\n \nWe study random XY and (dimerized) XX spin-1/2 quantum spin chains\nat their quantum phase transition driven by the anisotropy and\ndimerization, respectively. Using exact expressions for\nmagnetization, correlation functions and energy gap, obtained by the\nfree fermion technique, the critical and off-critical\n(Griffiths-McCoy) singularities are related to persistence\nproperties of random walks. In this way we determine exactly the\ndecay exponents for surface and bulk transverse and longitudinal\ncorrelations, correlation length exponent and dynamical exponent.\n\n\n\n \nWe study random XY and (dimerized) XX spin-1/2 quantum spin chains\nat their quantum phase transition driven by the anisotropy and\ndimerization, respectively. Using exact expressions for\nmagnetization, correlation functions and energy gap, obtained by the\nfree fermion technique, the critical and off-critical\n(Griffiths-McCoy) singularities are related to persistence\nproperties of random walks. In this way we determine exactly the\ndecay exponents for surface and bulk transverse and longitudinal\ncorrelations, correlation length exponent and dynamical exponent.\n\n\n

Feb 15, 20002 citations

Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

Ferenc Iglói, Róbert Juhász, Heiko Rieger

We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities, and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random-walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confirmed by numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems $(L<~512)$ are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution shows multiscaling character. In the Griffiths phase, which is an extended part of the off-critical region, average autocorrelations have a power-law form with a nonuniversal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities, and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random-walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confirmed by numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems $(L<~512)$ are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution shows multiscaling character. In the Griffiths phase, which is an extended part of the off-critical region, average autocorrelations have a power-law form with a nonuniversal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.

arXiv:cond-mat/9911480Physical review. B, Condensed matterMay 1, 200051 citations

Critical exponents of random Ising-like systems in general dimensions

Yurij V. Holovatch, Mykola A. Shpot

Journal of Statistical PhysicsFeb 1, 199238 citations

Critical exponents of the 3D XY model from cluster update Monte Carlo

Martin Hasenbusch B, Steffen Meyer

Physics Letters BMay 1, 199081 citations

Infinite randomness with continuously varying critical exponents in the random XYZ spin chain

Brenden Roberts, Olexei I. Motrunich

We study the antiferromagnetic XYZ spin chain with quenched bond randomness, focusing on a critical line between localized Ising magnetic phases. A previous calculation using the spectrum-bifurcation renormalization group, and assuming marginal many-body localization, proposed that critical indices vary continuously. In this work, we solve the low-energy physics using an unbiased numerically exact tensor network method named the ``rigorous renormalization group.'' We find a line of fixed points consistent with infinite-randomness phenomenology, with indeed continuously varying critical exponents for average spin correlations. A self-consistent Hartree--Fock-type treatment of the $z$ couplings as interactions added to the free-fermion random XY model captures much of the important physics including the varying exponents; we provide an understanding of this as a result of local correlation induced between the mean-field couplings. We solve the problem of the locally correlated XY spin chain with an arbitrary degree of correlation and provide analytical strong-disorder renormalization group proofs of continuously varying exponents based on an associated classical random walk problem. This is also an example of a line of fixed points with continuously varying exponents in the equivalent disordered free-fermion chain. We argue that this line of fixed points also controls an extended region of the critical interacting XYZ spin chain.We study the antiferromagnetic XYZ spin chain with quenched bond randomness, focusing on a critical line between localized Ising magnetic phases. A previous calculation using the spectrum-bifurcation renormalization group, and assuming marginal many-body localization, proposed that critical indices vary continuously. In this work, we solve the low-energy physics using an unbiased numerically exact tensor network method named the ``rigorous renormalization group.'' We find a line of fixed points consistent with infinite-randomness phenomenology, with indeed continuously varying critical exponents for average spin correlations. A self-consistent Hartree--Fock-type treatment of the $z$ couplings as interactions added to the free-fermion random XY model captures much of the important physics including the varying exponents; we provide an understanding of this as a result of local correlation induced between the mean-field couplings. We solve the problem of the locally correlated XY spin chain with an arbitrary degree of correlation and provide analytical strong-disorder renormalization group proofs of continuously varying exponents based on an associated classical random walk problem. This is also an example of a line of fixed points with continuously varying exponents in the equivalent disordered free-fermion chain. We argue that this line of fixed points also controls an extended region of the critical interacting XYZ spin chain.

arXiv:2107.12937Physical review. B./Physical review. BDec 27, 20219 citations

(L<~512)

are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution shows multiscaling character. In the Griffiths phase, which is an extended part of the off-critical region, average autocorrelations have a power-law form with a nonuniversal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.

z

couplings as interactions added to the free-fermion random XY model captures much of the important physics including the varying exponents; we provide an understanding of this as a result of local correlation induced between the mean-field couplings. We solve the problem of the locally correlated XY spin chain with an arbitrary degree of correlation and provide analytical strong-disorder renormalization group proofs of continuously varying exponents based on an associated classical random walk problem. This is also an example of a line of fixed points with continuously varying exponents in the equivalent disordered free-fermion chain. We argue that this line of fixed points also controls an extended region of the critical interacting XYZ spin chain.