￼

Resolving Berezinskii-Kosterlitz-ThoulessTransition in 2D XY model with TNR+CFT

Masaki Oshikawa
ISSP, UTokyo1￼just appeared in arXiv

Atsushi Ueda
2

￼

Prototype: Berezinskii-Kosterlitz-Thouless Transition
Canonical model: 2D classical XY model“easy” to simulate byMonte Carlo?
3

fi

￼

nite-size scaling of Tc

4

fi

￼

nite-size scaling of Tcl = ln bL<latexit sha1_base64="uOFgcZK/6E/e1QrQRkVQloKR2uc=">AAAC83ichVLLSsNAFD3Gd31V3QhugqXiqkyLqAiCqAsXiq9WC1YkiVMNTh7kUdDgD/gBunDhA12I+BVu/AEXfoK4rODGhTdpQLSoMyRz75l77p0zd1Rb6K7H2HOD1NjU3NLa1p7o6Ozq7kn29q27lu9ovKBZwnKKquJyoZu84Ome4EXb4YqhCr6h7s+G+xsV7ri6Zea9A5tvGcquqZd1TfEIKgl5Si4JM1DlhaPtZIplWDTkeiMbGynEY9lKvqCEHVjQ4MMAhwmPbAEFLs1NZMFgE7aFgDCHLD3a5zhCgrg+RXGKUAjdp/8ueZsxapIf5nQjtkZVBH0OMWWk2RO7ZVX2yO7YC/v4NVcQ5QjPckCrWuNye7vneGDt/V+WQauHvS8WMdK/clSaxj+6PJQxEenRSZ8dIaFSrVajcnhaXZtcTQfD7Iq9ksZL9sweSKVZedNuVvjq2R9n/qo+R36tCw5Zi7GKJWJwQkIvvMeAIvPRfQbxmggnPYLsz5bXG+u5THYsk1sZTU3PxM+hDYMYwgj1fBzTmMcyCnQKGyc4x4XkS2fSlXRdC5UaYk4/vg3p/hPNzqic</latexit>

4

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
4

￼

Kosterlitz RG Flowvortex fugacity

spin-wave stiffness

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<latexit sha1_base64="qe3LyRw6cm4ObJGx25DQBstIXZ4=">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</latexit><latexit sha1_base64="UPvc5Evv45kYPn/PxqhMc76JJbc=">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</latexit>

dg
=
dl

g2

5

￼

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+1j

BKT transition at Δ=1 (SU(2) symmetric point)
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) symmetry6◆Determination of the critical point from thenite-size spectrum [Okamoto-Nomura 1994]
BKT transition can be identi ed by SU(2) symmetryof the nite-size spectrum!!
State-operator correspondence in CFT<latexit sha1_base64="IRiuewqffmCBwNZvG+DYExprLQA=">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</latexit>

En

2⇡
E0 =
L

X

xn +

m

BKT transition

cnnm ym L2 xm + . . .

!

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

fi

7
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

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 4<latexit sha1_base64="ufJtawlijhwYi4YoL59SOFHNib8=">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</latexit>

single vortex op.

8

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

￼

Level Spectroscopy for 2D Stat Mech
Level spectroscopy has been developed for quantum 1D
1D quantum Hamiltonian
Transfer matrix for 2D stat mechContinuous spin: series expansion of Boltzmann weightTransfer matrix still “too large” to be diagonalized
we utilize Tensor Network Renormalization
9￼

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 direction10￼

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

This procedure eliminates
logarithmic corrections
to all orders in g

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

extra degeneracy
forming SU(2) triplet〜 BKT transition11￼

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=∞
12

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 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 Dcf.) “ xed point tensor” from TNR
fi

fi

13
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
too small for L=32
BUT….

14

￼

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!

14

too small for L=32
BUT….

￼

Estimates of Tc

15

Energy levels vs marginal perturbations E
n<latexit sha1_base64="s67a/M2b/iu9viQyHBbpL/0IvZ0=">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</latexit>

2⇡
E0 =
xn
L

[Lukyanov 1998]

estimate yK & yV from the nite-size energy levels
Less accuracy than Tc, but then can apply to larger systems(up to L=512)
16
fi

￼

Extraction of Couplings￼

Visualization of Kosterlitz RG Flow!

17

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(stay tuned!)
fl

18
fi

￼

Conclusions