Solving a Macroscopic Model for High-Temperature Superconductivity

ORNL researchers settle 20-year dispute in solid-state physics

Thomas Schulthess

The potential impact of a major

breakthrough in superconductivity

research is difficult to

overstate. Currently, electrical

utilities must transmit electricity in

excess of thousands of volts ? despite

the fact that actual electrical usage is

typically in volts or hundreds of volts

? simply because so much energy is

otherwise lost during transmission. If

we were able to build large-scale power

grids with superconducting materials

? that is, with materials that can carry

a current with zero resistance ? we

could generate and transmit power

over long distances at much smaller

voltages. This would eliminate enormous

amounts of wasted energy, render

entire segments of today?s electrical

transmission systems unnecessary,

allow us to systematically exploit alternative

energy sources, and dramatically

reduce the cost of energy worldwide.

Of course, this ideal has not yet been

achieved, primarily because no materials

currently exist that can become superconductive

at room temperatures.

Historically, scientists had to cool conventional

conductors close to absolute

zero to produce the phenomenon. Then,

in the 1980s, researchers discovered a

new class of materials that become

superconductive at much higher temperatures,

kicking off a new era of

exploration in the field. Although still

requiring very cold temperatures, these

new materials could be cooled with liquid

nitrogen, which is much easier and

less expensive to produce than the liquid

helium required by the old materials.

Today, many scientists and engineers

are engaged in research to develop practical

high-temperature superconductors

for power transmission and many other

applications.However, the phenomenon

of high-temperature superconductivity

is still poorly understood.While a

microscopic theory explaining conventional

superconductivity has existed for

half a century, most scientists agree that

it is not applicable to this new class of

materials. The most promising model

proposed to describe high-temperature

superconductivity, called the twodimensional

(2-D) Hubbard model, has

been unproven and controversial for

many years.

Now, thanks to new techniques developed

by a research team at the National

Center for Computational Sciences at

Oak Ridge National Laboratory

(ORNL), the 2-D Hubbard model has

finally been solved. By explicitly proving

that the model describes high-temperature

superconductivity, we have helped

to settle a debate that has raged for two

decades, and have opened the door to a

much deeper understanding of this phenomenon.

Even more significantly, the

work may mark an important step

toward the development of a canonical

set of equations for understanding the

behavior of materials at all scales.

## Model for High-Temperature Superconductivity

**Модераторы:** mike@in-russia, varlash

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### Model for High-Temperature Superconductivity

С уважением, Морозов Валерий Борисович

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Understanding superconductivity

In normal materials, electrical resistance

causes energy loss during conduction

of a current. Superconductive

materials conduct electricity with no

resistance when cooled below a certain

temperature. Superconductivity is the

result of a transition to a new phase in a

material ? a macroscopic quantum

phenomenon ? in which a macroscopic

number of electrons (on the scale of

1023) condense into a coherent quantum

state. If this state is that of a current

flowing through a material, the current

will flow, theoretically indefinitely (provided

the material is kept in the superconducting

state), and the material will

be able to transmit electric power with

no energy loss.

Conventional superconductors were

first discovered in the early 20th century,

when Dutch physicist Heike Kamerlingh

Onnes cooled mercury to four degrees

Kelvin (-269 degrees Celsius), and

observed that the material?s electrical

resistance dropped to zero. Research into

conventional superconductors progressed

for several decades until 1957, when three

American physicists ? John Bardeen,

Leon Cooper and John Schrieffer ?

advanced the first strong mathematical

explanation for conventional superconductivity,

which came to be known as

BCS Theory. (This work won the physicists

the Nobel Prize in 1972.)

In the 1980s, a new class of superconductive

materials was discovered, which

become superconductive at much higher

temperatures. These new ceramic

materials ? copper-oxides, or cuprates,

combined with various other elements

? achieve superconductivity at temperatures

as high as 138 degrees Kelvin,

representing a major jump toward

room-temperature superconductors.

Physicists quickly recognized that superconductivity

in these new materials,

while ultimately producing the same

effect as conventional superconductivity,

was a very different phenomenon. As a

result, BCS Theory, which had served as

the standard model for describing

superconductivity for decades, was simply

not adequate to describe it.

In normal materials, electrical resistance

causes energy loss during conduction

of a current. Superconductive

materials conduct electricity with no

resistance when cooled below a certain

temperature. Superconductivity is the

result of a transition to a new phase in a

material ? a macroscopic quantum

phenomenon ? in which a macroscopic

number of electrons (on the scale of

1023) condense into a coherent quantum

state. If this state is that of a current

flowing through a material, the current

will flow, theoretically indefinitely (provided

the material is kept in the superconducting

state), and the material will

be able to transmit electric power with

no energy loss.

Conventional superconductors were

first discovered in the early 20th century,

when Dutch physicist Heike Kamerlingh

Onnes cooled mercury to four degrees

Kelvin (-269 degrees Celsius), and

observed that the material?s electrical

resistance dropped to zero. Research into

conventional superconductors progressed

for several decades until 1957, when three

American physicists ? John Bardeen,

Leon Cooper and John Schrieffer ?

advanced the first strong mathematical

explanation for conventional superconductivity,

which came to be known as

BCS Theory. (This work won the physicists

the Nobel Prize in 1972.)

In the 1980s, a new class of superconductive

materials was discovered, which

become superconductive at much higher

temperatures. These new ceramic

materials ? copper-oxides, or cuprates,

combined with various other elements

? achieve superconductivity at temperatures

as high as 138 degrees Kelvin,

representing a major jump toward

room-temperature superconductors.

Physicists quickly recognized that superconductivity

in these new materials,

while ultimately producing the same

effect as conventional superconductivity,

was a very different phenomenon. As a

result, BCS Theory, which had served as

the standard model for describing

superconductivity for decades, was simply

not adequate to describe it.

С уважением, Морозов Валерий Борисович

- morozov
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### Mystery of high-temperature superconductors

Mystery of high-temperature

superconductors

Within a few years of the discovery of

high-temperature superconductors, a

number of physicists suggested several

new mechanisms to describe this phenomenon

based on the 2-D Hubbard

model. This model is derived from

chemical structures, and purports to

describe superconductivity with a few

microscopically derivable parameters:

the probability that carriers (electrons

or holes) hop from one site

to another on a lattice of atoms,

an energy penalty for two carriers

to occupy the same site at once

(one can easily imagine that two

electrons repel each other)

the concentration of carriers.

However, there was disagreement

within the scientific community about

whether the model encompassed a

superconducting state in the typical

parameter and temperature range of the

cuprate superconductors and, as a

result, whether the model was appropriate

at all. (If the model did not include a

high-temperature superconducting

state, then it was not an appropriate

description of reality.)

At the time, the 2-D Hubbard model

was unsolvable due to the scale of the

computation required. Since superconductivity

is a macroscopic effect, any simulation

would need to encompass a lattice

on the scale of 1023 sites.At the same

time, since the model also must describe

the behavior of individual electrons hopping

from site to site and interacting with

each other, the simulation also had to

include calculations on the scale of a few

lattice spacings at a time. In short, the

model presented a difficult multi-scale

problem, making it extremely computationally

complex ? if not impossible ?

to solve numerically. Because of these difficulties,

the questions about the 2-D

Hubbard model remained unresolved for

more than a decade.

superconductors

Within a few years of the discovery of

high-temperature superconductors, a

number of physicists suggested several

new mechanisms to describe this phenomenon

based on the 2-D Hubbard

model. This model is derived from

chemical structures, and purports to

describe superconductivity with a few

microscopically derivable parameters:

the probability that carriers (electrons

or holes) hop from one site

to another on a lattice of atoms,

an energy penalty for two carriers

to occupy the same site at once

(one can easily imagine that two

electrons repel each other)

the concentration of carriers.

However, there was disagreement

within the scientific community about

whether the model encompassed a

superconducting state in the typical

parameter and temperature range of the

cuprate superconductors and, as a

result, whether the model was appropriate

at all. (If the model did not include a

high-temperature superconducting

state, then it was not an appropriate

description of reality.)

At the time, the 2-D Hubbard model

was unsolvable due to the scale of the

computation required. Since superconductivity

is a macroscopic effect, any simulation

would need to encompass a lattice

on the scale of 1023 sites.At the same

time, since the model also must describe

the behavior of individual electrons hopping

from site to site and interacting with

each other, the simulation also had to

include calculations on the scale of a few

lattice spacings at a time. In short, the

model presented a difficult multi-scale

problem, making it extremely computationally

complex ? if not impossible ?

to solve numerically. Because of these difficulties,

the questions about the 2-D

Hubbard model remained unresolved for

more than a decade.

С уважением, Морозов Валерий Борисович

- morozov
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Solving the 2-D Hubbard model

In the 1990s, one of the Center?s team

members,Mark Jarrell of the University

of Cincinnati, finally began to break this

deadlock with the development of

dynamical cluster approximation

(DCA), an extension of dynamical mean

field theory to systematically include

non-local correlations.Mean field theory

is the standard tool used in statistical

physics to describe multi-scale phenomena,

such as the interaction between an

entire system of particles and an individual

particle in the system. Dynamical

mean field theory is a quantum version

of this theory, allowing the study of

quantum fluctuations in one atom that

is embedded within many other atoms.

Jarrell?s innovation was a technique for

embedding a cluster of atoms ? on

which the many-body problem is solved

rigorously with quantum Monte Carlo

(QMC) simulations ? into a mean

field, allowing for the description of the

cluster?s interactions with the macroscopic

soup of particles in the system.

Using DCA and QMC techniques, the

team was finally able to simulate a correlated

system encompassing the macroscopic

scale, as well as the scale of a

small cluster of atoms within the system.

Jarrell worked with Thomas Maier of

ORNL to perform simulations using

these techniques with the QMC/DCA

code to solve for small clusters (foursites)

on an infinite lattice, using a

super-scalar computing system at

ORNL. The results were promising,

reproducing the phase diagram of the

cuprates, and demonstrating that the

model does achieve a superconductive

state at higher temperatures.However,

the work was not conclusive because it

also produced a magnetically ordered

state at finite temperatures that seemed

to be an artifact of the mean field

approximation. (Magnetic ordering at

finite temperature in the 2-D Hubbard

model is forbidden by a mathematical

theorem, the Mermin-Wagner theorem.)

Since the magnetic ordering was a

result of the very small clusters used in

the simulations, it remained to be seen if

the superconductivity was also an artifact

of the small clusters and the meanfield

approximation.However, the

super-scalar computer system used at

the time was simply not adequate to run

QMC simulations for larger clusters.

In the 1990s, one of the Center?s team

members,Mark Jarrell of the University

of Cincinnati, finally began to break this

deadlock with the development of

dynamical cluster approximation

(DCA), an extension of dynamical mean

field theory to systematically include

non-local correlations.Mean field theory

is the standard tool used in statistical

physics to describe multi-scale phenomena,

such as the interaction between an

entire system of particles and an individual

particle in the system. Dynamical

mean field theory is a quantum version

of this theory, allowing the study of

quantum fluctuations in one atom that

is embedded within many other atoms.

Jarrell?s innovation was a technique for

embedding a cluster of atoms ? on

which the many-body problem is solved

rigorously with quantum Monte Carlo

(QMC) simulations ? into a mean

field, allowing for the description of the

cluster?s interactions with the macroscopic

soup of particles in the system.

Using DCA and QMC techniques, the

team was finally able to simulate a correlated

system encompassing the macroscopic

scale, as well as the scale of a

small cluster of atoms within the system.

Jarrell worked with Thomas Maier of

ORNL to perform simulations using

these techniques with the QMC/DCA

code to solve for small clusters (foursites)

on an infinite lattice, using a

super-scalar computing system at

ORNL. The results were promising,

reproducing the phase diagram of the

cuprates, and demonstrating that the

model does achieve a superconductive

state at higher temperatures.However,

the work was not conclusive because it

also produced a magnetically ordered

state at finite temperatures that seemed

to be an artifact of the mean field

approximation. (Magnetic ordering at

finite temperature in the 2-D Hubbard

model is forbidden by a mathematical

theorem, the Mermin-Wagner theorem.)

Since the magnetic ordering was a

result of the very small clusters used in

the simulations, it remained to be seen if

the superconductivity was also an artifact

of the small clusters and the meanfield

approximation.However, the

super-scalar computer system used at

the time was simply not adequate to run

QMC simulations for larger clusters.

С уважением, Морозов Валерий Борисович

- morozov
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Employing a vector supercomputer

Frequently, an application cannot take

advantage of a massively parallel system

because its physical domain is too small

to be distributed over a large number of

processors. In ORNL?s case, the

QMC/DCA code (specifically, an individual

Markov chain in the Monte Carlo

simulation) does not scale well on scalar

processors, and the algorithm the team

was using became too computationally

inefficient when expanded beyond foursite

clusters. The algorithm includes a

step at which the system must perform

an outer-product. As the vectors became

larger, the cache of the scalar system was

not able to keep up, causing the performance

to decline drastically. At that

point, adding parallel processors offered

no advantages. The team needed a different

type of supercomputer.

Given the limited scalability of the

application, the researchers needed a

system with very high performance and

high memory bandwidth on the processors

to supply as many floating-point

operations as possible on fewer processors.

They felt the best approach to this

problem was to employ a system with

the fastest vector processors available.

The team began using the Cray X1 system

and later moved to the Cray X1E

system at ORNL.

The X1E supercomputer is a dual

processor implementation of the X1 system.

With the doubling of the number

of processors on a board and a higher

clock speed, the X1E is able to deliver

more than twice the performance in the

same package as the X1. It is binary

compatible to the X1 system and uses

multi-streaming vector processors

(MSP) to achieve very high performance

per processor. The multi-streaming

architecture gives the programmer yet

another level of parallelism to facilitate a

higher sustained peak performance.

ORNL?s Cray X1E used in this study

contains 1024 MSP units each capable

of 18 GFLOPS.

It was necessary to perform two

major areas of computation:

level-3 basic linear algebra subprograms

(BLAS), which are highly

computationally-intensive and typically

achieve a very high fraction

of peak performance (80 to 90

percent)

level-2 BLAS, which are extremely

memory bandwidth-limited.With

the ability to fetch and store vector

arguments directly into the vector

registers, the Cray X1 and Cray

X1E systems offered much higher

memory bandwidth than typical

massively parallel systems.

Thanks to the system?s vector processors

and its high memory bandwidth,

calculations could be performed with

clusters of up to 30 atoms with only

minimal changes to the code. This scale

turned out to be enough to show that

the magnetic order that had plagued the

earlier work disappears monotonically as

the clusters grow larger, while the superconductivity

effect remains. In short, the

team proved for the first time that the

2-D Hubbard model does include a

high-temperature superconductive state

and is an accurate representation of the

phenomenon ? settling a 20-year

dispute in solid-state physics.

Frequently, an application cannot take

advantage of a massively parallel system

because its physical domain is too small

to be distributed over a large number of

processors. In ORNL?s case, the

QMC/DCA code (specifically, an individual

Markov chain in the Monte Carlo

simulation) does not scale well on scalar

processors, and the algorithm the team

was using became too computationally

inefficient when expanded beyond foursite

clusters. The algorithm includes a

step at which the system must perform

an outer-product. As the vectors became

larger, the cache of the scalar system was

not able to keep up, causing the performance

to decline drastically. At that

point, adding parallel processors offered

no advantages. The team needed a different

type of supercomputer.

Given the limited scalability of the

application, the researchers needed a

system with very high performance and

high memory bandwidth on the processors

to supply as many floating-point

operations as possible on fewer processors.

They felt the best approach to this

problem was to employ a system with

the fastest vector processors available.

The team began using the Cray X1 system

and later moved to the Cray X1E

system at ORNL.

The X1E supercomputer is a dual

processor implementation of the X1 system.

With the doubling of the number

of processors on a board and a higher

clock speed, the X1E is able to deliver

more than twice the performance in the

same package as the X1. It is binary

compatible to the X1 system and uses

multi-streaming vector processors

(MSP) to achieve very high performance

per processor. The multi-streaming

architecture gives the programmer yet

another level of parallelism to facilitate a

higher sustained peak performance.

ORNL?s Cray X1E used in this study

contains 1024 MSP units each capable

of 18 GFLOPS.

It was necessary to perform two

major areas of computation:

level-3 basic linear algebra subprograms

(BLAS), which are highly

computationally-intensive and typically

achieve a very high fraction

of peak performance (80 to 90

percent)

level-2 BLAS, which are extremely

memory bandwidth-limited.With

the ability to fetch and store vector

arguments directly into the vector

registers, the Cray X1 and Cray

X1E systems offered much higher

memory bandwidth than typical

massively parallel systems.

Thanks to the system?s vector processors

and its high memory bandwidth,

calculations could be performed with

clusters of up to 30 atoms with only

minimal changes to the code. This scale

turned out to be enough to show that

the magnetic order that had plagued the

earlier work disappears monotonically as

the clusters grow larger, while the superconductivity

effect remains. In short, the

team proved for the first time that the

2-D Hubbard model does include a

high-temperature superconductive state

and is an accurate representation of the

phenomenon ? settling a 20-year

dispute in solid-state physics.

С уважением, Морозов Валерий Борисович

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Looking ahead

While solving the 2-D Hubbard

model represents an important step forward

in understanding the physics of

high-temperature superconductivity, we

still have much work to do in this area.

The next step is to demonstrate that a

generalized version of this model can

not only describe superconductivity in

principle, but also can accurately reflect

the behavior of specific materials and

explain why different materials become

superconductive at different temperatures.

One potential outcome of gaining

this level of understanding would be the

ability to design and produce even higher-

temperature superconductive compounds

that could be used in a variety

of applications ? from electric grids to

quantum supercomputers.

However, putting aside the practical

engineering possibilities, we believe this

work could have even more profound

ramifications by serving as a first step

toward a true canonical solution for

quantum problems in complex materials.

In the field of computational fluid

dynamics, for example, the Navier-

Stokes equations provide us with the

ability to solve a broad range of engineering

problems. For materials sciences,

chemistry and nanoscience, the

many-body Schrödinger equation plays

a similar role.However, in contrast to

Navier-Stokes, no canonical approach

currently exists to solve the many-body

Schrödinger equation. Finding one

would change solid-state physics forever,

and greatly expand the role of computation

in scientific discovery.

Thomas Schulthess is Group

Leader, Computational Material

Sciences and Nanomaterials Theory

Institute, ORNL. He can be reached at

editor@ScientificComputing.com.

While solving the 2-D Hubbard

model represents an important step forward

in understanding the physics of

high-temperature superconductivity, we

still have much work to do in this area.

The next step is to demonstrate that a

generalized version of this model can

not only describe superconductivity in

principle, but also can accurately reflect

the behavior of specific materials and

explain why different materials become

superconductive at different temperatures.

One potential outcome of gaining

this level of understanding would be the

ability to design and produce even higher-

temperature superconductive compounds

that could be used in a variety

of applications ? from electric grids to

quantum supercomputers.

However, putting aside the practical

engineering possibilities, we believe this

work could have even more profound

ramifications by serving as a first step

toward a true canonical solution for

quantum problems in complex materials.

In the field of computational fluid

dynamics, for example, the Navier-

Stokes equations provide us with the

ability to solve a broad range of engineering

problems. For materials sciences,

chemistry and nanoscience, the

many-body Schrödinger equation plays

a similar role.However, in contrast to

Navier-Stokes, no canonical approach

currently exists to solve the many-body

Schrödinger equation. Finding one

would change solid-state physics forever,

and greatly expand the role of computation

in scientific discovery.

Thomas Schulthess is Group

Leader, Computational Material

Sciences and Nanomaterials Theory

Institute, ORNL. He can be reached at

editor@ScientificComputing.com.

С уважением, Морозов Валерий Борисович

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