We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix... more We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z1,..., zr of the Bethe ansatz equations, the size of the determinants is of order r×r. We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.
The exact 2-spinon part of the dynamic spin structure factor $S_{xx}(Q,\omega)$ for the one-dimen... more The exact 2-spinon part of the dynamic spin structure factor $S_{xx}(Q,\omega)$ for the one-dimensional $s$=1/2 $XXZ$ model at $T$=0 in the antiferromagnetically ordered phase is calculated using recent advances by Jimbo and Miwa in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The 2-spinon excitations form a 2-parameter continuum consisting of two partly overlapping sheets in $(Q,\omega)$-space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\pi/2)$ and a smooth minimum with a gap at the zone center $(Q=0)$. The 2-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the 2-spinon transition rates, the two regimes $0 \leq Q < Q_\kappa$ (near the zone center) and $Q_\kappa < Q \leq \pi/2$ (near the zone boundary) must be distinguished, where $Q_\kappa \to 0$ in the Heisenberg limit and $Q_\kappa \to \pi/2$ in the Ising limit. The resulting 2-spinon part of $S_{xx}(Q,\omega)$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0 < Q_\kappa \leq \pi/2$, by contrast, the 2-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. Existing perturbation studies have been unable to capture the physics of the regime $Q_\kappa < Q \leq \pi/2$. However, their line shape predictions for the regime $0 \leq Q < Q_\kappa$ are in good agreement with the new exact results if the anisotropy is very strong.
We present exact and explicit results for the thermodynamic properties (isochores, isotherms, iso... more We present exact and explicit results for the thermodynamic properties (isochores, isotherms, isobars, response functions, velocity of sound) of a quantum gas in dimensions D>=1 and with fractional exclusion statistics 0<=g<=1 connecting bosons (g=0) and fermions (g=1). In D=1 the results are equivalent to those of the Calogero-Sutherland model. Emphasis is given to the crossover between boson-like and fermion-like features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. The full isochoric heat capacity and the leading low-T term of the isobaric expansivity in D=2 are independent of g. The onset of Bose-Einstein condensation along the isobar occurs at a nonzero transition temperature in all dimensions. The T-dependence of the velocity of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential.
We analyze the problem of microwave absorption by the Heisenberg-Ising magnet in terms of shifted... more We analyze the problem of microwave absorption by the Heisenberg-Ising magnet in terms of shifted moments of the imaginary part of the dynamical susceptibility. When both, the Zeeman field and the wave vector of the incident microwave, are parallel to the anisotropy axis, the first four moments determine the shift of the resonance frequency and the line width in a situation where the frequency is varied for fixed Zeeman field. For the one-dimensional model we can calculate the moments exactly. This provides exact data for the resonance shift and the line width at arbitrary temperatures and magnetic fields. In current ESR experiments the Zeeman field is varied for fixed frequency. We show how in this situation the moments give perturbative results for the resonance shift and for the integrated intensity at small anisotropy as well as an explicit formula connecting the line width with the anisotropy parameter in the high-temperature limit.
Having introduced the magnon in part I and the spinon in part II as the relevant quasi-particles ... more Having introduced the magnon in part I and the spinon in part II as the relevant quasi-particles for the interpretation of the spectrum of low-lying excitations in the one-dimensional (1D) s=1/2 Heisenberg ferromagnet and antiferromagnet, respectively, we now study the low-lying excitations of the Heisenberg antiferromagnet in a magnetic field and interpret these collective states as composites of quasi-particles from a different species. We employ the Bethe ansatz to calculate matrix elements and show how the results of such a calculation can be used to predict lineshapes for neutron scattering experiments on quasi-1D antiferromagnetic compounds. The paper is designed as a tutorial for beginning graduate students. It includes 11 problems for further study.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are a... more The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
As part of a study that investigates the dynamics of the s=1/2 XXZ model in the planar regime |De... more As part of a study that investigates the dynamics of the s=1/2 XXZ model in the planar regime |Delta|<1, we discuss the singular nature of the Bethe ansatz equations for the case Delta=0 (XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin-fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the formulas to two-spinon transition rates for chains with up to N=4096 sites.
We consider linear arrays of cells of volume V_{c} populated by monodisperse rods of size σV_{c},... more We consider linear arrays of cells of volume V_{c} populated by monodisperse rods of size σV_{c},σ=1,2,..., subject to hardcore exclusion interaction. Each rod experiences a position-dependent external potential. In one application we also examine effects of contact forces between rods. We employ two distinct methods of exact analysis with complementary strengths and different limits of spatial resolution to calculate profiles of pressure and density on mesoscopic and microscopic length scales at thermal equilibrium. One method uses density functionals and the other statistically interacting vacancy particles. The applications worked out include gravity, power-law traps, and hard walls. We identify oscillations in the profiles on a microscopic length scale and show how they are systematically averaged out on a well-defined mesoscopic length scale to establish full consistency between the two approaches. The continuum limit, realized as V_{c}→0,σ→∞ at nonzero and finite σV_{c}, connects our highest-resolution results with known exact results for monodisperse rods in a continuum. We also compare the pressure profiles obtained from density functionals with the average microscopic pressure profiles derived from the pair distribution function.
We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external f... more We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field $h_{3}$ and a small periodic field of strength $h_{1}$. For the case of a transverse staggered field a gap opens, which scales with $h_{1}^{\epsilon_{1}}$, where $\epsilon_{1}=\epsilon_{1}(h_{3})$ is given by the critical exponent $\eta_{1}(M(h_{3}))$ defined through the transverse structure factor of the model at $h_{1}=0$. For the case of a longitudinal periodic field with wave vector $q=\pi/2$ and strength $h_{q}$ a plateau is found in the magnetization curve at $M=1/4$. The difference of the upper- and lower magnetic field scales with $h_{3}^{u}-h_{3}^{l}\sim h_{q}^{\epsilon_{3}}$, where $\epsilon_{3}=\epsilon_{3}(h_{3})$ is given by the critical exponent $\eta_{3}(M(h_{3}))$ defined through the longitudinal structure factor of the model at $h_{q}=0$.
We consider the individual excitations of the antiferromagnetic spin-1/2 chain in the Luttinger L... more We consider the individual excitations of the antiferromagnetic spin-1/2 chain in the Luttinger Liquid formalism. The inclusion of Umklapp scattering and other irrelevant operators introduces an interaction between quasiparticles, which lifts the degeneracy in the linearized spectrum. So far this effect has been systematically understood only for the lowest excited state in each sector. We now show for a number of low lying excitations how rotations in the degenerate subspaces diagonalize the Umklapp term perturbatively. Our results are verified by taking advantage of the Bethe solution of the spin chain to perform an exact finite size scaling analysis for individual excited levels. From this we can identify the correspondence of quantum numbers between the bosonic and Bethe states. We also make contact to the well known quantum numbers at the non-interacting free fermion point.
We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix... more We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z1,..., zr of the Bethe ansatz equations, the size of the determinants is of order r×r. We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.
The exact 2-spinon part of the dynamic spin structure factor $S_{xx}(Q,\omega)$ for the one-dimen... more The exact 2-spinon part of the dynamic spin structure factor $S_{xx}(Q,\omega)$ for the one-dimensional $s$=1/2 $XXZ$ model at $T$=0 in the antiferromagnetically ordered phase is calculated using recent advances by Jimbo and Miwa in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The 2-spinon excitations form a 2-parameter continuum consisting of two partly overlapping sheets in $(Q,\omega)$-space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\pi/2)$ and a smooth minimum with a gap at the zone center $(Q=0)$. The 2-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the 2-spinon transition rates, the two regimes $0 \leq Q < Q_\kappa$ (near the zone center) and $Q_\kappa < Q \leq \pi/2$ (near the zone boundary) must be distinguished, where $Q_\kappa \to 0$ in the Heisenberg limit and $Q_\kappa \to \pi/2$ in the Ising limit. The resulting 2-spinon part of $S_{xx}(Q,\omega)$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0 < Q_\kappa \leq \pi/2$, by contrast, the 2-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. Existing perturbation studies have been unable to capture the physics of the regime $Q_\kappa < Q \leq \pi/2$. However, their line shape predictions for the regime $0 \leq Q < Q_\kappa$ are in good agreement with the new exact results if the anisotropy is very strong.
We present exact and explicit results for the thermodynamic properties (isochores, isotherms, iso... more We present exact and explicit results for the thermodynamic properties (isochores, isotherms, isobars, response functions, velocity of sound) of a quantum gas in dimensions D>=1 and with fractional exclusion statistics 0<=g<=1 connecting bosons (g=0) and fermions (g=1). In D=1 the results are equivalent to those of the Calogero-Sutherland model. Emphasis is given to the crossover between boson-like and fermion-like features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. The full isochoric heat capacity and the leading low-T term of the isobaric expansivity in D=2 are independent of g. The onset of Bose-Einstein condensation along the isobar occurs at a nonzero transition temperature in all dimensions. The T-dependence of the velocity of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential.
We analyze the problem of microwave absorption by the Heisenberg-Ising magnet in terms of shifted... more We analyze the problem of microwave absorption by the Heisenberg-Ising magnet in terms of shifted moments of the imaginary part of the dynamical susceptibility. When both, the Zeeman field and the wave vector of the incident microwave, are parallel to the anisotropy axis, the first four moments determine the shift of the resonance frequency and the line width in a situation where the frequency is varied for fixed Zeeman field. For the one-dimensional model we can calculate the moments exactly. This provides exact data for the resonance shift and the line width at arbitrary temperatures and magnetic fields. In current ESR experiments the Zeeman field is varied for fixed frequency. We show how in this situation the moments give perturbative results for the resonance shift and for the integrated intensity at small anisotropy as well as an explicit formula connecting the line width with the anisotropy parameter in the high-temperature limit.
Having introduced the magnon in part I and the spinon in part II as the relevant quasi-particles ... more Having introduced the magnon in part I and the spinon in part II as the relevant quasi-particles for the interpretation of the spectrum of low-lying excitations in the one-dimensional (1D) s=1/2 Heisenberg ferromagnet and antiferromagnet, respectively, we now study the low-lying excitations of the Heisenberg antiferromagnet in a magnetic field and interpret these collective states as composites of quasi-particles from a different species. We employ the Bethe ansatz to calculate matrix elements and show how the results of such a calculation can be used to predict lineshapes for neutron scattering experiments on quasi-1D antiferromagnetic compounds. The paper is designed as a tutorial for beginning graduate students. It includes 11 problems for further study.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are a... more The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
As part of a study that investigates the dynamics of the s=1/2 XXZ model in the planar regime |De... more As part of a study that investigates the dynamics of the s=1/2 XXZ model in the planar regime |Delta|<1, we discuss the singular nature of the Bethe ansatz equations for the case Delta=0 (XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin-fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the formulas to two-spinon transition rates for chains with up to N=4096 sites.
We consider linear arrays of cells of volume V_{c} populated by monodisperse rods of size σV_{c},... more We consider linear arrays of cells of volume V_{c} populated by monodisperse rods of size σV_{c},σ=1,2,..., subject to hardcore exclusion interaction. Each rod experiences a position-dependent external potential. In one application we also examine effects of contact forces between rods. We employ two distinct methods of exact analysis with complementary strengths and different limits of spatial resolution to calculate profiles of pressure and density on mesoscopic and microscopic length scales at thermal equilibrium. One method uses density functionals and the other statistically interacting vacancy particles. The applications worked out include gravity, power-law traps, and hard walls. We identify oscillations in the profiles on a microscopic length scale and show how they are systematically averaged out on a well-defined mesoscopic length scale to establish full consistency between the two approaches. The continuum limit, realized as V_{c}→0,σ→∞ at nonzero and finite σV_{c}, connects our highest-resolution results with known exact results for monodisperse rods in a continuum. We also compare the pressure profiles obtained from density functionals with the average microscopic pressure profiles derived from the pair distribution function.
We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external f... more We study the one-dimensional spin-1/2 antiferromagnetic Heisenberg model exposed to an external field, which is a superposition of a homogeneous field $h_{3}$ and a small periodic field of strength $h_{1}$. For the case of a transverse staggered field a gap opens, which scales with $h_{1}^{\epsilon_{1}}$, where $\epsilon_{1}=\epsilon_{1}(h_{3})$ is given by the critical exponent $\eta_{1}(M(h_{3}))$ defined through the transverse structure factor of the model at $h_{1}=0$. For the case of a longitudinal periodic field with wave vector $q=\pi/2$ and strength $h_{q}$ a plateau is found in the magnetization curve at $M=1/4$. The difference of the upper- and lower magnetic field scales with $h_{3}^{u}-h_{3}^{l}\sim h_{q}^{\epsilon_{3}}$, where $\epsilon_{3}=\epsilon_{3}(h_{3})$ is given by the critical exponent $\eta_{3}(M(h_{3}))$ defined through the longitudinal structure factor of the model at $h_{q}=0$.
We consider the individual excitations of the antiferromagnetic spin-1/2 chain in the Luttinger L... more We consider the individual excitations of the antiferromagnetic spin-1/2 chain in the Luttinger Liquid formalism. The inclusion of Umklapp scattering and other irrelevant operators introduces an interaction between quasiparticles, which lifts the degeneracy in the linearized spectrum. So far this effect has been systematically understood only for the lowest excited state in each sector. We now show for a number of low lying excitations how rotations in the degenerate subspaces diagonalize the Umklapp term perturbatively. Our results are verified by taking advantage of the Bethe solution of the spin chain to perform an exact finite size scaling analysis for individual excited levels. From this we can identify the correspondence of quantum numbers between the bosonic and Bethe states. We also make contact to the well known quantum numbers at the non-interacting free fermion point.
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Papers by Michael Karbach