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arXiv:2205.14456 · Physical review. B./Physical review. B

Multicritical point with infinite fractal symmetries

Nayan E. Myerson-Jain, Kaixiang Su, Cenke Xu

Published Sep 19, 2022• 3 citations
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Abstract

Recently, ``Pascal's triangle model'' constructed with $\mathrm{U}(1)$ rotor degrees of freedom was introduced and it was shown that ($\mathit{i}$) this model possesses an infinite series of fractal symmetries and ($\mathit{ii}$) it is the parent model of a series of ${Z}_{p}$ fractal models each with its own distinct fractal symmetry. In this work we discuss a multicritical point of Pascal's triangle model that is analogous to the Rokhsar-Kivelson point of the better known quantum dimer model. We demonstrate that the expectation value of the characteristic operator of each fractal symmetry at this multicritical point decays as a power law of space and this multicritical point is shared by the family of descendent ${Z}_{p}$ fractal models. Afterwards, we generalize our discussion to a $(3+1)d$ model termed ``Pascal's tetrahedron model'' that has both planar and fractal subsystem symmetries. We also establish a connection between Pascal's tetrahedron model and the $\mathrm{U}(1)$ Haah's code.

DOI
10.1103/physrevb.106.115130

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Recently, Pascal's triangle model'' constructed with $\mathrm{U}(1)$ rotor degrees of freedom was introduced and it was shown that ($\mathit{i}$) this model possesses an infinite series of fractal symmetries and ($\mathit{ii}$) it is the parent model of a series of ${Z}_{p}$ fractal models each with its own distinct fractal symmetry. In this work we discuss a multicritical point of Pascal's triangle model that is analogous to the Rokhsar-Kivelson point of the better known quantum dimer model. We demonstrate that the expectation value of the characteristic operator of each fractal symmetry at this multicritical point decays as a power law of space and this multicritical point is shared by the family of descendent ${Z}_{p}$ fractal models. Afterwards, we generalize our discussion to a $(3+1)d$ model termed Pascal's tetrahedron model'' that has both planar and fractal subsystem symmetries. We also establish a connection between Pascal's tetrahedron model and the U(1)\mathrm{U}(1)U(1) Haah's code.
We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries-termed "Pascal's triangle symmetries"-which provides a natural U(1) generalization of a spin-(1/2) system with Sierpinski triangle fractal symmetries introduced in Newman et al., [Phys. Rev. E 60, 5068 (1999).PLEEE81063-651X10.1103/PhysRevE.60.5068]. The Pascal's triangle symmetry gives rise to exact degeneracies, as well as a manifold of low-energy states which are absent in the Sierpinski triangle model. Breaking the U(1) symmetry of this model to Z_{p}, with prime integer p, yields a lattice model with a unique fractal symmetry which is generated by an operator supported on a fractal subsystem with Hausdorff dimension d_{H}=ln(p(p+1)/2)/lnp. The Hausdorff dimension of the fractal can be probed through correlation functions at finite temperature. The phase diagram of these models at zero temperature in the presence of quantum fluctuations, as well as the potential physical construction of the U(1) model, is discussed.
Abstract. For any classical statistical-mechanics model with a discrete state space, and endowed with a dynamics satisfying detailed balance, it is possible to generalize the Rokhsar-Kivelson point for the quantum dimer model. That is, a quantum Hamiltonian can be constructed (on the same state space), such that the ground state wavefunction coincides with the classical equilibrium distribution. Furthermore the excited eigenstates correspond to classical relaxation modes, which (in cases with a symmetry or conserved quantity) permits extraction of the dispersion law of long-wavelength excitations. The mapping is natural mainly when the states have equal weight, as is typical of a highly frustrated model. Quantum and classical correlation functions are related by analytic continuation to the imaginary time axis. The quantum dimer model [1] of Rokhsar and Kivelson (RK) was inspired by the resonating-valence bond state of a quantum antiferromagnet, which had just been proposed as a starting point for the explanation of high-temperature superconductivity. That was not to be, but models of this type are of interest among
We show that the Rokhsar-Kivelson (RK) point in quantum dimer models (QDM) can emerge in realistic quantum Ising spin systems. Specifically, we investigate the J1J_1J1​-J2J_2J2​-J3J_3J3​ transverse field Ising model on the triangular lattice with large-scale quantum Monte Carlo simulations. We find that the multicritical point in the phase diagram corresponds to the RK point of the QDM on the honeycomb lattice. We further measure the spectral functions and identify three branches of quadratic dispersions. In the phase diagram, we also find a sequence of incommensurate states, which meet the microscopic ingredients for the 'Cantor deconfinement' scenario. Our study provides a promising direction to realise the RK deconfinement in experimental platforms such as magnetic materials or programmable Rydberg arrays.
We identify a quantum critical point with fractal symmetry whose effective theory eludes the renormalization group framework. We consider the Newman-Moore model with three-body interaction subjected to an external transverse field, which exhibits a Kramers-Wannier type self-duality and a fractal Z2Z_2Z2​ symmetry with Ising charge conserved on a fractal subset of sites, i.e., on Sierpinski gaskets. Using large-scale quantum Monte Carlo simulations, we identify a continuous quantum phase transition between a phase with spontaneous fractal symmetry breaking and a paramagnetic phase. This phase transition is characterized by the emergence of a fractal scaling dimension d=ln⁡(3)/ln⁡(2)d=\ln(3)/\ln(2)d=ln(3)/ln(2) at the quantum critical point, where the power-law exponent of the correlation function is related to the fractal dimension of the Sierpinski triangle. We develop a field theory to elucidate such quantum criticality and denote the fractal scaling as a subsequence of UV-IR mixing, where the low energy modes at the critical point are manipulated by short-wavelength physics due to the fractal symmetry.
Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of $p\ifmmode\times\else\texttimes\fi{}1$ commensurate ordered phases for all integers $p\ensuremath{\ge}2$. The other sequence appears to consist of "generic" multicritical points terminating lines of nnn-phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.
Loop condensed phases are scale-invariant quantum liquid phases of matter. These phases include topologically ordered liquid phases such as the toric code as well as critical liquids such as the Rokhsar-Kivelson point of the quantum dimer model on the square lattice. To investigate the extent to which nonlocal geometric observables capture a signature of the nonlocal quantum order present in these phases, we compute geometric properties of such loop condensed states using directed loop Monte Carlo calculations. In particular, we investigate the loop condensed nature of ground state of the square lattice quantum dimer model at the Rokhsar-Kivelson point and compare with other loop condensed states on the square lattice, including those of the toric code and fully packed loop model. The common features of such liquids are a scale invariant distribution of loops and a fractal dimensionality of spanning loops. We find that the fractal dimension of the loop condensate of the square lattice quantum dimer model at the Rokhsar-Kivelson point is 3/23/23/2, which provides quantitative confirmation of the effective height model that is commonly used to describe this critical dimer liquid.