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Edge exponent in the dynamic spin structure factor of the Yang-Gaudin model
ZVONAREV, Mikhail, CHEIANOV, V. V., GIAMARCHI, Thierry
Abstract
The dynamic spin structure factor S(k,ω) of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function of ω it is shown to exhibit a power-law singularity at the threshold frequency defined by the energy of a magnon at given k. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.
ZVONAREV, Mikhail, CHEIANOV, V. V., GIAMARCHI, Thierry. Edge exponent in the dynamic spin structure factor of the Yang-Gaudin model. Physical Review. B, Condensed Matter , 2009, vol. 80, no. 20
DOI : 10.1103/PhysRevB.80.201102
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http://archive-ouverte.unige.ch/unige:36007
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1 / 1
Edge exponent in the dynamic spin structure factor of the Yang-Gaudin model
M. B. Zvonarev,1V. V. Cheianov,2and T. Giamarchi1
1DPMC-MaNEP, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva 4, Switzerland
2Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom 共Received 15 September 2009; published 10 November 2009兲
The dynamic spin structure factorS共k,兲of a system of spin-1/2 bosons is investigated at arbitrary strength of the interparticle repulsion. As a function ofit is shown to exhibit a power-law singularity at the threshold frequency defined by the energy of a magnon at givenk. The power-law exponent is found exactly using a combination of the Bethe ansatz solution and an effective-field theory approach.
DOI:10.1103/PhysRevB.80.201102 PACS number共s兲: 05.30.Jp, 02.30.Ik, 64.60.Ht
The remarkable progress achieved by the theory of one- dimensional 共1D兲 quantum fluids is rooted in the fact that dimensionality imposes severe constraints on the fluid’s low- energy excitation spectrum. Due to these constraints the in- vestigation of the low-energy dynamics of the fluid reduces to choosing the effective-field theory from a limited number of universality classes. Perhaps the most ubiquitous 共and most thoroughly investigated兲is the universality class called the Luttinger liquid.1 Other nontrivial examples include states with non-Abelian currents and spin-incoherent2–4 and ferromagnetic liquids.5–9For all such cases there exist well- developed analytical methods allowing one to calculate in- frared asymptotics of dynamical correlation function, spec- tral properties, and scaling dimensions of local observables.
In a series of recent papers5–19 it has been found that the dimensionality constraints and the resulting universality may extend far beyond the low-energy sector of the excitation spectrum of the fluid. It was shown that there exists a curve
−共k兲in the共k,兲 space at which spectral functions exhibit power-law singularities of the type
S共k,兲 ⯝c共k兲关−−共k兲兴关−−共k兲兴⌬共k兲. 共1兲 Here,共x兲 is the Heaviside step function,⌬共k兲 andc共k兲are some momentum-dependent functions, and −共k兲 is the en- ergy of the lowest excited state of the fluid at a given mo- mentumk. In a generic 1D fluid−共k兲⬎0 for allkexcept a discrete set of points defined by the Luttinger theorem. The spectrum of momentum-dependent anomalous exponents
⌬共k兲in Eq.共1兲is a natural generalization of the spectrum of scaling dimensions of the low-energy effective theory.
Understanding the structure of the spectrum of ⌬共k兲 will greatly advance the theory of 1D quantum fluids. There are several approaches to this problem. In Refs.10 and12 per- turbation theory was used to get⌬共k兲in a fermionic system.
In Refs. 13–15 an effective-field theory approach establishing a link with the mobile quantum impurity problem20–22 was proposed. This approach was comple- mented by Bethe ansatz 共BA兲 calculations for several inte- grable models: Calogero-Sutherland,11Heisenberg,16,17,19and Lieb-Liniger.18 Constraints on ⌬共k兲 implied by symmetries of microscopic Hamiltonian were discussed in Refs. 8 and 15.
In a recent work5 on the dynamical properties of a strongly repulsive ferromagnetic Bose gas, observable phe-
nomena such as spin trapping and Gaussian damping of spin waves were predicted and a link between these phenomena and the singular behavior关Eq.共1兲兴of the dynamic spin struc- ture factor was established. It was shown that, at infinite pointlike repulsion and fork→0,
⌬共k兲 ⯝− 1 +K
2
冉
kkF冊
2, 共2兲whereKis the Luttinger parameter andkF=0with0be- ing the average particle density. Assuming the validity of Eq.
共2兲 at a large but finite repulsion, a crossover between trapped and open regimes of spin propagation was character- ized completely. A different approach to the dynamics of the same system proposed in Ref. 6confirmed Eq. 共2兲. The ap- proach of Ref. 6 was further developed in Ref. 7, demon- strating that for infinite pointlike repulsion⌬共k兲has the form 共2兲for arbitrary k. In Ref.8 the small kexpansion of ⌬共k兲 was shown to have the form 共2兲 for arbitrary interparticle repulsion. However, the case of arbitrarykand interparticle repulsion remains unexplored.
In this Rapid Communication we investigate the behavior of ⌬共k兲and−共k兲for the dynamic spin structure factor of a ferromagnetic system of spin-1/2 bosons interacting through a pointlike repulsive potential of arbitrary strength. This sys- tem is described by the Yang-Gaudin model.23 Combining the Bethe ansatz with an effective-field theory, we obtain our main result: explicit expressions 共20兲–共22兲 for ⌬共k兲 at arbi- trary kand interparticle repulsion.
The Hamiltonian of the Yang-Gaudin model is
H=
冕
0 Ldx关x↑†x↑+x↓†x↓+g2兴, 共3兲
where ↑共↓兲共x兲,↑共↓兲† 共x兲 are canonical Bose fields satisfying periodic boundary conditions on a ring of circumference L and 共x兲 is the total particle density operator. We consider the dynamic spin structure factor
S共k,兲=
冕
dx dt ei共t−kx兲具⇑兩s+共x,t兲s−共0,0兲兩⇑典. 共4兲Here,s+共x兲=↑†共x兲↓共x兲is the local spin raising operator and s−共x兲=关s+共x兲兴†. The average in Eq. 共4兲is taken with respect to a fully polarized ground state 兩⇑典 of the Hamiltonian共3兲
satisfyings+共x兲兩⇑典= 0 for allx. In the spectral representation Eq. 共4兲takes the form
S共k,兲=
兺
f
␦„−Ef共k兲…兩具f,k兩s−共k兲兩⇑典兩2, 共5兲
where the sum is taken over the eigenstates 兩f,k典 of the Hamiltonian 共3兲 carrying the momentum k. The energies Ef共k兲are defined byH兩f,k典=Ef共k兲兩f,k典. The frequency−共k兲 in Eq. 共1兲is given by −共k兲= minfEf共k兲. Thus, the calcula- tion of−共k兲reduces to the analysis of the energy spectrum of excitations. The calculation of⌬共k兲directly from formula 共5兲is a far more difficult task. It requires the knowledge of the matrix element and their resummation procedure. For most integrable models, including Yang-Gaudin, such a cal- culation is beyond the reach of the existing theory. A way to bypass this problem is to combine the BA with an effective- field theory in a way similar to the derivation of scaling exponents in Luttinger liquids.24–29This is the route we take in our calculations.
We begin our analysis with a brief description of BA equations and a calculation of −共k兲. A general solution to the Yang-Gaudin model is given by the nested BA.23All the states兩f,k典in Eq.共5兲lie in the sector with thezprojection of the total spin given by Sz=N/2 − 1. In this sector Bethe’s wave functions are characterized by a set of quasimomenta 兵1, . . . ,N,其satisfying
Lj+
兺
k=1 N
共j−k兲= 2Ij+共2j− 2兲+. 共6兲
Here,共兲= 2 arctan共/g兲is the two-particle phase shift and Ij=nj−共N+ 1兲/2, wherenjare a set of distinct integers. The branch of共兲is chosen so that 共⫾⬁兲=⫾. The total en- ergy E and momentum P of a system are given by E=兺Nj=12j andP=兺Nj=1j, respectively. The quasimomentum
enters inEandPindirectly, through the solution of Eq.共6兲. In the limit =⬁ Bethe’s equations 共6兲 are identical to Be- the’s equations of the fully polarized system,30 Sz=N/2, which is equivalent to the Lieb-Liniger model.31,32The dis- tribution ofIjin the ground state of the model is
Ij=j−N+ 1
2 , j= 1, . . . ,N. 共7兲 Introducing the quasimomenta density 共j兲
= 1/关L共j+1−j兲兴 and taking the thermodynamic limit 0⬍0⬍⬁as N,L→⬁, one gets the integral equation
共兲− 1
2
冕
−⌳⌳d 共兲K共,兲=21 共8兲 for the quasimomenta in the state 共7兲and=⬁. The kernel K共,兲⬅K共−兲 is K共兲=共兲/= 2g/共g2+2兲. Note that 共兲 should satisfy 兰−⌳⌳d共兲=0. This formula to- gether with Eq. 共8兲 is used to get the value of the Fermi quasimomentum ⌳ as a function of the particle density0. The ground-state energy in the thermodynamic limit isE0=L
冕
−⌳⌳
d2共兲 共9兲
and the momentum of the ground state is zero.
Consider now the state characterized by a finite value of andIjgiven by their ground-state values关Eq.共7兲兴. This state is an excitation above the vacuum, which we shall call a magnon. Introducing the so-called shift function33 by F共j兩兲=共j−˜
j兲/共j+1−j兲, where j are ground-state quasimomenta and˜
j are those of the excited state, we get the following integral equation for F in the thermodynamic limit:
F共兩兲− 1
2
冕
−⌳⌳dK共,兲F共兩兲= −+共22− 2兲. 共10兲 The momentum of the excited state isk=
冕
−⌳⌳
d共兲关+共2− 2兲兴, 共11兲
and its energy above the ground state is
−共k兲= − 1
冕
−⌳⌳
d 共兲K共2− 2兲. 共12兲 Here,−is written as a function of the physical共observable兲 momentum k, which is related to the quasimomentum by the integral equation 共11兲. The quasienergy共兲is given by the solution of the integral equation
共兲− 1
2
冕
−⌳⌳d共兲K共,兲=2− 共13兲satisfying a condition 共⫾⌳兲= 0. The parameter entering Eq.共13兲is the chemical potential, defined by=共E0/N兲L, where E0 is found from Eq. 共9兲. One can show that at small k the dispersion law 共12兲 is parabolic 共Ref. 34兲,
−共k兲=k2/2mⴱ, with the effective mass satisfying mⴱ−1= −共g02兲−1兰−⌳⌳d 共兲.
Another way to excite the system is to create a particle- hole pair by moving one of the quantum numbers Ij in Eq.
共6兲outside of the ground-state distribution共7兲. Such excita- tions are analyzed in detail in Ref.32共see also Ref.33兲. In particular, at small momentum they are shown to be equiva- lent to sound waves propagating at the velocity
vs= 1
2共⌳兲
冏
共兲冏
=⌳. 共14兲
Any excitation in the N-particle sector with Sz=N/2 − 1 consists of several particle-hole pairs and one magnon. The exact lower bound of the particle-hole continuum32 and the dispersion curve of the magnon are illustrated in Figs. 1共a兲 and1共b兲. For all values of the coupling constantg the mag- non branch lies below the particle-hole continuum. It is thus the single magnon dispersion关Eq. 共12兲兴that gives the exact lower bound of the excitation spectrum.
While−共k兲is found using BA exclusively, in order to get
ZVONAREV, CHEIANOV, AND GIAMARCHI PHYSICAL REVIEW B80, 201102共R兲 共2009兲
201102-2
the threshold exponent⌬共k兲in Eq.共1兲for the function共4兲we need to combine the BA solution with a low-energy effective-field theory. To do so, we adapt the method pro- posed in Ref. 17 to the nested BA case discussed here. By doing so we reduce the initial problem to the problem of a Luttinger liquid minimally coupled to the infinitely heavy spin degree of freedom. The latter is solved explicitly by a unitary transformation. As a first step we introduce an auxil- iary microscopic theory with a local HamiltonianH˜ depend- ing on kas an external parameter and having the following properties: 共i兲 it conserves the total momentum, which will be denoted by q;共ii兲its excitation spectrum at q=kis gap- less; and共iii兲its structure factorS˜ satisfies
S˜共q,兲
S„q,−共k兲+…→1, q=k, →0. 共15兲 In integrable models H˜ can be constructed as a linear combination of a finite number of mutually commuting local integrals of motion. The eigenstates 兩f,q典 of H are at the same time the eigenstates of H˜; therefore, S˜共q,兲=兺f␦(−E˜f共q兲)兩具f,q兩s−共q兲兩⇑典兩2, where H˜兩f,q典
=E˜
f共q兲兩f,q典. Like forH, the low-energy spectrum ofH˜ con- sists of sound waves and a magnon. The energy of the mag- non is proportional to 共k−q兲2 as q→k. Condition 共15兲 re- quires that the velocities of the right- and left-moving sound waves be different and given by35
v⫾=vs⫾−共k兲/k, 共16兲 wherevsis given by Eq.共14兲.
The dynamics of sound waves is governed by the Lut- tinger Hamiltonian
H0=
兺
r=⫾
Hr, Hr= vr
4
冕
0 Ldx:关xr共x兲兴2:, 共17兲 where the operators r are chiral boson fields 关r共x兲,r⬘共x⬘兲兴=ir␦rr⬘sgn共x−x⬘兲 related to the micro- scopic particle density by
共x兲=0+共2兲−1
冑
K关x+共x兲−x−共x兲兴 共18兲 and the symbol :: stands for the boson normal ordering. In order to describe the low-energy magnon excitation we in- troduce the spin-density field˜s共x兲, related to the microscopic spin density by sz共x兲=˜sz共x兲+0/2 and s⫾共x兲=e⫾ikx˜s⫾共x兲, wheres⫾=sx⫾isyare the local spin-ladder operators of Eq.共4兲. Within the effective theory the operators˜s⫾are smooth spin-flip fields. Generally, a local spin flip may excite sound waves. Thus, an effective theory should contain a coupling between˜s and⫾. There is no general prescription on how to derive the corresponding coupling term microscopically.
Here, we construct it as the minimal local coupling respect- ing the SU共2兲symmetry of the microscopic theory, and van- ishing in the absence of magnon excitations36
Hi= −
兺
r=⫾
vrr
2
冕
0 Ldxxr共x兲s˜z共x兲. 共19兲
Other possible couplings involve higher gradient terms and higher harmonics of the density operator共18兲. Those are in- frared irrelevant and do not contribute to the critical expo- nents. The kinetic-energy density of the spin field is repre- sented by a higher gradient termx˜s+共x兲x˜s−共x兲that can also be neglected in the calculation of the critical exponents.37 The total Hamiltonian of the effective theory describing the dynamics near the threshold is thus given by Heff=H0+Hi. This Hamiltonian is diagonalized by a unitary transformation eiSHeffe−iS with S=共2兲−1兰0Ldx关++共x兲−−−共x兲兴s˜z共x兲. For the function共4兲this gives
⌬共k兲= − 1 + 1
42共+2+−2兲. 共20兲 What remains is to determine the coupling constants ⫾ in terms of the parameters of the microscopic theory. This is done by the comparison of the low-energy spectrum of the microscopic Hamiltonian H˜, found from the BA solution, and the spectrum of the effective Hamiltonian Heff.33 This procedure yields
r= 2rF共r⌳兩兲, r= ⫾1, 共21兲 where F is defined by the solution of the integral equation 共10兲. We solve this equation and find ⌬共k兲 numerically for different values of the coupling constant␥=g/0. For easier comparison with Eq.共2兲we represent our result in the form
⌬共k兲= − 1 +K
2
冉
kkF冊
2+共K− 1兲K 2␣共k兲, 共22兲 wherekF=0 andK=kF/vsis the Luttinger parameter cal- culated using Eq.共14兲.The function ␣共k兲 for different values of the Luttinger parameter is shown in Fig.2. One can see that at large values FIG. 1. Excitation spectrum in the model共3兲.共a兲and共b兲show
the dispersion curve−共k兲 关Eq.共12兲兴of the magnon共solid line兲and the exact lower bound of the particle-hole continuum共dashed line兲 for two values of␥=g/0.共c兲shows the␥-dependent exact lower boundEpof the particle-hole continuum atk=kF⬅0.共d兲shows the␥-dependent magnon energyEm=−atk=kF.
of ␥共or Kclose to 1兲the smallkexpansion derived in Ref.
5 is valid for all values ofk, in agreement with Ref.7. It is interesting to note that the leading correction to this result is of the second order, 共K− 1兲2⬃␥−2. At arbitrary interaction
strength ␣共k兲⬃k4as k→0; therefore, the smallkexpansion of ⌬共k兲 found in Ref. 5 remains valid for all values of ␥, confirming the general result of Ref. 8. Note that ␣共k兲also vanishes at k= 2kF.
The problem considered in the present work is directly related to the x-ray edge problem in the theory of the mobile impurity. In this context, the model 共3兲 was investigated in Ref.22. The approach of Ref.22exploits a transformation to the comoving reference frame and combines BA with an effective-field theory similar to ours. The method of Ref. 22 has recently been successfully applied to the Heisenberg model and later was shown19to produce results equivalent to the method of Ref.17used here. A direct comparison of the present work with Ref. 22is however not possible, because the latter is largely based on a Bethe ansatz solution38 vio- lating continuity conditions.
This work was supported in part by the Swiss National Science Foundation under MaNEP and Division II and by ESF under the INSTANS program.F
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36The vanishing of the Hamiltonian共19兲in the absence of magnon excitations to the leading order in gradient expansion is ensured by the operator identity共x兲= 2sz共x兲, valid in the fully polarized sector of the system’s Hilbert space and by Eq.共18兲.
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of Eq.共22兲is plotted for different values of the dimensionless cou- pling constant␥. The values of the Luttinger parameterKare indi- cated for each curve and correspond in increasing order to ␥=⬁, 1.65, 0.56, 0.238 and 0.109, respectively.
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201102-4
