Translation-Invariant Parent Hamiltonians of Valence Bond Crystals

Abstract

We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming S = 1/2 spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamltonians of the columnar-and the staggered-VBC on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the depart from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.Quantum magnets host a wealth of phases and exotic phenomena. The complex interplay between spin interactions and lattice topology may eventually prevent the stabilization of magnetic order. In particular, frustrated antiferromagnetic spin-1/2 interactions may favor the partition of the system into nearest-neighbor (NN) spin singlets, so-called valence-bonds (VB), covering the lattice in a periodic pattern or VB crystal (VBC) [1]. Eventually, VBs may resonate and recover translational invariance by forming a resonating-VB (RVB) spin liquid, as has been conjectured to occur in high-T c cuprates [2,3]. Particularly interesting is the zero-temperature quantum phase transition from the former state to an ordered AF phase [4], a transition that can be experimentally probed upon tuning external pressure [5] or magnetic field on various materials [6]. Under certain specific conditions, it has been argued that a class of VBC-AF transitions in two-dimensions (2D) are driven by the deconfinement of fractional excitations at a critical point [7], contrary to what is generally expected within Landau theory. To test these ideas, a family of antiferromagnetic Heisenberg Hamiltonians with additional four-and six-spin interactions favoring VBC order and amenable to quantum Monte Carlo (QMC) computations has been introduced during the last decade [8][9][10]. These models, socalled J −Q models, show unusual scaling behavior at the VBC-AF transition point [10][11][12][13][14] and, in some cases, a weak VBC signal [8,11]. In order to furnish this putative new class, it would be desirable to enlarge the number of Hamiltonians hosting VBC-AF transitions. In particular, antiferromagnetic Hamiltonians hosting an exact VBC ground-state (GS) shall provide a convenient test bed where the VBC phase is unambiguously defined. In this Letter, we present a general method to construct SU(2) symmetric and translation-invariant VBC parent Hamiltonians based on a canonical mapping that exactly identifies a VBC state with the vacuum of a new ...

Variational Quantum Simulation of Valence-Bond Solids

“…On the other hand, extensions of this work may include the implementation of error mitigation strategies based on symmetry [79], the description of low-energy excitations over the ground-state, for which HMFT has a well-stablished framework [35,36], or the benchmark of optimization routines and quantum hardware on exactly solvable models possessing VBS as exact ground-states [39,80]. From the experimental standpoint, we expect these results to motivate further development and refinement of the general tunable XY gates as key elements for variational quantum algorithms.…”

“…Generalization to n-body interactions appearing in e.g. ring-exchange models is straightforward, although tedious [33,37,39].…”

“…The method here introduced is built upon the grounds of hierarchical mean-field theory (HMFT), an algebraic framework based on the identification of clusters as the building blocks containing the essential symmetries and correlations to describe a system [32]. A cluster-Gutzwiller ansatz represents the lowest order approximation which, harnessed with a scaling analysis, allows to unveil quantum phases and phase transitions characterized by competition or coexistence of different orders, including VBS states [33][34][35][36][37][38][39]. More specifically, we present a quantum-assisted approach to HMFT (Q-HMFT) where the cluster wave function is provided by a U(1) symmetry-preserving PQC.…”

Classifying Data with Local Hamiltonians

“…finding a Hamiltonian H with H |ψ = 0 for all |ψ in some set), an ongoing field of research-e.g. with results on matrix product states [39,40], quantum error correction [41], or lattice crystals [42]. While related, we impose the additional constraint that we are given a fixed set of local interactions which we optimize, and that we simultaneously want to minimize overlap with another set of wave functions (the NO data); nevertheless, we hope that new insights from the research on parent Hamiltonians will help us to further develop our ideas.…”

Variational Quantum Simulation of Valence-Bond Solids

“…Generalization to n-body interactions appearing in e.g. ring-exchange models is straightforward [39,43,65]. The optimal energy E=E(θ ) with The wave function of the cluster is provided by a PQC whose parameters {θ} are tuned so as to optimize the energy density in the thermodynamic limit (2), which is an upper bound to the exact, E 0 .…”