￼

Tensor-Network Renormalization
Approach to
Topological Phase Transitions
Masaki Oshikawa (ISSP, UTokyo)

Novel Quantum States in Condensed Matter 2022
at Yukawa Institute for Theoretical Physics, Kyoto, Nov 18, 2022
1￼Atsushi Ueda

2

￼

3

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4

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“Topological Phase Transitions”
Prototype: Berezinskii-Kosterlitz-Thouless Transition
which is also closely related to “Haldane Gap”
Canonical model for the BKT transition:
2D classical XY model

5￼

BKT Transition
Classical 2D XY model
TBKT High-T (Disordered) PhaseT

Low-T (BKT) Phase

BKT
transition<latexit sha1_base64="Exl6FDVxTwAywdaDDopZmi3y5vo=">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</latexit>✓ ◆⌘1
h~si · ~sj i /r~sj = (cos ✓j , sin ✓j )h~si · ~sj i / exp

power-law decay

6

✓

r
⇠

exponential decay◆

- Classical spin model with positive Boltzmann weight
no sign problem- Just 2 dimensions- Ef cient cluster algorithms available
Easily studied with Monte Carlo, right?
7fi

￼

2D XY Modelfi￼nite-size scaling of Tcl = ln bL<latexit sha1_base64="uOFgcZK/6E/e1QrQRkVQloKR2uc=">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</latexit>

largest system:
L=65536
calculation using 256GPUs
8￼Vortex in the 2D XY model
XY spin goes back to itself by 2π-rotation
existence of defect (vortex)

vortex

attraction
9

antivortex￼

BKT Transition
Low-T phase：vortex and antivortex form a paircf.) formation of atoms by nuclei and electronsvortices are effectively absent at lengthscales larger than the pair size

✓ ◆⌘1
h~si · ~sj i /rHigh-T phase：vortices/antivortices dissociate from pairs andmove freelycf.) plasma state formed by dissociation of nuclei/electrons

h~si · ~sj i / exp
10

✓

r
⇠

◆

￼

Sine-Gordon Field Theory for BKT
1
2
L=
(@µ )
2⇡K
<latexit sha1_base64="OHi9Mpz0okYsSjU9nHuxFN1gVlg=">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</latexit>2

yK (@µ ) + yV V

✓ angle of the XY spin<latexit sha1_base64="K02RDQu7okc68qa1IoeaGQwSEVo=">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</latexit><latexit sha1_base64="XKKp+OCejRycrOfvU9XHFkJLwT4=">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</latexit>✓ ⇠ ✓ + 2⇡
<latexit sha1_base64="1ZA3OcnXiTO1hPvVLBZhhZcKxSA=">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</latexit>

dual (mutually non-local)<latexit sha1_base64="gOh2EzUWbCZ8zx+1UFIznqmDLGE=">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</latexit>⇠+⇡

scaling dimension 2/K
marginal at K=2
Single vortex creation / annihilation operator
1 2i
2i
V = cos 2 =e +e2<latexit sha1_base64="irnASgbH6OZ4pHyPmtnVV/nEr/Q=">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</latexit>

yVvortex fugacityyK

renormalization of Luttinger parameter K<latexit sha1_base64="psFGGRoDXHAmq0oDoPWVFB1VhrY=">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</latexit><latexit sha1_base64="24PKL9TEfOWvgrOSyO7DiGeC/Jc=">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</latexit>

11￼

Kosterlitz RG Flowvortex fugacity

spin-wave stiffness
Luttinger parameter K=2 (vortex operator marginal)
BKT transition: yV = yK = g<latexit sha1_base64="heVwN66Hi+W8FHQE+JVL+D1I6VA=">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</latexit>

slow decay
1
1
↓
g⇠ ⇠
l
ln
L
log-corrections
12<latexit sha1_base64="qe3LyRw6cm4ObJGx25DQBstIXZ4=">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</latexit><latexit sha1_base64="UPvc5Evv45kYPn/PxqhMc76JJbc=">AAADAHichVI7SwNBEJ6crxhfUQsFm8MQsTFsgqgIQlALGzGaJ8QY7i6beHgv7i6BeKSx9A9YWKkIiorY2tr4ByzyE4JlBBsL5y4HoiFml7ud+Xa+mf12ltck0TAJqXuYnt6+/gHvoG9oeGR0zD8+kTLUsi7QpKBKqp7hOYNKokKTpmhKNKPplJN5iab5ow17P12huiGqSsKsajQncyVFLIoCZyKU90/tF3VOsApsqWYVpBq7xi6wpYMIm/cHSIg4g203wq4RAHfEVH8D9qEAKghQBhkoKGCiLQEHBs4shIGAhlgOLMR0tERnn0INfMgtYxTFCA7RI/yX0Mu6qIK+ndNw2AJWkfDTkclCkLyRW9Ikr+SONMhXx1yWk8M+SxVXvsWlWn7sdDr+2ZUl42rC4Q8LGcGOHB6n3EWXCUVYcfSIqE9zEFup0KpROT5rxlf3gtYcuSTvqPGC1MkLqlQqH8L1Lt07/+fMP9U30W91QUdr21WxgwyKiO3Z92hhZMK5T8tdffbERxD+2/J2IxUJhZdCkd3FQHTdfQ5emIFZmMeeL0MUtiAGSUfXFdzDA3PC3DCPzFMrlPG4nEn4NZjnb223rNo=</latexit>

dg
=
dl

g2

￼

BKT Transition in S=1/2 XXZ Chain<latexit sha1_base64="y7Sh/nnmqcTNeywhSV4MIXIU+bI=">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</latexit>

H=

X

y y
x x
Sj Sj+1 + Sj Sj+1 +z z
Sj Sj+1jSingle vortex creation/annihilation operatorj~~j+1
cos 2 ⇠ ( 1) Sj · S<latexit sha1_base64="w1khD5Y4pRp7v3ha+6qExP4MN2Q=">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</latexit><latexit sha1_base64="Jgj59xnOM32fqTKQjr3TD/pbF0c=">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</latexit>

sin 2 ⇠ (

j

z
1) Sj

Forbidden in Hamiltonian by the translation symmetry!
(Haldane 1980 → “Haldane conjecture”
related to Lieb-Schultz-Mattis theorem etc.)
The leading (most relevant) perturbation is thus
double vortex creation/annihilation op. cos 4<latexit sha1_base64="yPcMg9DcYkakrcKv+6J9NnymhqA=">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</latexit>

13￼

BTK Transition in S=1/2 XXZ Chain
1
2
L=
(@µ )
2⇡K
<latexit sha1_base64="OHi9Mpz0okYsSjU9nHuxFN1gVlg=">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</latexit>

2

V = cos 4<latexit sha1_base64="NrOOg6iwyGCFsOcqwRk+fTSwOzU=">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</latexit>yK (@µ ) + yV V

double vortex
fugacity

spin-wave stiffness

Luttinger parameter K=1/2(double vortex operator marginal)
14

<latexit sha1_base64="y7Sh/nnmqcTNeywhSV4MIXIU+bI=">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</latexit>

H=

X

y y
x x
Sj Sj+1 + Sj Sj+1 +z z
Sj Sj+1j

BKT transition at Δ=1 (SU(2) symmetric point)
IR xed point at the BKT transition:
Free boson (Tomonaga-Luttinger Liquid) at K=1/2
equivalent to Level 1 SU(2) Wess-Zumino-Witten
Effective theory in the vicinity of the BKT transition<latexit sha1_base64="Uizla/YfLFmTJWSulr/cF/b24yM=">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</latexit>

WZW
L
R
L = Lk=1 + gJ · J + t

yV = g + t, yK = gt

BKT transition

t=0<latexit sha1_base64="JOcvBEJRiBfZaOWRJmUZFyHE5Tw=">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</latexit>

✓

1 L R
J+ J+
2

1 L R
L R
J J+ + Jz Jz
2

SU(2) symmetry
15

fi

￼

BKT Transition in S=1/2 XXZ Chain◆

Determination of the critical point from thenite-size spectrum [Okamoto-Nomura 1994, …]“Double vortex” BKT transition at K=1/2 can be identi ed
by SU(2) symmetry of the nite-size spectrum!
State-operator correspondence andFinite-Size Scaling in CFT[Cardy 1984, 1986]<latexit sha1_base64="IRiuewqffmCBwNZvG+DYExprLQA=">AAADP3ichVJNaxQxGH47VltXa7d6EXoJLisV6ZJdxIogFLXgoWK/dlvo1GEmzW5DM5khk126hv0D/gEPPbXgofgXvIgX/4CHXr1IqYhQQQQPvjM7UGqpTZjJ8z7v+yR5kgSxFImhdH/AuTB48dLQ8OXClasj10aLY9cbSdTWjNdZJCO9EvgJl0LxuhFG8pVYcz8MJF8ONp+k+eUO14mI1JLpxnwt9FtKNAXzDVJesTHjKTJJZjxKHhG3qX1ma24sena2V3Alb5oJsoUVd4mbtEMvJMyzSoU90kU8+9LWJrc8jDAt1yOTEFeL1oa5Q7xiiVZo1shpUM1BCfI2FxUPwIV1iIBBG0LgoMAgluBDgn0VqkAhRm4NLHIakcjyHHpQQG0bqzhW+Mhu4r+F0WrOKozTOZNMzXAViZ9GJYEy/Uz36BH9RN/RA/rnzLlsNke6ly6OQV/LY2/09c3FX+eqQhwNbByrUFE+UxNgD8/xZaAJDzI/Av3FGZM6Zf01Oq/eHC0+XCjb23SXHqLHHbpPP6JL1fnJ3s7zhe3/7Pl49acY929BI3qeu3iBCo5MGqXnaLFyKTtPm4+FtOMjqP575adBo1ap3q/U5u+Vph/nz2EYxuEWTOCdT8E0PIM5qOMu3sMXOIRvzgfnq/Pd+dEvdQZyzQ040ZzffwFr98R6</latexit>

En

2⇡
E0 =
L

BKT transition

xn +

X
m

cnnm ym L2 xm + . . .

!

Energy levels form SU(2) singlet, triplet, …
fi

fi

16
fi

￼

Level Spectroscopy￼

1D S=1/2 XXZ vs 2D Classical XY
1
2
L=
(@µ )
2⇡K
<latexit sha1_base64="OHi9Mpz0okYsSjU9nHuxFN1gVlg=">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</latexit>2

yK (@µ ) + yV V

Classical XY

S=1/2 XXZ

<latexit sha1_base64="gOh2EzUWbCZ8zx+1UFIznqmDLGE=">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</latexit>

K=2

K=1/2 (SU(2)1 WZW)⇠+⇡V ⇠ cos 2<latexit sha1_base64="lk/5TGeGyCni/n2SO2cwDT132hk=">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</latexit>

V ⇠ cos 4<latexit sha1_base64="s7K1zxgJrcr45X21kpDUZWnFabs=">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</latexit>

2 !
<latexit sha1_base64="Xnl3DTsF/uii33QML+G3eIV/E3I=">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</latexit>

double vortex op.

cos 2✓, sin 2✓, sin<latexit sha1_base64="F4tuXT9DJnghzCg1OUrYxV5IOpg=">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</latexit>

SU(2) triplet (degenerate at BKT)xyzn ⇠ cos ✓, n ⇠ sin ✓, n ⇠ sin 2<latexit sha1_base64="fGyFl9Dv3jroIshG2s8aE5FQZpY=">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</latexit>

single vortex op.

17

half-vortex op.
(eigenstate under
antiperiodic b.c.)
Nomura-Kitazawa 1998

￼

Level Spectroscopy for 2D Stat Mech
Level spectroscopy has been developed for quantum 1D,
but why not for 2D stat mech models (such as XY)??
1D quantum Hamiltonian
Transfer matrix for 2D stat mechContinuous spin: series expansion of Boltzmann weightthe series may betruncated to -15≦n≦15

Transfer matrix still “too large” to be diagonalized
we utilize Tensor Network Renormalization18(We used Loop-TNR)￼TNR Construction of Transfer Matrix

after n steps, a single tensor represents
a square block of linear size L = √2ncontract horizontal indicestransfer matrix in vertical direction19￼

Identifying Tc with Level Crossing“spin-wave”
excited states
under periodic b.c.

cos 2✓, sin 2✓<latexit sha1_base64="KLTyz3tl5t8wS/rAlaG/vCPuCo0=">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</latexit>

ground state under
antiperiodic b.c.

sin<latexit sha1_base64="jISPJ4q6XAozj1dKrryT4c3O9oU=">AAAC8XichVLLSsNAFD3Gd3206kZwI5aKqzItouKqqAs3olbbilYkiaMdmhdJWqjBH3AtiIiiggvpZ7jxB1z4CcVlBTcuvEkDoqLOkMy9Z+65d87cUSxNOC5jz21Se0dnV3dPb6Svf2AwGhsazjtmxVZ5TjU1095SZIdrwuA5V7ga37JsLuuKxgtKedHfL1S57QjT2HRrFt/V5UNDHAhVdgnaLjrC8IpWSRzvxeIsyYIx/tNIhUYc4VgzYw0UsQ8TKirQwWHAJVuDDIfmDlJgsAjbhUeYTZYI9jmOESFuhaI4RciElul/SN5OiBrk+zmdgK1SFY0+m5jjSLAnds+a7JHVWYO9/5rLC3L4Z6nRqrS43NqLnoxuvP3L0ml1UfpkESPxK0ehqf+jy8UB5gI9gvRZAeIrVVs1qkdnzY35bMKbZLfshTTesGf2QCqN6qt6t86zF3+c+bP6EvmtLthkrYQqVonBCfE9/x49itwM7tML14g/6RGkvrf8p5FPJ1MzyfT6dDyzED6HHoxhAlPU81lksIw15OgUBk5xiSvJkc6la+m2FSq1hZwRfBlS/QNZFaje</latexit>

This procedure eliminates
logarithmic corrections
to all orders in g
20

extra degeneracy
forming SU(2) triplet〜 BKT transition￼

Remaining Finite-Size EffectLevel crossing point
weakly depends on thesystem size L

L=4,8,16,32

Effect of irrelevant
perturbations<latexit sha1_base64="U5padq/cEAOlBDd6Jhrhoj9bWFE=">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</latexit>

22

T , T̄ , T T̄ , . . .
T: holomorphic part of theenergy-momentum tensor

1
T ⇠ Tc + const. 2
L<latexit sha1_base64="6ljYSleorfrsDdYry07ya+xRy/k=">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</latexit>⇤

Extrapolate to L=∞
21

Our nal estimate
Tc=0.892943(2)

22
fi

￼

Dependence on Bond Dimension D

Finite bond dimension D

nite “correlation length”⇠D ⇠ 0.3D
<latexit sha1_base64="QLmdaB6NVFEUp2jsVV0nu0WN5hs=">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</latexit><latexit sha1_base64="ER2qYPiTabOtVy1K9DJHTaXzzmU=">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</latexit>6 = ⇣q⌘12cc +1[Pollmann et al. 2008]⇠D > L low-energy nite-size spectrum almost exact!
<latexit sha1_base64="MOWIBwwj3NQG3DySSfOuCdB4jUQ=">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</latexit>⇠D < L low-energy spectrum still reasonably accurate,<latexit sha1_base64="pKfyJiu+XyDe0WPdzjt+VBVkGsY=">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</latexit>

but some error due to the nite D

fi

fi

23
fi

￼

Effect of Finite Bond-Dimension￼

Tc dependence on D
D=48 gives ξ〜54
enough for up to L=32
D=28 gives ξ〜26

∼10-4

comparable to the error
in the best existing estimates!

24

too small for L=32
BUT….

(Loop-)TNR is often used to construct the “ xed point”
tensor, which would describe the large scale behaviors,by iterating TNR many times
This approach has given accurate results for large systems,
but small errors due to the nite bond dimension remain[A. Ueda & M.O., in preparation]

In our approach, we study the spectrum of nite-size
systems with TNR. TNR is almost exact when the systemsize is less than the effective correlation length.
TNR calculation of the nite-size spectrum+ Level Spectroscopy → better accuracy
fi

fi

fi

25
fi

￼

Error in (Loop-)TNR

￼Estimates of Tc

26

You must have seen this diagram many times….
vortex fugacity

spin-wave stiffness

but have you really seen the RG ow?
27
fl

￼

Kosterlitz RG Flow￼“vacuum energy” under the twisted boundary condition
with the twist angle θ [notation clash…]

28

￼

Extrapolating Lukyanov’s Result
In our notations, the energy levels would be given by<latexit sha1_base64="s67a/M2b/iu9viQyHBbpL/0IvZ0=">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</latexit>

En<latexit sha1_base64="4F87Fz2q6uArUomRMLbOL7XKCjw=">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</latexit>

e

±i2✓

<latexit sha1_base64="TfL4knBdbtBHBPiu2zTqGqGE4e4=">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</latexit>

e

2⇡
E0 =
xn
L

±i

They form two doublets, and no triplet is formed
even on the BKT transition line yK = yV ?!
<latexit sha1_base64="2YapQ5TF+dr3tD/Rq3ksRyDJwYw=">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</latexit>

The two states corresponding to e±i are mixed by thevortex perturbation cos 2 and split into two levelssccorresponding to V1/2
= sin , V1/2 = cos<latexit sha1_base64="TfL4knBdbtBHBPiu2zTqGqGE4e4=">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</latexit><latexit sha1_base64="uahqJwRS7IDApveaHADV1GTEIWw=">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</latexit><latexit sha1_base64="BCp2HOm3cwAtjIIBcIUXa9jWQRM=">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</latexit>

29

￼

Energy Levels up to 2nd Order-s , xV cysplit between xV1/2
should
be
odd
in
V
1/2
SU(2) triplet should be formed on
the BKT transition line yK = yV
uniquely determines the energy levels up to O(y2)<latexit sha1_base64="1WXBU//LQ1XA/EP5H/NdC8epojk=">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</latexit><latexit sha1_base64="pod2BKbvE8F6QSPHVnbdl/aTnK4=">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</latexit><latexit sha1_base64="2YapQ5TF+dr3tD/Rq3ksRyDJwYw=">AAADRXichVLNatRQFP4mWm1jtVOlUHAzOExxNdypoiIIRV0UROzfTAttCUl624bmj+TO4HiZF/AFXLiq4EL7GC70BVz0EURctdhNQb9kAmUsHW9I7jnfOd/5y3Fi30uVEIcl49LlkStXR8fMa+PXb0yUJ2+20qiduLLpRn6UrDl2Kn0vlE3lKV+uxYm0A8eXq87es8y+2pFJ6kXhiurGcjOwd0Jv23NtRcgqT3UtvRHYate1ff2i16s8qXStllWuirrIT+W80CiEKoqzEE2WxrGBLURw0UYAiRCKsg8bKZ91NCAQE9uEJpZQ8nK7RA8muW16SXrYRPf43aG2XqAh9SxmmrNdZvH5JmRWUBPfxSdxJL6JA/FDnDKWidqF8bbQYd6YtsGYrweimhdWpHNW1lGXt9OvQMbWxNvp5ZP/sgLeCrtnrCG1akZ3yBg+HYVtPMo78NhRnCPZvNx+js6bd0fLj5dqekZ8ED85qX1xKL5wVmHn2P24KJfeD6n5LPtz6v1/mVB6WXTxigxJJNOyuWl6ruTz08WdzdHkLjX+3ZzzQmu23nhQv7d4vzr3tNiqUdzGHdzl5jzEHOaxgCar0NjHZxwYX41fxrHxu+9qlArOLQwc489fDu66+A==</latexit>30

We can estimate yK & yV from the nite-size energy levels
Less accuracy than Tc, but we can apply to larger systems(up to L=512)
31
fi

￼

Obtaining Running Coupling Constants￼

Visualization of Kosterlitz RG Flow!
L=16
32
64
128
256
512

32

TNR + Level Spectroscopy ( nite size scaling of CFT)
allows- super accurate determination of BKT critical point- visualization of Kosterlitz RG ow by extraction ofrunning coupling constants from the spectrum
for continuous valued 2D classical spin system such as XY
model
Future: extension/application to more nontrivial
systems & unknown physics

fl

33
fi

￼

Conclusions (BKT)￼Classical Heisenberg Modelcontinuum limit1L=(@µ~n)22g<latexit sha1_base64="ZMPGnBIe8myaBlvZIfr7j9TsWd4=">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</latexit><latexit sha1_base64="4t5PI4Es32gRtronPGN7LEwN0ek=">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</latexit>

2~n = 1

O(3) Nonlinear Sigma Model

coupling g corresponds to temperature
Asymptotic freedom disordered at any T>0<latexit sha1_base64="fAlyIUxBWXqjYczjWY4JEaKKikA=">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</latexit>

dg
g2
=
+ ...
d log L
2⇡

Supported also by factorizable S-matrix (Zamolodchikov2)
BUT…
34

But not easy toprove/disprove
dif cult to distinguishmassive withvery large
correlation length
frommassless…
35fi￼Really??

￼

Z2 vortex-driven transition?
Nematic liquid crystal:symmetric rod-like molecules(no distinction between head & tail)Lebwohl-Lascher 1972
Kawamura-Miyashita 1984

Target space = RP2
2⇡1 (RP ) = Z2
<latexit sha1_base64="FRBb4RzrB8Oka0pYcg1qLVrPWBI=">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</latexit>

Z2 vortex!

BKT-like transition driven by the Z2 vortices?
But the RG equation also implies asymptotic freedom..
36

T→0
Spontaneous Symmetry
Breaking of SO(3)2 independent
Nambu-Goldstone modes
c=2
Con rmed by TNR

37
fi

￼

UV Fixed Pointc〜1.94 < 2 no candidate CFT
does notconverge

likely to support“asymptotic freedom”
RG ow c=2 → c=0
38
fl

￼

Heisenberg Model at Intermediate T￼2
RP Model at Intermediate T

c〜1.92 < 2 no candidate CFT

Z2 vortex dissociationtransition(?)

39

Spectrum does not converge
well even at T≦T*
c〜1.92 < 2
SU(2) level 4
c=2 but smallest scaling dim2
xj=1 = = 0.666 . . .
3<latexit sha1_base64="rgTlfI9esxKw5CBV2Nbrtoe56ZA=">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</latexit>

0.2

likely to support “asymptotic freedom”
RG ow c=2 → c=0 , but with a strong crossover at T*
40
fl

￼

RP2 Model at T≦T*

Both Heisenberg & RP2 Models
show c=2 at the UV xed point for T→0consistently with 2 (would-be) Nambu-Goldstone modes
At intermediate temperatures, the effective central charge
exhibit a “plateau” at c〜1.9
but there is no appropriate candidate CFT
and scaling dimensions do not converge
Our results support the “asymptotic freedom”
scenario with the single disordered phase in T>0
41
fi

￼

2
Conclusions (Heisenberg & RP )

Dif culty in distinguishing very large but nite correlation
length from criticality
Even when (system size) < (correlation length)
the Hamiltonian/transfer matrix spectrum
gives useful information!

fi

42
fi

￼

2
Discussion (Heisenberg & RP )

TNR + Level Spectroscopy ( nite size scaling of CFT)
allows- super accurate determination of BKT critical point- visualization of Kosterlitz RG ow by extraction ofrunning coupling constants from the spectrum- identify the UV xed point for
the Heisenberg & RP2 models- clarify(?) the crossover inthe Heisenberg & RP2 models
Future: extension/application to more nontrivial
systems & unknown physics
fl

fi

43
fi

￼

Conclusions (Overall)

￼

44

￼

application to
more
controversial
problems45￼46