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arXiv:2104.10564

Entanglement scaling for $λϕ_2^4$

Bram Vanhecke, Frank Verstraete, Karel Van Acoleyen

Published Apr 21, 2021• 1 citation
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Abstract

We study the $λϕ^4$ model in $0+2$ dimensions at criticality, and effectuate a simultaneous scaling of UV and IR physics. We demonstrate that the order parameter $ϕ$, the correlation length $ξ$ and quantities like $ϕ^3$ and the entanglement entropy exhibit useful double scaling properties. The calculations are performed with boundary matrix product state methods on tensor network representations of the partition function, though the technique is equally applicable outside the realm of tensor networks. We find the value $α_c=11.09698(31)$ for the critical point, improving on previous results.

DOI
10.48550/arxiv.2104.10564

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Entanglement scaling for λϕ24λϕ_2^4λϕ24​
We study the λϕ4λϕ^4λϕ4 model in 0+20+20+2 dimensions at criticality, and effectuate a simultaneous scaling of UV and IR physics. We demonstrate that the order parameter ϕϕϕ, the correlation length ξξξ and quantities like ϕ3ϕ^3ϕ3 and the entanglement entropy exhibit useful double scaling properties. The calculations are performed with boundary matrix product state methods on tensor network representations of the partition function, though the technique is equally applicable outside the realm of tensor networks. We find the value αc=11.09698(31)α_c=11.09698(31)αc​=11.09698(31) for the critical point, improving on previous results.
We study the $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ model in 0+20+20+2 dimensions at criticality, focusing on the scaling properties originating from the UV and IR physics. We demonstrate that the entanglement entropy, the correlation length $\ensuremath{\xi}$ and order parameters $\ensuremath{\phi}$ and ${\ensuremath{\phi}}^{3}$ exhibit distinctive double scaling properties that prove a powerful tool in the data analysis. The calculations are performed with boundary matrix product state methods on tensor network representations of the partition function to which the entanglement scaling hypothesis is applied, though the technique is equally applicable outside the realm of tensor networks. We find the value ${\ensuremath{\alpha}}_{c}=11.09698(31)$ for the critical point, improving on previous results.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.
We report on tensor renormalization group calculations of entanglement entropy in one-dimensional quantum systems. The reduced density matrix of a Gibbs state can be represented as a 1+11 + 11+1-dimensional tensor network, which is analogous to the tensor network representation of the partition function. The HOTRG method is used to approximate the reduced density matrix for arbitrary subsystem sizes, from which we obtain the entanglement entropy. We test our method in the quantum Ising model and obtain the entanglement entropy of the ground state by taking the size of time direction to infinity. The central charge ccc is obtained as c=0.49997(8)c = 0.49997(8)c=0.49997(8) for a bond dimension Dcut=96D_\text{cut}=96Dcut​=96, which agrees with the theoretical value c=1/2c=1/2c=1/2 within the error.
Simulating strongly correlated quantum systems in continuous space belongs to the most challenging and long-concerned issues in quantum physics. This work investigates the quantum entanglement and criticality of the ground-state wave functions of infinitely many coupled quantum oscillators (iCQOs). The essential task involves solving a set of partial differential equations (Schr\"odinger equations in the canonical quantization picture) with infinitely many variables, which currently lacks valid methods. By extending the imaginary-time evolution algorithm with translationally invariant functional tensor network, we simulate the ground state of iCQOs with the presence of two- and three-body couplings. We determine the range of coupling strengths where there exists a real ground-state energy (dubbed as physical region). With two-body couplings, we reveal the logarithmic scaling law of entanglement entropy (EE) and the polynomial scaling law of correlation length against the virtual bond dimension $\ensuremath{\chi}$ at the dividing point of physical and nonphysical regions. These two scaling behaviors are signatures of criticality, according to the previous results in quantum lattice models, but were not reported in continuous-space quantum systems. The scaling coefficients result in a central charge c=1c=1c=1, indicating the presence of free boson conformal field theory (CFT). We further show that the presence of three-body couplings, for which there are no analytical or numerical results, breaks down the CFT description at the dividing point. Our work uncovers the scaling behaviors of EE in the continuous-space quantum many-body systems. These results provide solid numerical evidence supporting the high efficiency of TN in representing the continuous-space quantum many-body wave functions in the thermodynamic limit, and meanwhile suggest an efficient approach to study the entanglement properties and criticality in continuous space.
We investigate the entanglement spectrum in HOTRG---tensor renormalization group (RG) method combined with the higher order singular value decomposition---for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by ``doubling'' of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to D=64D=64D=64, where DDD is the cutoff dimension of tensors. At the critical point, we also find that a nontrivial DDD scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.