)ei✓(~r) ! (~r)eReplace derivatives by “covariant derivative”
⇣

⌘

(~r)

covariantderivative
⇣
⌘

(~r)

~ (~r)r(~r) ! (~r)e

i✓(~
r)

~ r) ! A(~~ r) + r✓(~
~ r)
A(~

~
r

~
r

~ r)
iA(~

~ r)
iA(~

i✓(~
r)

is gauge invariant26￼

Meaning of Covariant Derivative@j (~r) = lim(~r + ~ej )

(~r)

!0

(@j

iAj ) (~r) = lim(~r + ~ej )e~ r )· ~
iA(~
ej

(~r)

!0

“parallel transport”
Even when there were no vector potential initially,
we can introduce a non-zero vector potential bya gauge transformation = local change of the phase
Before comparing wavefunctions at two points, we need
the corresponding phase change (“parallel transport’’)27￼Path Integral
extra phase ✓ Z
exp i

P

~ r) · d~rA(~◆

due to the parallel transport
along the path

✓ Z~ r) · d~r
exp i
A(~

Z

P

I

@S

~ r) · d~r =
A(~

Z

S

P0

~ r) · d~rA(~~ · d~n
rotA
28

◆

✓ I
= exp i@S~ r) · d~rA(~

Stokes’ theorem◆

~ = rotA~ (“curvature” = magnetic eld) is gauge invariantB⇣⌘~ 0 = rot A~ 0 + r✓~~rotA
= rotAI@S~ r) · d~r =
A(~

Z

S

~ · d~n =
rotA

Z

S

~ · d~n =B(S)phase difference = magnetic ux through the enclosed area
Only the gauge-invariant magnetic (and electric)
eld is physical
Vector potential has a gauge ambiguity and must be
unphysical (just a mathematical trick) — right?
fl

fi

29
fi

￼

Gauge Invarianceparticles do not touch
the magnetic eld directly
no effect within classical mechBut quantum interference isstill affectedx

Aharonov-Bohm effect

Quantum system de ned on the annulus does dependon the ux, except when the Aharonov-Bohm phase is= 2⇡ ⇥ integer
fi

fi

30

fl

￼

Aharonov-Bohm EffectI have implicitly chosen the units so thate=1~=1
Covariant derivative
✓

e
exp i
~

I

@S

~ r) · d~rA(~

kinetic momentum

⇣

◆

~
i~r

~ r)
eA(~


= exp 2⇡i

⌘

(~r)

(S)
0

h
15
= 4.136 ⇥ 10
Wb
0 =
e

(twice the “unit ux quantum” commonly used insuperconductivity literature) 31
fl￼

Unit Flux Quantum￼Spectrum of the Hamiltonian
the spectrum
is identical!

Energy=0

Neverthelessgenerally depends
on Φ (AB effect)H(= 2⇡) 6= H(
32

= 0)

= 2⇡(=

0)If the Aharonov-Bohm ux is an integral multiple ofthe unit ux quantum it can be eliminated by
a topologically nontrivial (“large”) gauge transformation(~r) ! (~r)ephase is multivalued
but wavefunctionis uniquex✓(~r) = 2⇡
Lx

i✓(~
r)

For a many-body Hamiltonian on a latticeH(1= 2⇡) = Ux

Ux = exp

H(!

= 0)Ux

2⇡i X
xn~r
Lx
~
r

identical to “LSM twist”
operator!

fl

33
fl

￼

Large Gauge Transformation

Spinless Hubbard in d dimensions have
- U(1) symmetry (particle # conservation)- Translation symmetry
as exact symmetries.
However, U(1) gauge transformation together with thetranslation produces an extra phase factor
Ux

1

Tx Ux = Tx exp

2⇡i X
n~r
Lx
~
r

!

<latexit sha1_base64="Bf5U7ykPMNNvxqgzinlBXAvPOvo=">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</latexit>= Tx exp (2⇡iLy ⌫)corresponding to“mixed ’t Hooft anomaly” in eld theory
34
fi

￼

“Anomalous Symmetry”

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

35
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 Z36

￼

Crystallographic Symmetries

37

Neither the LSM “slow twist” or U(1) ux insertion
works for discrete symmetries, but
the generalization of the LSM holds!
[1D]
MPS-based “proof” Chen-Gu-Wen 2011
Field-theory argument Fuji 2014
Mathematical proof Ogata-Tachikawa-Tasaki 2020
[2D and higher]
Po-Watanabe-Jian-Zalatel 2017, Else-Thorngren 2020
Watanabe-Po-Vishwanath-Zalatel 2015
Yao-M.O. 2021
38
fl

￼

LSM for Discrete Symmetry

￼

Example: XYZ model“XYZ” spin model on the square lattice of size L1 ✕ L2
<latexit sha1_base64="hIPcZBN69rFL6qHIrXfp1+Q1v4k=">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</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
odd number of “spin 1/2” per unit cell
→ ground-state degeneracy or gapless spectrum

39

CZ| "#i =| "#i
CZ| #"i =| #"i
CZ| ##i =fl￼

CZ| ""i =| ""i<latexit sha1_base64="VRGEX6DzBdBkjEaoHlaEzwgcOoE=">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</latexit>| ##idoes not have thestandard Z2 symmetry(spin ip)￼

Anomalous Z2 Symmetry

111110 111111- 111111111111
x

CZ<latexit sha1_base64="SdkPqX4IknJorysZnwdByW1+nYY=">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</latexit>

Sym.

Commutes with Sym.
Hamiltonian!

- 000001000000- 000000000000
CZ<latexit sha1_base64="SdkPqX4IknJorysZnwdByW1+nYY=">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</latexit>

41

x

￼

Consequences
The system is “ingappable” (gapless or g.s. degeneracy)

G.S. energy per site

1/L2
42

Exact Diagonalization
with Python
up to L=21
consistent with
Conformal Field Theory(quantum critical)

- Metals are gapless and conducting, but this is rather
nontrivial from statistical mechanics point of view
- Gaplessness (≒ quantum criticality, conductivity) is-achieved either by ne-tuning or some“protection”
Lieb-Schultz-Mattis (LSM)-type theorems provide
powerful and general constraints, which often
protect quantum criticality in translation invariant& charge conserving systems
Active topic of current research with numerous
generalizations
43
fi

￼

Summary