Emergent Conformal Boundaries from Finite-Entanglement Scaling in Matrix Product States.

The use of finite entanglement scaling with matrix product states (MPS) has become a crucial tool for studying one-dimensional critical lattice theories, especially those with emergent conformal symmetry. We argue that finite entanglement introduces a relevant deformation in the critical theory. As a result, the bipartite entanglement Hamiltonian defined from the MPS can be understood as a boundary conformal field theory with a physical and an entanglement boundary. We are able to exploit the symmetry properties of the MPS to engineer the physical conformal boundary condition. The entanglement boundary, on the other hand, is related to the concrete lattice model and remains invariant under this relevant perturbation. Using critical lattice models described by the Ising, Potts, and free compact boson conformal field theories, we illustrate the influence of the symmetry and the relevant deformation on the conformal boundaries in the entanglement spectrum.

Extracting the Speed of Light from Matrix Product States.

We provide evidence that the spectrum of the local effective Hamiltonian and the transfer operator in infinite-system matrix product state simulations are identical up to a global rescaling factor, i.e., the speed of light of the system, when the underlying system is described by a 1+1 dimensional CFT. We provide arguments for this correspondence based on a path integral point of view. This observation turns out to yield very precise estimates for the speed of light in practice, confirming exact results to high precision where available, but also allowing us to finally determine the speed of light of the non-integrable, critical SU(2) Heisenberg chains with half-integer spin S>1/2 with unprecedented accuracy. We also show that the same technology applied to doped Hubbard ladders provides highly accurate velocities for a range of dopings. Combined with measurements of compressibilities we present new results for the Luttinger liquid parameter in the Luther-Emery regime of doped Hubbard ladders, outperforming earlier approaches based on the fitting of real-space correlation functions.

Spin, Charge, and η-Spin Separation in One-Dimensional Photodoped Mott Insulators.

We show that effectively cold metastable states in one-dimensional photodoped Mott insulators described by the extended Hubbard model exhibit spin, charge, and η-spin separation. Their wave functions in the large on-site Coulomb interaction limit can be expressed as |Ψ⟩=|Ψ_{charge}⟩|Ψ_{spin}⟩|Ψ_{η-spin}⟩, which is analogous to the Ogata-Shiba states of the doped Hubbard model in equilibrium. Here, the η-spin represents the type of photo-generated pseudoparticles (doublon or holon). |Ψ_{charge}⟩ is determined by spinless free fermions, |Ψ_{spin}⟩ by the isotropic Heisenberg model in the squeezed spin space, and |Ψ_{η-spin}⟩ by the XXZ model in the squeezed η-spin space. In particular, the metastable η-pairing and charge-density-wave (CDW) states correspond to the gapless and gapful states of the XXZ model. The specific form of the wave function allows us to accurately determine the exponents of correlation functions. The form also suggests that the central charge of the η-pairing state is 3 and that of the CDW phase is 2, which we numerically confirm. Our study provides analytic and intuitive insights into the correlations between active degrees of freedom in photodoped strongly correlated systems.

Continuous symmetry breaking in a two-dimensional Rydberg array.

Spontaneous symmetry breaking underlies much of our classification of phases of matter and their associated transitions1-3. The nature of the underlying symmetry being broken determines many of the qualitative properties of the phase; this is illustrated by the case of discrete versus continuous symmetry breaking. Indeed, in contrast to the discrete case, the breaking of a continuous symmetry leads to the emergence of gapless Goldstone modes controlling, for instance, the thermodynamic stability of the ordered phase4,5. Here, we realize a two-dimensional dipolar XY model that shows a continuous spin-rotational symmetry using a programmable Rydberg quantum simulator. We demonstrate the adiabatic preparation of correlated low-temperature states of both the XY ferromagnet and the XY antiferromagnet. In the ferromagnetic case, we characterize the presence of a long-range XY order, a feature prohibited in the absence of long-range dipolar interaction. Our exploration of the many-body physics of XY interactions complements recent works using the Rydberg-blockade mechanism to realize Ising-type interactions showing discrete spin rotation symmetry6-9.

Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms.

Quantum simulation using synthetic systems is a promising route to solve outstanding quantum many-body problems in regimes where other approaches, including numerical ones, fail1. Many platforms are being developed towards this goal, in particular based on trapped ions2-4, superconducting circuits5-7, neutral atoms8-11 or molecules12,13. All of these platforms face two key challenges: scaling up the ensemble size while retaining high-quality control over the parameters, and validating the outputs for these large systems. Here we use programmable arrays of individual atoms trapped in optical tweezers, with interactions controlled by laser excitation to Rydberg states11, to implement an iconic many-body problem-the antiferromagnetic two-dimensional transverse-field Ising model. We push this platform to a regime with up to 196 atoms manipulated with high fidelity and probe the antiferromagnetic order by dynamically tuning the parameters of the Hamiltonian. We illustrate the versatility of our platform by exploring various system sizes on two qualitatively different geometries-square and triangular arrays. We obtain good agreement with numerical calculations up to a computationally feasible size (approximately 100 particles). This work demonstrates that our platform can be readily used to address open questions in many-body physics.

Quantum Monte Carlo Simulation of the Chiral Heisenberg Gross-Neveu-Yukawa Phase Transition with a Single Dirac Cone.

We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and ß=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.

Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks.

We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-Abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries.

Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories.

The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for (2+1)D systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyze the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a Z_{2} topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.

Chiral Spin Liquids in Triangular-Lattice SU(N) Fermionic Mott Insulators with Artificial Gauge Fields.

We show that, in the presence of a π/2 artificial gauge field per plaquette, Mott insulating phases of ultracold fermions with SU(N) symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by an approximate ground space of N low-lying singlets for periodic boundary conditions, and by chiral edge states described by the SU(N)_{1} Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for N between 3 and 9, and by a parton construction based on a set of N Gutzwiller projected fermionic wave functions with flux π/N per triangular plaquette. Experimental implications are briefly discussed.

"Light-cone" dynamics after quantum quenches in spin chains.

Signal propagation in the nonequilibrium evolution after quantum quenches has recently attracted much experimental and theoretical interest. A key question arising in this context is what principles, and which of the properties of the quench, determine the characteristic propagation velocity. Here we investigate such issues for a class of quench protocols in one of the central paradigms of interacting many-particle quantum systems, the spin-1/2 Heisenberg XXZ chain. We consider quenches from a variety of initial thermal density matrices to the same final Hamiltonian using matrix product state methods. The spreading velocities are observed to vary substantially with the initial density matrix. However, we achieve a striking data collapse when the spreading velocity is considered to be a function of the excess energy. Using the fact that the XXZ chain is integrable, we present an explanation of the observed velocities in terms of "excitations" in an appropriately defined generalized Gibbs ensemble.

Majorana edge States in atomic wires coupled by pair hopping.

We present evidence for Majorana edge states in a number conserving theory describing a system of spinless fermions on two wires that are coupled by pair hopping. Our analysis is based on a combination of a qualitative low energy approach and numerical techniques using the density matrix renormalization group. In addition, we discuss an experimental realization of pair-hopping interactions in cold atom gases confined in optical lattices.

Entanglement spectrum of the two-dimensional Bose-Hubbard model.

We study the entanglement spectrum (ES) of the Bose-Hubbard model on the two-dimensional square lattice at unit filling, both in the Mott insulating and in the superfluid phase. In the Mott phase, we demonstrate that the ES is dominated by the physics at the boundary between the two subsystems. On top of the boundary-local (perturbative) structure, the ES exhibits substructures arising from one-dimensional dispersions along the boundary. In the superfluid phase, the structure of the ES is qualitatively different, and reflects the spontaneously broken U(1) symmetry of the phase. We attribute the basic low-lying structure to the "tower of states" Hamiltonian of the model. We then discuss how these characteristic structures evolve across the superfluid to Mott insulator transition and their influence on the behavior of the entanglement entropies. We briefly outline the implications of the ES structure on the efficiency of matrix-product-state based algorithms in two dimensions.

Shell-filling effect in the entanglement entropies of spinful fermions.

We consider the von Neumann and Rényi entropies of the one-dimensional quarter-filled Hubbard model. We observe that for periodic boundary conditions the entropies exhibit an unexpected dependence on system size: for L=4 mod 8 the results are in agreement with expectations based on conformal field theory, while for L=0 mod 8 additional contributions arise. We explain this observation in terms of a shell-filling effect and develop a conformal field theory approach to calculate the extra term in the entropies. Similar shell-filling effects in entanglement entropies are expected to be present in higher dimensions and for other multicomponent systems.

Fractional Chern insulators in topological flat bands with higher Chern number.

Lattice models forming bands with higher Chern number offer an intriguing possibility for new phases of matter with no analogue in continuum Landau levels. Here, we establish the existence of a number of new bulk insulating states at fractional filling in flat bands with a Chern number C = N > 1, forming in a recently proposed pyrochlore model with strong spin-orbit coupling. In particular, we find compelling evidence for a series of stable states at ν = 1/(2N + 1) for fermions as well as bosonic states at ν = 1/(N + 1). By examining the topological ground state degeneracies and the excitation structure as well as the entanglement spectrum, we conclude that these states are Abelian. We also explicitly demonstrate that these states are nevertheless qualitatively different from conventional quantum Hall (multilayer) states due to the novel properties of the underlying band structure.

Boundary-locality and perturbative structure of entanglement spectra in gapped systems.

The entanglement between two parts of a many-body system can be characterized in detail by the entanglement spectrum. Focusing on gapped phases of several one-dimensional systems, we show how this spectrum is dominated by contributions from the boundary between the parts. This contradicts the view of an "entanglement Hamiltonian" as a bulk entity. The boundary-local nature of the entanglement spectrum is clarified through its hierarchical level structure, through the combination of two single-boundary spectra to form a two-boundary spectrum, and finally through consideration of dominant eigenfunctions of the entanglement Hamiltonian. We show consequences of boundary-locality for perturbative calculations of the entanglement spectrum.

Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice.

Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Néel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.

Unveiling the nature of three-dimensional orbital ordering transitions: the case of e(g) and t(2g) models on the cubic lattice.

We perform large scale finite-temperature Monte Carlo simulations of the classical e(g) and t(2g) orbital models on the simple cubic lattice in three dimensions. The e(g) model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent ν ≈ 0.66(1) is close to the 3D XY value, the exponent η ≈ 0.15(1) differs substantially from O(N) values. At T(c) a U(1) symmetry emerges, which persists for T < T(c) below a crossover length scaling as Λ âˆ¼ ξ(a), with an unusually small a ≈ 1.3. Finally, for the t(2g) model we find a first order transition into a low-temperature lattice-nematic phase without orbital order.

Re-examining the directional-ordering transition in the compass model with screw-periodic boundary conditions.

We study the directional-ordering transition in the two-dimensional classical and quantum compass models on the square lattice by means of Monte Carlo simulations. An improved algorithm is presented which builds on the Wolff cluster algorithm in one-dimensional subspaces of the configuration space. This improvement allows us to study classical systems up to L=512. Based on this algorithm, we give evidence for the presence of strongly anomalous scaling for periodic boundary conditions which is much worse than anticipated before. We propose and study alternative boundary conditions for the compass model which do not make use of extended configuration spaces and show that they completely remove the problem with finite-size scaling. In the last part, we apply these boundary conditions to the quantum problem and present a considerably improved estimate for the critical temperature which should be of interest for future studies on the compass model. Our investigation identifies a strong one-dimensional magnetic ordering tendency with a large correlation length as the cause of the unusual scaling and moreover allows for a precise quantification of the anomalous length scale involved.

Disentangling entanglement spectra of fractional quantum Hall states on torus geometries.

We analyze the entanglement spectrum of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thin-torus limit.

Effect of rare fluctuations on the thermalization of isolated quantum systems.

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.