非平衡動的クラスター理論の開発
新学術領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」
平成２５年度第１回研究会
辻直人
東京大学大学院理学系研究科物理学専攻
1

公募研究「A03: 非平衡動的平均場理論の展開と低次元強相関系への応用」
研究目的：本研究では非平衡動的平均場理論を発展させ、これまで理論計算が困難とされてきた低次元強相関系の長時間非平衡ダイナミクスに応用する。
・非平衡動的クラスター理論
・二次元量子系の非平衡ダイナミクス

連携研究者：青木秀夫（東京大学理学系研究科物理学専攻）
岡隆史（東京大学工学系研究科物理工学専攻）
研究協力者：Philipp Werner (University of Fribourg)Peter Barmettler (University of Geneva)
Martin Eckstein (University of Hamburg)2

動的平均場理論 (DMFT)
Metzner,Vollhardt, ’89, Georges, Kotliar, ’92.

imp
Glat(i)=G[ ](i n )nii

(i n )
自己無撞着な方程式系

解きたい格子模型

lati j (i n ) =ij

imp(i n )

有効不純物模型

自己エネルギーの周波数依存性(動的相関)。モット転移を記述。無限次元極限で厳密。第一原理計算との組み合わせ。

3

formation. Because of coupling between lattice
vibrations (phonons) to the electrons, electronicenergy is gained if a phonon with a wave vector

rs superior energy and angle resoluable to detect the effect of collective
e form of dispersion kinks or spectral

transient equilibrium beyond 1 ps. The mode associated with this oscillation is related to theamplitude mode but is not quite the same.

電子相関と非平衡ing
ent
teronfor0.8R (×10-1)

0 ps
0.12 eV

www.sciencemag.org~
Photon energy (eV)Nph (photons/ET)(!R) of ET-F2 TCNQ for the 1.55 eV pump and the photon
densities of N~ ph % 0:003 and 0:095 ph:=ET. The pump andprobe lights are polarized parallel to a. The upper solid line
represents the polarized R spectrum along a. (b) Experimental(circles) and calculated (solid lines) time profiles of !R at

Okamoto et al., PRL ’07

VOL 321019 SEPTEMBER 2008 -4

Schmitt et al., Science ’08

FIG. 2 (color online). (a) Photoinduced reflectivity changes
光誘起モット絶縁体金属転移0.40.20.034.84.022(c)1

電荷秩序の融解
SCIENCE

1(b)

b(c)

0.6f (THz)δEprobe(tgate=t0) (arb. units)∆R (×10-2)∆R (×10-2)∆R (×10-2)and) oftheatoroto-∆R (×10-2)

Detail of theg. 1A′ withitions (white
電子相関が生み出す多種多様な非平衡ダイナミクス
me-resolvedn (B) to (D)・光誘起相転移 (金属絶縁体, 超伝導, 磁性, 電荷秩序, 格子構造, ... )a function ofndicated cut
・秩序パラメーター (Higgs, Nambu-Goldstone, topological defects, ...)line) of phontensity is(Prethermalization, nonthermal fixed point, ...)
week ending
functionof L E T T・孤立量子系の熱平衡化問題
REVIEWERS
19 JANUARY 2007
position [(E)4momentum
(a)
~ (b)
N
ph(a) τpump/τΔ=0.57
nateas a func=0.095
delays. All
0.12 eVollected at3
s= 2 mJ/cm2. ~~Nph=0.095
Nphin (E) to (I)=0.0032nJ/cmor bars indi0.12 eV
9.6
eance to the28.5
g~g sample
7.9
Nph=0.003
7.2
Delay time (ps)h is a good×5
6.4r the error
5.6-2024tpp (ps)6805165
Pump Energ

超伝導体におけるヒッグスモードの観測
Figure 2

Matsunaga, et al., arXiv:1305.0381

4librium magnitude, whereas the gap at f = 27°
decreases by only 20%. This may indicate different dynamics inside and outside the Fermi arc,
which is reported to end rather abruptly at f =30° for samples of this doping (13, 14), although
Figure4. A visualizationof quasiparticle
states arein a two-dimensional
studiesfarther fromthe nodeneeded. Gapd-wave superconductor. The square depicts a region of momentum
recoveryrates
are illustrated
in dotted
Fig. 2F,
wherespacethat includes
all quasiparticle
states. Thelines show
the thelocationofthe2D
Fermi
surface,andthewhiteregions
indicate
curves from (E) have been inverted and rescaled“nodal” directions along which the magnitude of the superconducting
gap
funnels, withchange.
their peculiarThebanana-shaped
crossbyvanishes.
their Themaximum
initial recovery
sections, depict the quasiparticle dispersion relation. At the apex ofisthe
slowerfor statesnode (0.9 T
arate
funnel,quasiparticle
energy iscloser
zero, andto
thethelow-energy−1
excitations
nearthat
specialpoint
areknown
asnodalquasiparticles.
0.6 ps ) than for states farther from the node
The high-energy states, with wavevectors rotated approximately 45

非局所相関効果

above the energy gap that often accompanies theonset of ordered phases. In the second class, the goal
is to manipulate the order parameter itself. Usually,
photoexcitation weakens the degree of order, but forsystems with coexisting and competing order parameters, one phase’s loss can be another’s gain.

Fig. 2. Evolution of the superconducting gap after puand higher (B) fluence. The gap is obtained by fittingby halving the distance between positive and negativ
fitting details, see supplementary materials (15). (C annormalized by its equilibrium value versus pump-prob
magnitude, inverted and normalized by maximal cha

Lessons from chemistry
特に一次元や二次元の低次元強相関系では、非局所的な電子相関効果が重要な役割を果たす。

Although measurements with resolution on electronic time scales were enabled by the remarkable
Ti:sapphire laser, a pulse of visible light seemed tobe a rather blunt instrument when unleashed on acorrelated electron system such as a high-Tc superconductor. To skeptics, injecting photons, each delivering 1.5 eV of energy—a hundred times larger
than the characteristic energy scale for superconductivity, kBTc—was like detonating a nuclear
device in a public library reading room. Beginningin the early 1990s, though, a few believers began toaim their lasers at high-Tc superconductors.3
Initial results showed that a phase transition tothe superconducting state reveals itself in a small
but clearly discernible ∆R signal. The fact that one
can probe the onset of superconductivity in thenear-IR evidently means that a quantum material
such as YBCO changes color when it enters thatstate, although the 0.01% change in reflectivity ofvisible light is too subtle for the unaided eye. The
mechanism for the color change in highly correlated
superconductors remains a mystery and an active
field of research to this day.4
Whatever the underlying mechanism, reflectivity changes could be exploited to probe superconductivity on subpicosecond time scales. It wasassumed, and later confirmed, that absorption of a1.5-eV photon promotes an electron to a state highabove the Fermi level, from which it decays rapidly
by creating an avalanche of broken Cooper pairs(quasiparticles) and phonons. The ∆R(t) signal isthen directly proportional to the nonequilibrium
quasiparticle density, ∆n(t).
To illustrate what can be learned from timeresolved measurements of ∆n(t), consider a simpler
system—a buffered solution of, for example,
sodium chloride dissolved in water. By measuringionic concentration as a function of temperature,
one can determine the free-energy difference between dissociated NaCl and its crystalline phase.
And yet such equilibrium measurements tell us
nothing about the strength of the interaction between the ions in solution; that information is hidden in the kinetics of the reaction Na+ + Cl− → NaCl.
The distinction between energetics and kinetics
applies equally to superconductors, in which therole of solid NaCl is played by the condensate of
Cooper-paired electrons; the solvated ions correspond to the gas of quasiparticles that coexists withthe condensate. In equilibrium, quasiparticles evaporate from the condensate and, at the same rate, undergo the reverse process of recombining as quasiparticle pairs. When a quasiparticle pair jumps backinto the condensate, its binding energy is released
to the system. In conventional superconductors,
that energy takes the form of a quantum of lattice
vibrational energy, or phonon—the same excitation

・異方的超伝導	  (d波,	  p波,	  ...)
・擬ギャップ...

Orenstein, Physics Today ’12.

目標:	  degrees from the nodal point, are antinodal quasiparticles. (Image
courtesy of Christopher Smallwood, University of California, Berkeley.)Fig. 3. Quasiparticle recombination dynamics versus
pump fluence and crystal momentum. ARPES data
correspond to intensity change above EF (∆I) as
integrated between the blue and black double arrows
in Fig. 1C. Time-resolved reflectivity rates correspondto fractional change in reflectivity. (A) Nodal decaycurves at 2, 5, and 10 mJ/cm2, normalized to the sameamplitude. (B) Analogous off-nodal decay curves (f =31°). (C) Overlay of nodal and off-nodal curves at thesame fluence. (D) Initial decay rate g0 versus fluence,obtained by fitting decay curves at short times [forthat provides the glue that binds electrons into∆I (t)≳ In∆Ihigh-Tto the convolution0 /2) materials,
where the bind- of a Gaussian
Cooperpairs.
−g0)
ingandinteraction
isunknown,
identifying
the0(t−tbosonthefunction
f(t)
=
∆I
eQ(t − t0), where0(or bosons) created by quasiparticle recombination
t0 are additional fit parameters. Timeis a∆Itantalizing
0 and prospect.
Figure 3 shows the change in reflectivity obresolved reflectivity rates were multiplied by 3/2 intained from experiments that measure the kineticsof order
quasiparticle
recombination.
The datareveal that depth of theto takethe finite
penetration
the analogy between quasiparticle dynamics andprobe
into account(15).
theoptical
reaction ratesof physicalchemistry isquite prec5(1)	  非平衡ダイナミクスにおける非局所電子相関効果を明cise and quantitative. As discussed above, ∆R(t) is
proportional to the photogenerated quasiparticle
density, ∆n(t). Each curve in figure 3a corresponds
to a different laser power, which determines thenumber of excess quasiparticles immediately after
photoexcitation, ∆n(t = 0). When plotted on double
logarithmic axes, the curves acquire a “knee” thatshifts to shorter time delay as the initial quasiparticle concentration increases. The inverse of thetime at which the knee occurs is the characteristic
decay rate γ0. Moreover, for times greater than 1/γ0 ,the curves straighten out and become parallel, indicating a power law decay that is independent of∆n(t = 0). All those features of the data are predictedby a simple chemical rate equation, known assecond-order kinetics, d/dt ∆n(t) = −β[∆n(t)]2, which
describes a decay process requiring the interactionof a pair of particles.

らかにする。低次元(特に二次元)強相関系の非平衡現象を解析する。

(2)	  そのために非局所ゆらぎを取り入れた非平衡理論手法の確立を目指す。

1138The dreaded bottleneck

A seemingly paradoxical aspect of quasiparticle recombination in high-Tc superconductors is that thepair-wise kinetics are in accord with calculations
based on Bardeen-Cooper-Schrieffer theory,6 while
results on more conventional superconductors are

d波超伝導体の緩和過程

1 JUNE 2012

VOL 336SCIENCEwww.scien

Smallwood et al., Science ’12.
5非平衡量子多体系の計算手法
非平衡多体系に対して様々な計算手法が使われてきている。各々に対して長所短所がある。
・厳密対角化 (近似なし、システムサイズが小さい有限系)
・時間依存ハートリー・フォック近似 (平均場近似)
・量子ボルツマン方程式 (勾配近似、摂動論、緩和時間近似)
・時間依存密度汎関数理論 (局所密度近似など、現実の物質系)
・時間依存密度行列繰り込み群 (一次元系、行列積状態)
・非平衡動的平均場理論 (無限次元系、自己エネルギーの局所近似)....
自己エネルギーの局所近似をクラスターに拡張して、短距離の非局所相関を取り込んだ方法論を開発する。有限次元系を扱えるようにする。
ー 非平衡動的クラスター理論

6非平衡動的平均場理論
Schmidt, Monien ’02, Freericks, Turkowski, Zlatic ’06.

latimpḠii (t, t ) = G [G](t, t )(t, t )kky

自己無撞着な方程式系

ttkx
latk (t, t ) =imp(t, t )

自己エネルギーの時間依存性(動的相関)。無限次元極限で厳密。熱力学極限がとれる。短時間スケールと長時間スケールの両方を記述。

7

動的クラスター理論
Maier, Jarrell, Pruschke, Hettler, RMP ’05

latclcl
GḠlat(t,(t,tt))==GG](t,tt ))I J](t,iKjIKJ[ Kk̃Kky

自己無撞着な方程式系

kxlat
clcl(t,t)
(i,j)
(t,t)
=(t,t)IJ
lat
KK+ k̃
(t,t)
=ij

0

(i, j)

clustercluster

自己エネルギーの動的相関とともに短距離の非局所相関をも取り込む。
Nc →	  ∞で厳密。実空間形式(CDMFT)と波数空間形式(DCA)。

8

Hubbard 模型†Ji j ci c jH(t) =ni ni+ U(t)iji

強相関電子系の典型的な模型であるHubbard模型に適用する。時間変化は、最も簡単な設定である「相互作用クエンチ」を考える。(冷却原子系への応用。)
相互作用パラメーターをクエンチした後、どのように系が緩和していくかを観察する。prethermalizationやthermalizationの物理。
まず一次元Hubbard模型に対してBenchmarkを行う。時間依存密度行列繰り込み群(DMRG)や通常の摂動論との比較を行う。
同様の設定で二次元Hubbard模型に適用して、Brillouine zone内で波数に依存した緩和ダイナミクスを見る。

9

Cluster solver
ここではfirst trialとして最もシンプルな二次摂動(IPT)を用いる。弱結合領域に適用。

1
K (t, t ) = U(t)U(t ) 2
Nc K

1

K 2 +K 3 =K

G K1 (t, t )G K2 ¯ (t , t)G K3 ¯ (t, t )K3

その他のsolverの候補:
ー量子モンテカルロ法。負符号問題により非常に短時間K2のシミュレーションに制約。
ー強結合展開(NCA)。ヒルベルト空間の次元に制約。小さなクラスターサイズ。

KtK1tー高次の弱結合摂動論。計算コストの問題。
ー厳密対角化。 Arrigoni, Knap, von der Linden, PRL ’13,Gramsch, Balzer, Eckstein, Kollar, arXiv:1306.6315.

10

DMFTの結果: 一次元Hubbard(a) 二重占有率 d=⟨n↑ n↓⟩  、(b)運動量分布関数 n(k,t)、および(c)その跳びΔn。
(a)

U=0(b)

20.25
0Æ10.150Æ20.250Æ30.210Æ4

a

U=0.5
U=1
U=1.5d(t)

dHtL

0.20

PRL 103, 056403 (2009)0.17

U=20.106810t(c) 1.00.25

0Æ10.60.21

Dn0Æ2

prethermalization plateau

0.40.20.0046t

80.170.130Æ30Æ42P H Y S I C AU=3L REV

PRL103, 056403 (2009)cf.
無限次元Hubbard模型

0.8

U=2.50.13

aU=0.5b10.2

U=1

U=80.8

U=1.50.6

U=2
U=2.5
U=310
U=0.5

U=1dmed
U=1.5
dth

0.146

U=50.2

U=2.5 U=4c0

U=3.3
U=3

10.8 Eckstein, Kollar, Werner, PRL ’09.
U=1.5

U=18 U

U=2U=60.401

U=0.5d4∆n(t)

2d(t)

0

U=823t4d110DCAの結果: 一次元Hubbard
二重占有率 d=⟨n↑ n↓⟩  と運動量分布関数の跳びΔn。
1.00

0.25

U=01

0.23

0.96

0.22

0.94

0.21

DMFTSH2L
DCA HNc=2L
DCA HNc=4L
DCA HNc=8L
DCA HNc=16L

0.20
0.19
0.18
0DMRG
1234t

U=00.98

DndHtL

0.24

0.92
0.90
0.88
0.86

510

DMFT
SH2L
DCA HNc=16L
DCA HNc=32L
DCA HNc=64L
246810t

二重占有率に関してはDMRGとDCAはNc=16でよく一致する。ΔnについてはNcをより大きくとる必要がある。
一次元系では、DCA+IPTはNc →	  ∞で通常の二次摂動 (Σ(2)) と一致してしまう。一次元系では熱平衡へ緩和しないため？
12DCAの結果: 二次元Hubbard
二重占有率 d=⟨n↑ n↓⟩  と運動量分布関数の跳びΔn。DCA: Nc=16x160.25

1.00

U=010.98

U=0Hpê2, pê2L

1Hpê4, 3pê4LHp, 0L

0.24
0.96

0.23
dDn

0.94
0.92

0.22
0.90

DCADCASH2L

0.21

DMFT
0.20
02468SH2L0.88

DMFT0.86

10024t6810t

二重占有率は瞬時に緩和するのに対して運動量分布は緩和が非常に遅い。しかも緩和時間が運動量に依存する。(π,0)が(π/2, π/2)よりも速く緩和する。DCAは通常の二次摂動(Σ(2))からはずれていく。 Σ(2)は熱平衡化しない。
13緩和時間の波数依存性準粒子の緩和時間を自己エネルギーの虚部から見積もる。正方格子はhalf fillingでvan Hove特異点をもつ。フェルミ面付近で非フェルミ流体的な振る舞い:

1kIm R (k, )Im R (k, )/準粒子の緩和時間は波数空間で強い非等方性を示す。DCAや Σ(2) の結果とconsistent。

14DCAの結果: 二次元Hubbard
相互作用 U を変えたときの結果。 DCA: Nc=16x16

U=01.021.0

Hpê2, pê2L4

DCAHpê4, 3pê4L

0.9

U=0Hp, 0LSH2L
DMFT

0.8

Hpê2, pê2L
Hpê4, 3pê4LHp, 0L0.8

Dn

Dn

0.6

0.7

0.40.6

DCA0.5DMFT

0.40SH2L20.246t8

100.00246810t

三段階の緩和プロセス:
prethermalization (波数空間で等方的) →	  準粒子緩和(非等方的) →	  thermalization (等方的)15まとめ非平衡動的クラスター理論を、簡単なsolver	  (IPT)を使って実装した。
一次元と二次元のHubbard模型に適用して時間発展を計算した。DMFT,	  DMRG, Σ(2)との比較を行った。
一次元ではDCAと Σ(2) はNc →	  ∞で一致し、熱平衡化しない。二次元では波数空間で非等方的な準粒子緩和がprethermalizaLonとthermalizaLonの間に見られた。

今後の課題
・k点数、クラスターサイズの系統的評価。
・大規模並列化、メモリの問題。
・より適用範囲の広いcluster	  solverの開発(higher-­‐order	  PT,	  NCA,	  ED,	  ...)。
・光パルス電場の物理。
・秩序相(CDW,	  d-­‐wave	  SC,	  ....)への応用。
16