- Email: [email protected]
“Solidification” in a soluble model of bosons on a one-dimensional lattice: The “Boson-Hubbard chain”
Volume 80A, number 4
PHYSICS LETTERS
8 December 1980
"SOLIDIFICATION" IN A SOLUBLE MODEL OF BOSONS ON A ONE-DIMENSIONAL LATTICE: THE "BOSON-HUBBARD CHAIN" F.D.M. HALDANE Institut Laue-Langevin, 38042 Grenoble, France Received 15 September 1980
Lieb and Liniger's soluble model of the 1-D Bose gas is put on a lattice, becoming a "boson-Hubbard model". It remains soluble by the Bethe ansatz. When the coupling exceeds a critical value (U/2rcT) > 2x/'3fir,a gap is present when the density n bosom per site is 1.
Lieb and Liniger [1] (LL) have introduced a soluble model of a one-dimensional Bose fluid: nonrelativistic bosom with a finite-strength repulsive delta-function interaction, which is soluble by the Bethe-ansatz technique [2] and its modern "Quantum InverseScattering Transform" reformulation [3,4]. I report an interesting variant of the model that does not seem to have been previously considered, obtained by putting it on a lattice, so it becomes a "bosonHubbard model". It too is easily solved by the Bethe ansatz, the relevant equations being obtained by a marriage of LL's bose gas analysis with the fermion Hubbard chain analysis of Lieb and Wu [5] (LW). The main new feature of this model is the appearance of long-range density-wave order (solidification) and the opening of a gap in the excitation spectrum when the characteristic 1-D Bose fluid "Kohn-anomaly" wavevector "2kF" = 2nn is equal to a reciprocal lattice vector (i.e., when the boson density n per site is integral) and the coupling is sufficiently strong. I establish that in the case n = 1, this solidification occurs for dimensionless coupling u = U/2rrT > 2Vr3br = 1.102... ; for weaker coupling the system apparently remains liquid when n = I. The model is given by M---r
+ h.c. + v l
bi+Na=bi .
Tt,,t,, , l
(1)
Combining the LW and LL analyses, it is easily shown that the Bethe ansatz that diagonalises this model is given by N a Nn,
Nak(nm) = 2 . I n + ~ ~ Otsin(k (m)) -- sin(k(,m'))], n'=l rn'=l IRe(k(m))l
N=~Nn, n
E = ~ ] - 2Tcos(k(nm)), n,m
( - 1 ) In = - ( - 1 ) N .
p = ~k (m), n,m
tan[½0(x)] =
-x/(½rru), (2)
Here N n can be interpreted as the "occupation" of the nth of the N a modes, where N a is the number of sites, and each mode n corresponds to a different one of the N a allowed values o f I n in the range - ~ N . < 1n < ½Na. Every complex pseudomomentum k~nm) occurs with its conjugate (k(nm)) * = k(nm'), ensuring that the total momentum is real. The dimensionless coupling has been chosen to be u = U/21rT, as this is the appropriate choice for the fermion model. For low densities at least, the choice of ground state quantum numbers N n will mirror that of the continuum model of LL: N n = 1 for [1hi < ½g, and zero otherwise, with no multiply occupied modes. In the limit U ~ *% it is well known that the system becomes equivalent to a free spinless fermion system, and this construction of the ground state is valid for 281
V o l u m e 80A, n umb er 4
PHYSICS LETTERS
all densities n ~< I. With this choice of quantum numbers, the kn(1) are real, and lie in a segment [ - k 0 , k0] of the real line [-rr, rr]. Passing to the thermodynamic limit in the standard way [1,5], the equation for the ground state density of the k n is 2fro(k) = 1 + 21r cos(k) ko x f dk' K ( s i n ( k ) - s i n ( k ' ) ; u)p(k'),
-ko ko
n = f dkp(k), E/2NaT=-f d cos(k)o(k), -ko -ko where the kernel K(x; u) of this inhomogeneous Fredholm equation of the second kind is
K(x; u) = 7r-16/(x 2 + ~ 2) ,
~ = g1~ u .
(4)
To study the case where k 0 > ½rr, it is useful to notice that o(k) can be written in the form 2zrp(k) = 1 + c o s ( k ) f (sin(k)), f ( x ) = (1 - x 2 ) - 1 / 2 - ' f ( x ) ,
(5)
where
f(x) = [(1 -- X2) -1/2 - g(x)] Xo
+f
dyK(x-y;u)f(V),
x0=sin(k0) ,
(6)
--X 0
!
g(x) = 2 f dy (1 - y 2 ) - l / 2 K ( x - y; u) -- i[Z(x + i5) -- •(x --
i6)1
,
(7)
and Z(w) = (w 2 - 1)1/2. The density n per site is given by 1
n = l - N
XO
f
dx f ( x ) .
case, o(k) is positive definite in [ - k 0 , k0] , as required for physical consistency, and n is a monotonically increasing function of ko, reaching 1 as k 0 ~ n. When k 0 = % x 0 = 0, and p(k) = (1/2zr)[1 + cos(k)g(sin(k))]. This picture breaks down when there is a solution to the equation
g(Xc(U)) = [1 -- Xc(U)2 ] - 1 / 2 .
(3)
ko
8 December 1980
(S)
--X 0
The integral equation (6) is of a very well-behaved type: the kernel is symmetric and positive definite, and all the eigenvalues of the associated homogeneous equation are real and greater than ~lr/tan-l(xo/ 6) > 1. The iterative solution to eq. (6) is thus convergent. Thus if the inhomogeneous term is positive definite in the range [-Xo, Xo] , so is the solution f (x). The solution is thus very well behaved if g(x) < (1 - x 2 ) - 1 / 2 throughout the range [ - 1 , 1]. In this
For large enough u, this equation has no solution, and the solution of eq. (6) is well behaved for all k 0 ~ rr. A solution to eq. (9) first appears at x c = 0; when 6 = X/~ and u = 2x/3/rr, g(0) = 1. For smaller values of u, there is a single solution Xc(U), which increases from 0 at u = 2x/~/~r to 1 as u ~ 0. This value x c constitutes a lower bound to values o f x 0 in eq. (6) giving rise to acceptable physical solutions: i f x 0 = x c in eq. (6), the solution f(x) is negative clef'mite in [ - x 0 , Xo] , corresponding to n > 1 in eq. (8) - a clear contradiction, since the construction of the ground state was only valid for n ~< 1. In addition,f(xo) is negative definite, so P(ko) < 0. For u < 2w/3/rr the solution of type (3) must break down when k 0 increases to a critical upper value kS, where k S < rr - sin-1 (Xc). This limit presumably corresponds to n -+ 1, but there seems to be no obvious way to identify k8 from the integral equation without resorting to numerical analysis. An alternative way the above construction of the ground state might break down in the weak coupling regime is that it might become energetically favourable to have multiple occupancy of the modes with small I i before completing the Filling of modes with large I i. When the ground state is described by (3), it is simple to calculate the sound velocity, by studying the change of energy by taking a particle out of the mode w i t h i n =/max = max(in,N n _- 1), and replacing it in a mode with 1n = I max + ~I, 1 ~ AI ~ N a. The sound velocity is
Vs/2T= [2rr0(ko)]-I ko X [sin(ko)+cos(ko)f
dksin(k)y(sin(k))],
-k 0
y(x) = ~ [K(x - Xo) - K ( x + x0) ] Xo +
f -X 0
282
(9)
d.x'K(x - x')y(x').
(10)
Volume 80A, number 4
PHYSICS LETTERS
Evidently, the sound velocity vanishes as n --* 1, u > 2x/3/Tr (k 0 --* 7r). Based on the similarity with the weak-field behaviour of the easy-axis antiferromagnetic chain studied by Yang and Yang [6] (among others), and the known behaviour as U ~ ~, I conelude that the system becomes insulating, with a gap for elementary excitations, and long-range densitywave order, as n ~ 1, for u > 2x/3/~r. For weaker coupling, the system presumably remains fluid. It should be noted that this type of behaviour (a critical coupling below which no gap opens when n is integral) is expected [7] from a theory of the lowenergy structure of the 1-D Bose fluid based on an effective hamiltonian derived from the Luttinger model [8,9] which suggests that the critical behaviour associated with the opening of this gap is essentially similar to that around the isotropic point of the uni. axially anisotropic antiferromagnetic chain [6] in a weak field; however, if the weak coupling, n ~ 1 behaviour of the boson lattice fluid turns out to involve multiple occupancy of long-wavelength modes, the situation would become more complicated. Fluctuation effects in one dimension prevent true solidification of a Bose fluid with short.range forces, though for strong coupling it has a natural tendency to form a density wave with a reciprocal lattice vector "2kF" = 2nn characterizing current excitations (obtained here by taking a particle from the occupied mode with the most negative value of 1n and placing it in the mode w i t h i n =/max + 1). However, as ex-
8 December 1980
emplified by this model, solidification can occur ff this fundamental wavevector is commensurate with an underlying periodicity of the system, provided by the lattice. To summarise: I have noted that an interesting generalisation of Lieb and Liniger's [1] soluble model of a Bose fluid is a "boson-Hubbard model"; the original model can be recovered as the low density limit of this. Interesting new features arise in the high density state n ~ 1, where n is the boson density per site. The details of how the general ground state is constructed by populating the various boson modes remains to be elucidated. When the coupling exceeds a critical lower limit U/21rT > 2x/~/Ir = 1.102 .... the system evidently becomes insulating with a gap in its spectrum as n ~ q . References
[1] E.H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605. [2] H. Bethe, Z. Phys. 71 (1931) 205. [3] D.R. Creamer, H.B. Thaeker and D. Wilkinson, Phys. Rev. D21 (1980) 1523. [4] L.D. Faddeev, Steklov Mathematical Institute report, Leningrad, 1979, unpublished. [5] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. [6] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 312. [7] F.D.M. Haldane, to be published. [8] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [9] D.C. Mattis and E.H. Lieb, J. Math. Phys. 6 (1965) 304.
283
PHYSICS LETTERS
8 December 1980
"SOLIDIFICATION" IN A SOLUBLE MODEL OF BOSONS ON A ONE-DIMENSIONAL LATTICE: THE "BOSON-HUBBARD CHAIN" F.D.M. HALDANE Institut Laue-Langevin, 38042 Grenoble, France Received 15 September 1980
Lieb and Liniger's soluble model of the 1-D Bose gas is put on a lattice, becoming a "boson-Hubbard model". It remains soluble by the Bethe ansatz. When the coupling exceeds a critical value (U/2rcT) > 2x/'3fir,a gap is present when the density n bosom per site is 1.
Lieb and Liniger [1] (LL) have introduced a soluble model of a one-dimensional Bose fluid: nonrelativistic bosom with a finite-strength repulsive delta-function interaction, which is soluble by the Bethe-ansatz technique [2] and its modern "Quantum InverseScattering Transform" reformulation [3,4]. I report an interesting variant of the model that does not seem to have been previously considered, obtained by putting it on a lattice, so it becomes a "bosonHubbard model". It too is easily solved by the Bethe ansatz, the relevant equations being obtained by a marriage of LL's bose gas analysis with the fermion Hubbard chain analysis of Lieb and Wu [5] (LW). The main new feature of this model is the appearance of long-range density-wave order (solidification) and the opening of a gap in the excitation spectrum when the characteristic 1-D Bose fluid "Kohn-anomaly" wavevector "2kF" = 2nn is equal to a reciprocal lattice vector (i.e., when the boson density n per site is integral) and the coupling is sufficiently strong. I establish that in the case n = 1, this solidification occurs for dimensionless coupling u = U/2rrT > 2Vr3br = 1.102... ; for weaker coupling the system apparently remains liquid when n = I. The model is given by M---r
+ h.c. + v l
bi+Na=bi .
Tt,,t,, , l
(1)
Combining the LW and LL analyses, it is easily shown that the Bethe ansatz that diagonalises this model is given by N a Nn,
Nak(nm) = 2 . I n + ~ ~ Otsin(k (m)) -- sin(k(,m'))], n'=l rn'=l IRe(k(m))l
N=~Nn, n
E = ~ ] - 2Tcos(k(nm)), n,m
( - 1 ) In = - ( - 1 ) N .
p = ~k (m), n,m
tan[½0(x)] =
-x/(½rru), (2)
Here N n can be interpreted as the "occupation" of the nth of the N a modes, where N a is the number of sites, and each mode n corresponds to a different one of the N a allowed values o f I n in the range - ~ N . < 1n < ½Na. Every complex pseudomomentum k~nm) occurs with its conjugate (k(nm)) * = k(nm'), ensuring that the total momentum is real. The dimensionless coupling has been chosen to be u = U/21rT, as this is the appropriate choice for the fermion model. For low densities at least, the choice of ground state quantum numbers N n will mirror that of the continuum model of LL: N n = 1 for [1hi < ½g, and zero otherwise, with no multiply occupied modes. In the limit U ~ *% it is well known that the system becomes equivalent to a free spinless fermion system, and this construction of the ground state is valid for 281
V o l u m e 80A, n umb er 4
PHYSICS LETTERS
all densities n ~< I. With this choice of quantum numbers, the kn(1) are real, and lie in a segment [ - k 0 , k0] of the real line [-rr, rr]. Passing to the thermodynamic limit in the standard way [1,5], the equation for the ground state density of the k n is 2fro(k) = 1 + 21r cos(k) ko x f dk' K ( s i n ( k ) - s i n ( k ' ) ; u)p(k'),
-ko ko
n = f dkp(k), E/2NaT=-f d cos(k)o(k), -ko -ko where the kernel K(x; u) of this inhomogeneous Fredholm equation of the second kind is
K(x; u) = 7r-16/(x 2 + ~ 2) ,
~ = g1~ u .
(4)
To study the case where k 0 > ½rr, it is useful to notice that o(k) can be written in the form 2zrp(k) = 1 + c o s ( k ) f (sin(k)), f ( x ) = (1 - x 2 ) - 1 / 2 - ' f ( x ) ,
(5)
where
f(x) = [(1 -- X2) -1/2 - g(x)] Xo
+f
dyK(x-y;u)f(V),
x0=sin(k0) ,
(6)
--X 0
!
g(x) = 2 f dy (1 - y 2 ) - l / 2 K ( x - y; u) -- i[Z(x + i5) -- •(x --
i6)1
,
(7)
and Z(w) = (w 2 - 1)1/2. The density n per site is given by 1
n = l - N
XO
f
dx f ( x ) .
case, o(k) is positive definite in [ - k 0 , k0] , as required for physical consistency, and n is a monotonically increasing function of ko, reaching 1 as k 0 ~ n. When k 0 = % x 0 = 0, and p(k) = (1/2zr)[1 + cos(k)g(sin(k))]. This picture breaks down when there is a solution to the equation
g(Xc(U)) = [1 -- Xc(U)2 ] - 1 / 2 .
(3)
ko
8 December 1980
(S)
--X 0
The integral equation (6) is of a very well-behaved type: the kernel is symmetric and positive definite, and all the eigenvalues of the associated homogeneous equation are real and greater than ~lr/tan-l(xo/ 6) > 1. The iterative solution to eq. (6) is thus convergent. Thus if the inhomogeneous term is positive definite in the range [-Xo, Xo] , so is the solution f (x). The solution is thus very well behaved if g(x) < (1 - x 2 ) - 1 / 2 throughout the range [ - 1 , 1]. In this
For large enough u, this equation has no solution, and the solution of eq. (6) is well behaved for all k 0 ~ rr. A solution to eq. (9) first appears at x c = 0; when 6 = X/~ and u = 2x/3/rr, g(0) = 1. For smaller values of u, there is a single solution Xc(U), which increases from 0 at u = 2x/~/~r to 1 as u ~ 0. This value x c constitutes a lower bound to values o f x 0 in eq. (6) giving rise to acceptable physical solutions: i f x 0 = x c in eq. (6), the solution f(x) is negative clef'mite in [ - x 0 , Xo] , corresponding to n > 1 in eq. (8) - a clear contradiction, since the construction of the ground state was only valid for n ~< 1. In addition,f(xo) is negative definite, so P(ko) < 0. For u < 2w/3/rr the solution of type (3) must break down when k 0 increases to a critical upper value kS, where k S < rr - sin-1 (Xc). This limit presumably corresponds to n -+ 1, but there seems to be no obvious way to identify k8 from the integral equation without resorting to numerical analysis. An alternative way the above construction of the ground state might break down in the weak coupling regime is that it might become energetically favourable to have multiple occupancy of the modes with small I i before completing the Filling of modes with large I i. When the ground state is described by (3), it is simple to calculate the sound velocity, by studying the change of energy by taking a particle out of the mode w i t h i n =/max = max(in,N n _- 1), and replacing it in a mode with 1n = I max + ~I, 1 ~ AI ~ N a. The sound velocity is
Vs/2T= [2rr0(ko)]-I ko X [sin(ko)+cos(ko)f
dksin(k)y(sin(k))],
-k 0
y(x) = ~ [K(x - Xo) - K ( x + x0) ] Xo +
f -X 0
282
(9)
d.x'K(x - x')y(x').
(10)
Volume 80A, number 4
PHYSICS LETTERS
Evidently, the sound velocity vanishes as n --* 1, u > 2x/3/Tr (k 0 --* 7r). Based on the similarity with the weak-field behaviour of the easy-axis antiferromagnetic chain studied by Yang and Yang [6] (among others), and the known behaviour as U ~ ~, I conelude that the system becomes insulating, with a gap for elementary excitations, and long-range densitywave order, as n ~ 1, for u > 2x/3/~r. For weaker coupling, the system presumably remains fluid. It should be noted that this type of behaviour (a critical coupling below which no gap opens when n is integral) is expected [7] from a theory of the lowenergy structure of the 1-D Bose fluid based on an effective hamiltonian derived from the Luttinger model [8,9] which suggests that the critical behaviour associated with the opening of this gap is essentially similar to that around the isotropic point of the uni. axially anisotropic antiferromagnetic chain [6] in a weak field; however, if the weak coupling, n ~ 1 behaviour of the boson lattice fluid turns out to involve multiple occupancy of long-wavelength modes, the situation would become more complicated. Fluctuation effects in one dimension prevent true solidification of a Bose fluid with short.range forces, though for strong coupling it has a natural tendency to form a density wave with a reciprocal lattice vector "2kF" = 2nn characterizing current excitations (obtained here by taking a particle from the occupied mode with the most negative value of 1n and placing it in the mode w i t h i n =/max + 1). However, as ex-
8 December 1980
emplified by this model, solidification can occur ff this fundamental wavevector is commensurate with an underlying periodicity of the system, provided by the lattice. To summarise: I have noted that an interesting generalisation of Lieb and Liniger's [1] soluble model of a Bose fluid is a "boson-Hubbard model"; the original model can be recovered as the low density limit of this. Interesting new features arise in the high density state n ~ 1, where n is the boson density per site. The details of how the general ground state is constructed by populating the various boson modes remains to be elucidated. When the coupling exceeds a critical lower limit U/21rT > 2x/~/Ir = 1.102 .... the system evidently becomes insulating with a gap in its spectrum as n ~ q . References
[1] E.H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605. [2] H. Bethe, Z. Phys. 71 (1931) 205. [3] D.R. Creamer, H.B. Thaeker and D. Wilkinson, Phys. Rev. D21 (1980) 1523. [4] L.D. Faddeev, Steklov Mathematical Institute report, Leningrad, 1979, unpublished. [5] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. [6] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 312. [7] F.D.M. Haldane, to be published. [8] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [9] D.C. Mattis and E.H. Lieb, J. Math. Phys. 6 (1965) 304.
283