## Final Report Summary - CORTOPO (Correlations and Topology in Electronic Systems)

The goal of the project "Correlations and Topology in Electronic Systems", performed by the group of Dr. Erez Berg at the Weizmann Institute, Israel, is to deepen the understanding of collective phenomena that occur in systems of strongly interacting electrons at ultra-low temperatures. Such systems exhibit a fascinating interplay between quantum mechanics and many-body physics. Among the behaviors that emerge under such conditions are superconductivity, a collective state of electrons that conducts electricity with no resistance, as well as other forms of electronic order (such as spin and charge order). Even more interestingly, exotic ``topological'' forms of order may emerge, which cannot be characterized by any local order parameter; instead, they are described by a hidden, non-local ordering, encoded in the many-particle wavefunction.

In this research program, we have tackled two of the main outstanding problems in the field of theoretical condensed matter physics. The first is the problem of quantum criticality in metals; these are itinerant electron systems with strong interactions, which undergo a zero temperature transition from a plain metallic state into an ordered (spontaneously symmetry-broken) state. The properties of such a critical quantum fluid, which is just on the verge of ordering and therefore exhibits large fluctuations, is not well understood. The second problem we attached is the problem of topologically non-trivial phases of matter in the presence of strong interparticle interactions. The question is how do the phases familiar from the classification of non-interacting fermion systems get modified due to the Coulomb interactions between the electrons. Moreover, can the same Coulomb interactions be useful to realize novel topological phases that are not accessible without the interactions? On both problems, we have made significant progress by throwing in a fresh set of tools, which has not been attempted in this context previously.

The problem of quantum criticality in metals was attacked using a numerical technique, called Quantum Monte Carlo (QMC). This is a numerical technique that involves highly demanding, large-scale numerical simulations performed on supercomputer clusters, enabling solutions of models describing the physics of such systems. The advantage of this technique is that it is free of uncontrolled approximations, and thus gives the exact answer for the stated model. It has not applied before to this set of problems, because it was believed that the infamous ``fermion sign problem'' will prevent efficient simulations at low temperatures. Dr. Berg and his group were the first to show that a large class of models describing quantum critical points in metals can be formulated in a way that the fermion signs all cancel, enabling an efficient numerical solution down to extremely low temperatures.

Following this insight, the Berg group (in collaboration with three different groups across the world) have performed extensive simulations of three different types of metallic critical points: Ising nematic, antiferromagnetic, and ferromagnetic critical points. The simulations were shown to converge with respect to both statistical errors and finite size effects, enabling us to access the relevant physics. A rich variety of physical phenomena were observed in these models: in particular, superconductivity was found to emerge around the critical point, the the maximum superconducting transition temperature occurring very near the critical point; and the strong scattering of electrons at criticality leads to the breakdown of the traditional Fermi liquid theory. Some of the findings are consistent with the expectations from approximate field-theoretic treatments of the problem; others deviate substantially from the theoretical expectations. The behavior observed in our simulations is strikingly similar to that seen in many highly-correlated materials.

On a different front, the Berg group has made progress in the study of topological phases in condensed matter systems. The PI and his collaborators described a new way to produce a topological phase of matter called a "time-reversal-invariant" (TRI) topological superconductor in a quantum wire. This is a generalization of the time-reversal broken topological superconducting phase to the case in which no magnetic field is applied, and hence time-reversal symmetry is preserved. Several different physical mechanisms for the for- mation of TRI topological superconductors were proposed, either by controlling the relative phase of the superconductors coupled to the nanowire or by intrinsic Coulomb interactions in the wire. Concrete experimental signatures of the TRI topological state were studied. Several other groups followed up on this work and extended these ideas.

Topological superconductors states are elusive in nature; the same non-local properties that make them appealing also makes detecting them difficult. The recent reports of signatures of Majorana zero modes call for a unique "smoking gun" experiment that can unambiguously identify them. Dr. Berg has proposed several different experimental tests of Majorana zero modes. These include signatures in spin-resolved shot noise experiments, offering a robust pattern of current correlations that

is markedly distinct from that of non-topological states, and an experimental protocol that demonstrates the non-Abelian nature of Majoranas through a "fermion pumping" effect. The latter experiment, proposed in the context of systems of ultra-cold atoms, is now being considered by solid state experimentalists.

Finally, Dr. Berg and his collaborators proposed unique signatures that can be used to detect other topological states of matter, known as a quantum spin liquid. This state has been proposed several decades ago as the ground state of a frustrated magnetic insulator; recently, several potential candidate materials have been discovered. These systems lack any form of conventional order (such as long-range magnetic order); instead, they exhibit topological order, manifested in the fractionalized nature of their excitations. Dr. Berg found that, upon interfacing the quantum spin liquid with a superconductor, electrons injected into the spin liquid can leave some of their quantum numbers - spin, charge, or their fermionic statistics - at the boundary, and propagate coherently as fractionalized composite particles. Such a "coherent transmutation" would be a dramatic confirmation of the topological nature of the host material.

In this research program, we have tackled two of the main outstanding problems in the field of theoretical condensed matter physics. The first is the problem of quantum criticality in metals; these are itinerant electron systems with strong interactions, which undergo a zero temperature transition from a plain metallic state into an ordered (spontaneously symmetry-broken) state. The properties of such a critical quantum fluid, which is just on the verge of ordering and therefore exhibits large fluctuations, is not well understood. The second problem we attached is the problem of topologically non-trivial phases of matter in the presence of strong interparticle interactions. The question is how do the phases familiar from the classification of non-interacting fermion systems get modified due to the Coulomb interactions between the electrons. Moreover, can the same Coulomb interactions be useful to realize novel topological phases that are not accessible without the interactions? On both problems, we have made significant progress by throwing in a fresh set of tools, which has not been attempted in this context previously.

The problem of quantum criticality in metals was attacked using a numerical technique, called Quantum Monte Carlo (QMC). This is a numerical technique that involves highly demanding, large-scale numerical simulations performed on supercomputer clusters, enabling solutions of models describing the physics of such systems. The advantage of this technique is that it is free of uncontrolled approximations, and thus gives the exact answer for the stated model. It has not applied before to this set of problems, because it was believed that the infamous ``fermion sign problem'' will prevent efficient simulations at low temperatures. Dr. Berg and his group were the first to show that a large class of models describing quantum critical points in metals can be formulated in a way that the fermion signs all cancel, enabling an efficient numerical solution down to extremely low temperatures.

Following this insight, the Berg group (in collaboration with three different groups across the world) have performed extensive simulations of three different types of metallic critical points: Ising nematic, antiferromagnetic, and ferromagnetic critical points. The simulations were shown to converge with respect to both statistical errors and finite size effects, enabling us to access the relevant physics. A rich variety of physical phenomena were observed in these models: in particular, superconductivity was found to emerge around the critical point, the the maximum superconducting transition temperature occurring very near the critical point; and the strong scattering of electrons at criticality leads to the breakdown of the traditional Fermi liquid theory. Some of the findings are consistent with the expectations from approximate field-theoretic treatments of the problem; others deviate substantially from the theoretical expectations. The behavior observed in our simulations is strikingly similar to that seen in many highly-correlated materials.

On a different front, the Berg group has made progress in the study of topological phases in condensed matter systems. The PI and his collaborators described a new way to produce a topological phase of matter called a "time-reversal-invariant" (TRI) topological superconductor in a quantum wire. This is a generalization of the time-reversal broken topological superconducting phase to the case in which no magnetic field is applied, and hence time-reversal symmetry is preserved. Several different physical mechanisms for the for- mation of TRI topological superconductors were proposed, either by controlling the relative phase of the superconductors coupled to the nanowire or by intrinsic Coulomb interactions in the wire. Concrete experimental signatures of the TRI topological state were studied. Several other groups followed up on this work and extended these ideas.

Topological superconductors states are elusive in nature; the same non-local properties that make them appealing also makes detecting them difficult. The recent reports of signatures of Majorana zero modes call for a unique "smoking gun" experiment that can unambiguously identify them. Dr. Berg has proposed several different experimental tests of Majorana zero modes. These include signatures in spin-resolved shot noise experiments, offering a robust pattern of current correlations that

is markedly distinct from that of non-topological states, and an experimental protocol that demonstrates the non-Abelian nature of Majoranas through a "fermion pumping" effect. The latter experiment, proposed in the context of systems of ultra-cold atoms, is now being considered by solid state experimentalists.

Finally, Dr. Berg and his collaborators proposed unique signatures that can be used to detect other topological states of matter, known as a quantum spin liquid. This state has been proposed several decades ago as the ground state of a frustrated magnetic insulator; recently, several potential candidate materials have been discovered. These systems lack any form of conventional order (such as long-range magnetic order); instead, they exhibit topological order, manifested in the fractionalized nature of their excitations. Dr. Berg found that, upon interfacing the quantum spin liquid with a superconductor, electrons injected into the spin liquid can leave some of their quantum numbers - spin, charge, or their fermionic statistics - at the boundary, and propagate coherently as fractionalized composite particles. Such a "coherent transmutation" would be a dramatic confirmation of the topological nature of the host material.