We study the zero-temperature magnetization curve $(M\ensuremath{-}H$ curve) of the one-dimensional quantum antiferromagnet of spin one. The Hamiltonian H we consider is of the bilinear-biquadratic form: $H={\ensuremath{\sum}}_{i}f({s}_{i}\ensuremath{\cdot}{s}_{i+1})$ (+Zeeman term) where ${s}_{i}$ is the spin operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0<~\ensuremath{\beta}<1.$ We focus on validity of the $\ensuremath{\delta}$-function Bose-gas picture near the two critical fields: upper-critical field ${H}_{s}$ above which the magnetization saturates and the lower-critical field ${H}_{c}$ associated with the Haldane gap. As for the behavior near ${H}_{s},$ we take ``low-energy effective S matrix'' approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M\ensuremath{-}H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White's density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field ${H}_{c}.$ Comparing the PWFRG-calculated $M\ensuremath{-}H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\ensuremath{\beta}$ separated by a critical value ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0<\ensuremath{\beta}<{\ensuremath{\beta}}_{c},$ the effective Bose coupling c is positive but rather small. The small value of c makes the ``critical region'' of the square-root behavior $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, we find that in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ the square-root behavior transmutes to a different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\approx}1/4.$ In region II, ${\ensuremath{\beta}}_{c}<\ensuremath{\beta}<1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.We study the zero-temperature magnetization curve $(M\ensuremath{-}H$ curve) of the one-dimensional quantum antiferromagnet of spin one. The Hamiltonian H we consider is of the bilinear-biquadratic form: $H={\ensuremath{\sum}}_{i}f({s}_{i}\ensuremath{\cdot}{s}_{i+1})$ (+Zeeman term) where ${s}_{i}$ is the spin operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0<~\ensuremath{\beta}<1.$ We focus on validity of the $\ensuremath{\delta}$-function Bose-gas picture near the two critical fields: upper-critical field ${H}_{s}$ above which the magnetization saturates and the lower-critical field ${H}_{c}$ associated with the Haldane gap. As for the behavior near ${H}_{s},$ we take ``low-energy effective S matrix'' approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M\ensuremath{-}H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White's density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field ${H}_{c}.$ Comparing the PWFRG-calculated $M\ensuremath{-}H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\ensuremath{\beta}$ separated by a critical value ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0<\ensuremath{\beta}<{\ensuremath{\beta}}_{c},$ the effective Bose coupling c is positive but rather small. The small value of c makes the ``critical region'' of the square-root behavior $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, we find that in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ the square-root behavior transmutes to a different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\approx}1/4.$ In region II, ${\ensuremath{\beta}}_{c}<\ensuremath{\beta}<1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.

arXiv:cond-mat/9807266Physical review. B, Condensed matterMar 1, 199951 citations

Renormalization group approach to the interacting bose fluid

F.W. Wiegel

Physica A Statistical Mechanics and its ApplicationsApr 1, 197816 citations

Thermodynamics and renormalized quasiparticles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

Jan Krieg, Dominik Straßel, Simon Streib +2 more

We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature $T$ and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin- $1/2$ quantum $XY$ model with coupling $J$ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature $T$ and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-$1/2$ quantum $XY$ model with coupling $J$ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.

arXiv:1607.01015Physical review. B./Physical review. BJan 13, 20173 citations

Renormalization group theory of the three-dimensional dilute Bose gas

M. Bijlsma, Henk T. C. Stoof

We study the three-dimensional atomic Bose gas using renormalization group techniques. Using our knowledge of the microscopic details of the interatomic interaction, we determine the correct initial values of our renormalization group equations and thus obtain also information on nonuniversal properties. As a result, we can predict, for instance, the critical temperature of the gas and the superfluid and condensate density of the Bose-Einstein condensed phase in the regime na${\mathrm{\ensuremath{\Lambda}}}_{\mathit{th}}^{2}$\ensuremath{\ll}1, where the average interaction energy is small compared to the average kinetic energy. \textcopyright{} 1996 The American Physical Society.We study the three-dimensional atomic Bose gas using renormalization group techniques. Using our knowledge of the microscopic details of the interatomic interaction, we determine the correct initial values of our renormalization group equations and thus obtain also information on nonuniversal properties. As a result, we can predict, for instance, the critical temperature of the gas and the superfluid and condensate density of the Bose-Einstein condensed phase in the regime na${\mathrm{\ensuremath{\Lambda}}}_{\mathit{th}}^{2}$\ensuremath{\ll}1, where the average interaction energy is small compared to the average kinetic energy. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9607188Physical Review ADec 1, 1996127 citations

{s}_{i}

is the spin operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0<~\ensuremath{\beta}<1.$ We focus on validity of the $\ensuremath{\delta}$-function Bose-gas picture near the two critical fields: upper-critical field

{H}_{s}

above which the magnetization saturates and the lower-critical field

{H}_{c}

associated with the Haldane gap. As for the behavior near

{H}_{s},

we take ``low-energy effective S matrix'' approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M\ensuremath{-}H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White's density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field

{H}_{c}.

Comparing the PWFRG-calculated $M\ensuremath{-}H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\ensuremath{\beta}$ separated by a critical value ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0<\ensuremath{\beta}<{\ensuremath{\beta}}_{c},$ the effective Bose coupling c is positive but rather small. The small value of c makes the ``critical region'' of the square-root behavior $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, we find that in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ the square-root behavior transmutes to a different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\approx}1/4.$ In region II, ${\ensuremath{\beta}}_{c}<\ensuremath{\beta}<1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.

T

and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-

1/2

quantum

XY

model with coupling

J

in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.