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Physical review. D/Physical review. D.

Entanglement scaling for λϕ24

Bram Vanhecke, Frank Verstraete, Karel Van Acoleyen

Published Oct 20, 2022• 8 citations
Publisher

Abstract

We study the $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ model in $0+2$ 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.

DOI
10.1103/physrevd.106.l071501

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We study the \ensuremathλ\ensuremathϕ4\ensuremath{\lambda}{\ensuremath{\phi}}^{4}\ensuremathλ\ensuremathϕ4 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 \ensuremathξ\ensuremath{\xi}\ensuremathξ and order parameters \ensuremathϕ\ensuremath{\phi}\ensuremathϕ and \ensuremathϕ3{\ensuremath{\phi}}^{3}\ensuremathϕ3 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 \ensuremathαc=11.09698(31){\ensuremath{\alpha}}_{c}=11.09698(31)\ensuremathαc​=11.09698(31) for the critical point, improving on previous results.
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.
Entanglement scaling provides a powerful probe of universal properties at quantum critical points. In two dimensions, contributions originating from a corner-shaped bipartition exhibit a universal scaling, which is determined by the underlying conformal field theory. We develop a method to extract this corner entanglement entropy from projected entangled pair states directly in the thermodynamic limit. When applied to models at a quantum critical point, we show that the corner contribution exhibits scaling with the effective correlation length, in agreement with the hypothesis of finite-entanglement scaling. Our results for the corner coefficients are consistent with other methods, demonstrating the efficiency of our method for diagnosing strongly-correlated quantum critical points in two dimensions.
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.
In this paper, we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to (1+1)(1+1)(1+1)-dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant field theories describing the critical models under investigation. The algorithm is applied to three exemplary models; the cMPS version to the nonrelativistic Lieb-Liniger model and the relativistic massless boson, and MPS version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter. This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results.
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing aaa, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with DDD the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.
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.