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arXiv:cond-mat/0610499 · Physical Review Letters

Entropic Analysis of Quantum Phase Transitions from Uniform to Spatially Inhomogeneous Phases

Örs Legeza, Jenő Sólyom, L. Tincani, R. M. Noack

Published Aug 24, 2007• 65 citations
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Abstract

We propose a new approach to study quantum phase transitions in low-dimensional fermionic or spin models that go from uniform to spatially inhomogeneous phases such as dimerized, trimerized, or incommensurate phases. It is based on studying the length dependence of the von Neumann entropy and its corresponding Fourier spectrum for finite segments in the ground state of finite chains. Peaks at a nonzero wave vector are indicators of oscillatory behavior in decaying correlation functions and also provide significant information about certain relevant features of the excitation spectrum; in particular, they can identify the wave vector of soft modes in critical models.

DOI
10.1103/physrevlett.99.087203

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We propose a new approach to study quantum phase transitions in low-dimensional fermionic or spin models that go from uniform to spatially inhomogeneous phases such as dimerized, trimerized, or incommensurate phases. It is based on studying the length dependence of the von Neumann entropy and its corresponding Fourier spectrum for finite segments in the ground state of finite chains. Peaks at a nonzero wave vector are indicators of oscillatory behavior in decaying correlation functions and also provide significant information about certain relevant features of the excitation spectrum; in particular, they can identify the wave vector of soft modes in critical models.
We investigate three different types of quantum phase transition occurring in quasi one-dimensional systems theoretically and numerically. First, we study the band-insulator to Mott-insulator transition occurring in charge-transfer complexes, for which the half-filled one-dimensional ionic Hubbard model is considered to be the prototype model. The study is carried out by first deriving an effective spin-one model, and then studying the model numerically using the density matrix renormalization group. We perform a careful finite-size scaling analysis of the mass gaps, order-parameters, and relative susceptibility. We confirm the existence of two quantum critical points. Analysis of the critical exponents confirms that the band-insulator-to-spontaneously-dimerized phase transition belong to the 2D Ising class. The spontaneously dimerized phase undergoes a phase transition to the Mott-insulator which is an infinite-order. Second, we investigate the Mott metal-insulator transition for the half-filled Hubbard model with both nearest-neighbor t and next-nearest-neighbor t′ hopping terms. We study the model using the bosonization approach and density matrix renormalization group simulations. An effective low-energy Hamiltonian that describes the insulator-metal transition is derived. We present results of density matrix renormalization group calculations of spin and charge distribution in various sectors of the phase diagram. The numerical results support the picture derived from the effective theory and give evidence for the complete separation of the transitions involving the spin and the charge degrees of freedom. Finally, we investigate quantum phase transitions phases in low-dimensional fermionic and spin models that go from uniform to spatially inhomogeneous, i.e., dimerized, trimerized, or incommensurate, phases. We propose a new approach based on studying the length dependence of the von Neumann entropy and its corresponding Fourier spectrum for finite segments in the ground state of finite chains. Peaks at a nonzero wave vector are indicators of oscillatory behavior in decaying correlation functions and also provide significant information about certain relevant features of the excitation spectrum; in particular, they can identify the wave vector of soft modes in critical models.
We propose a new approach to study quantum phase transitions in low-dimensional lattice models. It is based on studying the von Neumann entropy of two neighboring central sites in a long chain. It is demonstrated that the procedure works equally well for fermionic and spin models, and the two-site entropy is a better indicator of quantum phase transition than calculating gaps, order parameters, or the single-site entropy. The method is especially convenient when the density-matrix renormalization-group algorithm is used.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.
We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.
The modulation is analyzed from the analytical properties of zeros of the free fermionic partition function on the complex plane of wave numbers. It is shown how these properties are related to the oscillations of correlation functions. This approach can be used for analysis of phase transitions with local or nonlocal order parameters, as well as for the disorder lines. We find an infinite cascade of disorder lines at finite temperature in the quantum XYXYXY chain (equivalent to free fermions). The well-known ground-state factorization on the disorder line, and consequently disentanglement, is shown to follow directly from analytical properties of this model on the complex plane. From the quantum-classical correspondence the results for the chain are used to detect the disorder lines in several frustrated two-dimensional Ising models. The present formalism can be applied to other fermionic models in two and three spatial dimensions. In particular, we find the temperature-dependent Fermi wave vector of oscillations in the degenerate gas of three-dimensional fermions, which naturally leads in the limit $T\ensuremath{\rightarrow}0$ to the definition of the Fermi energy as the surface of quantum criticality. The modulation is a very common phenomenon, and it occurs in a large variety of models. The important point is that all these modulation transitions can be related to the complex zeros of partition functions, as done in the present study.