Quantum Matter

The conventional Euclidean time Monte Carlo approach to Lattice Field Theories faces a major obstacle in the sign problem in certain parameter regimes, such as the presence of a nonzero chemical potential or a topological theta-term. Tensor Network States, a family of ansatzes for the efficient description of quantum many-body states, offer a promising alternative for addressing Lattice Field Theories in the Hamiltonian formulation. In particular, numerical methods based on Tensor Network states do not suffer from the sign problem which makes it possible to study scenarios which are not accessible with standard Monte Carlo methods. In this talk I will present some recent work demonstrating this capability using two (1+1)-dimensional models as a test bed. Studying the O(3) nonlinear sigma model at nonzero chemical potential and the Schwinger model with topological theta-term, I will show how Tensor Networks States accurately describe the low-energy spectrum and that numerical errors can be controlled well enough to make contact with continuum predictions.

We give rigorous analytical results on the temporal behavior of two-point correlation functions (also known as dynamical response functions or Green’s functions) in quantum many body systems undergoing unitary dynamics. Using recent results from large deviation theory, we show that in a large class of models the correlation functions factorize at late times -> , thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. This conclusion connects the behavior of correlation functions to that of the late-time fluctuations of quenched systems out of equilibrium. For auto-correlation functions such as (as well as the symmetrized and anti-symmetrized versions) we provide an upper bound on the timescale at which they reach that factorized late time value. Remarkably, this bound is a function of local expectation values only, and does not increase with system size. As such it constraints, for instance, the behavior of current auto-correlation functions that appear in quantum transport. We give numerical examples that show that this bound is a good estimate in chaotic models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the Kubo function of linear response theory, for which we give an analogous result. Joint work with Luis Pedro Garcia-Pintos and Jonathon Riddell

Key to characterizing universality in critical systems is the identification of the RG fixed point, which is very often a conformal field theory (CFT). We show how to use lattice operators that mimic the Virasoro generators of conformal symmetry to systematically extract, from a generic critical quantum spin chain, a complete set of the conformal data (central charge, scaling dimensions of primary fields, OPE coefficients) specifying a 2D CFT. We further show that, in the case of an extended superconformal symmetry, one can construct lattice operators that mimic the generators of the superconformal algebra, which allows us to identify superconformal primary states.

We discuss recent progress in theory and experiment on emergent topological phases in Kitaev materials. Here the competition between different anisotropic spin-exchange interactions may lead to a number of exotic phases of matter. We investigate possible emergence of quantum spin liquid, topological magnons, and topological superconductivity in two and three dimensional systems. We make connections to existing and future experiments.

It is an open question how well tensor network states in the form of an infinite projected entangled-pair states (iPEPS) tensor network can approximate gapless quantum states of matter. In this talk we address this issue for two different physical scenarios: (i) a conformally invariant (2+1)d quantum critical point in the incarnation of the transverse-field Ising model on the square lattice and (ii) spontaneously broken continuous symmetries with gapless Goldstone modes exemplified by the S=1/2 antiferromagnetic Heisenberg and XY models on the square lattice. We find that the energetically best wave functions display finite correlation lengths and we introduce a powerful finite correlation length scaling framework for the analysis of such finite bond dimension (finite-D) iPEPS states. The framework is important (i) to understand the mild limitations of the finite-D iPEPS manifold in representing Lorentz-invariant, gapless many-body quantum states and (ii) to put forward a practical scheme in which the finite correlation length ξ(D) combined with field theory inspired formulas can be used to extrapolate the data to infinite correlation length, i.e., to the thermodynamic limit. The finite correlation length scaling framework opens the way for further exploration of quantum matter with an (expected) Lorentz-invariant, massless low-energy description, with many applications ranging from condensed matter to high-energy physics.

In the first half, I will demonstrate an efficient and general approach for realizing non-trivial quantum states, such as quantum critical and topologically ordered states, in quantum simulators. In the second half, I will present a related variational ansatz for many-body quantum systems that is remarkably efficient. In particular, representing the critical point of the one-dimensional transverse field Ising model only requires a number of variational parameters scaling logarithmically with system size. Though optimizing the ansatz generally requires Monte Carlo sampling, our ansatz potentially enables a partial mitigation of the sign problem at the expense of having to optimize a few parameters.

We introduce an isometric restriction of the tensor-network ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D tensor network. Second, we introduce a 2D version of the time-evolving block decimation algorithm (TEBD2) for approximating the ground state of a Hamiltonian as an isometric tensor network, which we demonstrate for the 2D transverse field Ising model.

In the context of quantum spin liquids, it is long known that the condensation of fractionalized excitations will inevitably break certain physical symmetries sometimes. For example, condensing spinons will usually break spin rotation and time reversal symmetries. We generalize these phenomena to generic continuous quantum phase transitions between symmetry enriched topological orders driven by anyon condensation. We provide a generic rule to determine whether a symmetry is enforced to break across an anyon condensation transition. Using a dimensional reduction scheme, we establish a mapping between these symmetry-breaking anyon-condensation transitions in two spatial dimensions, and deconfined quantum criticality in one spatial dimension.

The Hallmark of strongly entangled quantum phases is an intrinsic impossibility to describe them locally in terms of microscopic degrees of freedom. Two popular methods that have been developed to analytically describe these exotic states are known as (1) ‘parton construction’ and (2) ‘coupled-wire approach’. The former provides a constructive route for determining which non-trivial phases may arise, in principle, for a given set of constituent degrees of freedom and symmetries. This capability comes at the expense of having very little predictive power what phases do arise, in practice, in any particular system. The latter technique, by contrast, yields explicit expressions of ground states, excitations as well as parent Hamiltonians in terms of microscopic degrees of freedom. The price to pay is a lack of flexibility, and each phase needs to be analyzed on a laborious case-by-case basis. I will show how recent understanding of two-dimensional dualities provides a natural link between the two approaches. Specifically, I will show how a wide range of parton mean-field states can be easily translated into explicit coupled-wire models, and how their universal properties can be obtained in a transparent manner.

We perform both numerical and theoretical studies on the phase diagram of the Kitaev materials in the presence of a magnetic field. We find that a new quantum spin liquid state with neutral Fermi surfaces emerges at intermediate field strengths, between the regimes for the non-Abelian chiral spin liquid state and for the trivial polarized state. We discuss the exotic field-induced quantum phase transitions from this new state with neutral Fermi surfaces to its nearby phases. We also theoretically study the field-induced quantum phase transitions from the non-Abelian chiral spin liquid to the symmetry-broken zigzag phase and to the trivial polarized state. Utilizing the recently developed dualities of gauge theories, we find these transitions can be described by critical bosons or gapless fermions coupled to emergent non-Abelian gauge fields, and the critical theories are of the type of a QCD3-Chern-Simons theory. We propose that all these exotic quantum phase transitions can potentially be direct and continuous in the Kitaev materials, and we present sound evidence for this proposal. Therefore, besides being systems with intriguing quantum magnetism, Kitaev materials may also serve as table-top experimental platforms to study the interesting dynamics of emergent strongly interacting quarks and gluons in 2+1 dimensions. Finally, we address the experimental signatures of these phenomena.