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The plaquette lattice. Thick lines denote interactions J +Q/2. The circles indicate four-spin terms of strength Q in the Hamiltonian (1). Small-sized numbers label spins within a plaquette, while the larger ones label plaquettes.
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We study the quantum phase diagram of the Heisenberg planar antiferromagnet
with a subset of four-spin ring exchange interactions, using the recently
proposed heirarchical mean-field approach. By identifying relevant degrees of
freedom, we are able to use a single variational anzatz to map the entire phase
diagram of the model and uncover the natur...
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... We build upon the grounds of hierarchical mean-field theory (HMFT) [38], an algebraic framework based on the use of clusters for which a Gutzwiller ansatz represents the lowest order approximation. Furnished with a scaling analysis, it allows to uncover ground-state phase diagrams in the thermodynamic limit characterized by co-existence and competition of different long-range orders (LROs) and quantum paramagnetic phases [39][40][41][42][43][44]. Aiming at overcoming the scaling limitations of HMFT, here we present its quantum-assisted approach, dubbed Q-HMFT, where the cluster wave function is generated via a PQC whose central element is a parameterized real two-qubit XY gate that efficiently generates valence-bonds on nearest-neighbor (NN) qubits. ...
... 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 . ...
We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational optimization to the U(1) subspace. We benchmark the method against the J 1 − J 2 Heisenberg model on the square lattice and uncover its phase diagram, which hosts long-range ordered Neel and columnar anti-ferromagnetic phases, as well as an intermediate valence-bond solid phase characterized by a periodic pattern of 2×2 strongly-correlated plaquettes. Our results show that the convergence of the algorithm is guided by the onset of long-range order, opening a promising route to synthetically realize frustrated quantum magnets and their quantum phase transition to paramagnetic valence-bond solids with currently developed superconducting circuit devices.
... 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. ...
... Generalization to n-body interactions appearing in e.g. ring-exchange models is straightforward, although tedious [33,37,39]. ...
We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit. The key ingredient is a tunable real XY gate allowing to generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates permit the description of magnetically ordered phases while efficiently restricting the variational optimization to the U(1) symmetric subspace. We benchmark the method against the paradigmatic J1-J2 Heisenberg model on the square lattice, for which the present hybrid ansatz is an exact realization of the cluster-Gutzwiller with 4-qubit clusters. In particular, we describe the Neel order and its continuous quantum phase transition onto a valence-bond solid characterized by a periodic pattern of 2x2 strongly-correlated plaquettes, providing a route to synthetically realize valence-bond solids with currently developed superconducting circuit devices.
... b) Plaquettes comprise more than one site and each site belongs to a single plaquette: with this choice it is possible to obtain reasonable estimates of the GS energy (up to corrections which scale with the plaquette boundaries) and short-range correlations (of the order of the plaquette size). c) Overlapping (i.e., entangled) plaquettes are used: this ansatz clearly produces, accounting for the entangled nature of a quantum GS, estimates of the energy and long-range correlations much more accurate than those computable with a WF whose weights are the product of non overlapping plaquettes [33, 34] as in a) and b). A variety of variational ansatze, as illustrated in [25], have a straigthforward representation in terms of EPS. ...
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N=324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ~64% of the classical value).
The criticality between the nematic and valence-bond-solid (VBS) phases was
investigated for the two-dimensional quantum S=1-spin model with the three-spin
and biquadratic interactions by means of the numerical diagonalization method.
It is expected that the criticality belongs to a novel universality class, the
so-called deconfined criticality, accompanied with unconventional critical
indices. In this paper, we incorporate the three-spin interaction, and adjust
the (redundant) interaction parameter so as to optimize the finite-size
behavior. Treating the finite-size cluster with N \le 20 spins, we estimate the
correlation-length critical exponent as \nu=0.88 (3).
We study the zero temperature phase diagram of two-dimensional hard-core
bosons on a square lattice with nearest neighbour and plaquette (ring-exchange)
hoppings, at arbitrary densities, by means of a hierarchical mean-field theory.
In the frustrated regime, where quantum Monte Carlo suffers from a sign
problem, we find a rich phase diagram where exotic states with nonzero
chirality emerge. Among them, novel insulating phases, characterized by nonzero
bond-chirality and plaquette order, are found over a large region of the
parameter space. In the unfrustrated regime, the hierarchical mean-field
approach improves over the standard mean-field treatment as it is able to
capture the transition from a superfluid to a valence bond state upon
increasing the strength of the ring-exchange term, in qualitative agreement
with quantum Monte Carlo results.
We consider the S=1/2 Heisenberg model with nearest-neighbor interaction J and an additional multispin interaction Q3 on the square lattice. The Q3 term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the Néel state to the VBS as a function of Q3/J . We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The Néel-VBS transition in this case is strongly first order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).
The J_1-J_2 model with the biquadratic (plaquette-four-spin) interaction was
simulated with the numerical-diagonalization method. Some limiting cases of
this model have been investigated thoroughly. Taking the advantage of the
extended parameter space, we survey the phase boundary separating the N'eel and
valence-bond-solid phases. According to the deconfined-criticality scenario,
the singularity of this phase boundary is continuous, accompanied with
unconventional critical indices. Diagonalizing the finite-size cluster with N
\le 36 spins, we observe a signature of continuous phase transition. Our
tentative estimate for the correlation-length critical exponent is \nu=1.1(3).
In order to elucidate a non-local character of criticality, we evaluated the
Roomany-Wyld \beta function around the critical point.
In a recent paper [ Phys. Rev. B 80 174403 (2009)] Kotov et al. studied the paramagnetic-to-antiferromagnetic transition in the J-Q model. Their findings were claimed to be in “fairly good agreement” with previous quantum Monte Carlo (QMC) results. In this Comment we show that the above claim is misleading and in reality their phase transition point is not only far from the corresponding QMC value but also lies in a region of parameter space not yet explored in the literature. We also show that their reference dimer state is unstable against formation of a plaquette condensate, which could in part explain the large fluctuations they found.
We study the three dimensional SU(2)-symmetric noncompact CP1 model, with two
charged matter fields coupled minimally to a noncompact Abelian gauge-field.
The phase diagram and the nature of the phase transitions in this model have
attracted much interest after it was proposed to describe an unusual continuous
transition associated with deconfinement of spinons. Previously, it has been
demonstrated for various two-component gauge theories that weakly first-order
transitions may appear as continuous ones of a new universality class in
simulations of relatively large, but finite systems. We have performed
Monte-Carlo calculations on substantially larger systems sizes than those in
previous works. We find that in some area of the phase diagram where at finite
sizes one gets signatures consistent with a single first-order transition, in
fact there is a sequence of two phase transitions with an O(3) paired phase
sandwiched in between. We report (i) a new estimate for the location of a
bicritical point and (ii) the first resolution of bimodal distributions in
energy histograms at relatively low coupling strengths. We perform a flowgram
analysis of the direct transition line with rescaling of the linear system size
in order to obtain a data collapse. The data collapses up to coupling constants
where we find bimodal distributions in energy histograms.
We propose a new ansatz for the ground-state wave function of quantum
many-body systems on a lattice. The key idea is to cover the lattice with
plaquettes and obtain a state whose configurational weights can be optimized by
means of a Variational Monte Carlo algorithm. Such a scheme applies to any
dimension, without any "sign" instability. We show results for various two
dimensional spin models (including frustrated ones). A detailed comparison with
available exact results, as well as with variational methods based on different
ansatzs is offered. In particular, our numerical estimates are in quite good
agreement with exact ones for unfrustrated systems, and compare favorably to
other methods for frustrated ones.
