Quantum Matter

Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here we employ classical variants of these concepts, in particular Renyi entropies and their associated mutual information, to identify topological order in classical systems.

Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical Renyi entropies. We demonstrate that an additional subleading O(1) contribution generically arises for all Renyi entropies S(n) with n ≥ 2 when driving the system towards a phase transition, e.g. into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme – typically used to extract the topological contribution to the Renyi entropies – does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from Renyi entropies requires an additional level of scrutiny to distinguish topological from non-topological O(1) contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a Renyi entropy perspective.

Solitons in a ferromagnet have interesting dynamics because atomic magnetic moments behave like little gyroscopes. A domain wall in a magnetic wire can be modeled as a bead on a string: it has two soft modes, position and orientation. This "bead" rotates when it is pushed and moves when twisted.

From that one can deduce that the angular momentum is proportional to the domain wall's coordinate and the linear momentum to its azimuthal angle. In fact, the definition of conserved momenta for magnetic solitons has been discussed in the scientific literature for 40 years. A naive attempt to derive them from the application of Noether's theorem yields unphysical, gauge-dependent answers. To resolve this problem, we exploit a similarity between the dynamics of a ferromagnetic soliton and that of a charged particle in a magnetic field. For the latter, canonical momentum is also gauge-dependent and thus unphysical; the physical momentum is the generator of magnetic translations, a symmetry combining physical translations with gauge transformations. We use this analogy to unambiguously define conserved momenta for ferromagnetic solitons.

While entanglement entropy of ground states usually follows the area law, violations do exist, and it is important to understand their origin. In 1D they are found to be associated with quantum criticality. Until recently the only established examples of such violation in higher dimensions are free fermion ground states with Fermi surfaces, where it is found that the area law is enhanced by a logarithmic factor.  In Ref. [1], we use multi-dimensional bosonization to provide a simple derivation of this result, and show that the logarithimic factor has a 1D origin. More importantly the bosonization technique allows us to take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids. The central result of our work is that Fermi liquid interactions do not alter the leading scaling behavior of the entanglement entropy, and the logarithmic enhancement of area law is a robust property of the Fermi liquid phase. In sharp contrast to the fermioic systems with Fermi surfaces, quantum critical (or gapless) bosonic systems do not violate the area law above 1D (except for the case discussed below). The fundamental difference lies in the fact that gapless excitations live near a single point (usually origin of momentum space) in such bosonic systems, while they live around an (extended) Fermi surface in Fermi liquids. In Ref. [2], we studied entanglement properties of some specific examples of the so called Bose metal states, in which bosons neither condense (and become a superfluid) nor localize (and

insulate) at T=0. The system supports gapless excitations around ``Bose surfaces", instead of isolated points in momentum space. We showed that similar to free Fermi gas and Fermi liquids, these states violate the entanglement area law in a logarithmic fashion. Compared to ground states, much less is known concretely about entanglement in

(highly) excited states. Going back to free fermion systems, in [3] we show that there exists a duality relation between ground and excited states, and the area law obeyed by ground state turns into a volume law for excited states, something that is widely expected but very hard to prove. Most importantly, we find in appropriate limits the reduced density matrix of a subsystem takes the form of thermal density matrix, providing an explicit example of the eigenstate thermalization hypothesis. Our work [3] explicitly demonstrates how statistical physics emerges from entanglement in a single eigenstate.

[1] Entanglement Entropy of Fermi Liquids via Multi-dimensional Bosonization, Wenxin Ding, Alexander Seidel, Kun Yang, Phys. Rev. X 2,

011012 (2012).

[2] Violation of Entanglement-Area Law in Bosonic Systems with Bose

Surfaces: Possible Application to Bose Metals, Hsin-Hua Lai, Kun Yang, N.

E. Bonesteel, Phys. Rev. Lett. 111, 210402 (2013).

[3] Entanglement entropy scaling laws and eigenstate thermalization in free fermion systems, Hsin-Hua Lai, Kun Yang, Phys. Rev. B 91,081110 (2015)

The evolution of many kinetic processes in 1+1 dimensions results in 2D directed percolative landscapes. The active phases of those models possess numerous hidden geometric orders characterized by distinct percolative patterns. From Monte-Carlo simulations of the directed percolation (DP) and the contact process (CP) we demonstrate the emergence of those patterns at specific critical points as a result of continuous phase transitions.These geometric transitions belong to the DP universality class and their nonlocal order parameters are the capacities of corresponding backbones. The multitude of conceivable percolative ordering patterns implies the existence of infinite cascades of such transitions in the models considered. We conjecture that such cascades of transitions is a generic feature of percolation as well as many other transitions with non-local order parameters.

The condensation of bosons can induce transitions between topological quantum field theories (TQFTs). This as been previously investigated through the formalism of Frobenius algebras and with the use of Vertex lifting coefficients. I will discuss an alternative, algebraic approach to boson condensation in TQFTs that is physically motivated and computationally efficient. With a minimal set of assumptions, such as commutativity of the condensation with the fusion of anyons, we can prove a number of theorems linking boson condensation in TQFTs with algebra extensions in conformal field theories and with the problem of factorization of completely positive matrices over the positive integers. I will present an algorithm for obtaining a condensed theory fusion algebra and its modular matrices. In addition, I will discuss how this formalism can be used to build multi-layer TQFTs which could be a starting point to build three-dimensional topologically ordered phases. Using this formalism, I will also give examples of bosons that cannot undergo a condensation transition due to topological obstructions.

In this talk, I will revise some of the aspects that lead isolated interacting quantum systems to thermalize.

In the presence of disorder, however, the thermalization process fails resulting in a phenomena where 

transport is suppressed known as many-body localization. Unlike the standard Anderson localization for 

non-interacting systems, the delocalized (ergodic) phase is very robust against disorder even for moderate

values of interaction. Another interesting aspect of the many-body localization phase is that under the time

evolution of the quenched disorder, information present in the initial state may survive for arbitrarily long times.

This was recently used as a probe of many-body localization of ultracold fermions in optical lattices

with quasi-periodic disorder [1]. Here, we will stress that this analysis may suffer from substantial finite-size effects 

after comparing with the numerical results in one-dimensional Hubbard chains [2].

References:

[1] - M.Schreiber, S. S. Hodgman,. P. Bordia,.H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, I. Bloch, Science 349, 842 (2015)

[2] - Rubem Mondaini and Marcos Rigol, Phys. Rev. A 92, 041601(R) (2015)

Topological aspects of physical systems, including the called topological states of matter, have become hot topics in the frontiers of physics in recent years. Here I would like to present a mathematically "popular" talk for professional physicists for a highlight or overview of how one can systematize knowledge of topological aspects of quantum field theories. Our starting points are Descent Equations and Gauge Structure in Configuration Space in Field Theory. (The audience needs only to know the meaning of "differential forms".)

In this talk, I will present our latest results on the development of algorithms for computer simulations based on tensor network states (TNS). The major part of this talk is concerned with finite projected entangled pair states (PEPS) in 2D: I will discuss their algorithmic properties as well as their performance for typical benchmark Hamiltonians [1, 2]. A minor part of this talk deals with the question whether TNS methods can be useful for density functional theory (DFT).

[1] M. Lubasch, J. I. Cirac, and M.-C. Ba\~{n}uls, New J. Phys. \textbf{16}, 033014 (2014).

[2] M. Lubasch, J. I. Cirac, and M.-C. Ba\~{n}uls, Phys. Rev. B \textbf{90}, 064425 (2014).

Modern numerical methods have revolutionized the practice of science, creating a third discipline between traditional theory and experiment. Perhaps the most widely known and successful technique has been the Monte Carlo method in general, and the Metropolis algorithm in particular. In this talk, I will present a new way of performing unbiased Monte Carlo simulations based on highly-accurate tensor network contractions. The resulting technique inherits the legendary precision of tensor networks without any of the variational bias. From a Monte Carlo point-of-view, the method can be seen as an aggressive multi-sampling technique where each sample may account for the vast majority of the entire partition function resulting a a drastic reduction in sample-to-sample variance (in contrast to standard
configuration-based Monte Carlo, where only a small subset of possible configurations are sampled). The presented results are all classical, though applications to quantum systems and the sign problem will be discussed.

The orbital angular momentum in a chiral superfluid has posed a paradox for several decades. For example, for the $p+ip$-wave superfluid of $N$ fermions, the total orbital angular momentum should be $N/2$ if all the fermions form Cooper pairs. On the other hand, it appears to be substantially suppressed from $N/2$, considering that only the fermions near the Fermi surface would be affected by the pairing interaction. To resolve the long-standing question, we studied chiral superfluids in a two-dimensional circular well, in terms of a conserved charge and spectral flows. We find that the total orbital angular momentum takes the full value $N/2$ in the chiral $p+ip$-wave superfluid, while it is strongly suppressed in higher-order ($d+id$ etc.) chiral superfluids. This surprising difference is elucidated in terms of edge states.

Matrix product operators form a natural language for describing topological quantum order. I will discuss how they arise as symmetries in PEPS, how anyon excitations arise as end points on them, and how the virtual indices of the MPO's provide a tensor product structure for the logical qubits in topological quantum computation.

Using the method of flux fusion anomaly test recently developed by M. Hermele and X. Chen (arXiv:1508.00573), we show that the possible ways of fractionalize crystal symmetry is greatly restricted if we assume the spin liquid has an SU(2) spin rotation symmetry and the spinon carries a half-integer spin. For a Z_2 spin liquid, under these assumptions the vison can only take the crystal symmetry fractionalization described by the Ising gauge theory. For a chiral spin liquid these assumptions imply that the spinon must also take fractionalized quantum numbers of crystal symmetries.