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Density Matrix Renormalization Group and Quantum Information applied to Quantum Critical Phenomena in One-Dimensional Systems

L. Tincani

Published Nov 25, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.17192/z2008.0777

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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 study the quantum phase transition from an insulator to a metal realized at t'=t'_c > 0.5t in the ground state of the half-filled Hubbard chain with both nearest-neighbor (t) and next-nearest-neighbor (t') hopping. The study is carried out using the bosonization approach and density matrix renormalization group calculations. An effective low-energy Hamiltonian that describes the insulator-metal transition is derived. We find that the gross features of the phase diagram are well-described by the standard theory of commensurate-incommensurate transitions in a wide range of parameters. We also obtain an analytical expression for the insulator-metal transition line t'_c(U,t). We argue that close to the insulator-metal transition line, a crossover to a regime corresponding to an infinite-order transition takes place.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 spin and charge degrees of freedom.
We study the zero-temperature phase diagram of the half-filled one-dimensional ionic Hubbard model. This model is governed by the interplay of the on-site Coulomb repulsion and an alternating one-particle potential. Various many-body energy gaps, the charge-density-wave and bond-order parameters, the electric as well as the bond-order susceptibilities, and the density-density correlation function are calculated using the density-matrix renormalization group method. In order to obtain a comprehensive picture, we investigate systems with open as well as periodic boundary conditions and study the physical properties in different sectors of the phase diagram. A careful finite-size scaling analysis leads to results which give evidence in favor of a scenario with two quantum critical points and an intermediate spontaneously dimerized phase. Our results indicate that the phase transitions are continuous. Using a scaling ansatz we are able to read off critical exponents at the first critical point. In contrast to a bosonization approach, we do not find Ising critical exponents. We show that the low-energy physics of the strong-coupling phase can only partly be understood in terms of the strong-coupling behavior of the ordinary Hubbard model.
We study the quantum phase transition from an insulator to a metal realized at ${t}^{\ensuremath{'}}={t}_{c}^{\ensuremath{'}}>0.5t$ in the ground state of the half-filled Hubbard chain with both nearest-neighbor (t)(t)(t) and next-nearest-neighbor $({t}^{\ensuremath{'}})$ hopping. The study is carried out using the bosonization approach and density-matrix renormalization-group calculations. An effective low-energy Hamiltonian that describes the insulator-metal transition is derived. We find that the gross features of the phase diagram are well described by the standard theory of commensurate-incommensurate transitions in a wide range of parameters. We also obtain an analytical expression for the insulator-metal transition line ${t}_{c}^{\ensuremath{'}}(U,t)$. 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 spin and charge degrees of freedom.
Strongly correlated electron systems are one of the most fascinating problems in current physics. The strong electron-electron interaction in these materials leads to the emergence of nontrivial elementary excitations (quasiparticles, QPs) above the ground state ranging from fractional spins in quasi-one-dimensional materials to magnetic monopole in the pyrochlore lattice. The condensation of these quasiparticles upon changing some external parameters may stabilize new exotic states of matter. Experimental measurements such as inelastic neutron scattering provide us with valuable information about the excitation spectrum of such systems which require microscopic models to be described. This thesis is devoted to a detailed analysis of the excitation spectrum and of the quantum phase transitions in the one-dimensional (1D) ionic Hubbard model (IHM). The IHM consists of a nearest-neighbor (n.n.) hopping, onsite Hubbard interaction, and an ionic (staggered) potential separating the odd and even sites energetically. The model exhibits two continuous phase transitions on increasing the Hubbard interaction identified by a low-energy effective field theory and confirmed by a rigorous density matrix renormalization group (DMRG) analysis after several attempts. The first transition occurs from band insulator (BI) phase to the 2-fold degenerate spontaneously dimerized insulator (SDI) phase. The transition is in the Ising universality class as is plausible from symmetry considerations. The SDI phase becomes unstable towards a quasi-long-range order Mott insulator (MI) phase at a second transition point resembling the Kosterlitz-Thouless (KT) transition in the frustrated Heisenberg chain. We employ continuous unitary transformations (CUT) to systematically map the IHM to effective Hamiltonians (almost) conserving the number of QPs in the system. Using an analysis in the BI regime where electrons and holes define QPs, the low-energy excitation spectrum of the model is quantitatively determined in the BI phase almost up to the first transition point. The transition from the BI to the SDI phase is signaled by the vanishing of an S=0 exciton mode at the total momentum K=\pi. The condensation of these excitons beyond the first transition point is described by a BCS-type-theory showing the stabilization of the SDI phase. The mean-field solution indicates no second phase transition to the quasi-long-range order MI phase. This is interpreted as the effect of strong quantum fluctuations in one dimension. We consider the IHM in the dimer limit where the uniform chain is separated into independent dimers. The different phases of the IHM are studied by increasing the interdimer hopping and reaching the uniform limit. This dimer limit satisfactorily produces the excitation spectrum of the BI phase confirming the vanishing of an S=0 exciton mode at the first transition point. It is found that the SDI-to-MI transition takes place by softening of a magnetic S=1 excitation, i.e., a triplon. We report rigorous results for the gapless triplon dispersion in the MI phase and discuss the binding effects in the 2-triplon sector.
The density-matrix renormalization group is used to study the phase diagram of the one-dimensional half-filled Hubbard model with on-site (U) and nearest-neighbor (V) repulsion and hopping t. A critical line V(c)(U) approximately U/2 separates a Mott insulating phase from a charge-density-wave phase. The formation of bound charge excitations for V>2t changes the phase transition from continuous to first-order at a tricritical point U(t) approximately 3.7t, V(t)=2t. A frustrating effective antiferromagnetic spin coupling induces a bond-order-wave phase on the critical line V(c)(U) for U(t)<U less, similar 7t.
We investigate the neutral-to-ionic insulator-insulator transition in one-dimensional materials by treating a strong-coupling effective model based on the ionic Hubbard model using the density-matrix renormalization group and finite-size scaling. The effective model, formulated in a spin-one representation, contains a single parameter. We carry out an extensive finite-size scaling analysis of the relevant gaps and susceptibilities to characterize the two zero-temperature transitions. We find that the transition from the ionic band-insulating phase to an intermediate spontaneously dimerized phase is Ising, and the transition from the dimerized phase to the Mott-insulating phase is Kosterlitz-Thouless, in agreement with the field-theory-based predictions.