Computational
cience AllSfor11

K computer Newsletter
October 2015 No.

Visualizing the Universe
in Any Material
Scientists in search of“scale-free” critical phenomena

Visualizing the Universe
in Any Material
Scientists in search of “scale-free” critical phenomena

Kenji Harada
Assistant Professor of the Graduate School of Informatics, Kyoto University

Imaginary time

2-dimensional spaceat imaginary time B

Electron, etc.

B2-dimensional spaceat imaginary time A

World lineA

In the image on the right, the horizontal direction represents 2-dimensional
space for electrons, etc. and the verticaldirection indicates imaginary time.
Imaginary time is measured in imaginary numbers. (An imaginary number,when squared, gives a negative result.)
Imaginary time is used to express
quantum ﬂuctuations numerically.

Fig. 1 Deconﬁned quantum critical phenomenon

Harada is a member of a team with a mission to discover new critical phenomena, for which they are making full use of K and other supercomputers.
The phenomena under investigation represent a form of phase transition,
where a substance in one state suddenly changes to another state under aparticular set of conditions. Such phenomena are not limited by physicalscale and are thought to occur in everything from elementary particle, thesmallest unit of matter, to the universe itself.

and the state of world line
Colored lines in the ﬁgure show the paths (world lines)of electrons and other particles in imaginary
time-space. Veriﬁcation of deconﬁned quantum
critical phenomena has its basis in data from world
line calculations.

Looking back at his ﬁrst involvement incomputers, Harada says, “In high school,
we built a computer graphics program as
an extracurricular project. During theendeavor, I encountered ray tracing, amethod of computing and tracking thelight reaching our eyes from reﬂectedobjects.”“Our CG program was simple because
the computerʼs performance was limited,
but I was deeply impressed by its potential to create a seemingly real world of
light beams and objects whose images
could not easily be distinguished fromreal objects.”
Today, Harada is engaged in simulating
inﬁnitesimally small elements, such asatoms and electrons, on supercomputers.
“Itʼs impossible to simulate the world of
quanta on current mainstream computers,” he says. “Even an electron is so unstable that it is continually ﬂuctuating. From
this fact, it appears that there occur in thequantum world various events that are
hard for us to imagine happening. Our job
is to ﬁnd ways to simulate a range of phenomena occurring in substances that are
in line with certain quantum properties.”
But this task entails an enormous
number of calculations. Imagine simulating a state where the number of atoms
has increased 10 times more than previously. The number of calculations soars bya margin of 10 raised to the power 10, notsimply by 10 times.”
To overcome this diﬃculty, Haradaʼsteam carefully chooses sample substances and phenomena that are easy to compute, and the group constantly reviews its
calculations and procedures so as toobtain the best possible results from aquantum-world simulation.
“Itʼs a diﬃcult but really interesting
assignment,” says Harada.

Two diﬀerent criticalphenomena occurringconcurrently
One kind of phenomenon Harada issimulating is called critical phenomena, aform of phase transition.
So what is phase transition? “For example,Intensity of the order

Diﬃculties in simulating
the quantum world

Néel ordered state

VBS state

Quantum critical point

Pressure, etc.

Fig. 2 Deconﬁned quantum critical phenomenon
Compared to the Néel ordered state where electron-spin directions alternate, these spins are paired up in theValence Bond Solid (VBS) state. If a substance shows no change in its overall state after conducting a certain operation, then it is “symmetrical” to the operation. In fact, the Néel ordered state shows no change even following a90-degree rotation of lattices. However, it does change when the electron-spin directions are both rotated at thesame time. Therefore, the Néel ordered state spontaneously breaks the rotational symmetry of spin directions, notthe 90-degree rotational symmetry of lattices. On the other hand, the VBS state breaks the symmetry of the lattices,not the spin direction of the electrons. Upon increasing the pressure applied to a sample substance, the broken
rotational symmetry of electron- spin directions in the Néel ordered state is switched to the broken 90-degree
rotational symmetry of lattices in the VBS state at the transition point. Given critical phenomena are interpreted as
representing spontaneous symmetry breaking, it is thought that two diﬀerent critical phenomena occur collectively and inmutually reverse directions at the transition point between Néel and VBS ordered phases when symmetry is broken.

a phase transition occurs when waterchanges to ice at 0℃, or when some substances turn magnetic at a certain level ofpressure or temperature. Phase transitionrefers to a situation where one substanceundergoes a sudden change in its properties,”
explains Harada. Is there a diﬀerence betweena phase transition, like the water-to-ice
change, and critical phenomena?
“To explain further, a phase transition
takes place̶water to gas, for instance̶
when a substanceʼs parameters such astemperature or pressure changes at a critical point in the boundary between thetwo phases or states,” says Harada. “At thiscritical point the two phases can become
indistinguishable, the boundary disappears and the substance undergoes adrastic change. The theory of critical phenomena seeks to explain this drasticchange. And because critical phenomena
are not limited in terms of physical scale,they are scale-free. The same kind of critical phenomena occurring in materials weunderstand well are also believed to happening at the level of elementary particles, the smallest units of matter, as well as
whatever else makes up our entire universe. Research on critical phenomena,
therefore, merits much attention.”In theory, substances are subject toinnumerable kinds of critical phenomena,but only a few have been reported fortwo- and three-dimensional substances.
A critical phenomenon, then, can bethought of as a buried diamond̶very
hard to discover. Against this backdrop,
Harada and his colleagues are investigating a phenomenon, called “deconﬁnedquantum criticality,” which takes place
very rarely.
Deconﬁned quantum criticality
involves electron spin and takes place atabsolute zero (-273.15℃). Electron spin,
the source of magnetic force generation,
has a direction. And it is when many electrons spin in the same direction that amaterial turns magnetic.
With a deconﬁned quantum criticalphenomenon, the pattern of electron-spin direction collapses suddenly
under a particular circumstance, and thenthe columnar arrangement of entangled
electron-pairs appears (see Fig. 2). This
indicates two diﬀerent critical phenomena are occurring at the same time, fromthe point of symmetry.
“To clarify,” says Harada, “imagine abucket of magnetized water. If you keep
applying pressure to the water, the liquidBefore he was fully aware of the fact, Harada had
become a scientist. The transformation happened in a
graduate school environment where he received muchencouragement from senior colleagues in diﬀerent
ﬁelds, and especially after he worked with a researchsupervisor who impressed with his vigor and passion inbuilding a parallel computer from scratch.

(Harada poses in front of Sekirei, a supercomputer at The
University of Tokyoʼs Institute for Solid State Physics)

changes to ice at a certain point whilelosing its magnetism. Two seemingly
unrelated phenomena apparently occur
at the same time. A deconﬁned quantum critical phenomenon is similar tothis, where two kinds of critical phenomenon occur in mutually reverse
directions simultaneously.”
There is much debate about deconﬁned quantum critical phenomenon,
which cannot be explained by conventional theories.

K has the power toverify deconﬁned
quantum criticality
In searching for incidents of deconﬁned quantum criticality, Haradaʼsgroup has made simulations of certain
materials and changed various parameters aﬀecting them, such as the level ofapplied pressure. The researchers eventually discovered that electrons in small
amounts of these materials exhibit achange in electron spin, a phenomenon
similar to deconﬁned quantum criticality (see Fig. 1 and Fig. 3).
To verify this apparent critical phenomenon, the group has to substantiate if the same electron-spin changeoccurs in a larger portion of these substances. The task, however, requires a
great number of calculations. So Harada
has employed the K computer, one of theworldʼs fastest supercomputers, and theSekirei supercomputer at The Universityof Tokyoʼs Institute for Solid State Physics.
“Thereʼs a high probability, about 80
percent, that computer-detectedchanges will be deconﬁned quantum
critical phenomena, while twenty percent of them may not be,” says Harada.
“We assume that at the very moment achange begins (the quantum criticalpoint), previously conﬁned virtual particles (spinons) are set free and start tomove about in these substances.
Spinons apparently help hook up thetwo diﬀerent phenomena and thencause them to occur at the same time.

Fig. 3 Part of the computing program to verify deconﬁned quantum criticality (based on the ALPS/looper)

RIKEN Advanced Institute for Computational Science (AICS)
7-1-26, Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo, 650-0047, Japan
TEL: + 81 78 940 5555 FAX: + 81 78 304 4964 E-mail: aics-koho@riken.jp http://www.aics.riken.jp/en/“Weʼll work to verify this unique phenomenon by simulating free-moving
spinons on supercomputers,” he added.

Powerful aid to researchon elementary particles
and the cosmos
Whatʼs the importance of critical phenomenon research? “Whenever mywife asks how my research will be
useful, Iʼm at a loss for an answer,”
Harada says with a chuckle.
Given the fact that deconﬁned quantum criticality occurs only at absolutezero, results of the related research areunlikely to lead directly to the development of equipment and materials used
for practical purposes. Nevertheless,they will certainly have a huge impact
on our understanding of physics.
Notes Harada: “Deconﬁned quantum
criticality has long been considered totake place in elementary particles.
Calculations based on the latest
elementary particle theories are more
complex than those of sample substances weʼve used. Such criticality,
therefore, has yet to be fully veriﬁed.”“But if it is veriﬁed for these substances, the phenomenon can be considered
to be happening across all physicalscales, from elementary particles to theuniverse itself, which will contribute
greatly to advancing our understandingof particle physics and cosmology. We
would be able to see aspects of the universe in the tiniest of substances. Thatʼs
one of the most alluring prospects ofconducting research on critical phenomena.”