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arXiv:2502.04165 · Physical Review A

Continuously varying critical exponents in an exactly solvable long-range cluster X Y model

Tian-Cheng Yi, Chengxiang Ding, Maoxin Liu, Liangsheng Li, Wen‐Long You

Published Feb 6, 2025• 3 citations
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Abstract

We investigate a generalized antiferromagnetic cluster $\mathit{XY}$ model in a transverse magnetic field, where long-range interactions decay algebraically with distance. This model can be exactly solvable within a free fermion framework. By analyzing the gap, we explicitly derive the critical exponents $\ensuremath{\nu}$ and $z$, finding that the relationship $\ensuremath{\nu}z=1$ still holds. However, the values of $\ensuremath{\nu}$ and $z$ depend on the decaying exponent $\ensuremath{\alpha}$, in contrast to those for the quantum long-range antiferromagnetic Ising chain. To optimize scaling behavior, we verify these critical exponents using correlation functions and fidelity susceptibility, achieving excellent data collapse across various system sizes by adjusting fitting parameters. Finally, we compute the entanglement entropy at the critical point to determine the central charge $c$, and find it also varies with $\ensuremath{\alpha}$. This study provides insights into the unique effect of long-range cluster interactions on the critical properties of quantum spin systems.

DOI
10.1103/physreva.111.023307

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We investigate a generalized antiferromagnetic cluster XY\mathit{XY}XY model in a transverse magnetic field, where long-range interactions decay algebraically with distance. This model can be exactly solvable within a free fermion framework. By analyzing the gap, we explicitly derive the critical exponents \ensuremathν\ensuremath{\nu}\ensuremathν and zzz, finding that the relationship \ensuremathνz=1\ensuremath{\nu}z=1\ensuremathνz=1 still holds. However, the values of \ensuremathν\ensuremath{\nu}\ensuremathν and zzz depend on the decaying exponent \ensuremathα\ensuremath{\alpha}\ensuremathα, in contrast to those for the quantum long-range antiferromagnetic Ising chain. To optimize scaling behavior, we verify these critical exponents using correlation functions and fidelity susceptibility, achieving excellent data collapse across various system sizes by adjusting fitting parameters. Finally, we compute the entanglement entropy at the critical point to determine the central charge ccc, and find it also varies with \ensuremathα\ensuremath{\alpha}\ensuremathα. This study provides insights into the unique effect of long-range cluster interactions on the critical properties of quantum spin systems.
We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as $1/{r}^{\ensuremath{\alpha}}$, where rrr denotes the distance between two spins and $\ensuremath{\alpha}$ is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of $\ensuremath{\alpha}$. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}<1$ we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime $\ensuremath{\alpha}\ensuremath{\ge}1$, and qualitative agreement with DMRG in the strong-long range regime $\ensuremath{\alpha}\ensuremath{\le}1$. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.
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The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
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We study the criticality of a long-range quantum ferromagnetic Ising chain with algebraically decaying interactions $1/{r}^{\ensuremath{\alpha}}$ via the fidelity susceptibility based on the exact diagonalization and the density-matrix renormalization-group techniques. We find that critical exponents change monotonically from the mean-field universality class to the short-range Ising universality class for intermediate $\ensuremath{\alpha}$, which are consistent with recent results obtained from renormalization-group techniques. In addition, we determine the critical values for $1.8\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$ from the finite-size scaling of the fidelity susceptibility. Our work provides very nice numerical data from the fidelity susceptibility for the quantum long-range ferromagnetic Ising chain.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size lll, the EE fits a critical behavior with a central charge close to c=1c=1c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins NNN, we find deviations which are rather consistent with the strong disorder fixed point central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.