- BY NATALIE WOLCHOVER, SIMONS SCIENCE NEWS

The strange properties of superconducting materials called “cuprates” (bismuth strontium calcium copper oxide is shown in this microscopy image), which cannot be described by known quantum mechanical methods, may correspond to properties of black holes in higher dimensions. *Image: Robert Goddard at the National High Magnetic Field Laboratory, Tallahassee, Fla.*

According to modern quantum theory, energy fields permeate the universe, and flurries of energy in these fields, called “particles” when they are pointlike and “waves” when they are diffuse, serve as the building blocks of matter and forces. But new findings suggest this wave-particle picture offers only a superficial view of nature’s constituents.

For decades, the surface-level description of the subatomic world has been sufficient to make accurate calculations about most physical phenomena. But recently, a strange class of matter that defies description by known quantum mechanical methods has drawn physicists into the depths below.

“I’ve grown up as a physicist just living on that flatland, that 2-D space,” said Subir Sachdev, a physics professor at Harvard University who studies these strange forms of matter. Now, there is a whole new dimension to explore, he said, and “you can think of the particles as just ending on that surface.”

Of all the strange forms of matter, cuprates — copper-containing metals that exhibit a property called high-temperature superconductivity — may be the strangest. In new research published online June 24 in the Journal of High Energy Physics, physicists at the University of California-Santa Barbara have explored the deeper phenomena that they claim are connected to the perplexing “surface-level” behavior of cuprates. By focusing their calculations on that underlying environment, the researchers derived a formula for the conductivity of cuprates that was previously known only from experiments.

“The amazing thing is you start with this theory and out you get the conductivity of these strange superconductors,” said Sachdev, who was not involved with the work.

The results bolster the evidence that this new way of looking at nature’s building blocks is real and that it is “strikingly literal,” said Jan Zaanen, a theoretical physicist at Leiden University in the Netherlands.

What’s more, the results could be seen as an unusual, indirect kind of evidence for string theory — a 40-year-old framework that weaves together quantum mechanics and gravity and is as mathematically elegant and profoundly explanatory as it is unproven.

With looming questions about the nature of dark matter, the mysterious substance thought to constitute 84 percent of the mass in the universe, and the search for a “theory of everything” that mathematically describes all of nature, researchers say the findings could have sweeping implications.

“There is a realistic chance that we will make enormous progress in fundamental physics in the next couple of years,” Zaanen said. “It’s moving very, very quickly.”

**Below the Surface**

If waves and particles are like the turbulence on the surface of a pond, the connection between that turbulence and events in the interior of the pond was first described by a mathematical principle discovered in 1997. In a landmark paper, Juan Maldacena, an Argentinian-American physicist then at Harvard University and now at the Institute for Advanced Study in Princeton, N.J., showed that events taking place in a 3-D region of space mathematically correspond to very different events taking place on that region’s 2-D boundary. (Events in 4-D also correspond to events in 3-D, and 5-D to 4-D and so on.)

Consider the 3-D interior and 2-D surface of the metaphoric pond. For the correspondence to work, the interior must be mathematically described by string theory, in which electrons, photons, gravitons and the rest of nature’s building blocks are invisibly small, one-dimensional lines, or “strings.” Mass and other macroscopic properties correspond to the strings’ vibrations, and interactions between different kinds of matter and forces come from the way strings split and connect. These strings live inside the pond.

Now, imagine that the 2-D surface of the pond is described by quantum mechanics. Particles are the splashes on the surface, and waves are the cascade of ripples from those splashes. On the surface of this imaginary pond, there is no force of gravity.

The holographic duality, discovered in 1997 by Juan Maldacena, says that events inside a region of space that involve gravity and are described by string theory are mathematically equivalent to events on the surface of that region that involve particles and are gravity-free. *Illustration: Annenberg Lerner 2013*

Maldacena’s discovery, known as the holographic duality, showed that events in the interior region, which involve gravity and are described by string theory, are mathematically translatable to events on the surface, which are gravity-free and described by quantum particle theories.

“To understand this relationship, the crucial aspect is when the gravity theory is easy to analyze, then the particles on the boundary” — or, in the pond analogy, the surface — “are interacting very strongly with each other,” Maldacena said. The converse is also true: When the particles are calm on the surface, as they are in most forms of matter, then the situation in the pond’s interior is extremely complicated.

That contrast is what makes the duality useful.

The strange class of materials that includes cuprates belongs in the first category; experiments suggest that particles in these materials interact so strongly with one another that they lose their individuality. Physicists say the particles are “strongly correlated.” The wavy ripples corresponding to each overlap so much that a kind of swarm effect is believed to occur. Strongly correlated matter can behave in diverse and unexpected ways that are difficult or in some cases impossible to describe with known quantum mechanical methods, said Sean Hartnoll, a physics professor at Stanford University. “You need a different way of looking at them than starting from single particle descriptions,” he said. “You don’t try to explain the ocean in terms of individual water molecules.”

If strongly correlated matter is thought of as “living” on the 2-D surface of a pond, the holographic duality suggests that the extreme turbulence on that surface is mathematically equivalent to still waters in the interior. Physicists can get at the surface-level behavior by studying the parallel, but much simpler, situation below. “You can compute things in that tranquil world,” Zaanen said.

In the mathematical parlance of the holographic duality, certain strongly correlated matter in 2-D corresponds, in 3-D, to a black hole — an infinitely dense object with an inescapable gravitational pull, which is mathematically simple. “These very complicated quantum mechanical collective effects are beautifully captured by black hole physics,” said Hong Liu, an associate professor of physics at the Massachusetts Institute of Technology. “For strongly correlated systems, if you put an electron into the system, it will immediately ‘disappear’ — you can no longer track it.” It’s like an object falling into a black hole.

**A Superconductive Model**

Increasingly over the past decade, studying the black hole equivalents of strongly correlated forms of matter has yielded groundbreaking results, such as a new equation for the viscosity of strongly interacting fluids and a better grasp of interactions between quarks and gluons, which are particles found in the nuclei of atoms.

Gary Horowitz, right, a physics professor at UC-Santa Barbara, and Jorge Santos, a post-doctoral researcher in Horowitz’s group, have modeled strange materials called cuprates as peculiarly shaped black holes in higher dimensions. *Photo: Courtesy of Gary Horowitz*

Now, Gary Horowitz, a string theorist at UC-Santa Barbara, and Jorge Santos, a post-doctoral researcher in Horowitz’s group, have applied the holographic duality to cuprates. They derived a formula for the conductivity of the metals, which are approximately 2-D, by studying related properties of what may be their counterpart in 3-D: an electrically charged, peculiarly shaped black hole.

The work took numerical virtuosity. In cuprates, a swarm of strongly correlated electrons moves through a fixed lattice of atoms. Modeling the metals with the holographic duality therefore required working the equivalent of a lattice into the structure of the corresponding black hole by giving it a corrugated outer surface, or horizon.

“When it comes to playing ball with black holes, you need Gary [Horowitz],” Zaanen said.

To determine a formula for the conductivity of cuprates, Horowitz and Santos had to study how light would interact with the complicated horizon of their black hole. The equations were too thorny to solve exactly, so they found approximate solutions using a computer. In their first paper detailing this approach, co-authored by Cambridge University physics professor David Tong and published in July 2012 in the Journal of High Energy Physics, they derived a formula that matched the conductivity of cuprates at high temperatures in response to an alternating current. In the new work, they extended the calculation down to the temperature range in which cuprates become superconductive, or conduct electricity with no resistance, and again found a close match with experimental measurements of real cuprates.

“It amazes me that such a simple gravity model is able to reproduce any feature of a real material,” Horowitz said. “So this is encouraging us to think harder.”

The accuracy of Horowitz and Santos’ model breaks down in some significant cases, such as for alternating currents with extremely high frequencies, but Sachdev said that considering how simple the corrugated black hole model is, “it couldn’t have worked any better.” Incorporating more of the microscopic details of cuprates into the structure of the black hole will probably deepen their congruence, he said.

Hartnoll, who recently used the holographic duality to model metal-insulator transitions in strongly correlated materials, hopes to build on the results by solving Horowitz and Santos’ equations exactly. “They have an input and an output; we’d like to decompress it and understand the critical steps in between,” he said. Doing so would reveal where the conductivity formula originates in the black hole environment, providing more insights about the corresponding forces at play inside cuprates.

**A New Duality**

Understanding the physics of cuprates could have important practical applications. Most metals start to superconduct when their temperature drops close to absolute zero. But, for reasons not completely understood, cuprates exhibit superconductivity at much more accessible temperatures, making them useful for devices ranging from high-power electrical cables to ship propulsion motors. Cuprates are brittle and expensive, however, and engineering better versions by tweaking their properties could lead to dramatic improvements in a range of technologies, from magnetically levitating vehicles and other devices to more efficient power grids.

There is also the potential for advancing fundamental physics. If the holographic duality yields increasingly accurate predictions about the behavior of cuprates and other strongly correlated materials, these materials can be conceived as, essentially, being black holes in higher dimensions.

“If we had a model which reproduced all the features of a material, it could be viewed as a theory of it — a very unusual kind of theory, but given the duality, it’s equivalent to any theory you would produce on the boundary, with the usual particles,” Horowitz said. “And it might just be a lot simpler.”

The computer-rendered surface, or horizon, of a black hole that was used in new research as a model of materials called cuprates. The undulations on the horizon correspond to the periodic lattice of atoms inside cuprates.*Illustration: Gary Horowitz and Jorge Santos*

The holographic duality echoes the wave-particle duality that led to the development of quantum mechanics. In the early 1900s, light, which was previously thought to be a wave, seemed perplexing in some experiments unless it was treated as particles, and electrons, thought to be particles, sometimes didn’t make sense unless they were conceived as waves. “The wave-particle duality was, when first proposed, a big surprise because these were two seemingly different concepts, and we learned that they are the same thing,” Horowitz said. The holographic duality “is more sophisticated, but it has that same feature,” he said. “You have two very different-seeming objects that turn out to be completely equivalent.”

But how does the holographic duality factor into our understanding of nature? Are the one-dimensional strings from the pond analogy real? Not necessarily, physicists say. In fact, the strings never factored into Horowitz and Santos’ calculations of the properties of the black hole they used as a model of cuprates. But the findings do give physicists a sense that “all these theories that we thought were different are actually all related,” Maldacena said. “It shows that string theory is not disconnected from the rest of physics.”

String theory may simply be the best mathematical language for grappling with certain aspects of reality, the physicists interviewed for this article said.

“Physics was traditionally reductionist; it wants to take something complicated and find out what the building blocks are,” Hartnoll explained. “The point is there’s not a unique way to do that: In some cases, electrons could be the building blocks, but in others, collective excitations of electrons are playing a more fundamental role than any of the individual electrons.

“We are trying to find the right building blocks to describe these strange phases of matter,” he said. “And they might be strings in one higher dimension.”

As physicists interpret what it means that particles in a strange, brittle metal mathematically correspond to strings and a peculiar black hole that exists — at least theoretically — in a higher dimension, the holographic duality enables them to “think differently about the mysteries in the laboratories,” Zaanen said. “And perhaps it’s not only about thinking differently; it’s about seeing the real, beautiful facts.”

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