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arXiv:2106.07940 · Physical review. B./Physical review. B

Frustrated mixed-spin ladders: Evidence for a bond-order wave phase between rung-singlet and Haldane phases

N. Ahmadi, Jahanfar Abouie, Reza Haghshenas, D. Baeriswyl

Published Nov 18, 2022• 2 citations
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Abstract

In frustrated spin ladders, the interplay of frustration and correlations leads to the familiar Haldane (H) and rung-singlet (RS) phases. The nature of the transition between these two phases is still under debate. In this paper we tackle this issue using tools of quantum information theory. We consider frustrated mixed-spin-(1, 1/2) ladders with antiferromagnetic leg, rung and diagonal couplings, and calculate various quantities, such as the entanglement entropy (EE), the Schmidt gap, and the level degeneracy of the entanglement spectrum (ES). We use two numerical techniques, the infinite time-evolving block decimation (iTEBD) and the density matrix renormalization group (DMRG). We demonstrate that there exists an intermediate phase in which the ES levels do not exhibit the characteristic degeneracies of the H and RS phases. To understand the underlying physics in this phase, we investigate short-range spin correlations along legs, rungs and diagonals and show that in this intermediate phase long-wavelength modulations occur, akin to bond-order waves.

DOI
10.1103/physrevb.106.174419

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In frustrated spin ladders, the interplay of frustration and correlations leads to the familiar Haldane (H) and rung-singlet (RS) phases. The nature of the transition between these two phases is still under debate. In this paper we tackle this issue using tools of quantum information theory. We consider frustrated mixed-spin-(1, 1/2) ladders with antiferromagnetic leg, rung and diagonal couplings, and calculate various quantities, such as the entanglement entropy (EE), the Schmidt gap, and the level degeneracy of the entanglement spectrum (ES). We use two numerical techniques, the infinite time-evolving block decimation (iTEBD) and the density matrix renormalization group (DMRG). We demonstrate that there exists an intermediate phase in which the ES levels do not exhibit the characteristic degeneracies of the H and RS phases. To understand the underlying physics in this phase, we investigate short-range spin correlations along legs, rungs and diagonals and show that in this intermediate phase long-wavelength modulations occur, akin to bond-order waves.
In frustrated spin ladders composed of antiferromagnetically coupled chains, homogeneous or inhomogeneous, the interplay of frustration and correlations causes the emergence of two phases, Haldane (H) phase and rung singlet (RS) phase, in which the transition between these two phases had been under debate. In this paper we investigate the ground state phase diagram of frustrated mixed-spin-(1, 1/2) ladders, using the notions from quantum information theory such as entanglement entropy, Schmidt gap and entanglement levels degeneracies, and also defining various local and nonlocal order parameters. Employing two numerical techniques, the infinite time-evolving block decimation (iTEBD) and density matrix renormalization group (DMRG) algorithms, we obtain the ground state phase diagram of the ladder, and demonstrate that there is an intermediate phase between RS and H phases, where the ground state is disordered and the entanglement spectrum follow no particular pattern.
We study the impact of the diagonal frustrating couplings on the quantum phase diagram of a two-leg ladder composed of alternating spin-1 and spin-12\frac{1}{2}21​ rungs. As the coupling strength is increased the system successively exhibits two gapped paramagnetic phases (a rung-singlet and a Haldane-type nondegenerate states) and two ferrimagnetic phases with different ferromagnetic moments per rung. The first two states are similar to the phases studied in the frustrated spin-12\frac{1}{2}21​ ladder, whereas the magnetic phases appear as a result of the mixed-spin structure of the model. A detailed characterization of these phases is presented using density-matrix renormalization-group calculations, exact diagonalizations of periodic clusters, and an effective Hamiltonian approach inspired by the analysis of numerical data. The present theoretical study was motivated by the recent synthesis of the quasi-one-dimensional ferrimagnetic material FeIIFeIII{\text{Fe}}^{\text{II}}{\text{Fe}}^{\text{III}}FeIIFeIII (trans-1,4-cyclohexanedicarboxylate) exhibiting a similar ladder structure.
The quantum phases of two-leg spin-12\frac{1}{2}21​ ladders with skewed rungs are obtained using exact diagonalization of systems with up to 26 spins and by density-matrix renormalization-group calculations to 500 spins. The ladders have isotropic antiferromagnetic (AF) exchange J2>0{J}_{2}>0J2​>0 between first neighbors in the legs, variable isotropic AF exchange J1{J}_{1}J1​ between some first neighbors in different legs, and an unpaired spin per odd-membered ring when ${J}_{1}\ensuremath{\gg}{J}_{2}$. Ladders with skewed rungs and variable J1{J}_{1}J1​ have frustrated AF interactions leading to multiple quantum phases: AF at small J1{J}_{1}J1​, either ferromagnetic or AF at large J1{J}_{1}J1​, as well as bond-order-wave phases or reentrant AF (singlet) phases at intermediate J1{J}_{1}J1​.
Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.
We investigate a frustrated four-leg spin-1/2 ladder using density matrix renormalization group calculations. The uniform system displays three regimes: short-range antiferromagnetic legs, short-range ferromagnetic legs, and an effective spin-2 Heisenberg chain, separated by a crossover and a first-order transition. The spin-2 regime is confirmed through its finite string order parameter, edge-localized excitations, and excellent agreement with a projected Sr = 2 effective Hamiltonian. Recasting the model as two frustrated two-leg ladders coupled by rung and diagonal interactions, we track how the trivial and Haldane phases of an isolated ladder evolve as interladder couplings are introduced. The resulting phase diagrams reveal crossover and first-order lines whose locations are captured by the spin-2 projection and show how singlet- and triplet-dominated regimes reorganize when two ladders merge into a four-leg structure, clarifying the emergence of effective spin-1 versusspin-2 behavior.
The frustrated ladder with alternate ferromagnetic(F) exchange −JF-J_F−JF​ and AF exchange JAJ_AJA​ to first neighbors and F exchange −JL-J_L−JL​ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of 2N2N2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S=0)(S = 0)(S=0) and the singlet-triplet gap εT\varepsilon_TεT​ is finite for the exchanges considered. Spin-1/2 string correlation functions g1(N)g_1(N)g1​(N) and g2(N)g_2(N)g2​(N) are defined for an even number NNN of consecutive spins in systems with two spins per unit cell; the ladder has string order g2(∞)>0g_2(\infty)> 0g2​(∞)>0 and g1(∞)=0g_1(\infty) = 0g1​(∞)=0. The minimum N∗N^*N∗ of g2(N)g_2(N)g2​(N) is related to the range of ground-state spin correlations. Convergence to g2(∞)g_2(\infty)g2​(∞) is from below, and g1(N)g_1(N)g1​(N) decreases exponentially for N≥N∗N \geq N^*N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JFJ_FJF​, JLJ_LJL​ and JAJ_AJA​ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J1−J2J_1-J_2J1​−J2​ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at JL=JF/2≤JAJ_L = J_F/2 \leq J_AJL​=JF​/2≤JA​, that is reversed on interchanging −JF-J_F−JF​ and JAJ_AJA​. Inversion symmetry is spontaneously broken in the dimer phase of the J1−J2J_1-J_2J1​−J2​ model where the Kekulé diagrams are the doubly degenerate ground states at J2/J1=1/2J_2/J_1 = 1/2J2​/J1​=1/2.