無序關聯性系統的大規模計算模擬

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科技部補助專題研究計畫成果報告

期末報告

無序關聯性系統的大規模計算模擬

計畫類別：個別型計畫計畫編號： MOST 104-2112-M-004-002-執行期間： 104年08月01日至105年10月31日執行單位：國立政治大學應用物理研究所計畫主持人：林瑜琤計畫參與人員：碩士班研究生-兼任助理人員：張太乙碩士班研究生-兼任助理人員：張鎮宇碩士班研究生-兼任助理人員：柯志緯大專生-兼任助理人員：林源青報告附件：移地研究心得報告出席國際學術會議心得報告

中　華　民　國　106　年　01　月　31　日

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中文摘要：在多自旋耦合極強極限下，均質J-Q鏈的基態將自發轉變成一價鍵固態。我們設計一針對J-Q鏈的強無序重整化群法(SDRG)用以探討耦合無序對自發性價鍵固態的影響。我們的結果顯示無序價鍵固態裂成交錯的二聚區間，成無晶形的價鍵固態，激發旋子位於疇壁。這無晶形價鍵固態與無序海森堡鏈的基態同為所謂的無序單態相態。中文關鍵詞：無序系統，無序單態基態，關聯函數，玻璃態

英文摘要： The ground state of a J-Q chain in the absence of disorder and in the limit of strong multi-spin (Q) couplings is a valence-bond solid (VBS) with spontaneous dimerization. We introduce a strong-disorder renormalization group

(SDRG) scheme to study the effect of quenched bond randomness on J-Q chains. Our results show that the

random VBS state breaks into alternating dimerized domains with singlets formed between spinons localized

at domain walls. This amorphous valence-bond solid at long distances is asymptotically a random-singlet state,

similar to the ground state of the random Heisenberg chain. 英文關鍵詞： disordered systems, random-singlet ground state,

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Large-scale numerical simulations of disordered and

correlated matter

Yu-Cheng Lin

January 30, 2017

1 Research results

Results accomplished under this project are summarized below:

1.1 Properties of the random-singlet phase

Logarithmic corrections to the power-law decaying spin correlations

We use unbiased zero-temperature quantum Monte Carlo (QMC) simulations to study the antiferromagnetic S = 1/2 Heisenberg chain with random couplings, calculating disorder-averaged spin and dimer correlations. Surprisingly, in light of the large number of previous

studies and the widely accepted notion that the r−2 asymptotic form, predicted by the

strong-disorder renormalization group (SDRG) analysis, for the mean spin-spin correlations is exact [1], our QMC results show that the SDRG method misses universal multiplicative logarithmic (log) corrections to the power-law decays (see Fig. 1). By studying different contributions to the spin-spin correlation functions in QMC calculations in the VB basis, we find that the log corrections arise from a contribution that is completely missing in the simple singlet-product ground state resulting from the SDRG method.

The approximate ground state resulting from the SDRG is described by a single set of

bipartite valence bonds, |ψ0i = |vi, with no bonds crossing each other, while the ground

state projected out by the QMC method is a superposition of VB states,

|ψ0i =

X

v

αv|vi , (1)

with non-negative coefficients αv that are determined stochastically. The matrix elements

needed hv0|Si· Sj|vi for computing the correlation function in the valence-bond basis are

finite only if sites i and j belong to the same loop in the transition graph of the overlap

between |vi and |v0i. In Fig. 2 we illustrate two different types of one-loop structures,

corresponding to |vi 6= |v0i and |vi = |v0_{i, in the transition graph of the overlap hv}0_{|vi. A}

loop of type (b), which is the only kind of loop appearing in an overlap hv|vi between same 1

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101 102 103 N 101 N 2 C(r max ) d=1, QMC d=2, QMC d=1, SDRG d=2, SDRG

Figure 1: The correlation functions at the largest distance rmax = N/2 of the random

Heisenberg chain of length N , multiplied by N2_{, shows the presence of a multiplicative}

logarithmic correction to the 1/N2 scaling for the QMC results, which increases with

dis-tance. The correction for the SDRG results shows a small enhancement for very small N and fast convergence to a constant, implying a small conventional subleading power-law correction. From Ref. [2].

(a)

(b)

Figure 2: Two different types of loop structures in the transition graph of the overlap

between two VB states |vi (upper, black bonds) and |v0i (lower, red bonds). The gray and

white sites belong to two different sublattices. Type (a) indicates the case where |vi 6= |v0i

and type (b) is a single-bond structure with |vi = |v0i. The correlation C(|i − j|) for any

pair of spins (i and j) located in the same loop is a finite constant. From Ref. [2].

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10 50 N 10-6 10-5 10-4 10-3 10-2 D * (N/2) d=1, QMC d=2, QMC d=1, SDRG d=2, SDRG d=5, SDRG d=10, SDRG

Figure 3: SDRG and QMC results for dimer correlations of the random-J model with

different disorder parameters d. The fitting forms are D∗(r) = αr−4+ βr−5 (SDRG) and

D∗(r) ∝ r−4ln(r/r0). From Ref. [2].

states (as in the SDRG ground state), contains only two sites in different sublattices sepa-rated by an odd number of lattice spacings. A loop of type (a) can have arbitrarily many sites greater than two. Thus the spin correlation function C(r) for even r is determined solely by loops of type (a).

The results are published in Ref. [2]. Dimer correlations

We have also investigated the dimer-dimer correlation function of the ground state of the random Heisenberg chain (the random-singlet state), which to our knowledge had not been previously considered, neither in SDRG nor QMC calculations. We have found numerically

that the SDRG dimer-dimer correlations decay as D∗(r) ≈ r−4; here D∗(r) is the staggered

dimer correlation function. The QMC results are very well accounted for by a multiplicative

log D∗(r) ∝ r−4lnσd_(r/r

0), with σd ≈ 1 (s. Fig. 3) [2]. The asymptotic r−4 decay of the

dimer correlation in the random-singlet (RS) state has been also tested by the principal investigator in the random XX chain using the mapping into a free fermion problem, which is also in an RS phase; the results are to be published.

1.2 Amorphous valence-bond solid

A chain, called the Q chain, described by

HQ = −

N

X

i=1

QiPi,i+1Pi+2,i+3Pi+4,i+5, (2)

where Qi > 0 and Pi,j is a singlet projector on two spins i, j, is a valence-bond solid (VBS)

at zero temperature when the system is clean (Qi ≡ Q ∀i). The Q chain combined with

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Figure 4: Qualitative AVBS ground state of an S = 1/2 spin chain. The gray and orange circles represent the two sublattices of the bipartite lattice. The short valence bonds (red and blue thick lines) form ordered domains, between which spinons localize. In the ground state the spinons freeze pairwise into long-bond singlets (indicated by the arches).

10 100 10-4 10-3 10-2 C(N/2) d=1, QMC d=2, QMC d=1, SDRG d=10, SDRG d=50, SDRG 10 _N 100 10-5 10-4 10-3 10-2 10-1 D * (N/2) d=1, QMC d=2, QMC d=1, SDRG d=10, SDRG d=50, SDRG d=100, SDRG (a) (b)

Figure 5: SDRG and QMC results for the spin (a) and dimer (b) correlations in the random-Q model. The functional forms fitted to the data (curves shown) are the same as in the corresponding cases in Fig. 1 and Fig. 3.

the Heisenberg coupling J (> 0),

HJ = −

N

X

i=1

JiPi,i+1, (3)

undergoes a transition to the standard critical antiferromagnet at J/Q ≈ 6 [3]. The

dimer-ization transition is in the same universality class as that in the well-studied J1-J2

Heisen-berg chain.[5, 6, 7] An important question is how disorder affects such a transition and the VBS state. In the latter, one can expect an amorphous VBS (AVBS) with alternating do-mains of the two different dimerization patterns (which differ by a translation of one lattice unit), and a simple valence-bond picture suggests that the domain walls between these do-mains should contain S = 1/2 spin degrees of freedom—localized spinons—corresponding to long valence bonds between different domain walls as illustrated in Fig. 4.

The generalization of the SDRG to the random-Q model (2) with three singlet projectors

(also called Q3 interactions) is non-trivial, as the multi-spin interaction generates various

terms of the forms Pi,i+1Pi+2,i+3 (Q2 interactions) and Pi,j (J interactions) under SDRG,

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with several different cases in the perturbative treatment of the decimated operators. The principal investigator worked out the completely new scheme for the SDRG of the J-Q model. The technical details of the method is described in Ref. [2].

Our SDRG and QMC results show consistently that both the random-J and the random-Q models are asymptotically governed by the RS fixed point. Thus, in a J -Q model, we do not expect any phase transition as a function of the ratio J/Q, unlike the clean system where there is a dimerization transition of the same universality class as

in the J1-J2 Heisenberg chain [3, 4]. Although there is no phase transition in the sense

of asymptotic, the AVBS can still be considered as a state of matter different from the Heisenberg-RS, because it possesses a length-scale—that of VBS domains (see Fig. 5(b))— which is not present at the RS fixed-point alone, but which can be made arbitrarily large by tuning interactions in the AVBS state.

The physics of the VBS and AVBS also applies to spin chains coupled to phonons. In the classical limit, any spin-phonon coupling leads to dimerization (the spin-Peierls distortion), while at finite phonon frequency a critical coupling is required [8, 9, 10]. The AVBS state we have identified and characterized in Ref. [2] should be relevant to

quasi-one-dimensional spin-phonon materials, e.g., CuGeO3 [11] and TiOCl [12]. RS scaling due to

localized spinons should be detectable using NMR [13, 14], and it would then be desirable to also calculate temperature dependent magnetic properties.

1.3 Transport modeling

In this subproject we applied the study of disordered interacting systems to problems in our daily lives. We have simulated highway traffic flow problems using cellular automaton models [15]. One of our main results is that for three-lane the asymmetric lane change rule in general leads to a higher flow than the symmetric lane change rule, or a mixed lane change rule which is used in Taiwan (see Fig. 6). Here by the so-called asymmetric lane change rule vehicles have to drive on the right lane (the default lane) except when overtaking overtaking/passing slower vehicles; in this system middle-lane ”hogging” is prohibited. By the mixed rule only the left lane is the passing lane, whereas the middle lane and the right lane are regarded as two symmetric lanes.

This subproject is interdisciplinary research aimed at providing guidance directed at those involved in designing highway transportation systems. From the point of view of physics, jamming process is a kind of phase transition, with similarities to a glass transition which remains challenging [16].

References

[1] D. S. Fisher, Phys. Rev. B 50, 3799 (1994).

[2] Y.-R. Shu, D.-X. Yao, C.-W. Ke, Y.-C. Lin, and A. W. Sandvik, Phys. Rev. B 94, 174442 (2016).

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0 0.2 0.4 0.6 0.8 1 ρ 0 0.2 0.4 0.6 0.8 1 J asymmetric model mixed model symmetric model

Figure 6: A fundamental density-flow diagram for the asymmetric, symmetric and mixed three-lane models.

[3] A. Banerjee and K. Damle, J. Stat. Mech.: Theor. Exp. 2010, P08017 (2010). [4] Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011).

[5] I. Affleck, Phys. Rev. Lett. 55, 1355 (1985).

[6] K. Nomura and K. Okamoto, Phys. Lett. A 169, 433 (1992). [7] S. Eggert, Phys. Rev. B 54, R9612 (1996).

[8] A. W. Sandvik and D. K. Campbell, Phys. Rev. Lett. 83, 195 (1999). [9] G. S. Uhrig and H. J. Schulz, Phys. Rev. B 54, R9624 (1996).

[10] H. Suwa and S. Todo, Phys. Rev. Lett. 115, 080601 (2015).

[11] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, 3651 (1993).

[12] J. Zhang, A. W¨olfel, M. Bykov, A. Sch¨onleber, S. van Smaalen, R. K. Kremer, and

H. L. Williamson, Phys. Rev. B 90, 014415 (2014). [13] T. Shiroka et al., Phys. Rev. B 88, 054422 (2013). [14] T. Shiroka et al., Phys. Rev. Lett. 106, 137202 (2011).

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[15] K. Nagel and M. Schreckenberg, Journal de Physique I. 2 2221 (1992). [16] B. Giulio, Nature Physics, 3 222 (2007).

2 Graduate students supervision

One master’s thesis supported by this project grant has been published online at http://thesis.lib.nccu.edu.tw/:

• Random singlets in an amorphous valence-bond solid: a strong-disorder renormaliza-tion group study, June. 2016, Chih-Wei Ke

Another thesis under this project studies jamming transition in traffic flow, being carried out by a student with bachelor degrees in computer science at NCCU.

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STATPHYS26 satellite meeting:

Renormalization Group Theory of Disordered Systems,

Ecole Normale Supérieure, Paris, July 25-27, 2016

In addition to the main event of STATPHYS26, a series of satellite meetings was held in that period preceding and following the conference. The satellite meeting on Renormal-ization Group Theory of Disordered Systems was closely related to the main theme of this project. Therefore, I applied for attending this satellite meeting after the main conference. The talks arranged in this meeting were of excellent quality. I was in particular inter-ested in generalizations of the strong-disorder renormalization group (SDRG) to dynamics and energy excitations. In addition to the workshop, I discussed with some participants intensively about our resent research results. In particular, I started to write a draft of a paper on quench dynamics of disordered quantum spin chains, which studies the dynamics of the order parameter; my co-author of the paper Ferenc Iglói was one of the speakers for the workshop. Below I include the abstract of the paper:

By means of free fermionic techniques combined with multiple precision arithmetic we study the time evolution of the average magnetization, m(t), of the random transverse-field Ising chain after global quenches. We observe different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered

initial state, the relaxation is logarithmically slow described by m(t) ∼ lnat, and in a

finite sample of length L the average magnetization saturates at a size-dependent plateau

mp(L) ∼ L−b; here the two exponents satisfy the relation b/a = ψ = 1/2. Starting from

a fully disordered initial state, the magnetization stays at zero for a period of time until

t = td with ln td∼ Lψ and then starts to increase until it saturates to an asymptotic value

mp(L) ∼ L−b

0

, with b0 ≈ 1.5. For both quenching protocols, finite-size scaling is satisfied

in terms of the scaled variable ln t/Lψ_{. Furthermore, the distribution of long-time limiting}

values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through semi-classical theory.

This paper has been online published in arXiv:1611.09495, and will appear in New Journal of Physics.

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RENORMALIZATION GROUP THEORY OF DISORDERED SYSTEMS

Room: Conf. IV, 2nd floor

Physics Department of the Ecole Normale Superieure 24 rue Lhomond, Paris

Monday 25: Functional RG for random systems, random manifolds, avalanches

9:30 - 10:15 T. Giamarchi -- Thermal effects for disordered systems

10:15 - 11:00 P. Le Doussal Functional Renormalisation Group and Avalanches

11:00 - 11:30 COFFEE BREAK

11:30 - 12:15 K. Wiese Non-Conventional Fixed Points in the RG of Disordered Systems and

Sandpiles.

12:15 - 14:00 LUNCH BREAK

14:00 - 14:45 M. Tissier -- Nonperturbative approach to the Random-Field Ising Model

14:45 - 15:30 I. Balog -- Hysteresis transition in the Random field Ising model

16:00 - 16:45 A. Fedorenko -- Classical and quantum dynamics of random field systems

16:45 - 17:30 G. Pruessner -- Field Theory of the Manna Model

17:30 - 18:15 T. Vojta -- Infinite-noise criticality: Nonequilibrium phase transitions in

fluctuating environments

Tuesday 26: RG for spin and structural glasses

9:30 - 10:15 M. Moore -- Do the de Almeida-Thouless and Gardner replica symmetry

breaking transitions exist in three dimensions?

10:15 - 11:00 G. Parisi -- Fat Diagrams

11:30 - 12:15 F. Ricci-Tersenghi -- Critical properties of Ising and XY models on random

graphs

14:00 - 14:45 M. Angelini -- Real space Renormalization Group for Spin-Glasses:

Migdal-Kadanoff vs Topological expansion

14:45 - 15:30 P. Urbani -- The Gardner transition in finite dimensions

16:00 - 16:45 C. Cammarota Few NPRG steps forward on the way for a theory of glass

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16:45 - 17:30 S.Yaida -- A nontrivial critical fixed point for replica-symmetry-breaking

transitions

17:30 - 18:15 S. Franz -- Interfaces and Renormalization Group in Disordered Dyson Lattices

Wednesday 27: RG for out-of-equilibrium systems

9:30 - 10:15 J. Sethna -- Bifurcation theory and the renormalization group: Nonlinear scaling

for logarithms, resonances, and exponentials

10:15 - 11:00 L. Canet -- Spatiotemporal velocity-velocity correlation function in fully

developed turbulence

RG for quantum disordered systems, Anderson and many body localisation, Bose glass

11:30 - 12:15 G. Refael -- Strong disorder Wegner flow and 1d localization transitions

14:00 - 14:45 F. Igloi -- Random quantum systems with long-range interactions

14:45 - 15:30 I. Gruzberg -- Multifractality and scaling dimensions at Anderson transitions:

exact results.

16:00 - 16:45 M. Tarzia -- Critical properties of Anderson localization in high-dimension

16:45 - 17:30 R. Vasseur -- Renormalization group approaches for many-body localization

and eigenstate phase transitions

19:30 CONFERENCE DINNER & CONCERT Bellevilloise

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26th IUPAP International conference on Statistical Physics,

STATPHYS26, 2016

First I would like to acknowledge the financial support from the Ministry of Science and Technology (Grant No. MOST 104-2112-M-004-002), which enabled my participation in STATPHYS26 held in Lyon (France) from July 18 to 22, 2016.

My talk entitled ”Properties of the random-singlet phase: from the disordered Heisenberg chain to an amorphous valence-bond solid” was contributed to Session: Quantum fluids and condensed matter. I presented results from my recent study, in collaboration with colleagues at SYSU (Guangzhou, China) and BU (Boston, USA), on the random J-Q chain, using a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties. Our results have published in: Phys. Rev. B 94, 174442 (2016), after the conference.

The conference STATPHYS attracted 1300 participants with more than 50 nationalities. The pressure on selected oral presentations was high. My presentation has become the only one oral presentation from Taiwan.

My talk slides are included in this report.

After this main conference in Lyon, I attended a satellite meeting on Renormalization Group Theory of Disordered Systems held in Paris.

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Ju ly 19, 2016 / S ta tP hys 26 / L yo n

Properties of the random-singlet phase

from the disordered Heisenberg c

hain to an amorphous

va

len

ce

-b

on

d s

oli

d

Y u-Cheng L in / N at io nal Cheng chi U niv ers it y / T aiwan P ro p er ties o f th e r an d o m -s in g let p h as e: f ro m th e d is o rd er ed Heis en b er g ch ain to an am o rp h o u s v alen ce-b o n d s o lid arXiv :1603.04362 Co llab orat ors : C h ih -W ei K e (N at io nal Cheng chi U niv ers it y, T aiwan) Y u -R o n g S h u , Dao -Xin Y ao (S un Y at -S en U niv ers it y, China ) A n d er s W . S an d v ik (B os to n U niv ers it y, U SA ) ✤

RG

p

ro

ced

ur

e fo

r

str

ong-disor

der RG

str ong es t c oup ling ef fec tiv e c oup ling H = X i J i ~S i ·_~S j Rand om -s ing let s tat e: a no n-c ro ss ing v alenc e-b ond s tat e ⌦ ˜J = JJ 0 2⌦ < ⌦ D as g up ta, M a(1980)

random-singlet state

✤

F

ix

ed

-p

oint

d

is

trib

ut

io

n o

f

✤

N

um

b

er o

f ac

tiv

e s

p

ins

˜J P ( ˜J)= ˜J 1 +1 / = ln( ⌦ ), ⌦ = m ax { ˜J} !1 ˜J P ( ˜J) n ⌦ ⇠ 1 ln 2 ⌦ ` ⇠ ln 2 ⌦ infinit e-rand om nes s D . F is her (1994)

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random-singlet state

✤

U

nc

onv

ent

io

nal d

y

nam

ic

al s

caling

✤

M

ean s

p

in-s

p

in c

orr

elat

io

n func

tio

n

⌧ ⇠ exp( p ⇠) z = 1 d y nam ic al ex p onent C (r )=[ h ~S i ·_~S i+ r i] av ⇠ n 2 ⌦ C (r ) ⇠ 1 r 2 D . F is her (1994)

ener

gy-length scaling

10 -2 10 -1 10 0 n Ω 10 -2 100 10 2 -ln(Ω) ⇠ 1/ p n ⌦

`

⇠

ln

2

⌦

-400 -300 -200 -100 0 -500 ln( Ω f₎ 10 -6 10 -4 10 -2 P( ln(Ωf ) ) L = 1024 L = 2048 L = 4096 L = 8192 -6 -5 -4 -3 -2 -1 0 ln( Ω f _{) /} L 1/2 10 -6 10 -4 10 -2 100 P( ln(Ωf ) / L 1/2 ) L = 1024 L = 2048 L = 4096 L = 8192 infinit e-rand om nes s fix ed p oint

valence-bond (VB) states

✤

B

ip

art

it

e V

B

s

tat

e:

a s et o f s ing let s b et ween t wo d if fer ent sub lat tic es o n a b ip art it e lat tic e. ✤

V

B

b

as

is

:

expectation values

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expectation values

Sing

let

p

ro

jec

to

r:

valence-bond pr

ojector QMC

valence-bond pr

ojector QMC

T ian h e-2

log-corr

ection to C(r)

!

in t he Rand om -S ing let S tat e 10 100 L 10 -4 10 -3 10 -2 10 -1 C(L / 2) d=1 , QMC d=2 , QMC d=1 , SDRG d=2 , SDRG QMC SD RG C (r ) ⇠ 1 r 2 C (r ) ⇠ ln a (r ) r 2 a ⇡ 0. 5 D . F is her (1994) ⇡ (J )= 1 d J 1 /d 1 , for 0 <J  1 C (|i j|)=( 1) |i j| h ~S i ·_~S j_i

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log-corr

ection to C(r)

!

in t he Rand om -S ing let S tat e 10 100 1000 L 10 1 L2C(L / 2) d=1 , QMC d=2 , QMC d=1 , SDRG d=2 , SDRG QMC SD RG C (r ) ⇠ ln a (r ) r 2 a ⇡ 0. 5 J. Ho y os , et . al (2007) C (r ) ⇠ 1 r 2 ✓ 1+ r r 0 r ◆ ⇠ ✓ 1+ A L ↵ ◆ ⇡ (J )= 1 d J 1 /d 1 , for 0 <J  1 C (|i j|)=( 1) |i j| h ~S i ·_~S j i ⇠ p ln( L /B )

inside

QMC Q M C/S D RG C (|i j|)=( 1) |i j| h ~S i ·_~S j_i 3 4 i j i j |v i i j 3 8 |v i hv 0 | |v i hv 0 | 3 16 hv |

dimer corr

elation function

B i ⌘ ~S i ·_~S i+1 i i +1 j j +1 (a) i i +1 j j +1 (b) D (|i j|)= hB i B j_i i i +1 j j +1 (c) i i +1 j j +1 (d) 9 16 3 16 16 ₃ 9 32 QMC Q M C/S D RG Q M C/S D RG X

dimer corr

elation function

!

in t he Rand om -S ing let S tat e 10 100 L 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 D*(L / 2) d=1 , QMC d=2 , QMC d=1 , SDRG d=2 , SDRG d=5 , SDRG d=10 , SDRG D ⇤ (r )=  D (r ) 1 2 D (r 1) 1 2 D (r + 1) ( 1) r r D ⇤ (r ) ⇠ ln b (r ) r 4 QMC b ⇡ 1 SD RG D ⇤ (r ) ⇠ 1 r₄

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JQ chain

s

✤

T

he JQ

3

c

hain:

✤

T

he “c

lean” c

as

e:

 

sp o n tan eo u s lat tic e s y m m et ry b reak ing when J i ⌘ J ,Q i ⌘ Q , 8i Q /J & 0. 1645 Q /J liq uid lik e V B s olid (Q /J ) c H JQ 3 = X i J i_P i,i +1 X i Q i_P i,i +1 P i+2 ,i +3 P i+4 ,i +5 J i_,Q i > 0, 8i

JQ chain

s

✤

T

he JQ

3

c

hain:

✤

W

it

h

b

ond

rand

om

nes

s:

 

Q /J rand om s ing let p has e H JQ 3 = X i J i_P i,i +1 X i Q i_P i,i +1 P i+2 ,i +3 P i+4 ,i +5 J i_,Q i > 0, 8i

?

SDRG for the

random

JQ chain

Q Q Q/ 16 Q/ 4 hv 0 |Q PP 0 P 00 |v 0 i 1s t c orr ec tio n Q Q 0 Q Q 0 Tr unc at io n o f c oup ling s

SDRG for the

random

JQ chain

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SDRG for the

random

JQ chain

2nd c orr ec tio n G enerat io n o f ef fec tiv e c oup ling s Q Q J Q 0 Q 0 Q 0 Q 0 ˜J or ˜J = 1 8 QJ ⌦ ˜J = 1 8 QQ 0 ⌦

GS of the random Q chain

!

10 -2 10 -1 10 0

n

Ω 10 -2 10 ₀ 10 ₂

-ln(Ω)

random Q-chain random J-chain

⇠

1/

p

n

⌦ infinit e-rand om nes s fix ed p oint

`

⇠

ln

2

⌦

`

⇠

1/n

⌦

GS of the random Q chain

!

10 100 L 10 -4 10 -3 10 -2 10 -1 C(L / 2) d=1 , QMC d=2 , QMC d=1 , SDRG d=10 , SDRG d=50 , SDRG sp in c orr elat io n func tio n QMC C (r ) ⇠ ln a (r ) r₂ SD RG a ⇡ 0. 5 C (r ) ⇠ 1 r 2

GS of the random Q chain

!

10 100 L 10 -4 10 -3 10 -2 10 -1 D*(L / 2) d=1 , QMC d=2 , QMC d=1 , SDRG d=10 , SDRG d=50 , SDRG d=100 , SDRG d im er c orr elat io n func tio n QMC SD RG D ⇤ (r ) ⇠ ln b (r ) r 4 b ⇡ 1 D ⇤ (r ) ⇠ 1 r₄ D ⇤ (r )=  D (r ) 1 2 D (r 1) 1 2 D (r + 1) ( 1) r

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amorphous valence-bond solid

G ro und s tat e o f t he rand om Q -c hain A ls o a rand om -s ing let p has e

conclusions

✤ L og arit hm ic m ult ip lic at iv e c orr ec tio n t o in t he rand om -s ing let (RS ) p has e ✤ D im er c orr elat io n func tio n in t he RS p has e ✤ T he q uant um p has e t rans it io n in t he JQ c hain is d es tro y ed b y s tro ng d is or d er C (r ) ⇠ 1 /r 2 as o p p os ed t o an ad d it iv e c orr ec tio n fo und b y S D RG

C

(r

)

⇠

p

ln(

r)

r

2

D

(r

)

⇠

ln(

r)

r

4 P ro p er ties o f th e r an d o m -s in g let p h as e: f ro m th e d is o rd er ed Heis en b er g ch ain to an am o rp h o u s v alen ce-b o n d s o lid arXiv :1603.04362 Co llab orat ors : C h ih -W ei K e (N at io nal Cheng chi U niv ers it y, T aiwan) Y u -R o n g S h u , Dao -Xin Y ao (S un Y at -S en U niv ers it y, China ) A n d er s W . S an d v ik (B os to n U niv ers it y, U SA )

for mor

e details

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科技部補助計畫衍生研發成果推廣資料表

日期:2017/01/25

科技部補助計畫

計畫名稱: 無序關聯性系統的大規模計算模擬計畫主持人: 林瑜琤計畫編號: 104-2112-M-004-002- 學門領域: 其他凝體－理論

無研發成果推廣資料

(22)

104年度專題研究計畫成果彙整表

計畫主持人：林瑜琤計畫編號： 104-2112-M-004-002-計畫名稱：無序關聯性系統的大規模計算模擬成果項目量化單位質化（說明：各成果項目請附佐證資料或細項說明，如期刊名稱、年份、卷期、起訖頁數、證號...等）　　　　　　　國內學術性論文期刊論文 0 篇研討會論文 0 專書 0 本專書論文 0 章技術報告 0 篇其他 0 篇智慧財產權及成果專利權發明專利申請中 0 件已獲得 0 新型/設計專利 0 商標權 0 營業秘密 0 積體電路電路布局權 0 著作權 0 品種權 0 其他 0 技術移轉件數 0 件收入 0 千元國外學術性論文期刊論文 1 篇 Physical Review B, 94, 174442(14 pages) (2016) 研討會論文 1

26th IUPAP International conference on Statistical Physics, (July 18-22, 2016), Lyon, France 約1300與會者，來自臺灣的唯一獲選演講專書 0 本專書論文 0 章技術報告 0 篇其他 1 篇 arXiv:1611.09495,

to appear in New Journal of Physics 智慧財產權及成果專利權發明專利申請中 0 件已獲得 0 新型/設計專利 0 商標權 0

(23)

營業秘密 0 積體電路電路布局權 0 著作權 0 品種權 0 其他 0 技術移轉件數 0 件收入 0 千元參與計畫人力本國籍大專生 1 人次碩士生 3 博士生 0 博士後研究員 0 專任助理 0 非本國籍大專生 0 碩士生 0 博士生 0 博士後研究員 0 專任助理 0 其他成果（無法以量化表達之成果如辦理學術活動、獲得獎項、重要國際合作、研究成果國際影響力及其他協助產業技術發展之具體效益事項等，請以文字敘述填列。）　　本計畫研究結果受邀演講發表於 the 10th

International Conference on Computational Physics, 16-20 Jan 2017, Macao

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科技部補助專題研究計畫成果自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）、是否適

合在學術期刊發表或申請專利、主要發現（簡要敘述成果是否具有政策應用參考

價值及具影響公共利益之重大發現）或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

□達成目標

■未達成目標（請說明，以100字為限）

　　□實驗失敗

　　□因故實驗中斷

　　■其他原因

說明：

原計畫規劃為三年。但因僅被核准一年計畫，所以雖達到很好的研究結果，僅

能算達到原申請計畫的1/3多。儘管如此，此一年計畫的研究成果達到國際學

術影響力、且具持續性及長遠參考價值。

2. 研究成果在學術期刊發表或申請專利等情形（請於其他欄註明專利及技轉之證

號、合約、申請及洽談等詳細資訊）

論文：■已發表　□未發表之文稿　□撰寫中　□無

專利：□已獲得　□申請中　■無

技轉：□已技轉　□洽談中　■無

其他：（以200字為限）

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價值

（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性，以500字

為限）

1. 學術成就上，本研究計畫首度發現一重要無序量子相變點普適性外的修正

因子及其原因（相關論文已發表於 PRB）。研究成果也由計畫主持人演講發表

於2016年於法國 Lyon 舉行的 IUPAP StatPhys26 大會，此發表也是來自臺灣

的唯一獲選演講（大會演講獲選率僅約30%）。論文結果也是國際計算物理

ICCP10 的邀請演講，計畫主持人為被邀請講者。

2. 本計畫首度發展出處理一多自旋耦合鏈的 strong disorder RG 方法，此

方法將可延展至許許多多相關量子系統，處理棘手的無序效應。

3. 本計畫開啓一新的國際合作，得以藉目前全球最快的電腦叢集系統、位於

廣州的「天河二號」完成相關的QMC計算，共同發表論文。這個合作主要建立

在計畫主持人長年對無序系統領域的研究成果。

4. 參與本計畫的碩士生得以實質參與發表國際期刊論文，尤其可藉國際合作

增加其知識廣度。

4. 主要發現

本研究具有政策應用參考價值：■否　□是，建議提供機關

（勾選「是」者，請列舉建議可提供施政參考之業務主管機關）

(25)

無序關聯性系統的大規模計算模擬

科技部補助專題研究計畫成果報告

期末報告