PHYSICAL REVIEW E 91 , 022108 (2015)
Scaling phenomena driven by inhomogeneous conditions at first-order quantum transitions M assim o C am postrini,1 Jacopo N esp o lo ,1 A ndrea Pelissetto,2 and Ettore V icari1 1Dipartimento di Fisica dell’Universita di Pisa and INFN, Largo Pontecorvo 3, 1-56127 Pisa, Italy 2Dipartimento di Fisica dell’Universita di Roma ‘‘La Sapienza” and INFN, Sezione di Roma 1, 1-00185 Roma, Italy (Received 24 November 2014; published 9 February 2015) We investigate the effects of smooth inhomogeneities at first-order quantum transitions (FOQTs), such as those arising in the presence of a space-dependent external field, which smooths out the discontinuities of the low-energy properties at the transition. We argue that a universal scaling behavior emerges in the space transition region close to the point in which the external field takes the value for which the homogeneous system undergoes the FOQT. We verify the general theory in two model systems. We consider the quantum Ising chain in the ferromagnetic phase and the e/-state Potts chain for q = 10, investigating the scaling behavior which arises in the presence of an additional inhomogeneous parallel and transverse magnetic field, respectively. Numerical results are in full agreement with the general theory. DOI: 10.1103/PhysRevE.91.022108
PACS number(s): 05.30.Rt, 64.60.fd, 64.60.De
I. INTRODUCTION T he theories o f classical and quantum phase transitions [1 -3 ] generally apply to hom ogeneous system s. However, hom ogeneity is often an ideal lim it o f experim ental conditions. Inhom ogeneous conditions generally sm ooth out the singular ities at phase transitions. This is also expected at first-order transitions, w hich are characterized by discontinuities in the therm odynam ic quantities in the classical case or in the ground-state properties in the quantum case. In the presence o f sm ooth inhom ogeneities, we may sim ultaneously observe different phases in different space regions, separated by crossover regions in w hich the system goes from one phase to another. For exam ple, this behavior is observed in typical cold-atom experim ents [4], in w hich the atom s are constrained in a lim ited space region by an inhom ogeneous (usually harm onic) trap, w hich effectively m akes-the chem ical potential space dependent. T he effects o f the inhom ogeneous conditions have been m uch investigated at continuous transitions [4-46], For suffi ciently sm ooth inhom ogeneities, classical or quantum system s, at classical (finite-tem perature) or quantum (zero-tem perature) transitions, show a universal scaling behavior in a space region w hose size is controlled by the typical length scale £ o f the inhom ogeneity. T he scaling behavior is universal and som ew hat analogous to the standard finite-size scaling (FSS) occurring in hom ogeneous system s [47-50], In particular, it only depends on the universality class o f the transition occurring in the hom ogeneous system . T here is, however, a crucial difference betw een the usual FSS and that observed in the presence o f an inhom ogeneity. In the latter case, there are indeed two relevant length scales. Besides the correlation length £ there is the length scale £, related to £ by a nontrivial pow er law at the critical point, £ ~ l e , w here 0 is a universal exponent depending on the hom ogeneous universality class and on som e general features o f the external space-dependent field [19,25], Scaling phenom ena also em erge at first-order classical transitions in the presence o f a tem perature gradient [46] or a space-dependent external field. They are observed in the transi tion region in w hich the space-dependent tem perature assum es values close to the critical tem perature o f the hom ogeneous 1539-3755/2015/91 (2)/022108( 11)
system . T herm odynam ic quantities show a universal scaling behavior, characterized by nontrivial pow er laws, w hich is quite sim ilar to that observed at continuous transitions. In this paper we study the effects o f inhom ogeneous conditions at first-order quantum transitions (FO Q Ts). FOQTs are o f great interest, as they occur in a large num ber o f quantum m any-body system s, such as quantum H all sam ples [51], itinerant ferrom agnets [52], heavy ferm ion m etals [53—55], and so on. They also occur in m ulticom ponent cold-atom system s in optical lattices, w ith spin-orbit coupling and synthetic gauge fields. These system s show several phases, som e o f them separated by FO Q Ts [56-62]. We investigate the scaling behavior arising w hen one o f the m odel param eters sm oothly depends on space. We put forward a scaling theory, w hich describes the low -energy properties in the crossover space region w here the system changes phase. We verify this scaling theory in two relatively sim ple quantum m any-body system s, the quantum Ising and Potts chains in the presence o f space-dependent m agnetic fields. N um erical results are in full agreem ent w ith the theoretical predictions. The paper is organized as follow s. In Sec. II we define the quantum Ising and Potts chains w ith a space-dependent m agnetic field h x and present num erical results for their behavior around the spatial point w here h x = 0, the value at w hich the FO Q T occurs in the hom ogeneous case. In Sec. Ill w e present a general theory for the scaling behavior in the crossover region close to the transition spatial point. In Sec. IV we check the scaling theory by analyzing num erical results for the Ising and the Potts chain w ith q = 10. Finally, in Sec. V w e draw our conclusions.
II. QUANTUM ISING AND POTTS CHAINS In order to m ake the discussion concrete, w e first define the quantum m odels that we use as theoretical laboratories to study FO Q Ts in the presence o f a spatial inhom ogeneity. We consider the Ising chain in the ordered phase in the presence o f a parallel m agnetic field coupled to the order-param eter spin operator and the quantum (/-state Potts chain w ith q = 10 in the presence o f a transverse m agnetic field. In the hom ogeneous case, both m odels undergo a FOQT. For each
022108-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 022108 (2015)
CAMPOSTRINI, NESPOLO, PELISSETTO, AND VICARI model, we compute several quantities using the density matrix renormalization-group (DMRG) method [63]. Some details of the DMRG implementation can be found in Refs. [64,65], where we presented numerical studies of the same models in homogeneous conditions. From the numerical point of view, simulations of the inhomogeneous system do not present any additional difficulty.
Hamiltonian with a boundary term [64], LG-1 tf/,FO B C = ~
X ]
ax )(Jx l\
x = -H \
LG
-8
£
ax3> +
( ° H .G -
a w )-
(6 )
x = - \l \
A. The quantum Ising chain We consider a quantum Ising chain of size 2L + 1 with a space-dependent parallel magnetic field hx coupled with the order-parameter spin operator. Its Hamiltonian is
Hi = - J J 2
ai1)cri+i
The homogeneous Ising chain, i.e., model (1) with a uniform magnetic field hx = h, has a continuous transition at g = 1, h = 0 , belonging to the two-dimensional Ising universality class. This quantum critical point separates a paramagnetic (g > 1) and a ferromagnetic (g < 1) phase. In the ferromag netic phase g < 1, the parallel magnetic field h drives a FOQT at h — 0, with a discontinuity of the magnetization, i.e., of the ground-state expectation value of a ^ \ Indeed, we have [66]
x = -L
m± -
-g
53 ct*3) ” 5Z hx °x1J*
x = —L
x = —L
lim/,-*o±limz.->co(o^,)) = ±rnc,
(7)
w mc = ( l - g 2) ]/&.
where a xa> are the Pauli matrices, g ^ 0 is the transverse magnetic field, J is a coupling that will be taken equal to 1 in the following, and hx is a space-dependent magnetic field, which we write as
(8)
Therefore, in the presence of an inhomogeneous field which vanishes changing sign atx = 0, such as that defined in Eq. (3), the point x = 0 effectively corresponds to the spatial point which separates the two oppositely magnetized phases, with
(tK0)lt> = m + and
Scaling phenomena driven by inhomogeneous conditions at first-order quantum transitions M assim o C am postrini,1 Jacopo N esp o lo ,1 A ndrea Pelissetto,2 and Ettore V icari1 1Dipartimento di Fisica dell’Universita di Pisa and INFN, Largo Pontecorvo 3, 1-56127 Pisa, Italy 2Dipartimento di Fisica dell’Universita di Roma ‘‘La Sapienza” and INFN, Sezione di Roma 1, 1-00185 Roma, Italy (Received 24 November 2014; published 9 February 2015) We investigate the effects of smooth inhomogeneities at first-order quantum transitions (FOQTs), such as those arising in the presence of a space-dependent external field, which smooths out the discontinuities of the low-energy properties at the transition. We argue that a universal scaling behavior emerges in the space transition region close to the point in which the external field takes the value for which the homogeneous system undergoes the FOQT. We verify the general theory in two model systems. We consider the quantum Ising chain in the ferromagnetic phase and the e/-state Potts chain for q = 10, investigating the scaling behavior which arises in the presence of an additional inhomogeneous parallel and transverse magnetic field, respectively. Numerical results are in full agreement with the general theory. DOI: 10.1103/PhysRevE.91.022108
PACS number(s): 05.30.Rt, 64.60.fd, 64.60.De
I. INTRODUCTION T he theories o f classical and quantum phase transitions [1 -3 ] generally apply to hom ogeneous system s. However, hom ogeneity is often an ideal lim it o f experim ental conditions. Inhom ogeneous conditions generally sm ooth out the singular ities at phase transitions. This is also expected at first-order transitions, w hich are characterized by discontinuities in the therm odynam ic quantities in the classical case or in the ground-state properties in the quantum case. In the presence o f sm ooth inhom ogeneities, we may sim ultaneously observe different phases in different space regions, separated by crossover regions in w hich the system goes from one phase to another. For exam ple, this behavior is observed in typical cold-atom experim ents [4], in w hich the atom s are constrained in a lim ited space region by an inhom ogeneous (usually harm onic) trap, w hich effectively m akes-the chem ical potential space dependent. T he effects o f the inhom ogeneous conditions have been m uch investigated at continuous transitions [4-46], For suffi ciently sm ooth inhom ogeneities, classical or quantum system s, at classical (finite-tem perature) or quantum (zero-tem perature) transitions, show a universal scaling behavior in a space region w hose size is controlled by the typical length scale £ o f the inhom ogeneity. T he scaling behavior is universal and som ew hat analogous to the standard finite-size scaling (FSS) occurring in hom ogeneous system s [47-50], In particular, it only depends on the universality class o f the transition occurring in the hom ogeneous system . T here is, however, a crucial difference betw een the usual FSS and that observed in the presence o f an inhom ogeneity. In the latter case, there are indeed two relevant length scales. Besides the correlation length £ there is the length scale £, related to £ by a nontrivial pow er law at the critical point, £ ~ l e , w here 0 is a universal exponent depending on the hom ogeneous universality class and on som e general features o f the external space-dependent field [19,25], Scaling phenom ena also em erge at first-order classical transitions in the presence o f a tem perature gradient [46] or a space-dependent external field. They are observed in the transi tion region in w hich the space-dependent tem perature assum es values close to the critical tem perature o f the hom ogeneous 1539-3755/2015/91 (2)/022108( 11)
system . T herm odynam ic quantities show a universal scaling behavior, characterized by nontrivial pow er laws, w hich is quite sim ilar to that observed at continuous transitions. In this paper we study the effects o f inhom ogeneous conditions at first-order quantum transitions (FO Q Ts). FOQTs are o f great interest, as they occur in a large num ber o f quantum m any-body system s, such as quantum H all sam ples [51], itinerant ferrom agnets [52], heavy ferm ion m etals [53—55], and so on. They also occur in m ulticom ponent cold-atom system s in optical lattices, w ith spin-orbit coupling and synthetic gauge fields. These system s show several phases, som e o f them separated by FO Q Ts [56-62]. We investigate the scaling behavior arising w hen one o f the m odel param eters sm oothly depends on space. We put forward a scaling theory, w hich describes the low -energy properties in the crossover space region w here the system changes phase. We verify this scaling theory in two relatively sim ple quantum m any-body system s, the quantum Ising and Potts chains in the presence o f space-dependent m agnetic fields. N um erical results are in full agreem ent w ith the theoretical predictions. The paper is organized as follow s. In Sec. II we define the quantum Ising and Potts chains w ith a space-dependent m agnetic field h x and present num erical results for their behavior around the spatial point w here h x = 0, the value at w hich the FO Q T occurs in the hom ogeneous case. In Sec. Ill w e present a general theory for the scaling behavior in the crossover region close to the transition spatial point. In Sec. IV we check the scaling theory by analyzing num erical results for the Ising and the Potts chain w ith q = 10. Finally, in Sec. V w e draw our conclusions.
II. QUANTUM ISING AND POTTS CHAINS In order to m ake the discussion concrete, w e first define the quantum m odels that we use as theoretical laboratories to study FO Q Ts in the presence o f a spatial inhom ogeneity. We consider the Ising chain in the ordered phase in the presence o f a parallel m agnetic field coupled to the order-param eter spin operator and the quantum (/-state Potts chain w ith q = 10 in the presence o f a transverse m agnetic field. In the hom ogeneous case, both m odels undergo a FOQT. For each
022108-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 022108 (2015)
CAMPOSTRINI, NESPOLO, PELISSETTO, AND VICARI model, we compute several quantities using the density matrix renormalization-group (DMRG) method [63]. Some details of the DMRG implementation can be found in Refs. [64,65], where we presented numerical studies of the same models in homogeneous conditions. From the numerical point of view, simulations of the inhomogeneous system do not present any additional difficulty.
Hamiltonian with a boundary term [64], LG-1 tf/,FO B C = ~
X ]
ax )(Jx l\
x = -H \
LG
-8
£
ax3> +
( ° H .G -
a w )-
(6 )
x = - \l \
A. The quantum Ising chain We consider a quantum Ising chain of size 2L + 1 with a space-dependent parallel magnetic field hx coupled with the order-parameter spin operator. Its Hamiltonian is
Hi = - J J 2
ai1)cri+i
The homogeneous Ising chain, i.e., model (1) with a uniform magnetic field hx = h, has a continuous transition at g = 1, h = 0 , belonging to the two-dimensional Ising universality class. This quantum critical point separates a paramagnetic (g > 1) and a ferromagnetic (g < 1) phase. In the ferromag netic phase g < 1, the parallel magnetic field h drives a FOQT at h — 0, with a discontinuity of the magnetization, i.e., of the ground-state expectation value of a ^ \ Indeed, we have [66]
x = -L
m± -
-g
53 ct*3) ” 5Z hx °x1J*
x = —L
x = —L
lim/,-*o±limz.->co(o^,)) = ±rnc,
(7)
w mc = ( l - g 2) ]/&.
where a xa> are the Pauli matrices, g ^ 0 is the transverse magnetic field, J is a coupling that will be taken equal to 1 in the following, and hx is a space-dependent magnetic field, which we write as
(8)
Therefore, in the presence of an inhomogeneous field which vanishes changing sign atx = 0, such as that defined in Eq. (3), the point x = 0 effectively corresponds to the spatial point which separates the two oppositely magnetized phases, with
(tK0)lt> = m + and
