￼

Harvard CMSA
Quantum Matter Seminar
Aug 19 (Thu) 8:30pm- (EDT)

Conformal Field Theory andModern Numerical Approachto Condensed Matter Physics

Masaki Oshikawa
ISSP, University of Tokyo1

￼

Conformal Field Theory
Scale Invariance at the Critical Point
Locality of the Hamiltonian/Action
Invariance under conformal transformation(locally different scale transformation)
1+1 Dimension:conformal transformation complex analytic function∞-dimensional “symmetry” (Virasoro Algebra)
What can we deduce from the ∞-dimensional symmetry?
“Conformal Field Theory (CFT)”2

Also known as “Tomonaga-Luttinger Liquid”
in condensed matter physics!
1
2
L=
(@µ )
2⇡K

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⇠

K
2
L=
(@µ ✓)
2⇡<latexit sha1_base64="NpBDUM+LKyFRciPbuiYxvTdx4n4=">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</latexit><latexit sha1_base64="Tm8eQ9ij24yBam6JQKqqAB46tUA=">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</latexit>“T duality”+⇡✓ ⇠ ✓ + 2⇡
<latexit sha1_base64="AhXeiclsgWuoiaxzWc/koznFl4s=">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</latexit>

K: “Luttinger parameter”p˜⌘ pcorresponds to the✓˜ ⌘ K✓
K“compacti cation radius”p⇡✓˜ ⇠ 2⇡ K
in hep-th literature˜⇠ ˜+ p<latexit sha1_base64="bf4/DKknj1sYjVPXbp96mr5CBRY=">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</latexit><latexit sha1_base64="LrYTNrxyxafISqNqkvx4SGQzT1Y=">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</latexit><latexit sha1_base64="mgLJdUfTdNssOjd2HHZtGGTpp5M=">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</latexit>

K

fi

￼

Free Boson CFT

3￼This Talk
I will discuss some of the applications of the CFT
to condensed matter/statistical physics (in 1+1 Dim)
focusing on
free boson CFT (aka Tomonaga-Luttinger Liquid)
Most of the “theory” part is rather old
But we are seeing interesting developments,
thanks to recent advances in numerical algorithms

4

￼

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 /rpower-law decay~sj = (cos ✓j , sin ✓j )h~si · ~sj i / exp
5

✓

r
⇠

exponential decay◆￼Prototype: Berezinskii-Kosterlitz-Thouless Transition
Canonical model: 2D classical XY model6

￼

Vortex in the 2D XY model
XY spin goes back to itself by 2π-rotation
existence of defect (vortex)

vortex

attraction
7

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
8

✓

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>

dual (mutually non-local)
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=">AAADMXichVI9T9tQFD1xSwFDC7QLUlGFilIxRS9FgqoTKh26VCWEBCRAkW0eqYVjW7YTESxGFqSuZWACiaFi7MLepX+gQzbEhjJSiYWB4w8JBUR4lv3uPfee++Wru5bpB0K0MsqTpz3Pevv61YHB5y+Ghkdeln2n7hmyZDiW4y3rmi8t05alwAwsuex6UqvpllzSN+ci+1JDer7p2ItB05VrNa1qmxumoQWEis1KuTI8IXIiPuP3hXwqTMyOnRbau29O552RzCBWsQ4HBuqoQcJGQNmCBp/PCvIQcImtISTmUTJju8QOVHLr9JL00Ihu8lultpKiNvUoph+zDWax+HpkjiMr/olf4lL8FSfiQlwzlorsg/HW0WBel7bOmFsdUdUHKwpjVtRRk7eeVCDdytDeaPHqUVaNd4Dvt6wutYaMrpPRfToBNvAh7sBkR26MRPMykhyN7f3L4seFbPhOHIk2J3UoWuIPZ2U3/hvHBblw0KXm2+yfqSf/0qP0Ne3iGxmSSKRFcwvpuRjPL0zvaI4qdyl/d3PuC+X3ufx0bqrApfqE5PThNd5ikpszg1l8wTxKrKKKH/iJfeW30lLOlPPEVcmknFfoOEr7BhthtvQ=</latexit><latexit sha1_base64="24PKL9TEfOWvgrOSyO7DiGeC/Jc=">AAADP3ichVJLS9xQFP4mfaip7Yy6ESxFKlO6Gu60YIsrUReFInXUUUFlSOJVg3mR3Bk6hll20z/gwlWLLqR/wG3ppn+gC39BKUIpWCiFLvrlATIVpzck95zvnO+8cszAsSMlxFlBu3Hz1u2+/gH9zuDde8XS0PBK5DdDS9Yt3/HDNdOIpGN7sq5s5ci1IJSGazpy1dybTeyrLRlGtu8tq3YgN11jx7O3bctQhBqlYrsRb7iG2rUMJ37Z6TRKE6Ii0jN+VajmwsT0/dPa+ZsHpwv+UGEQG9iCDwtNuJDwoCg7MBDxWUcVAgGxTcTEQkp2apfoQCe3SS9JD4PoHr871NZz1KOexIxStsUsDt+QzHGUxRdxIi7EZ/FBfBN/GEtH+dp4W2gxb0Bbd8zXXVH1ayuKU1bSUZu3mVUgg0bx7ejSr/+yXN4Ku5esHrXGjG6S0Xs6Ctt4nnZgs6MgRZJ5WVmO1v7BxdLUYjl+JN6Lc07qnTgTnzgrr/XTOq7JxcMeNV9mn6Oe/cuQ0nzexSsyJJFES+YW03M5nV+c38kcde5S9d/NuSqsPKlUJytPa1yqGWSnH2N4iMfcnGeYxgssoJ5u1yGOcKx91L5q37UfmatWyDkj6Dra77/zJry4</latexit>

9￼

Kosterlitz RG Flowvortex fugacity

spin-wave stiffness
Luttinger parameter K=2 (vortex operator marginal)
10Free boson CFT is exactly solvable (Gaussian free eld)
Actually, the BKT transition was predicted andthe Kosterlitz RG ow was derived without
CFT techniques
Nevertheless, numerical study of the BKT transitionis rather challenging (even though it is “well-understood”)
and CFT ideas turn out to be practically very useful!

fi

11
fl

￼

Why CFT?

- Classical spin model with positive Boltzmann weight
no sign problem- Just 2 dimensions- Ef cient cluster algorithms available
Easily studied with Monte Carlo, right?
12
fi

￼

2D XY Modelfi￼nite-size scaling of Tc

13

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>

13

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
13

￼

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=">AAADAHichVJLS8NAEB7juz4a9aDgJVgqnspGREUQRD0IIj76UFAJSVzb0LxItoUaevHoH/DgSUVQVMSrVy/+AQ/9CcVjBS8enDxAVNRdkp35dr6Z/XZWsXXNZYTUmrjmlta29o7OWFd3T2+c7+vPuVbJUWlWtXTL2VJkl+qaSbNMYzrdsh0qG4pON5Xigr+/WaaOq1lmhlVsumvIeVPb11SZISTxgxUpJ8wKFWnHkFlBlXVvuYp+XuITJEWCIfw0xMhIQDTWLL4OO7AHFqhQAgMomMDQ1kEGF+c2iEDARmwXPMQctLRgn0IVYsgtYRTFCBnRIv7z6G1HqIm+n9MN2CpW0fFzkClAkjyTK9IgT+Sa1Mn7r7m8IId/lgquSsilthQ/Gkq//csycGVQ+GQhI/krR8Fp/KOLwT5MB3o01GcHiK9UDWuUD44b6ZmNpDdKzsgLajwlNfKIKs3yq3qxTjdO/jjzZ/VF9MMuOGitRCpWkUER8T3/Hj2MzAT36UVrzJ/4CMTvLf9p5MZT4mRqfH0iMTcfPYcOGIYRGMOeT8EcLMEaZANd53ADt9whd8ndcfdhKNcUcQbgy+AePgBPuK02</latexit>

1
1
g⇠ ⇠
l
ln L
<latexit sha1_base64="qe3LyRw6cm4ObJGx25DQBstIXZ4=">AAADEXichVK/SyNBFP7cU0/jr9zZCDbBELEKs0G846pwZ2Gh+DMqGAm76yQOzv5gdxPILfsPWNhYWFgpWMi111goos3B1Rb+CXKlBzYWvt0siIo6w+6898373ptv3uiOFJ7P2HWb8qG9o/NjV3eqp7evfyD96fOyZ9ddg5cMW9ruqq55XAqLl3zhS77quFwzdclX9K0f0f5Kg7uesK0lv+nwdVOrWaIqDM0nqJIeq2XKnjAz5aqrGYEaBjJ8BpSlFUyHYSWdZXkWj8xLQ02MLJIxZ6dvUMYGbBiowwSHBZ9sCQ0ezTWoYHAIW0dAmEuWiPc5QqSIW6coThEaoVv0r5G3lqAW+VFOL2YbVEXS5xIzgxy7Ysfslv1hv9gNu381VxDniM7SpFVvcblTGdgeWrx7l2XS6mPzkUWM3Kscnab5ji4fVXyN9QjS58RIpNRo1Wj83Ltd/LaQC0bZIftHGg/YNbsklVbjv3E0zxf23zjzY/VJ8ltdcMmaSVTMEoMTEnnRPQYUuRTfZ5CsqWjSI1Cft/ylsVzIqxP5wvx4tvg9eQ5dGMYIxqjnX1DEFOZQolPs4DdOcabsKifKuXLRClXaEs4gngzl7wNGdrVJ</latexit><latexit sha1_base64="UPvc5Evv45kYPn/PxqhMc76JJbc=">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</latexit>

dg
=
dl

g2

14

slow decay↓log-corrections

We already know the exact exponents at the BKTtransition, thanks to RG (or CFT)
But for “practical” purposes, determination of the criticalpoint (which is non-universal) can be important
The “brute-force” calculation already gives Tc for 3 digits,
which might be enough in practice
However, in order to elucidate more complicated systems,
it is highly desirable to develop a more ef cient method(even the universality class can be controversial!)
15
fi

￼

Should We Care?

Interestingly, the dif culty of the logarithmic corrections
in numerics was “solved” rst for 1+1D quantum systems
In condensed matter physics, a quantum system
are usually de ned in terms of a Hamiltonian
In CFT, we can formulate the spectrum of the Hamiltonianby a conformal mapping between
a cylinder and in nite plane

fi

fi

fi

16
fi

￼

CFT for 1+1D Quantum Systems

The conformal mapping also implies a one-to-one
correspondence between quantum states and operators
Energy eigenvalue of a nite system (under periodic b.c.)<latexit sha1_base64="LvtCQ8wuibRxhAUGoyzTvItOBm4=">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</latexit>

En

2⇡
E0 =
xn
L

[Cardy 1984]

L: system size (length)
xn: scaling dimension of the (primary) operator
Scaling dimensions (critical exponents)
can be extracted from the spectrum!
17
fi

￼

State-Operator Correspondence<latexit sha1_base64="MdeI4zyETkBgEKHo1mOwSNKcXas=">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</latexit>

H = HCFT +<latexit sha1_base64="IRiuewqffmCBwNZvG+DYExprLQA=">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</latexit>

En

X

yn

n

2⇡
E0 =
L

Z

xn +

[Cardy 1986]n (r) drX

cnnm ym L

2 xm

+ ...

m

clmn : Operator Product Expansion coef cient<latexit sha1_base64="d5dSbwwV/gbrUQGrv4btxOIN8/w=">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</latexit>

l

m ⇠

X

clmn

n

n

The coef cients of the perturbations can be also
extracted from the nite-size spectrum!
fi

fi

18
fi

￼

Spectrum of Perturbed CFT!

￼

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>

19￼

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=">AAADP3ichVLPSxRhGH52KrVJ3bUuQSFLstFp+VajIgikPHQRXXVXwZVlZvzUD2dnhpnZJRs8dukf6ODJyIP4D3gVL/4DHfwLIoQIDCLo0DM/QFZx+4aZ732f933eX/Oanq2CUIjTnHbj5q2+/oHb+p3BoeF8YeRuPXDbviVrlmu7/pJpBNJWjqyFKrTlkudLo2XactHcfBPbFzvSD5TrLIRbnlxpGeuOWlOWERJqFvL1V8WG5QbR02LD21DbzcKYKIvkFK8KlUwYm3x4WD37MHo4647kBtHAKlxYaKMFCQchZRsGAj7LqEDAI7aCiJhPSSV2iW3o5LbpJelhEN3kd53acoY61OOYQcK2mMXm65NZREl8FfviXJyIA/Fd/GUsHaVr462iw7webd0x33VF1a+tKEpYcUdbvM20Auk18x/vz//+L6vFO8TGBatHrRGjm2T0nk6INbxIOlDsyEuQeF5WmqPz/tP5/Mu5UvRYfBZnnNSuOBXHnJXT+WXtVeXcTo+aL7JPUU//pU9pOutihgxJJNbiuUX0XEjmF2V3PEedu1S5vDlXhfp4ufKsPFHlUr1GegbwAI/whJvzHJN4i1nUku3awRfsaUfaN+2H9jN11XIZ5x66jvbnH/yKu6w=</latexit>yK (@µ ) + yV V

double vortex
fugacity

spin-wave stiffness

Luttinger parameter K=1/2(double vortex operator marginal)20

<latexit sha1_base64="y7Sh/nnmqcTNeywhSV4MIXIU+bI=">AAADSHichVJNaxNBGH679aON1aZ6KXhZDCmVQpgtYkUQivbQi9g2TVvo1mV3nSTTzn6wOwkmS/6Af8CDJwUP4s8QQbx76E8ovQitFMGDz04Wi5baGXbneZ/3fWbmmRkvliJVjO2PGKOXLl+5OjZeujZx/cZkeermRhp1Ep83/EhGyZbnplyKkDeUUJJvxQl3A0/yTW/vSZ7f7PIkFVG4rnox3wncViiawncVKKfs2IGr2r4rs+WB+cgs2WkncHZNW/KmmjXrz18iyP/Z7pw1MOeAe5rp/WHsJS6VC6qvE/0iYSei1VZ3nXKF1Zhu5llgFaBCRVuJygdk0wuKyKcOBcQpJAUsyaUUfZssYhSD26EMXAIkdJ7TgErQdlDFUeGC3cO/hWi7YEPE+ZypVvtYReJLoDSpyr6xD+yIfWEf2QH7de5cmZ4j30sPozfU8tiZfDVdP7lQFWBU1D5VQVE9V+OhBxf4UtSkB9qPgL9YM7lTf7hGt//6qP5wrZrNsHfsEB7fsn32CS7D7rH/fpWvvfnPnk9XX0I8vIUE6Gnh4hkUHEwe5eeYoXJdn2dWjKW84xFY/175WbAxX7Pu1+ZX71UWHxfPYYxu0x2axZ0v0CIt0wo1sIvPdEjH9MP4anw3Toyfw1JjpNDcor/aqPEbQq3HGg==</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
21

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

22
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>

S=1/2 XXZ

2yK (@µ ) + yV V

S=1 XXZClassical XY

Nomura-Kitazawa 1998

K=2

K=1/2 (SU(2)1 WZW)V ⇠ cos 2<latexit sha1_base64="lk/5TGeGyCni/n2SO2cwDT132hk=">AAAC+3ichVLLSsNAFD3Gd31V3QhuiqXiqkyLqLgSdeFGfLVVsFKSONbBvEjSoob+gD/gQhBUxOfWL3DjD7joJ4hLBTcuvEkDolI7QzL3nrnn3jlzR7E04biMVZuk5pbWtvaOzkhXd09vX7R/IOeYJVvlWdXUTHtDkR2uCYNnXeFqfMOyuawrGl9X9ub8/fUytx1hGhn3wOJbulw0xI5QZZegQnQgF8s7Qo/lVdPx0nlrV1QK0ThLsmDE/hqp0IgjHMtm9AV5bMOEihJ0cBhwydYgw6G5iRQYLMK24BFmkyWCfY4KIsQtURSnCJnQPfoXydsMUYN8P6cTsFWqotFnEzOGBHtmN+yNPbE79sI+6+byghz+WQ5oVWpcbhX6jobWPhqydFpd7H6ziJGoy1Fo6g10udjBVKBHkD4rQHylaq1G+fD4bW16NeGNsnP2ShrPWJU9kkqj/K5ervDVk3/O/F19nvxaF2yyFkMVS8TghPief48eRWaC+/TCNeJPegSp3y3/a+TSydREMr0yHp+ZDZ9DB4YxgjHq+SRmsIBlZOkU+zjFFa6linQh3Ur3tVCpKeQM4seQHr4AUDqrxw==</latexit>

V ⇠ cos 4<latexit sha1_base64="Xnl3DTsF/uii33QML+G3eIV/E3I=">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</latexit>

double vortex op.

hidden SU(2) triplet

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.

2 !

<latexit sha1_base64="s7K1zxgJrcr45X21kpDUZWnFabs=">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</latexit>23

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

half-vortex op.
(eigenstate under
antiperiodic b.c.)￼24

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:“in nite dimensional local Hilbert space”
can be discretized, but still the dimension is toolarge for exact diagonalization
25
fi

￼

Level Spectroscopy for 2D Stat Mech￼

Atsushi Ueda
26

￼

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 Renormalization
27(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 direction28

￼

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=">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</latexit>

This procedure eliminates
logarithmic corrections
to all orders in g
29

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=∞
30

Our nal estimate
Tc=0.892943(2)

31
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

32
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….

33

￼

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!

33

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

34
fi

￼

Error in (Loop-)TNR

￼Estimates of Tc

35

￼

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

spin-wave stiffness36You must have seen this diagram many times….
vortex fugacity

spin-wave stiffness

but have you really seen the RG ow?
36
fl

￼

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

37

￼

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>

38￼

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=">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</latexit>39

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)
40
fi

￼

Obtaining Running Coupling Constants￼

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

41

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

42
fi

￼

Conclusions (Part 1)

Critical system in 1+1D with a boundary:invariance under conformal mapping which
preserves the boundary: “real analytic function”
Still in nite dimensional symmetry(Virasoro algebra)
For a given CFT, there are several conformally invariant
boundary conditions, which correspond toxed points of “boundary RG ow”

fl

fi

43
fi

￼

Boundary CFT

(imaginary)“time”|Bi<latexit sha1_base64="ahcxMqu2/0OIt2oICU5yuS7rUL4=">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</latexit>

boundary condition

boundary state<latexit sha1_base64="Qri2nODjHH4pd1HyJ0Oqhwv/Kx8=">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</latexit>

8n : (Ln<latexit sha1_base64="kPVN+WaBpHi22AVR2qwG9bKfTHE=">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</latexit>

|Bi =
fi

￼

Boundary StateL̄n )|Bi = 0

Ishibashi state for|hii
the primary eld h<latexit sha1_base64="dim/P/smjui4WIzqPqPTwrRBlYI=">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</latexit>XhhcB |hii
44

￼

BCFT Partition Function<latexit sha1_base64="sJuecTQabxXFBl5gfsGXSsxbM7c=">AAADM3ichVI9T9tQFD1xW6Au5XNBYokapeoUvVSoRUyIMrCgQkIACRCyzYM+4diW/RJBrf4BFkZAnYrEgDr3F3TpH+gQdau6INQJJJYOPXYsoRSRPst+9557z/3ytQNXRVqIVs548PBRT2/fY/NJ/9OBwaHhkeXIb4SOrDm+64erthVJV3myppV25WoQSqtuu3LF3n2T2FeaMoyU7y3p/UBu1K0dT20rx9KEauu21NbmUEGURHryd4VyJhSQnQV/ONePdWzBh4MG6pDwoCm7sBDxWUMZAgGxDcTEQkoqtUt8gElug16SHhbRXX53qK1lqEc9iRmlbIdZXL4hmXkUxXdxLq7EN/FZXIg/jGWieG+8LTSZN6CtM+ZeR1Tz3orilJV0tM/bblcgg83Bg7HqzX9Zdd4a725ZXWqNGd0mo/t0NLYxmXag2FGQIsm8nHaO5vujq+pUpRg/F6fikpP6JFriK2flNa+ds0VZ+dil5tvss9Tb/zKkNJ918ZYMSSTRkrnF9FxK5xdndzJHk7tU/ndz7grLL0vlV6WJxYnC9Ey2VX0YxzO84Oa8xjTmsIAaq1A4xDFOjC/GD+On8avtauQyzig6jvH7LwI6tCw=</latexit>

open string
channelclosed string channel

A

B
L<latexit sha1_base64="/R61AG8i/3h7bo6ieGSfop9YWj4=">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</latexit><latexit sha1_base64="m7YCVY3tghYCgAnBdTBqEoarXYw=">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</latexit>

ZAB = hB|e LH |Ai =<latexit sha1_base64="BYjlHjaExUE2F9OZGLrNFcbQF6k=">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</latexit>

ZAB =

X
j

X

(ciA )⇤ ciB<latexit sha1_base64="ZZEFmrbNDBKvXR2EqSRALf2WqTk=">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</latexit>i (q̃)iq̃ ⌘ e 4⇡L/modular transformation<latexit sha1_base64="iC5oUcawQK4yC66FMXSO5dSH5jo=">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</latexit>

j
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j (q)
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ZAB = hB|e<latexit sha1_base64="BYjlHjaExUE2F9OZGLrNFcbQF6k=">AAADUnichVJLa9RQFP5m4qPG0U51I7gJHUbqZrgjRUUQanXhRuxr2mKnhiS9be80L5PMYA3di3/AhSsFFyK41b0bf4Au+hPEZQVBXPjlAWUsHW9I7jnfOd955dihq+JEiP1KVTtx8tTpsTP62dq58+P1iQvLcdCPHNlxAjeIVm0rlq7yZSdRiStXw0hanu3KFXvnbmZfGcgoVoG/lOyGct2ztny1qRwrIWTWJx+Z6Z3ZPeO20Y37ntkz/Me9Auk628rsTT25atYboiXyYxwV2qXQQHnmgolKDV1sIICDPjxI+Egou7AQ81lDGwIhsXWkxCJKKrdL7EEnt08vSQ+L6A6/W9TWStSnnsWMc7bDLC7fiEwDTfFNvBMH4ot4L76LP4ylo3lsvA0MmDekbTjm06Go+rEVpTkr62iXt11UIENz/MWlxV//ZXm8E2wfskbUmjK6Tcbo6STYxM28A8WOwhzJ5uUUOQbPXh4s3lpoplfEG/GDk3ot9sVnzsof/HTezsuFVyNqPsx+j3rxLyNKD8ouHpIhiWRaNreUnkv5/NLyzuaoc5fa/27OUWH5Wqt9vTU9P92YmS23agyXMYkpbs4NzOA+5tBhFc/xAR/xqfq1+luraFrhWq2UnIsYOlrtL0//vCA=</latexit>

ZAB =

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i ⇤ i
(cA ) cBi (q̃)q̃ ⌘ e4⇡L/modular transformation<latexit sha1_base64="iC5oUcawQK4yC66FMXSO5dSH5jo=">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</latexit>jnABq ⌘ e ⇡ /L

j (q)

j
nAB 2 Z 0<latexit sha1_base64="kaohjCO73PBKZYN8JPLuX3u0WEI=">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</latexit><latexit sha1_base64="ZZEFmrbNDBKvXR2EqSRALf2WqTk=">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</latexit>ij# of primary eld jwith the boundary conditions
A&B
46

fi

￼

Cardy Condition

￼

47

Rational CFT:
CFT with a nite number of primary elds
(representation of Virasoro algebra)<latexit sha1_base64="5s273Jy/pc4tPxwkY02RmwLBsPM=">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</latexit>

1= finite1

reduced to a nite-dimensional problem

Irrational CFT:
CFT with an in nite number of primary elds<latexit sha1_base64="gI/diFj9+jNsoQextCxhhLfFUjE=">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</latexit>

1=1
1

more dif cult as a CFTClassi cation of conformal b.c. : generally unknown
fi

fi

fi

fi

fi

fi

48
fi

￼

Rational or Irrational?

1
2
L=
(@µ )
2⇡K
<latexit sha1_base64="7tlJNZaCxJR7wqSaqo5xHI+Mfw4=">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</latexit>

⇠

K
2
L=
(@µ ✓)
2⇡<latexit sha1_base64="NpBDUM+LKyFRciPbuiYxvTdx4n4=">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</latexit><latexit sha1_base64="Tm8eQ9ij24yBam6JQKqqAB46tUA=">AAADZHichVJPaxNBFH/J+qeu0aYWQRBkMUTqJcyGYIsgFPUgqNh/aQvdGmank2bo7O4wOwnWIV/ALyDoScGD+CUEL+rdQ7+AIB4rePHg281CiaVxlt157/fe7/3bFyopUkPIQansnDp95uzUOfd85cLF6erMpfU06WvG2yyRid4MacqliHnbCCP5ptKcRqHkG+Hevcy+MeA6FUm8ZvYV347obiy6glGDUKfaCiJqeoxK+2jo3fGCrqbM+kPbDJTwHg69uUBRbQSVnSDqe4HqiZtPm51qjTRIfrzjgl8INSjOUjJTqkAAO5AAgz5EwCEGg7IECik+W+ADAYXYNljENEoit3MYgovcPnpx9KCI7uF3F7WtAo1Rz2KmOZthFomvRqYHdfKNvCeH5DP5QH6QPxjLhfqJ8XZggHkV2sZjPhuL6p5Ykc1ZWUf7eIejCrjqTL+4svr7v6wIbwO9I9aEWi1GD5ExeToGurCQdyCwI5Uj2bzYKMfg+cvD1dsrdXuDvCU/cVJvyAH5hLOKB7/Yu2W+8npCzUfZ76M++pcapcdFF0+QwRHJtGxuFj3X8vnZ4s7m6OIu+f9uznFhvdnwbzVay63a4t1iq6bgKlyHOdyceViEB7AEbaziFXyEL/C1/N2pOLPO5ZFruVRwZmHsONf+AmViwsU=</latexit>“T duality”+⇡✓ ⇠ ✓ + 2⇡
<latexit sha1_base64="AhXeiclsgWuoiaxzWc/koznFl4s=">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</latexit>

K: “Luttinger parameter”

In nite number of primary elds include<latexit sha1_base64="88BOWaPmY9i5oPKtieYs7xpYcks=">AAADWnichVI7b9NQFP4S82jdQtPCgMRyRRSEoIpuqgoqpgoYWBB9Ja3UlMh2T5ur+oV9Y1Gs/AEmNgYmkBgQOytILPwBhkr9AxVjkVgYOHYsVaFquJZ9z/nO+c7Lxw5dFWspD0pl49z5CxfHxs2JyUuXpyrTM6046EUONZ3ADaIN24rJVT41tdIubYQRWZ7t0rq99zCzrycUxSrw1/R+SFueteurHeVYmqFO5Xark/qzXl+06XlPJYKepXNCCV+0w64Sd1j0RFt3SVv9TqUq6zI/4rTQKIQqirMUTJcm0cY2AjjowQPBh2bZhYWYn000IBEytoWUsYglldsJfZjM7bEXsYfF6B5/d1nbLFCf9SxmnLMdzuLyGzFToCZ/yI/yWH6Xn+SR/MOxTNTOjLeNhPOGbBuO+WIoqnlmRWnOyjra59seVEBhZ+rVtdXf/2V5fGt0T1gjak05us2M0dPR2MFC3oHijsIcyeblDHIkL98cr95fqaU35Xv5kyf1Th7IbzwrP/nlfFimlbcjaj7J/oj1wb+MWHpSdPGUGcRIpmVzS9lzLZ9fWtzZHE3epca/m3NaaM3VG3fr88vz1cUHxVaN4Tpu4BZvzj0s4jGW0OQqXuMzvuBr+dAoG+PGxMC1XCo4VzB0jKt/AQWwvgk=</latexit>

Vn,m ⌘ en, m 2 Z

2in +im✓<latexit sha1_base64="p78YxQ/xo/XY5YkGPYivSIgfD1s=">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</latexit>

General classi cation of irrational CFT: unknown
fi

fi

49
fi

￼

Free Boson: Irrational CFT!

￼Tomonaga-Luttinger Liquid
S. Tomonaga

In 1D, all the low-energy excitations in
the interacting electron system can be
described in terms of “phonons” (free bosons)equivalent to (quantized) vibrating “string”
1
2
L=
(@µ )
2⇡K
<latexit sha1_base64="Tm8eQ9ij24yBam6JQKqqAB46tUA=">AAADZHichVJPaxNBFH/J+qeu0aYWQRBkMUTqJcyGYIsgFPUgqNh/aQvdGmank2bo7O4wOwnWIV/ALyDoScGD+CUEL+rdQ7+AIB4rePHg281CiaVxlt157/fe7/3bFyopUkPIQansnDp95uzUOfd85cLF6erMpfU06WvG2yyRid4MacqliHnbCCP5ptKcRqHkG+Hevcy+MeA6FUm8ZvYV347obiy6glGDUKfaCiJqeoxK+2jo3fGCrqbM+kPbDJTwHg69uUBRbQSVnSDqe4HqiZtPm51qjTRIfrzjgl8INSjOUjJTqkAAO5AAgz5EwCEGg7IECik+W+ADAYXYNljENEoit3MYgovcPnpx9KCI7uF3F7WtAo1Rz2KmOZthFomvRqYHdfKNvCeH5DP5QH6QPxjLhfqJ8XZggHkV2sZjPhuL6p5Ykc1ZWUf7eIejCrjqTL+4svr7v6wIbwO9I9aEWi1GD5ExeToGurCQdyCwI5Uj2bzYKMfg+cvD1dsrdXuDvCU/cVJvyAH5hLOKB7/Yu2W+8npCzUfZ76M++pcapcdFF0+QwRHJtGxuFj3X8vnZ4s7m6OIu+f9uznFhvdnwbzVay63a4t1iq6bgKlyHOdyceViEB7AEbaziFXyEL/C1/N2pOLPO5ZFruVRwZmHsONf+AmViwsU=</latexit><latexit sha1_base64="osw/gUtQfNUdG/FZxLNFaFeaVKA=">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</latexit>

1 @2
v 2 @t2

@2
=
@x2

50

￼

Tomonaga-Luttinger Liquid
The free boson theory appears trivial--but it contains rich physics!
Luttinger parameter K represents the strengthof quantum fluctuation, and reflects theelectron interactions
K=0
Wigner
crystalK<1
repulsiveinteraction

K=1
free fermion
51

K>1
attractive
interaction

K=∞supercondutor￼Junction of 2 quantum wires
Kane-Fisher (1992), Furusaki-Nagaosa (1993)

tunneling
For non-interacting electrons:
transmission probability:
continuous function of the tunneling amplitudeconductance also depends continuouslyon the tunneling amplitude
However, we will see that the interaction
fundamentally changes the physics!
52

￼

Field theory on the junction“fold” the system at the junction
two-component boson field theory

charge conservation at the junction“kills” the total charge modeboundary problem of a single-component
free boson field theory (c=1 CFT)53

￼Boundary condition of free boson
In the low-energy limit, the boundary
condition approaches to a
conformally invariant boundary condition
Known conformally invariant boundary conditions:
Neumann b.c.
Dirichlet b.c.

Disconnected
wires
Perfect transmission(merged toasingle
wire)
54

￼

Phase diagram for 2 wires
backscattering

perfect wire
(Dirichlet for θ)
Non-Fermi Liquidbehavior for g ≠ 1 !

disconnected(Neumann for θ)

K=1
K
tunneling55￼Complete Set of BC?
From the RG analysis, there was no indication that
nontrivial conformally invariant b.c. exists
The conformally invariant boundary conditions forthe c=1 free boson CFT are exhausted by
Dirichlet, Neumann (and their SU(2) rotation?)

56

fi

￼

Is this really physical? (Dirichlet/Neumann has nonnegative integer coef cients)57

￼

Phase diagram for 2 wires
backscattering

perfect wire
(Dirichlet for θ)
Non-Fermi Liquidbehavior for g ≠ 1 !

disconnected(Neumann for θ)

K=1
K
tunneling58

￼“conformal mapping”

59

￼

60

<latexit sha1_base64="hTHAs8xYrRazLScPz19pZrvHic4=">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</latexit>

Effective system size = “max chord distance”

l⇡

Need to construct the “mirror image” of theboundary (junction) carefully
At short lengthscale, the boundary is not yet
conformally invariant (boundary RG ow)
Crossover of correlation functions (UV → IR)cannot be studied faithfully
which are based on the conformal mapping
Calculation without the conformal mapping?
61
fl

￼

IssuesWindow size
2L ≦ 400

fi

62
fi

￼

in nite DMRG
+ nite window

￼

63

￼

64

￼

Junction of 2 Inequivalent TLLsv↵ , K↵

tunneling

α

β

v ,K
<latexit sha1_base64="pDr4Xtt8buXpagtgE7RC9sfk67U=">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</latexit><latexit sha1_base64="f3wt8tPyzqCms5f0emwtf+j2oaM=">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</latexit>

Boundary CFT by rescaling↵<latexit sha1_base64="LZ1x10CHtM4lGrW2lqhtmx0N0JM=">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</latexit>↵zµ = vµ ⌧ + ix
<latexit sha1_base64="ZIFln+rwUWK09uUmj+jf17m6Pbs=">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</latexit>xzµ = ⌧ + i
vµhJ (y, 0)J (x, 0)i =<latexit sha1_base64="Wn4DTgTGnasmAufy6uQmkIwsskE=">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</latexit><latexit sha1_base64="8+34+gB6I2MepgH0AXBEjVSnwx8=">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</latexit>↵hJ (y, 0)J (x, 0)i =
65

G v v
2⇡ 2 (x + y)2
G
1
y 2
x
2
2⇡ ( v↵ + v )￼Junction of 2 Inequivalent TLLs

66

￼

Field theory on a 3-wires junction“fold” the system at the junction
3-component boson field theory

charge conservation at the junction“kills” the total charge modeboundary problem of a 2-component
free boson field theory (c=2 CFT) of
67

For c ≧ 2 free boson CFT,
there are nontrivial conformally invariant boundary
conditions (satisfying the original Cardy condition)other than Neumann/Dirichlet!
oblique compacti cation
lattice for c=2~✓~ ⇠ ✓~ + 2⇡ R
<latexit sha1_base64="M6WLg0OU33OctYLLjjfcsq/ASnU=">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</latexit>~ 2⇤
R<latexit sha1_base64="dtgrG1F6QqbGGYD+GjX8UjLARMc=">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</latexit>

68
fi

￼

Non-trivial Boundary Conditions

Necessity from the RG argument
Af eck-Ludwig 1991, Furusaki-Nagosa 1993,Yi-Kane 1997 etc.

when the “compacti cation lattice”
is oblique, Dirichlet & Neumanncan be stable or unstable
simultaneouslyintermediate RG xed point= nontrivial conformallyinvariant boundary cond.
fi

fi

69
fl

￼

RG argument

Given a consistent conformal b.c. A ZAA =<latexit sha1_base64="m5itXnZjLp+bSRDYyFlj46gNhbM=">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</latexit>

XinAA i (q)i

new consistent b.c. B can be constructed by fusion
fusion with primary eld kusing modular S-matrixiSiicB = cA ki
S0<latexit sha1_base64="PnJIJ4qq/5HekeRv750l75TRGMQ=">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</latexit><latexit sha1_base64="lkQLB+KaMdfD4wu868TcJc0AOCg=">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</latexit>i (q̃) =

X

j
Si

j (q)

j

<latexit sha1_base64="pMvcmsHrIzc027MLE7abWD08AYA=">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</latexit>

i
nAB =

X

Verlinde formulaX<latexit sha1_base64="nzc1oGQWNoiM+sxdarGJ9J8/CbM=">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</latexit>ji
Njk nAA

j

j

i i
S
i j
k Sl
Sj Nkl =S0i

fusion rule coef cient for OPE
70fii
Njk 2 Z 0<latexit sha1_base64="TbXjoT7V3cbQucIHLogyfJjDNsY=">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</latexit>fi￼Fusion Constructionc=2 free boson CFT witha special compacti cation latticeadmits conformal embedding
Af eck-M.O.-Saleur 2000
1
7
4
2= +
+
2 10 5<latexit sha1_base64="7JyHVSp7ZG4rihAF3Fj6HkQnBQE=">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</latexit>

Ising + tricritical Ising + Potts

fi

71
fl

￼

Nontrival c=2 Partition Function

Af eck-M.O.-Saleur 2000
6 4
2= +
5 5
<latexit sha1_base64="gPA1OXYCpTp0sMu4x0B0EAweABE=">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</latexit><latexit sha1_base64="itYd3RmVJINEDanLEzP0QVrfH4s=">AAAEXXicnVK/b9NAFH5uXSgmtA0MILGciBIlAqXnpKUVElIFGVgo6Y+0FU1r2ddra9Wxje1ElFP+AVYGBiaQGBB/AStd+AcYOjEjxiKxMPB8dlqF0iC4KLn3vfe+7757Oct37DCi9FAZGlZHzp0fvaBdzFwaG5/IXl4JvXbAeIN5jhesWWbIHdvljciOHL7mB9xsWQ5ftfbux/XVDg9C23OXo32fb7TMHdfetpkZYcrIKrceG6JW6xaflMhdUiDFJtu1Dbo5TW6SCpGggqB0XKhjQYbVzXqJNJtaARPFXq+oTk53JTmBswns0UUlxscSYkbCkoYyaGN+/v9sxD09+Wr332z1GvRJvc/Zn40aEzlapnKR04GeBjlIV93LKhlowhZ4wKANLeDgQoSxAyaE+FkHHSj4mNsAgbkAI1vWOXRBQ24buzh2mJjdw98dROtp1kUca4aSzfAUB78BMgnk6Wf6jh7RT/Q9/Up/opYG+TP1tqCD5/pY69d82qeqnelISFZ8o33crcQB943x59eWfvyV1cI9gt0T1gCvAtUtZAyeTgTbMCtvYOONfJmJ58WSMzrPXh4t3VnMiwJ9Q7/hpF7TQ3qAs3I739nbBb74aoDnk9NriJP/MsDoYXqLR8jgmIlRPDeBnctyfiLd4zlq+Jb031/O6WClUtZvl6cWpnJz99JXNQrX4QYU8eXMwBw8gDo0gCkvlA/KR+Vg+Is6ombUsaR1SEk5V6BvqVd/AT6CBKQ=</latexit>

Z3(5) + (Potts = Z3(4) )ZDD (q) =( 50 + 2 52 )( P
0 +
+ (2

ZN N (q) =(

5
3/5 +

P
3)
5
P
8/5 )( 2/5 +55P
0 + 2 2 )( 0 ++ (2 53/5 +

P
7/5 )

P
P
3 + 2 2/3 )

5
P
)(2
8/5
1/15 +

P
2/5 +

P
7/5 )

In either embedding, we can apply fusion constructionin each sector to possibly obtain new boundary states
72
fl

￼

Alternative Embedding

fl￼

Af eck-M.O.-Saleur 2000

73

Presumably incomplete
In fact, the conformal embedding works only fora special compacti cation (Luttinger parameter)
but the similar conformal boundary condition should
exist for a range of parameters

Classi cation of conformallyinvariant boundary conditions
for irrational CFTs?

fi

74
fi

￼

“Zoo” of Conformal B.C.

￼

9 6
3= +
5 5
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4 6
3=2+1=1+ +5 5<latexit sha1_base64="2XAUYxnNAsWZUvWAuBNUp+mpgVA=">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</latexit>

75

￼

76

￼

“K” = Y (3-channel Kondo,Yi-Kane)“C” = F (Chiral)77￼

Where’s the Fermi statistics?
electrons are fermions !
can we really reduce the junction to the
problem of the free boson?
Answer: YES for the junction of 2 wires(same result for the hard-core boson)
However, the Fermi statistics does playa crucial role in the junction of 3 wires
78

￼

Fermi statistics is essential!

Fermi statistics
(-1)

79

￼

80

￼

81

￼

Conjectured
RG Flow Diagramfor 1 < K < 3chiral χ±
exact solution(chiral rotation ofDirichlet/Neumann)82

Chiral xed point
agree w/BCFT prediction

83
fi

￼

Rahmani, Hou et al. 2012

￼

M Fixed Point
Rahmani, Hou et al. 2012
g=K
<latexit sha1_base64="e6swP2aqGLLPBBjuKZDsU99N2Wg=">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</latexit>

conjecture<latexit sha1_base64="HeIpkNv8SP8e8UDM5s4tWzE339k=">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</latexit>

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Chiral F.P. in iDMRG+Window
C.-Y. Lo, Y.-J. Kao, P. Chen et al., preliminary unpublished85￼M Fixed Point in iDMRG+Window
Rahmani et al. conjecture<latexit sha1_base64="HeIpkNv8SP8e8UDM5s4tWzE339k=">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</latexit>

C.-Y. Lo, Y.-J. Kao, P. Chen et al.,preliminary unpublished
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Dp Fixed Point

C.-Y. Lo, Y.-J. Kao, P. Chen et al.,preliminary unpublished88Classi cation of conformally invariant
boundary conditions of irrational CFTs:
open problem (generally unknown)
In particular, c≧2 free boson CFTs
(multicomponent TLLs) have various nontrivial
conformal boundary conditions with many applications
A few known exact solutions, and many morepresumed to exist from RG arguments
Re ned numerical methods can help verifying
predictions and discovering more
fi

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Conclusions (Part 2)