[PPT] - HOTRG study on partition function zeros in the p-state clock model PowerPoint Presentation

SLIDE 1

HOTRG study on partition function zeros in the p-state clock model

Dong-Hee Kim

Tensor Network States: Algorithms and Applications @ NCCU, 4-6 Dec. 2019

Gwangju Institute of Science and Technology, Korea

Dept. Physics and Photon Science

SLIDE 2

Outline

1. Fisher zero characterization of a phase transition
2. Monte Carlo: Numerical issues due to the stochastic nature
3. Higher-Order Tensor Renormalization Group

— Numerical methods of computing the partition function — How large systems can we consider for Fisher zeros?

[D.-H. Kim, PRE 96, 052130 (2017)] [S. Hong and D.-H. Kim, arXiv:1906.09036]

It depends on the type of phase transition: BKT has an issue. — BKT transitions in the p-state clock model? — Characterization of the two BKT transitions in the p-state clock model — Finite-size scaling analysis: logarithmic corrections — Fisher-zero determination of the BKT transition temperature

Department of Physics & Photon Science

SLIDE 3

Department of Physics & Photon Science

p-state clock model in square lattices

θ = 2πn p n = 0, 1, 2, .., p − 1

: “clock” spin Z(p) symmetry XY model: U(1) symmetry

p → ∞

Berezinskii-Kosterlitz-Thouless transition

Emergent U(1) symmetry:

BKT transition occurs even at finite p!

Z(p) broken disordered

2nd order

Z(p) broken disordered

x x

critical region

BKT BKT

critical

x

BKT

disordered

2 ≤ p ≤ 4 5 ≤ p < ∞

p → ∞

Review: “40 years of BKT theory”, ed. by J. V. Jose

Hp = −J X

hi,ji

cos(θi − θj)

Emergent symmetry in the 2D p-state clock model

SLIDE 4

Department of Physics & Photon Science

Z(p) broken disordered

2nd order

Z(p) broken disordered

x x

critical region BKT BKT

critical

x

BKT

disordered

2 ≤ p ≤ 4 5 ≤ p < ∞

p → ∞

(Villain approximation, self-dual, …)

There are approximations involved!

; see Borisenko et al., PRE 83, 041120 (2011).

Numerical results with the “exact” p-state clock model:

Lapilli/Pfeifer/Wexler, PRL 2006:

(mainly for the high-T transition)

It’s not the BKT transition for p < 8!

(helicity modulus)

Hwang, PRE 80, 042103 (2009):

(Fisher zero study)

Indeed, it doesn’t look like BKT for p=6.

Baek/Minnhagen/Kim PRE 2010:

(helicity modulus, more rigorously)

Noop, IT IS the BKT for p = 6! but…

Baek/Minnhagen PRE 2010:

p = 5 looks strange…

Baek et al, PRE 2013. Kumano et al, PRE 2013. Borisenko et al, PRE 2011. Chatelain, JSM 2014: DMRG

? Debates and remaining issues

review: “40 years of BKT theory”, ed. by J. V. Jose

SLIDE 5

Fisher zero test on p=5 and 6 ?

Hwang, PRE 80, 042103 (2009): the first Fisher zero calculation for p=6 Wang-Landau Monte Carlo calculations up to L=28

3.4
3.2
3
2.8
2.6
2.4
ln[L]
3.4
3.2
3
2.8

ln[Im(a1)]

slope = 0.67(1)

(b)

said it looks like the second-order(!) transition. Baek, Minnhagen, Kim, PRE 81, 063102 (2010) No, it’s BKT at p=6 (helicity modulus). Baek, Minnhagen, PRE 82, 031102 (2010)

0.2 0.4 0.6 0.8 0.9 1 1.1 1.2 1.3 Υ T (d) q=6 L=8 32 128 512

p=6

BKT

0.2 0.4 0.6 0.8 0.8 1 1.2 1.4 Υ T (c) q=5 L=8 32 128 512

p=5

BKT(?) Yes, it is BKT but with residual symmetry.

0.4 0.8 1.2 1.6 0.85 0.95 1.05 1.15 1.25

[cf. Baek et al., PRE 88, 012125, (2013)] Kumano et al., PRE 88, 104427 (2013). Chatelain, JSM P11022 (2014)

SLIDE 6

p-state clock model Helicity modulus Fisher zero

p=6

BKT

2nd. order ?

(WL calc. - Hwang 2009)

p=5

BKT

(with a new definition of helicity modulus)

?

Disagreements & Questions

The only previous Fisher calculations in the p-state clock models:

Wang-Landau method gives very accurate results, usually.
L=28: too small. WL simulations can be done for much larger ones for p=6;
cf. Lee-Yang zeros in the XY model: up to L=256 with MC + histogram reweighting.
Q. Are there any fundamental issues with the BKT transition?

Department of Physics & Photon Science

Hwang, PRE 80, 042103 (2009): p=6, up to L=28 with Wang-Landau method It disagreed with the helicity modulus results but has not been re-examined.

SLIDE 7

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In this talk, … What was wrong with Fisher zeros at the BKT transitions? Monte Carlo noises become unbearable, very quickly. This is a “feature” of BKT; it’s unavoidable within MC. Can we improve the situation? Yes, with HOTRG, to some extent. Logarithmic subleading-order corrections are essential. Better finite-size-scaling analysis can be done.

SLIDE 8

Zeros of Partition Function: the “Fisher” zeros

Z(β) = X

E

g(E) exp[−βE] = 0

β = βc

F = −kBT ln Z

Re[𝞬] Im[𝞬]

x 𝞬c

as L increases

Finite-Size-Scaling behavior

f the “leading” Fisher zero

A tool to study a phase transition without an order parameter

Lots of works have done. For a review, see   Bena et al., Int. J. Mod. Phys. B 19, 4269 (2005).

(complex temperature, no external field)

Phase transition: singular free energy! At,

No real solution exists in finite-size systems, but …

Im[z1] ∼ L−1/ν

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|Re[z1] − zc| ∼ L−1/ν∗

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Department of Physics & Photon Science

M. E. Fisher (1965)

SLIDE 9

Department of Physics & Photon Science

1. solving polynomial equation
2. graphical solution + minimization

Given that we have g(E) … for equally spaced discrete energies

Z = X

n

gne−n✏ = X gnzn

polynomial solver

Y

i

(z − zi)

quick and easy, and works the best with exact g(E).

Z(β) = ZR(β) + iZI(β)

from exact counting, histogram reweighing, Wang-Landau, or … map of zeros map of zeros find intersections minimize |Z| fine tuning

slow, manual-geared, cumbersome, but allows error analysis!

Numerical strategy to compute Fisher zeros

SLIDE 10

1. solving polynomial equation

Z = X

n

gne−n✏ = X gnzn Y

i

(z − zi)

2. Graphical search + refinement

Z(β) = ZR(β) + iZI(β) ZR(β) Z(βR) = X

E

P(βR; E) cos(βIE) ZI(β) Z(βR) = X

E

P(βR; E) sin(βIE)

Find zeros. Find zeros.

0.1 0.2 0.3 0.4 0.5 0.7 0.75 0.8 0.85 0.9 Im[β] Re[β]

1. Search for the cross point

2. Minimize |Z| for refinement

We need the energy distribution at a real T!

Monte Carlo with histogram reweighting Wang-Landau sampling method

Numerical strategy to compute Fisher zeros

Department of Physics & Photon Science

˜ Z ≡ Z(β) Z(βR)

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SLIDE 11

Leading Fisher zero behavior vs. type of phase transition

[Denbleyker et al., PRD 2014]

(XY: 𝜉=1/2)

System-size scaling behavior of the leading zero :

Im[β1] ∼ L−1/ν

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Im[β1] ∼ L−d

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Im[β1] ∼ [ln bL]−1− 1

ν

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Second-order: First-order: BKT: Can we compute the leading Fisher zero in a large enough system?

Ising model (2D) : It can be done up to L=256. (my own test, unpublished) Potts model (2D) : it reached L=128 long time ago. [PRE 65, 036110 (2002)] XY model : up to L=128 with HOTRG. [Denbleyker et al., PRD 89, 016008 (2014)] Clock model : up to L=32 with WL. [DHK, PRE 2017] well-established! well-established! indirectly examined;

nly for XY.

Department of Physics & Photon Science

SLIDE 12

Test: 2D Ising model

10-3 10-2 10-1 22 23 24 25 26 27 28 29 Im[z1] L

(c)

Ising: Im[z1] ~ L−1/ν ν=0.991 10-4 10-3 10-2 22 23 24 25 26 27 28 29 Re[z1] − zc L

(d)

Ising: Re[z1]−zc ~ L−λ λ=0.952

0.02 0.04 0.06 0.35 0.4 0.45 0.5 Im[z] Re[z]

(e) L=32, Ising

0.01 0.02 0.03 0.35 0.4 0.45 0.5 Re[z]

(f) L=64, Ising

. 4 . 8 . 1 2 0.35 0.4 0.45 0.5 Re[z]

(g) L=128, Ising

. 2 . 4 . 6 0.35 0.4 0.45 0.5 Re[z]

(h) L=256, Ising

z = exp[2β]

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“leading” zero

Im[z1] ∼ L−1/ν

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|Re[z1] − zc| ∼ L−1/ν∗

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ν ≈ ν∗ ≈ 1

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L=256: Parallel replica-exchange WL [Vogel et al., PRL 2013] Polynomial Solver + WL density of states

Department of Physics & Photon Science

SLIDE 13

10-2 10-1 8 16 32 64 128 Im[β1] L L-6/5 10-4 10-3 10-2 10-1 8 16 32 64 128 Im[β1] L L-2

Test: 2D q-state Potts model

Department of Physics & Photon Science

q=10 q=3 First-order transition Second-order transition Graphical solutions based on WL density of states

Im[β1] ∼ L−1/ν

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Im[β1] ∼ L−d

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SLIDE 14

0.05 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Im[z]

(a) L=8, six-state clock

0.05 0.1 0.3 0.4 0.5 0.6 0.7 0.8

(b) L=16, six-state clock

0.02 0.04 0.06 0.3 0.4 0.5 0.6 0.7 0.8

(c) L=32, six-state clock

0.01 0.02 0.03 0.3 0.4 0.5 0.6 0.7 0.8

(d) L=64, six-state clock

Re[z] Re[z] Re[z] Re[z]

2D p-state clock model

Department of Physics & Photon Science

0.1 0.2 0.4 4 8 16 32 Im[β1] L

p=6 (high-T)

p=6: a usual WL algorithm works fine. p≠6: [DHK, PRE 96, 052130 (2017): p=5, 8, 10, (12)] (WL DOS is available for up to L=256)

Z(p) broken disordered

x x

critical region BKT BKT

(irregular energy spacing problem resolved for WL) High T: the error blows up for L > 32. Low T: the error blows up for L > 16. T

No FSS!

SLIDE 15

XY model : leading Fisher zeros calculations using HOTRG

A. Denbleyker, Y. Liu, Y. Meurice, M. P

. Qin, T. Xiang, Z. Y. Xie, J. F . Yu, and H. Zou,

Phys. Rev. D 89, 016008 (2014).

Im[β1] ∼ [ln(bL)]−1− 1

ν

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|βc − Re[β1]| ∼ [ln(bL)]− 1

ν

<latexit sha1_base64="/1qYLPlKlzT+JGTzn4A5aUsjE=">ACKXicbVDLSgMxFM34tr6qLt0Ei1AXLTMq6LgxoULFdsKM2PJpHdqMkMSUYo4/yOG3/FjYKibv0R08dCqwcCh3Pu5eacKOVMG9f9cKamZ2bn5hcWS0vLK6tr5fWNlk4yRaFJE56oq4ho4ExC0zD4SpVQETEoR3dHg/89h0ozRJ5afophIL0JIsZJcZKnXLjPojAkA7FNRwIYm6UyC+g8EeqF97jQDOB/YDLanS6G17ntSBWhOZekQcyK4pOueLW3SHwX+KNSQWNcdYpvwTdhGYCpKGcaO17bmrCnCjDKIeiFGQaUkJvSQ98SyURoMN8mLTAO1bp4jhR9kmDh+rPjZwIrfsispODMHrSG4j/eX5m4qMwZzLNDEg6OhRnHJsED2rDXaAGt63hFDF7F8xvSG2CGPLdkSvMnIf0lr+7t1/fODyqNxriOBbSFtlEVegQNdAJOkNRNEDekKv6M15dJ6d+dzNDrljHc20S84X9+Av6bN</latexit>

Im[β1] ∝ |βc − Re[β1]|

3 2

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L = 4, 8, 16, 32, 64, 128 For a small Im[𝜸],

SLIDE 16

Department of Physics & Photon Science

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

square: XY model (HOTRG, PRD2014)

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

triangle: p=10

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

cross: p=8

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

circle: p=6

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

plus: p=5

High-temperature transition

L=4 L=8 L=16 L=32

SLIDE 17

Department of Physics & Photon Science

0.1 0.2 0.3 0.4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Im[β] Re[β]

p=5 p=6 p=8 p=10 XY

0.1 0.15 0.2 0.85 0.9 The leading zeros for p=6,8,10 falls onto those of the XY limit.

p>5 is like XY: p>5 is probably then BKT.

p=5 shows systematic difference from the larger p’s. The same singular form

f free energy emerges

for p>5.

Strong form of universality?

at a high-temperature transition

SLIDE 18

Department of Physics & Photon Science

an arc-like trajectory

Low-temperature transition

0.1 0.2 0.3 0.4 0.5 0.78 0.79 0.80 0.81 0.82 0.83 0.84

L = 4 L = 6 L = 8 L = 12 L = 16

Im[β1](1 − cos 2π

p )

Re[β1](1 − cos 2π

p )

p = 5 p = 6 p = 8 p = 10

Still, no clues for what they are…

SLIDE 19

Numerical visibility of Fisher zeros

Department of Physics & Photon Science

| ˜ Z|

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: amplitude of a normalized partition function

0.35 0.40 0.45 0.50 Re[β] 0.00 0.02 0.04 0.06 0.08 Im[β]

1 0−4 10−4 10−3 1 0−2 10−1

0.4325 0.4365 0.025 0.030

0.003 0.006 . 9

O X The leading Fisher zero

SLIDE 20

Scaling behavior of the tolerable error level

Department of Physics & Photon Science

~ Max. tolerable error

1
0.5

0.5 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Z(β) / Z(Re[β0]) Im[β] Re[Z] Im[Z]

exp[−1 2σ2

Eβ2 I]

Envelope function (Gaussian DOS) approx.

SLIDE 21

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2 1.4 specific heat

(a)

six-state clock T1 T2

L=8 L=16 L=32 L=64 L=128 L=256

exp[−1 2σ2

Eβ2 I]

Scaling behavior of the tolerable error level

approx. max. error tolerance

vs.

Wang-Landau fluctuations

constant

exp[−1 2σ2

Eβ2 I] → exp[−Ldc∗ Lβ2 I

2β2

R

]

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at a pseudo-transition point

BKT: ∼ exp[−L2(ln bL)−6]

(exponential decay) Going to a large L is not possible  under any finite fluctuations

1st order: ∼ exp[−L2dL−2d] = O(1)

:GOOD

2nd order: ∼ exp[−Ld+α/νL−2/ν] = O(1)

:GOOD

Department of Physics & Photon Science

∼ L−a

<latexit sha1_base64="layFruIs6P0vHyi7XRrExLmH4EQ=">AB8nicbVDJSgNBEO1xjXGLevTSGAQvhkX9Bjw4sFDBLPAJIaeTk/SpJehu0YIQz7DiwdFvPo13vwbO8kcNPFBweO9KqrqRYngFnz/21taXldWy9sFDe3tnd2S3v7DatTQ1mdaqFNKyKWCa5YHTgI1koMIzISrBkNbyZ+84kZy7V6gFHCOpL0FY85JeCksG25xHeP2SkZd0tlv+JPgRdJkJMylHrlr7aPU1TyRQawNAz+BTkYMcCrYuNhOLUsIHZI+Cx1VRDLbyaYnj/GxU3o41saVAjxVf09kRFo7kpHrlAQGdt6biP95YQrxdSfjKkmBKTpbFKcCg8aT/3GPG0ZBjBwh1HB3K6YDYgFl1LRhRDMv7xIGmeV4LxyeX9RrlbzOAroEB2hExSgK1RFt6iG6ogijZ7RK3rzwHvx3r2PWeuSl8coD/wPn8AtsiQ4g=</latexit>

∼ exp[aL−ω]

<latexit sha1_base64="wYg4SQWwza3Omud9UWsp7fLI=">ACAHicbVC7TsNAEDyHVwivAUFzYkIiYbI5iEoI9FQUASJPCTbROfLJjnlzrbuzojIcsOv0FCAEC2fQcfcElcQMJIK41mdrW7E8ScKW3b31ZhYXFpeaW4Wlpb39jcKm/vNFWUSAoNGvFItgOigLMQGpDu1YAhEBh1YwvBr7rQeQikXhnR7F4AvSD1mPUaKN1CnveYoJ7MFj7JKb+/TYiwT0SeZ3yhW7ak+A54mTkwrKUe+Uv7xuRBMBoacKOU6dqz9lEjNKIes5CUKYkKHpA+uoSERoPx08kCGD43Sxb1Imgo1nqi/J1IilBqJwHQKogdq1huL/3luonuXfsrCONEQ0umiXsKxjvA4DdxlEqjmI0MIlczciumASEK1yaxkQnBmX54nzZOqc1o9vz2r1Gp5HEW0jw7QEXLQBaqha1RHDURhp7RK3qznqwX6936mLYWrHxmF/2B9fkDcxSWUQ=</latexit>

Ising model: weak 2nd. order:

SLIDE 22

0.2 0.4 0.6 0.8 1 2 4 6 8 10 0.05 0.1 0.15 0.6 0.8 1 1.2 ˜ Z∗ 0.1 0.2 1.6 2 2.4 ˜ Z∗ 0.01 0.02 0.03 0.04 1.6 1.8 2 2.2

10−14 10−10 10−6 10−2 1.6 1.8 2.0 2.2

0.2 0.4 0.6 0.8 1 8 16 32 64 128 10−3 10−2 10−1 8 16 32 64 128 10−2 10−1 8 16 32 64 128 10−16 10−12 10−8 10−4 100 8 16 32 64 128 ten-state Potts model Ising model three-state Potts model five-state clock model

(a) (b) (c) (d)

| ˜ Z| L2Im[β] LIm[β] L6/5Im[β] L = 8 L = 16 L = 32 L = 64 L = 128 Im[β]/(βc −Re[β1])3/2 ˜ Z∗ L Q1st(L) L QIsing(L) L Q2nd(L) L upper lower Aexp[−aLx]

1st vs. 2nd. vs. BKT transitions

Department of Physics & Photon Science

[S. Hong and D.-H. Kim, arXiv:1906.09036]

System-size scaling of numerical visibility of the leading Fisher zero.

∼ L−a

<latexit sha1_base64="layFruIs6P0vHyi7XRrExLmH4EQ=">AB8nicbVDJSgNBEO1xjXGLevTSGAQvhkX9Bjw4sFDBLPAJIaeTk/SpJehu0YIQz7DiwdFvPo13vwbO8kcNPFBweO9KqrqRYngFnz/21taXldWy9sFDe3tnd2S3v7DatTQ1mdaqFNKyKWCa5YHTgI1koMIzISrBkNbyZ+84kZy7V6gFHCOpL0FY85JeCksG25xHeP2SkZd0tlv+JPgRdJkJMylHrlr7aPU1TyRQawNAz+BTkYMcCrYuNhOLUsIHZI+Cx1VRDLbyaYnj/GxU3o41saVAjxVf09kRFo7kpHrlAQGdt6biP95YQrxdSfjKkmBKTpbFKcCg8aT/3GPG0ZBjBwh1HB3K6YDYgFl1LRhRDMv7xIGmeV4LxyeX9RrlbzOAroEB2hExSgK1RFt6iG6ogijZ7RK3rzwHvx3r2PWeuSl8coD/wPn8AtsiQ4g=</latexit>

∼ exp[aL−ω]

<latexit sha1_base64="wYg4SQWwza3Omud9UWsp7fLI=">ACAHicbVC7TsNAEDyHVwivAUFzYkIiYbI5iEoI9FQUASJPCTbROfLJjnlzrbuzojIcsOv0FCAEC2fQcfcElcQMJIK41mdrW7E8ScKW3b31ZhYXFpeaW4Wlpb39jcKm/vNFWUSAoNGvFItgOigLMQGpDu1YAhEBh1YwvBr7rQeQikXhnR7F4AvSD1mPUaKN1CnveYoJ7MFj7JKb+/TYiwT0SeZ3yhW7ak+A54mTkwrKUe+Uv7xuRBMBoacKOU6dqz9lEjNKIes5CUKYkKHpA+uoSERoPx08kCGD43Sxb1Imgo1nqi/J1IilBqJwHQKogdq1huL/3luonuXfsrCONEQ0umiXsKxjvA4DdxlEqjmI0MIlczciumASEK1yaxkQnBmX54nzZOqc1o9vz2r1Gp5HEW0jw7QEXLQBaqha1RHDURhp7RK3qznqwX6936mLYWrHxmF/2B9fkDcxSWUQ=</latexit>

∼ exp[−aL−x]

<latexit sha1_base64="FbkfoTUAzTn6Fein7wR8OmkrvA0=">AB/HicbVDLSsNAFJ3UV62vaJduBovgpiXxgS4Lbly4qGAf0MQymU7aoTOTMDORhlB/xY0LRdz6Ie78G6dtFtp64MLhnHu5954gZlRpx/m2Ciura+sbxc3S1vbO7p69f9BSUSIxaeKIRbITIEUYFaSpqWakE0uCeMBIOxhdT/32I5GKRuJepzHxORoIGlKMtJF6dtlTlEOPjONuFd0+ZNXxO/ZFafmzACXiZuTCsjR6NlfXj/CSdCY4aU6rpOrP0MSU0xI5OSlygSIzxCA9I1VCBOlJ/Njp/AY6P0YRhJU0LDmfp7IkNcqZQHpMjPVSL3lT8z+smOrzyMyriRBOB54vChEdwWkSsE8lwZqlhiAsqbkV4iGSCGuTV8mE4C6+vExapzX3rHZxd16p1/M4iuAQHIET4IJLUAc3oAGaAIMUPINX8GY9WS/Wu/Uxby1Y+UwZ/IH1+QMXKpRp</latexit>

Potts (q=10) Potts (q=3) Ising Clock (p=5); HOTRG

SLIDE 23

Higher-Order Tensor Renormalization Group (HOTRG)

Xie, Chen, Qin, Zhu, Yang & Xiang, PRB 86, 045139 (2012).

Department of Physics & Photon Science

T

(n)

(a) M

(n)

T

(n+1)

(b) i T

(n)

T

(n)

x1 x2 x'1 x'2 y y' T

(n+1)

x x' y y' x1 x2 x'1 x'2 x x' y y' U

(n+1)

U

(n+1)

M

(n)

M (n)

xx0yy0 =

X

i

T (n)

x1x0

1yiT (n)

x2x0

2iy0

<latexit sha1_base64="xMzyhHVO5WBN7RoxPH8gRSj32gc=">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</latexit>

1. Contraction

T (n+1)

xx0yy0 =

X

ij

UixM (n)

ijyy0U ∗ jx0

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2. HOSVD
3. Truncation

x = x1 ⊗ x2

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x0 = x0

1 ⊗ x0 2

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M (n)

xx0yy0 =

X

ijkl

SijklU L

xiU R x0jU U ykU D y0l

<latexit sha1_base64="q512sAW/RQzXvlXM698wCcGN+r4=">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</latexit>

(D2xD2xDxD)

U : (D2 x Dc)

(DcxDcxDxD)

cutoff

x → y → x → y → · · ·

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SLIDE 24

HOSVD in practice

Department of Physics & Photon Science

Q. How can we get U?

M (n)

xx0yy0 =

X

ijkl

SijklU L

xiU R x0jU U ykU D y0l

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1. U = UL
2. U = UR

T (n+1)

xx0yy0 =

X

ij

UixM (n)

ijyy0U ∗ jx0

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y y' x1 x2 x'1 x'2 x x' U

(n+1)

U

(n+1)

M

(n)

Ax,x0yy0 ≡ Mxx0yy0

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Ax0,xyy0 ≡ Mxx0yy0

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(reordering for L)

AA† = UΛU †

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(reordering for R) (D2xD2 matrix diagonalization) Between L and R, choose the one with the smaller residual. Pick Dc largest eigenvalues and corresponding eigenvectors for U.

✏L,R = X

i>Dc

ΛL,R

i

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SLIDE 25

p-state clock model at a complex temperature

Z(β) = Y

i

X

θi

exp h β X

hi,ji

cos(θi − θj) i = Tr Y

i

Txix0

iyiy0 i

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Initial local tensor:

Txx0yy0 = q Ix(β)Ix0(β)Iy(β)Iy0(β)δmod(x+y−x0−y0,p),0

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eβ cos θ =

∞

X

n=−∞

In(β)einθ

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expansion with

Recipe for XY model: A. Denbleyker et al., PRD 89, 016008 (2014). c.f. XY : δx+y−x0−y0,0

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Issue with complex temperature

T (n+1)

xx0yy0 =

X

ij

UixM (n)

ijyy0U ∗ jx0

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Department of Physics & Photon Science

AA† = UΛU †

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Invariant under x <-> x’ & y <-> y’ If U is complex, it breaks the symmetry. Fix: orthogonal transformation

Re[AA†] = UΛU T

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SLIDE 26

p-state clock model at complex temperature : leading Fisher zeros

0.1 0.2 0.8 0.9 1

(a) (b) (c) (d)

0.1 0.2 0.3 0.4 1.14 1.16 1.18 1.2 0.1 0.2 0.3 0.8 0.9 1 0.2 0.4 0.6 1.56 1.59 1.62 1.65 upper transitions lower transitions p = 5 p = 5 p = 6 p = 6 Im[β] Re[β] Dc = 40 Dc = 50 Dc = 60 Dc = 70 Re[β] Dc = 40 Dc = 50 Dc = 60 Dc = 70 Im[β] Re[β] Dc = 40 Dc = 50 Dc = 60 Dc = 70 Re[β] Dc = 40 Dc = 50 Dc = 60 Dc = 70

Department of Physics & Photon Science

Z(p) broken disordered

x x

critical region

BKT(?) BKT(?)

Dc = 40, 50, 60, 70 are tested L = 8, 16, 32, 64, 128

Are they BKT?

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|βc − Re[β1]| ∼ [ln(bL)]− 1

ν

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A. Denbleyker et al., PRD 2014.
H. Zou, PhD Thesis 2014.

limiting case The imaginary part of 𝝄 cannot be constant unless L = ∞. We may need this.

[M. Hasenbusch, JPA 38, 5869 (2005)]

ξL(β) L = a0 + a1 1 ln L + O[(ln L)2]

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SLIDE 28

Department of Physics & Photon Science

Finite-Size-Scaling Ansatz with Logarithmic Corrections

(∆βx ± iβy)−ν ' a ln bL + i ⇣ c0 c1 ln L ⌘

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Corrected FSS form:

∆βx = |βc − Re[β1]|

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βy = Im[β1]

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Determination of Tc!

0.6 0.8 1 0.2 0.3 0.4 0.5

(a) (b) (c) (d)

1 1.1 1.2 1.3 1.4 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.8 0.9 0.2 0.3 0.4 0.5 upper transitions lower transitions p = 5 p = 5 p = 6 p = 6 Im[(∆βx −iβy)−ν] βc −2δ βc −δ βc βc +δ βc +2δ βc −2δ βc −δ βc βc +δ βc +2δ Im[(∆βx −iβy)−ν] 1/lnL βc −2δ βc −δ βc βc +δ βc +2δ 1/lnL βc −2δ βc −δ βc βc +δ βc +2δ to Eq. (22) with the zero data for L ≥ Lmin. β high

c

(p = 5) β low

c

(p = 5) β high

c

(p = 6) β low

c

(p = 6) reference 1.088(12) 1.47(4) [28] 1.1111 1.4706 [29] 1.1101(7) 1.4257(22) [30] 1.0510(10) 1.1049(10) [31] 1.1086(6) [36] 1.0593 1.1013 1.106(6) 1.4286(82) [37] 1.058(19) 1.094(14) [38] 1.0504(1) 1.1075(1) [40] 1.059 1.097 1.106 1.441 Lmin = 8 1.058 1.101 1.106 1.444 Lmin = 16

It agrees well with other method.

SLIDE 29

Finite-Size-Scaling Ansatz with Logarithmic Corrections

Department of Physics & Photon Science

0.025 0.05 0.1 0.2 0.4 2 3 4 5 6

(lnbL)−3

0.05 0.1 0.2 0.4 2 3 4 5 6

(lnbL)−2

0.05 0.1 0.2 0.4 0.8 2 3 4 5 6

(lnbL)−3

0.05 0.1 0.2 0.4 2 3 4 5 6

(lnbL)−2

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can be non-monotonic.

An arc-like trajectory is observed in a certain range of 𝛚L.

0.2 0.4 0.6 1.15 1.2 Re[β] Im[β] p=5 0.2 0.4 0.6 0.8 1 1.6 1.65 Im[β] p=6 0.4 0.8 1.2 1.6 2.7 2.75 2.8 Im[β] p=8 0.8 1.2 1.6 2 2.4 4.15 4.2 4.25 4.3 4.35 Im[β] p=10

[WL data at low T]

SLIDE 31

Finite-Size-Scaling Ansatz with Logarithmic Corrections

Department of Physics & Photon Science

0.01 0.1 1 0.05 0.1 0.2 0.4

βy ∝ ∆β 3/2

x

0.025 0.05 0.1 0.2 0.4 0.04 0.06 0.08 0.1

(e) (f)

βy ∝ ∆β 3/2

x

βy ∆βx p = 5 p = 6 XY rpβy rp∆βx p = 5 p = 6

∆βx = w1β

1 1+ν

y

+ w2βy + w3β

2−

1 1+ν

y

+ O

β

3−

2 1+ν

y

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Correction to the trajectory: Upper transition Lower transition With HOTRG data, p=5 fits well with the BKT scenario.

SLIDE 32

Summary and Conclusions

Department of Physics & Photon Science

Fisher-zero characterization of phase transitions in the p-state clock model

HOTRG vs. Wang-Landau MC

2. HOTRG: Leading Fisher zeros are computed up to L = 128.
1. p=5 : it now fits well to the BKT trajectory with HOTRG data

— Better FSS analysis is done using more accurate data + ansatz w/ correction. — BKT transition points are located using the logarithmic scaling behavior.

3. Corrections to the previous WL results:
2. The arc-like trajectory at the lower transition is now explained

by the BKT ansatz with the finite-size corrections.

[D.-H. Kim, PRE 96, 052130 (2017)] [S. Hong and D.-H. Kim, arXiv:1906.09036]

1. WL MC is destined to fail because of the non-diverging specific heat.