arXiv:cond-mat/0308176 · Physical Review B

Energy correlations in random transverse field Ising spin chains

Gil Refael, Daniel S. Fisher

Published Aug 17, 2004• 12 citations

Abstract

The end-to-end energy-energy correlations of random transverse field quantum Ising spin chains are computed using a generalization of an asymptotically exact real space renormalization group (RG) previously introduced. Away from the critical point, the average energy-energy correlations decay exponentially with a correlation length that is the same as that of the spin-spin correlations. The typical correlations, however, decay exponentially with a characteristic length, proportional to the square root of the primary correlation length. At the quantum critical point, the average correlations decay subexponentially as $\overline{{C}_{L}}\ensuremath{\sim}{e}^{\ensuremath{-}\mathrm{const}\phantom{\rule{0.2em}{0ex}}{L}^{1∕3}}$, whereas the typical correlations decay faster, as $\ensuremath{\sim}{e}^{\ensuremath{-}K\sqrt{L}}$, with $K$ a random variable with a universal distribution. The critical energy-energy correlations behave very similarly to the smallest gap, computed previously; this is explained in terms of the RG flow and the excitation structure of the chain. In order to obtain the energy correlations, an extension of the previously used methods was needed; here, this was carried out via RG transformations that involve a sequence of unitary transformations.

DOI: 10.1103/physrevb.70.064409

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arXiv:cond-mat/0308176 · Physical Review B

Energy correlations in random transverse field Ising spin chains

Gil Refael, Daniel S. Fisher

Published Aug 17, 2004• 12 citations

arXiv HTML PDF

Abstract

\overline{{C}_{L}}\ensuremath{\sim}{e}^{\ensuremath{-}\mathrm{const}\phantom{\rule{0.2em}{0ex}}{L}^{1∕3}}

, whereas the typical correlations decay faster, as

\ensuremath{\sim}{e}^{\ensuremath{-}K\sqrt{L}}

, with

K

a random variable with a universal distribution. The critical energy-energy correlations behave very similarly to the smallest gap, computed previously; this is explained in terms of the RG flow and the excitation structure of the chain. In order to obtain the energy correlations, an extension of the previously used methods was needed; here, this was carried out via RG transformations that involve a sequence of unitary transformations.

DOI: 10.1103/physrevb.70.064409

Related papers

Critical behavior of random transverse-field Ising spin chains

Daniel S. Fisher

A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.

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The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at $T=0.$ We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation $C(L)$ for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation $C(L)$ decays algebraically as $1/L$ in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at $T=0.$ We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation $C(L)$ for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation $C(L)$ decays algebraically as $1/L$ in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.

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A renormalization-group analysis of the spin-1/2 transverse field Ising model with quenched randomness is presented; it become exact asymptotically near the zero temperature ferromagnetic phase transition. The spontaneous magnetization is found to vanish with an exponent \ensuremath{\beta}=1/2(3- \ensuremath{\surd}5 ), while in the disordered phase the typical and average spin correlations are found to decay with different correlation lengths, which diverge with exponents \ensuremath{\nu}\ifmmode \tilde{}\else \~{}\fi{}=1 and \ensuremath{\nu}=2, respectively.A renormalization-group analysis of the spin-1/2 transverse field Ising model with quenched randomness is presented; it become exact asymptotically near the zero temperature ferromagnetic phase transition. The spontaneous magnetization is found to vanish with an exponent \ensuremath{\beta}=1/2(3- \ensuremath{\surd}5 ), while in the disordered phase the typical and average spin correlations are found to decay with different correlation lengths, which diverge with exponents \ensuremath{\nu}\ifmmode \tilde{}\else \~{}\fi{}=1 and \ensuremath{\nu}=2, respectively.

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Ferenc Iglói, István A. Kovács

The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$ , and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.

arXiv:1712.07467Physical review. B./Physical review. BMar 21, 20184 citations

Exact renormalization of the random transverse-field Ising spin chain in the strongly ordered and strongly disordered Griffiths phases

Ferenc Iglói

Fisher's [Phys. Rev. B $51,$ 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.Fisher's [Phys. Rev. B $51,$ 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.

arXiv:cond-mat/0108350Physical review. B, Condensed matterJan 16, 200260 citations

\overline{{C}_{L}}\ensuremath{\sim}{e}^{\ensuremath{-}\mathrm{const}\phantom{\rule{0.2em}{0ex}}{L}^{1∕3}}

, whereas the typical correlations decay faster, as

\ensuremath{\sim}{e}^{\ensuremath{-}K\sqrt{L}}

, with

K

The random transverse-field Ising chain is the simplest disordered model presenting a quantum phase transition at

T=0.

We compare analytically its finite-size scaling properties in two different ensembles for the disorder: (i) the canonical ensemble, where the disorder variables are independent and (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation

C(L)

for a chain of length L. At criticality, each observable decays typically as ${e}^{\ensuremath{-}w\sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable w are different in the two ensembles, in particular, in their asymptotic behaviors. As a consequence, the dependence on L of averaged observables differs in the two ensembles. For instance, the correlation

C(L)

decays algebraically as

1/L

in the canonical ensemble, but subexponentially as ${e}^{\ensuremath{-}{\mathrm{cL}}^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $\ensuremath{\nu}=2$ in both ensembles, but the following observables are governed by the exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1$ in the microcanonical ensemble, instead of the exponent $\ensuremath{\nu}=2$ in the canonical ensemble: (a) In the disordered phase, the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation; (b) in the ordered phase, the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.

d=1,2

, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In

d=1

the results are related to dimer-dimer correlations in the random antiferromagnetic

XX

chain and have been tested by numerical calculations using free-fermionic techniques.

Fisher's [Phys. Rev. B

51,

6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.