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arXiv:cond-mat/0702384 · Physical Review B

Effect of temperature on quantum criticality in the frustrated two-leg Heisenberg ladder

Brandon W. Ramakko, Mohamed Moussadek Azzouz

Published Aug 16, 2007• 16 citations
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Abstract

The antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals is studied using the Jordan-Wigner transformation and bond-mean-field theory. The inclusion of all three couplings introduces frustration to the system and depending on their relative strengths the ladder can adopt one of three possible magnetically disordered gapped states. The phase diagram found in this mean-field approach is in very good agreement with the one calculated by Weihong et al. [Phys. Rev. B 57, 11439 (1998)] using the Lanczos exact diagonalization method. By analyzing the ground-state energy, we study quantum criticality when the coupling parameters are varied at zero temperature. We study the effect of temperature on the phase boundaries and find that the system shows thermally induced criticality for some values of the rung and diagonal coupling constants. All the phase transitions encountered in this system occur between disordered phases and are all caused by frustration.

DOI
10.1103/physrevb.76.064419

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The antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals is studied using the Jordan-Wigner transformation and bond-mean-field theory. The inclusion of all three couplings introduces frustration to the system and depending on their relative strengths the ladder can adopt one of three possible magnetically disordered gapped states. The phase diagram found in this mean-field approach is in very good agreement with the one calculated by Weihong et al. [Phys. Rev. B 57, 11439 (1998)] using the Lanczos exact diagonalization method. By analyzing the ground-state energy, we study quantum criticality when the coupling parameters are varied at zero temperature. We study the effect of temperature on the phase boundaries and find that the system shows thermally induced criticality for some values of the rung and diagonal coupling constants. All the phase transitions encountered in this system occur between disordered phases and are all caused by frustration.
This paper is about the frustration-induced criticality in the antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals, and also about the effect of thermal fluctuations on this criticality. The method used is the bond-mean-field theory, which is based on the Jordan–Wigner transformation in dimensions higher than one. In this paper, we will summarize the main results presented in a talk, and report on new results about the couplings and temperature dependences of the spin susceptibility.PACS Nos.: 75.10.Jm, 75.10.Dg, 75.50.Ee
Field-induced quantum (zero temperature) criticality described using parameters that are not associated with a symmetry-breaking long-range order is found in the spin-12\frac{1}{2}21​ antiferromagnetic three-leg Heisenberg ladder. These parameters represent the spin bond order, which is a consequence of the probability for spins on adjacent sites to be bound in a singletlike arrangement. They underwent two phase transitions, one at the lower critical field hc1{h}_{c1}hc1​ and the other at the upper critical field hc2{h}_{c2}hc2​. The field dependence of these bond parameters and of the magnetization is calculated for several values of the rung coupling in all regimes. This yields the coupling field phase diagram. It is found that the rung coupling must exceed a threshold value for the plateau, at one third of the saturation magnetization, to appear. The results obtained here within the bond mean-field theory compare well with existing numerical data.
We study the quantum critical behavior in the antiferromagnetic Heisenberg chain and two-leg Heisenberg ladder resulting from the application of an external magnetic field. In each of these systems a finite-temperature crossover line between two different ferromagnetic phases ends with a quantum critical point at zero temperature. Using the bond-mean-field theory, we calculate the field dependence of the magnetization and the mean-field spin bond parameters in both systems. For the Heisenberg chain, we recover the existing exact results and show in addition that the saturation of the zero-temperature magnetization at the field hc=2J{h}_{c}=2Jhc​=2J is accompanied by a quantum phase transition, where the bond parameter vanishes. Here JJJ is the exchange coupling constant along the chain. For the two-leg ladder, we also recover the known results, like the two magnetization plateaus, and show that at the upper critical field, which corresponds to the appearance of the saturation magnetization plateau, the chain and rung spin bond parameters vanish. The identification of the order parameters that govern the field-induced quantum criticality in the systems we study here constitutes an original contribution. Because no long-range order, which breaks symmetry, characterizes the bond order, the latter could be a proposal for the so-called hidden order. We calculate analytically the bond parameters in both systems as functions of the field in the low- and high-field limits at zero temperature. At nonzero temperatures, the calculation of the magnetization and bond parameters is carried out by solving the mean-field equations numerically.
The antiferromagnetic Heisenberg model on the three-leg ladder is studied using the generalized Jordan-Wigner transformation in dimensions higher than 1, and the bond-mean-field theory. The magnetic susceptibility and other thermodynamic quantities are analyzed as a function of the rung-to-leg coupling ratio $\ensuremath{\alpha}$ and temperature TTT. We fit the experimental susceptibility data of the three-leg material Sr2Cu3O5{\mathrm{Sr}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{5}Sr2​Cu3​O5​ of Azuma and co-workers with good agreement. One of the main findings of this work is the proposal that close to two-thirds of the spin degrees of freedom on each of the rungs of the ladder lock at low TTT for small $\ensuremath{\alpha}$, then collectively almost 2∕32∕32∕3 of the spin degrees of freedom on all the rungs freeze completely at low TTT for $\ensuremath{\alpha}$ greater than a threshold value. The approach developed here can be used to study the three-leg ladder for all values of $\ensuremath{\alpha}$, and is thus suitable for the description of the crossover regime between the weak- and strong-coupling regimes.
The antiferromagnetic Heisenberg two-leg ladder in the presence of frustration and an external magnetic field is a system that is characterized by two sorts of quantum criticalities, not only one. One criticality is the consequence of intrinsic frustration, and the other one is a result of the external magnetic field. So the behaviour of each of them in the presence of the other deserves to be studied. Using the Jordan–Wigner transformation in dimensions higher than one and the bond-mean-field theory we examine the interplay between the field-induced and frustration-induced quantum criticalities in this system. The present work could constitute a prototype for those systems showing multiple, perhaps sometimes competing, quantum criticalities. We calculate several physical quantities such as magnetization and spin susceptibility as functions of field and temperature.PACS Nos.: 75.10.Jm 73.43.Nq 75.10.Pq 75.10.Dg
Using density-matrix renormalization group calculations, we investigate the general phase diagram of the frustrated two-leg ladder with Heisenberg interactions along legs, rungs, and diagonals. We confirm that all antiferromagnetic gapped states belong to the same universality class as the Haldane phase. In a three-dimensional phase diagram, we determine a continuous surface with singularities in the string-order parameter or its first derivative, corresponding to a transition between two Haldane phases with different topological order. Some parts of this transition surface are critical with zero gap and vanishing string-order parameter. In the complementary parts, the transition is first order with finite gap and string order. The boundary of this surface with the ferromagnetic region is a critical end line, when the surface is critical, and a triple line anywhere else. Part of this boundary coincides with the exactly soluble model proposed by Dmitriev, Krivnov, and Ovchinnikov [Phys. Rev. B 56, 5985 (1997)].