Dimer-dimer Correlation Research Articles

Quantum dimer model on fullerenes: Resonance, scarring, and confinement

The earliest known examples of quantum superposition were found in organic chemistry -- in molecules that `resonate' among multiple arrangements of $\pi$-bonds. Small molecules such as benzene resonate among a few bond arrangements. In contrast, a large system that stretches over a macroscopic lattice may resonate among an extensive number of configurations. This is analogous to a quantum spin liquid as described by Anderson's resonating valence bond (RVB) picture. In this article, we study the intermediate case of somewhat large molecules, the C$_{20}$ and C$_{60}$ fullerenes. We build a minimal description in terms of quantum dimer models, allowing for local resonance processes and repulsion between proximate $\pi$-bonds. This allows us to characterize ground-states, e.g., with C$_{60}$ forming a superposition of 5828 dimer covers. Despite the large number of contributing dimer covers, the ground state shows strong dimer-dimer correlations. Going beyond the ground state, the full spectrum of C$_{60}$ shows many interesting features. Its Hilbert space is bipartite, leading to spectral reflection symmetry. It has a large number of protected zero-energy states. In addition, the spectrum contains many scar-like states, corresponding to localized dimer-rearrangement-dynamics. Resonance dynamics in QDMs can manifest in the behaviour of defects, potentially binding defects via an effective attractive interaction. To test this notion, we introduce pairs of vacancies at all possible separations. Resonance energy reaches its lowest value when the vacancies are closest to one another. This suggests confinement of monomers, albeit within a finite cluster. We discuss qualitative pictures for understanding bonding in fullerenes and draw connections with results from quantum chemistry.

Gapless quantum spin liquid and global phase diagram of the spin-1/2 [formula omitted] square antiferromagnetic Heisenberg model

The nature of the zero-temperature phase diagram of the spin-1/2J1-J2 Heisenberg model on a square lattice has been debated in the past three decades, and it remains one of the fundamental problems unsettled in the study of quantum many-body theory. By using the state-of-the-art tensor network method, specifically, the finite projected entangled pair state (PEPS) algorithm, to simulate the global phase diagram of the J1-J2 Heisenberg model up to 24×24 sites, we provide very solid evidences to show that the nature of the intermediate nonmagnetic phase is a gapless quantum spin liquid (QSL), whose spin-spin and dimer-dimer correlations both decay with a power law behavior. There also exists a valence-bond solid (VBS) phase in a very narrow region 0.56≲J2/J1≤0.61 before the system enters the well known collinear antiferromagnetic phase. We stress that we make the first detailed comparison between the results of PEPS and the well-established density matrix renormalization group (DMRG) method through one-to-one direct benchmark for small system sizes, and thus give rise to a very solid PEPS calculation beyond DMRG. Our numerical evidences explicitly demonstrate the huge power of PEPS for highly frustrated spin systems. Finally, an effective field theory is also proposed to understand the physical nature of the discovered gapless QSL and its relation to deconfined quantum critical point (DQCP).

Quantum Kasteleyn transition

Dimer models arise as effective descriptions in a variety of physical contexts, and provide paradigmatic examples of systems subject to strong local constraints. Here we present a quantum version of the venerable Kasteleyn model, which has an unusual phase transition from a dimer solid to a U(1) liquid. We show how the phase structure of the quantum model can be understood in terms of the quantum mechanics of one-dimensional strings and determine the exact value of the critical coupling. By constructing effective models to describe the properties of these strings, we calculate properties such as the dimer-dimer correlation function in the neighborhood of the transition. We also discuss the full phase structure of the model, in the ground state and at nonzero temperature.

Pinwheel valence bond crystal ground state of the spin- 12 Heisenberg antiferromagnet on the shuriken lattice

We investigate the nature of the ground-state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the $shuriken$ lattice by complementary state-of-the-art numerical techniques, such as variational Monte Carlo (VMC) with versatile Gutzwiller-projected Jastrow wave functions, unconstrained multi-variable variational Monte Carlo (mVMC), and pseudo-fermion/Majorana functional renormalization group (PF/PM-FRG) methods. We establish the presence of a quantum paramagnetic ground state and investigate its nature, by classifying symmetric and chiral quantum spin liquids, and inspecting their instabilities towards competing valence-bond-crystal (VBC) orders. Our VMC analysis reveals that a VBC with a pinwheel structure emerges as the lowest-energy variational ground state, and it is obtained as an instability of the U(1) Dirac spin liquid. Analogous conclusions are drawn from mVMC calculations employing accurate BCS pairing states supplemented by symmetry projectors, which confirm the presence of pinwheel VBC order by a thorough analysis of dimer-dimer correlation functions. Our work highlights the nontrivial role of quantum fluctuations via the Gutzwiller projector in resolving the subtle interplay between competing orders.

Topological sectors, dimer correlations, and monomers from the transfer-matrix solution of the dimer model

We solve the classical square-lattice dimer model with periodic boundaries and in the presence of a field t that couples to the (vector) flux, by diagonalizing a modified version of Lieb's transfer matrix. After deriving the torus partition function in the thermodynamic limit, we show how the configuration space divides into topological sectors corresponding to distinct values of the flux. Additionally, we demonstrate in general that expectation values are t independent at leading order, and obtain explicit expressions for dimer occupation numbers, dimer-dimer correlation functions, and the monomer distribution function. The last of these is expressed as a Toeplitz determinant, whose asymptotic behavior for large monomer separation is tractable using the Fisher-Hartwig conjecture. Our results reproduce those previously obtained using Pfaffian techniques.

Mean-field theory of interacting triplons in a two-dimensional valence-bond solid: Stability and properties of many-triplon states

We study a system of interacting triplons (the elementary excitations of a valence-bond solid) described by an effective interacting boson model derived within the bond-operator formalism. In particular, we consider the square lattice spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model, focus on the intermediate parameter region, where a quantum paramagnetic phase sets in, and consider the columnar valence-bond solid phase. Within the bond-operator theory, the Heisenberg model is mapped into an effective boson model in terms of triplet operators $t$. The effective boson model is studied at the harmonic approximation and the energy of the triplons and the expansion of the triplon operators $b$ in terms of the triplet operators $t$ are determined. Such an expansion allows us to performed a second mapping, and therefore, determine an effective interacting boson model in terms of the triplon operators $b$. We then consider systems with a fixed number of triplons and determined the ground-state energy and the spectrum of elementary excitations within a mean-field approximation. We show that many-triplon states are stable, the lowest-energy ones are constituted by a small number of triplons, and the excitation gaps are finite. For $J_2=0.48 J_1$ and $J_2=0.52 J_1$, we also calculate spin-spin and dimer-dimer correlation functions, dimer order parameters, and the bipartite von Neumann entanglement entropy within our mean-field formalism in order to determine the properties of the many-triplon state as a function of the triplon number. We find that the spin and the dimer correlations decay exponentially and that the entanglement entropy obeys an area law, regardless the triplon number. Moreover, only for $J_2=0.48 J_1$, the spin correlations indicate that the many-triplon states with large triplon number might display a more homogeneous singlet pattern than the columnar valence-bond solid.

Exact solution and correlations of a dimer model on the checkerboard lattice

We present analytic results for a special dimer model on the {\em non-bipartite} and {\em non-planar} checkerboard lattice that does not allow for parallel dimers surrounding diagonal links. We {\em exactly} calculate the number of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions, and determine all dimer-dimer correlations. The latter are found to vanish beyond a certain distance. We find that this solvable model, despite being non-planar, is in close kinship with well-known paradigm-setting planar counterparts that allow exact mappings to $\mathbb{Z}_2$ lattice gauge theory.

Critical colored-RVB states in the frustrated quantum Heisenberg model on the square lattice

We consider a family of SU(2)-symmetric Projected Entangled Paired States (PEPS) on the square lattice, defining colored-Resonating Valence Bond (RVB) states, to describe the quantum disordered phase of the J_1-J_2J1−J2 frustrated Heisenberg model. For J_2/J_1\sim 0.55J2/J1∼0.55 we show the emergence of critical (algebraic) dimer-dimer correlations – typical of Rokhsar-Kivelson (RK) points of quantum dimer models on bipartite lattices – while, simultaneously, the spin-spin correlation length remains short. Our findings are consistent with a spin liquid or a weak Valence Bond Crystal in the neighborhood of an RK point.

Non-integrable dimer models: Universality and scaling relations

In the last few years, the methods of constructive fermionic renormalization group have been successfully applied to the study of the scaling limit of several two-dimensional statistical mechanics models at the critical point, including weakly nonintegrable 2D Ising models, Ashkin-Teller, 8-Vertex, and close-packed interacting dimer models. In this note, we focus on the illustrative example of the interacting dimer model and review some of the universality results derived in this context. In particular, we discuss the massless Gaussian free field (GFF) behavior of the height fluctuations. It turns out that GFF behavior is connected with a remarkable identity (“Haldane” or “Kadanoff relation”) between an amplitude and an anomalous critical exponent, characterizing the large distance behavior of the dimer-dimer correlations.

Quantum criticality on a chiral ladder: An SU(2) infinite density matrix renormalization group study

In this paper we study the ground state properties of a ladder Hamiltonian with chiral $SU(2)$-invariant spin interactions, a possible first step towards the construction of truly two dimensional non-trivial systems with chiral properties starting from quasi-one dimensional ones. Our analysis uses a recent implementation by us of $SU(2)$ symmetry in tensor network algorithms, specifically for infinite Density Matrix Renormalization Group (iDMRG). After a preliminary analysis with Kadanoff coarse-graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground state properties match those of the universality class of a $c = 1$ conformal field theory (CFT) in $(1+1)$ dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-$1/2$ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an $SU(2)_1$ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct Matrix Product Operators for $SU(2)$-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.

Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study

Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the Fractional Quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by Exact Diagonalisation of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral Projected Entangled Pair States (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with Density Matrix Renormalisation Group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range "gossamer tails". We argue these tails are finite-$D$ artifacts of the chiral PEPS, which become irrelevant when the PEPS bond dimension $D$ is increased. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)$_2$ Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge $c=3/2$, as expected for a Moore-Read chiral spin liquid. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.

Spin- 12 Heisenberg antiferromagnet on the pyrochlore lattice: An exact diagonalization study

We present exact diagonalisation calculations for the spin-1/2 nearest neighbour antiferromagnet on the pyrochlore lattice. We study a section of the lattice in the $[111]$ direction and analyse the Hamiltonian of the breathing pyrochlore system with two coupling constants $J_1$ and $J_2$ for tetrahedra of different orientations and investigate the evolution of the system from the limit of disconnected tetrahedra ($J_2 = 0$) to a correlated state at $J_1=J_2$. We evaluate the low energy spectrum, two and four spin correlations, and spin chirality correlations for a system size of up to 36 sites. The model shows a fast decay of spin correlations and we confirm the presence of several singlet excitations below the lowest magnetic excitation. We find chirality correlations near $J_1=J_2$ to be small at the length scales available at this system size. Evaluation of dimer-dimer correlations and analysis of the nature of the entanglement of the tetrahedral unit shows that the triplet sector of the tetrahedron contributes significantly to the ground state entanglement at $J_1 = J_2$.

Transverse spin correlations of the random transverse-field Ising model

The critical behavior of the random transverse-field Ising model in finite dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$ and $3$ dimensions. At the critical point an algebraic decay of the form $\sim r^{-\eta_t}$ is found, with a decay exponent being approximately $\eta_t \approx 2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random AF XX-chain and have been tested by numerical calculations using free-fermionic techniques.

Z3 topological order in the quantum dimer-pentamer model

We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square lattice quantum dimer model as its configuration space comprises fully-packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully-packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local $Z_3$ gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a 9-fold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wavefunction and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that $Z_3$ electric charges are deconfined. Additionally, we introduce a $Z_3$ magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation and with correlations. These numerical results suggest that the ground state of the QDPM is a dimer liquid with $Z_3$ topological order.

Investigation of the chiral antiferromagnetic Heisenberg model using projected entangled pair states

A simple spin-$1/2$ frustrated antiferromagnetic Heisenberg model (AFHM) on the square lattice - including chiral plaquette cyclic terms - was argued [Anne E.B. Nielsen, German Sierra and J. Ignacio Cirac, Nature Communications $\bf 4$, 2864 (2013)] to host a bosonic Kalmeyer-Laughlin (KL) fractional quantum Hall ground state [V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. $\bf 59$, 2095 (1987)]. Here, we construct generic families of chiral projected entangled pair states (chiral PEPS) with low bond dimension ($D=3,4,5$) which, upon optimization, provide better variational energies than the KL ansatz. The optimal $D=3$ PEPS exhibits chiral edge modes described by the Wess-Zumino-Witten $SU(2)_1$ model, as expected for the KL spin liquid. However, we find evidence that, in contrast to the KL state, the PEPS spin liquids have power-law dimer-dimer correlations and exhibit a gossamer long-range tail in the spin-spin correlations. We conjecture that these features are genuine to local chiral AFHM on bipartite lattices.

Fermionic algebraic quantum spin liquid in an octa-kagome frustrated antiferromagnet

We investigate the ground state and finite-temperature properties of the spin-1/2 Heisenberg antiferromagnet on an infinite octa-kagome lattice by utilizing state-of-the-art tensor network-based numerical methods. It is shown that the ground state has a vanishing local magnetization and possesses a $1/2$-magnetization plateau with up-down-up-up spin configuration. A quantum phase transition at the critical coupling ratio $J_{d}/J_{t}=0.6$ is found. When $0<J_{d}/J_{t}<0.6$, the system is in a valence bond state, where an obvious zero-magnetization plateau is observed, implying a gapful spin excitation; when $J_{d}/J_{t}>0.6$, the system exhibits a gapless excitation, in which the dimer-dimer correlation is found decaying in a power law, while the spin-spin and chiral-chiral correlation functions decay exponentially. At the isotropic point ($J_{d}/J_{t}=1$), we unveil that at low temperature ($T$) the specific heat depends linearly on $T$, and the susceptibility tends to a constant for $T\rightarrow 0$, giving rise to a Wilson ratio around unity, implying that the system under interest is a fermionic algebraic quantum spin liquid.

Height fluctuations in interacting dimers

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb Z^2$, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbol{\xi}$ of $\mathbb Z^2$. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ converge to those of the massless Gaussian Free Field (GFF), asymptotically as $|{\boldsymbol{\xi}}-{\boldsymbol{\eta}}|\to \infty$. We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the fermionic language, the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ takes the form of a non-local operator, consisting of a sum of monomials along an {\it arbitrary} path connecting $\boldsymbol{\xi}$ and $\boldsymbol{\eta}$. As in the non-interacting case, this path-independence plays a crucial role in the proof.

Tensor-product state approach to spin-12squareJ1−J2antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticality

The ground state phase of spin-1/2 $J_1$-$J_2$ antiferromagnetic Heisenberg model on square lattice around the maximally frustrated regime ($J_2\sim 0.5J_1$) has been debated for decades. Here we study this model using the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point $J_2^{c_1}=0.572(5)J_1$ and critical exponents $\nu=0.50(8)$, $\eta_s=0.28(6)$. In the range of $0.572 < J_2/J_1 \leqslant 0.6 $ we find a paramagnetic ground state with exponentially decaying spin-spin correlation. Up to $24\times 24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent $\eta_p=0.24(1)$ and an anomalous columnar scaling exponent $\eta_c=0.28(1)$ at $J_2/J_1=0.6$. These results are consistent with a potential gapless $U(1)$ spin liquid phase. However, since the $U(1)$ spin liquid is unstable due to the instanton effect, a VBS order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point (DQCP) at $J_2^{c_1}$. Remarkably, all the observed critical exponents are consistent with the $J-Q$ model.

Non-symmetry-breaking ground state of theS=1Heisenberg model on the kagome lattice

The $S=1$ antiferromagnetic Heisenberg model on a Kagome lattice is studied using the density-matrix renormalization group method. To identify the ground state, we take four kinds of clusters into account; periodic, cylindrical, and two open ones. The hexagonal singlet solid (HSS) and triangular valence bond solid (TVBS) states are artificially generated by modulating edge shapes of the open clusters. We find that the energy par sites of the HSS state is $e_0=E_0/N =-1.41095$, which is readily lower than that of the TVBS state ($e_0=-1.3912 \pm 0.0025$). This agrees well with those of the cylindrical ($e_0=-1.40988$) and periodic ($e_0=-1.409 \pm 0.005$) clusters, where no assumption as to the ordering is posed. Thus, we conclude that the HSS picture is consistent to describe the ground state of the $S=1$ Kagome Heisenberg model. This is further confirmed by finding non-symmetry-breaking state in the calculations of the dimer-dimer correlation functions as well as the entanglement entropy of cylindrical clusters. Finally, we estimate the single-triplet energy gap: The HSS ground state has $\Delta=0.1919$, while the TVBS excited state has a larger one $\Delta=0.2797$.

Simplex valence-bond crystal in the spin-1 kagome Heisenberg antiferromagnet

We investigate the ground state properties of a spin-1 kagome antiferromagnetic Heisenberg model using tensor-network (TN) methods. We obtain the energy per site {$e_0=-1.41090(2)$ with $D^*=8$ multiplets retained (i.e., a bond dimension of $D=24$), and $e_0=-1.4116(4)$ from large-$D$ extrapolation,} by accurate TN calculations directly in the thermodynamic limit. The symmetry between the two kinds of triangles is spontaneously broken, with a relative energy difference of $\delta \approx$ 19\%, i.e, there is a trimerization (simplex) valence-bond order in the ground state. The spin-spin, dimer-dimer, and chirality-chirality correlation functions are found to decay exponentially with a rather short correlation length, showing that the ground state is gapped. We thus identify the ground state be a simplex valence-bond crystal (SVBC). We also discuss the spin-1 bilinear-biquadratic Heisenberg model on a kagome lattice, and determine its ground state phase diagram. Moreover, we implement non-abelian symmetries, here spin SU(2), in the TN algorithm, which improves the efficiency greatly and provides insight into the tensor structures.