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Physical review. B, Condensed matter

Correlation functions of one-dimensional quantum systems

Thierry Giamarchi, Heinz J. Schulz

Published Mar 1, 1989• 213 citations

Abstract

A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.

DOI: 10.1103/physrevb.39.4620

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A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.

Correlation functions of one-dimensional Tomonaga-Luttinger models with sine-Gordon-like extensions are computed using bosonisation and scaling techniques. The author shows that by expanding the full correlation functions instead of considering only their first cumulants, exponents for their asymptotic decay, logarithmic corrections as well as renormalised amplitudes can be obtained. Applied to the Hubbard model, the method produces for the first time correlation functions consistent with its symmetries, exact spectrum and numerical simulations.

We numerically investigate classical and quantum correlations in one-dimensional quantum critical systems. The infinite matrix product state (iMPS) representation is employed in order to consider an infinite-size spin chain. By using the infinite time-evolving block decimation algorithm, iMPS ground state wave functions are obtained at critical points for the transverse-field spin-

1/2

XY model. From the ground state wave functions, we calculate classical and quantum correlations and mutual information. All of the correlations are found to exhibit a power-law decay with the increments of the lattice distance for both the transition lines of the Ising universality class and the Gaussian universality class. Such power-law scaling behaviors of the correlations manifest the existence of diversing correlation lengths, which means scale invariance. The critical features of the correlations can be characterized by introducing a critical exponent of the power-law decaying correlations. Similar to the critical exponent

η

of the spin-spin correlation for the universality classes in the transverse-field XY model, we calculate the critical exponents of the two-spin classical and quantum correlations as well as that of the corresponding mutual information. All of the correlations have the same critical exponents, i.e.,

η^{I}=η^{C}=η^{D}

at a critical point, where the superscripts

I

,

C

, and

D

stand for mutual information, classical correlation, and quantum correlation, respectively. Furthermore, the critical exponent

η

of the spin-spin correlation is shown to relate to

η= η^α/2

with

α\in \{ I, C, D\}

.

The asymptotic dynamical correlation functions in one-dimensional spin chains are described by power laws. The corresponding exponents characterize different bulk and boundary critical behavior. We present results for the logarithmic contribution to the boundary correlations of an isotropic Heisenberg chain. The exponent of the logarithm, $\ensuremath{\lambda}=1$, is derived using a renormalization group technique. We confirm our analytical results by comparing with numerical quantum Monte Carlo data.

We study the properties of the double-frequency sine–Gordon model in the vicinity of the Ising quantum phase transition displayed by this model. Using a mapping onto a generalised lattice quantum Ashkin-Teller model, we obtain critical and nearly-off-critical correlation functions of various operators. We discuss applications of the double-sine-Gordon model to one-dimensional physical systems, like spin chains in a staggered external field and interacting electrons in a staggered potential. PACS: 71.10.Pm; 71.10.Fd; 75.10.Jm

The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.

We present explicit expressions for the correlation functions of interacting fermions in one dimension which are valid for arbitrary system sizes and temperatures. The result applies to a number of very different strongly correlated systems, including mesoscopic quantum wires, quantum Hall edges, spin chains and quasi-one-dimensional metals. It is, for example, possible to calculate Coulomb blockade oscillations from our expression and determine their dependence on interaction strength and temperature. Numerical simulations show excellent agreement with the analytical results.