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text: American Gelbvieh Association Revamps Its Carcass EPDs Sep 19, 2007 With the release of the Fall 2007 Sire Summary and EPDs the American Gelbvieh Association (AGA) announces significant changes to the EPDs for carcass traits Real Cowmen: Gelbvieh fits varied management strategies for these three producers Sep 10, 2007 In the southeastern United States you will find fall calving herds, spring calving herds, or herds with both calving seasons. Some producers wean off the cow and sell at market; other producers will Choosing The Stocker Business Sep 01, 2007 Gallery Ranch at Dewey, OK is the winner of the 2007 National Stocker Award's backgrounding/drylot division. American Angus Association launches new genetics company; introduces new heifer EPD Jul 30, 2007 The American Angus Association Board of Directors recently voted to authorize the formation of a new company. Angus Genetics Inc. (AGI) will conduct business as the Beef Improvement Company and will focus on the genetic evaluation of cattle traits Marston joins AHA, Hereford World Staff as Field Rep Jul 30, 2007 The American Hereford Association (AHA) and Hereford World announced that Andee Marston, Manhattan, Kan., has joined the Hereford team Continuing Commitment Jun 01, 2007 "The challenge to BIF is to remain dynamic and move into the 21st century with a program attuned to the technology of that era. Embryo transfer, cloning, sexed semen, gene splicing, super computers, satellite conferences, EPDs for carcass traits…"... Forty Years of Foresight Jun 01, 2007 January 1967 marked an important beginning for the beef industry. Cattle producers and researchers gathered in Denver for a meeting during the National Western Stock Show to take a step toward the future a step toward beef improvement. Those who... Measuring Doability Jun 01, 2007 Most of the efficiency work underway today is focused on growing cattle. While cow feed costs are of overriding importance in integrated beef-production Predicting "Doability" In Cattle May 04, 2007 There's something almost mystical about "doability" in cattle. We all think we know what it is when we see it -- steers that keep gaining during a Panhandle snowstorm, cows that Commentary: Industry Calls; Hereford Answers Apr 16, 2007 My excitement to be a representative of the Hereford breed abounded as I listened to experts address attendees of the Cattle Industry Annual Convention this winter Load More first previous � 22 23 24 25 26 27 28 29 30 next last Load More


text:
'Chapter One Hundred and A Dream' is a body of new work composed and performed by Swedish Vocalist Emilia Martensson, Slovenian Accordion Player Janez Dovc and Brazilian Percussionist Adriano Adewale. In their attempt to weave together these three different folkloric cultures, the ensemble has explored the breadth and depth of storytelling from each country and the different approaches to musical form and texture.
All three members of the ensemble are highly experienced improvisers and well versed in the traditional music of their respective countries. The approach to the material brings together all their experience from their musical heritage as well as their various musical endeavours as working musicians plus a modern twist using traditional instrumentation as well as electronics.
Live at Cankarjev Dom, Ljubljana
Emilia says:
I am so happy to announce that I will be recording a new album this Spring with a completely new project, which I have been developing during the last year. The project features Janez Dovc from Slovenia on Accordion and Electronics and Adriano Adewale from Brazil on Percussion. I received a commission from The London Jazz Festival to perform a set of original music at The Wigmore Hall and decided I wanted these compositions to be based on old Folk Stories from our 3 countries, Sweden, Slovenia and Brazil. The project was named 'Chapter One Hundred and a Dream' for this commission and short after I was also rewarded the 'Women Make Music' Funding Scheme brought forward by PRSF which has helped me to develop this project further.
This project has now turned in to a new solid Trio and we will soon be announcing our name!! We will be making our first Album together in April, which will be released later in the year on the London based label, 'Two Rivers'.


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She might be less than a day old, but Gravity Blue Smith is already on her way to becoming one of this year's biggest models. Well, kind of.
Famous model parents Lucky Blue Smith and Stormi Bree Henley welcomed their first daughter into the world Thursday morning. And they wasted no time in publishing the news on Instagram. Smith, who is 19 and has 3 million followers, referred to his daughter as "little princess," while Henley, who is 26 and has 425k followers, called her "little angel" and "the most perfect beautiful gift." We wonder how long it will be before Gravity has a significant Instagram following of her own.
Smith and Henley obviously announced their pregnancy via Instagram this past March, just a few days after they walked hand-in-hand down the Dolce & Gabbana runway. "Hey guys I just wanted to let you in on an important part of my personal life," he wrote in his announcement post. "I have some wonderful and surprising news—Stormi and I are having a baby!"
If you're concerned about the child's extremely creative name, just know that Gravity is in good company. Her father's three sisters (who are also models) are named Starlie Cheyenne Smith, Daisy Clementine Smith and Pyper America Smith. You should also know that the quartet of siblings perform together at festivals like Coachella in a surf-rock band, called The Atomics. Henley, who was crowned Miss Teen USA in 2009, also has a sister named Darbi; she is apparently not a model.
So it seems Smith and Henley actually took the safe route when it came to naming their daughter—it would be really bizarre if they named the infant something normal, like Jessica or Amanda.


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Condensed Matter > Statistical Mechanics
Title: Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
(Submitted on 6 Jul 2009 (v1), last revised 1 Oct 2009 (this version, v2))
Abstract: We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter < L_N > = \alpha_N, \sqrt{T} and the mean area <A_N> = \beta_N T for all T. The prefactors \alpha_N and \beta_N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implication in ecological context in estimating the home range of a herd of animals with population size N.
Submission historyFrom: Satya N. Majumdar [view email]
[v1] Mon, 6 Jul 2009 06:49:44 GMT (364kb)
[v2] Thu, 1 Oct 2009 06:57:03 GMT (364kb)
Link back to: arXiv, form interface, contact.


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By Chibuike Nwabuko
Abuja (Sundiata Post) – House of Representatives, on Thursday, took a decision to drag Mr. Godswill Akpabio before a Court of law, for making indicting comment against it.
Sundiata Post recalls that Senator Godwill Akpabio on Monday indicted the House of Representatives of having benefited 60 percent of the contracts from the Ministry of Niger Delta.
While appearing before the House Committee on Niger Delta Development Commission (NDDC) probing the alleged financial recklessness on Monday, Akpabio had told the Committee that about 60% of total contracts at the NDDC were awarded to Members of the National Assembly.
Irked by the comment, the Lawmakers passed a resolution the next day, giving Akpabio 48 hours to make public, the names and details of such lawmakers who were beneficiaries of the contacts.
At the expiration of the 48-hour ultimatum, Speaker of the House, Rep. Femi Gbajabiamila, in his opening speech at the plenary, ordered that the Minister be sued for making such "indicting" comment, which he has failed to substantiate.


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text: Wait your argument is actually that the federal government is going to copy and run the btc software as fedcoin? Is that serious? Complete with miners and the distribution protocol? And us citizens will then buy fedcoin? That's one of the dumber things you've posted in here, to say nothing of the fact that the fed govt is not going to set a hard capped currency. I bought my btc on an exchange. Not sure what that has to do with anything. Nearly 18.5m btc already exist. If the fed wanted to buy up all the bitcoin, there's not a whole lot left for it to get. Your thought exercise is also impossible since future btc is distributed through mining rewards. That is not something the fed can simply "buy up," but don't let that get in the way of whatever point you're trying to make. And lol at overthrowing the planet and lambos. Nice strawman for something I've never mentioned or advocated for. If any government actor is in the position of trying to shut down internet service in an attempt to stop btc, it will already be too late.


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Quantum phase transitions of correlated electrons in two dimensions
We review the theories of a few quantum phase transitions in two-dimensional correlated electron systems and discuss their application to the cuprate high temperature superconductors. The coupled-ladder antiferromagnet displays a transition between the Néel state and a spin gap paramagnet with a sharp exciton: we develop a careful argument which eventually establishes that this transition is described by the familiar O(3) field theory in 2+1 dimensions. Crucial to this argument is the role played by the quantum Berry phases. We illustrate this role in a one-dimensional example where our results can tested against those known from bosonization. On bipartite lattices in two dimensions, we study the influence of Berry phases on the quantum transition involving the loss of Néel order, and show that they induce bond-centered charge order (e.g. spin Peierls order) in the paramagnetic phase. We extend this theory of magnetic transitions in insulators to that between an ordinary -wave superconductor and one with co-existing spin-density-wave order. Finally, we discuss quantum transitions between superconductors involving changes in the Cooper pair wavefunction, as in a transition between and superconductors. A mean-field theory for this transition is provided by the usual BCS theory; however, BCS theory fails near the critical point, and we present the required field-theoretic extension. Our discussion includes a perspective on the phase diagram of the cuprate superconductors, in the context of recent experiments and the transitions discussed here.
keywords:Quantum phase transitions, Berry phases, Néel order, spin gap, bond charge order, superconductivity
Pacs:75.10.Jm, 74.72.-h, 71.27.+a, 71.10.Hf
Lectures at the International Summer School on Fundamental Problems in Statistical Physics X, August-September 2001, Altenberg, Germany
1 Introduction
The theory of quantum phase transitions has emerged in the last decade as a powerful tool for the description of the unconventional electronic properties of a variety of correlated systems. This article will provide an elementary overview of some theoretical models of quantum phase transitions in two spatial dimensions, along with a brief discussion of their experimental applications. One reason for our focus on two dimensions is, of course, our interest in the properties of the cuprate superconductors, whose interesting electronic excitations reside within a CuO square lattice. However, not coincidentally, two is also the spatial dimension in which many of the most natural class of non-trivial quantum critical points reside; three is the "upper-critical dimension" of many critical points, at and above which the physical properties are adequately described by a mean field theory (possibly extended with some low order fluctuation corrections).
This article is addressed to newcomers to the study of quantum phase transitions who desire an introduction to some of the experimentally relevant theoretical questions under current discussion. We will assume the reader is well-versed in the standard theory of classical critical phenomena, and so our focus will be on novel aspects of quantum phase transitions, including the role of quantum Berry phases and of excitations with fermionic statistics. Earlier reviews by this author [1, 2] provide detailed discussions of background material which we will occasionally refer the reader to; most of our discussion here goes beyond these reviews, and focuses on current theoretical and experimental developments.
We begin in Section 2 by describing the magnetic quantum transition in the coupled ladder antiferromagnet. High precision numerical studies of the critical properties of this transition are now available, and these serve as a useful check on the theoretical models. We map the critical fluctuations of the magnetic Néel order on to the O(3) field theory in 2+1 dimensions, along with additional Berry phase terms which reflect the commutation relations of the individual quantum spins. We illustrate the role of the Berry phases by first considering a one-dimensional antiferromagnet in Section 3, where the results can be compared with those obtained by bosonization methods. We return to our discussion of two-dimensional quantum spin models in Section 4, where we extend the methods developed in Section 3: we demonstrate that the Berry phases induce bond-centered charge order in the paramagnetic phase of antiferomagnets with full square lattice symmetry, while they can be safely neglected in the coupled ladder antiferromagnet. The remaining sections extend these theories to doped antiferromagnets with mobile charge carriers. Section 5 considers magnetic transitions in superconductors, while Section 6 describes transitions between BCS superconductors with differing pair wavefunctions.
2 Coupled ladder antiferromagnet
We will begin our discussion by describing the quantum phase transition in a simple two-dimensional model of antiferromagnetically coupled Heisenberg spins. At the microscopic level, this model does not describe the spin fluctuations in the cuprate superconductors. However, a fairly subtle argument, which we develop in the following sections and culminate in Section 5, shows that the universal properties of the simple critical point described in this section are identical to those of a magnetic ordering transition in the cuprate superconductors. The model presented here also provides a useful description of other transition metal oxides with a spin gap [3].
We consider the "ladder" Hamiltonian
where are spin-1/2 operators on the sites of the coupled-ladder lattice shown in Fig 1, with the links forming decoupled two-leg ladders while the links couple the ladders as shown.
The ground state of depends only on the dimensionless coupling , and we will describe the low temperature properties as a function of . We will restrict our attention to and .
We will begin with a physical discussion of the phases and excitations of the coupled ladder antiferromagnet, in Section 2.1. We will propose a quantum field-theoretical description of this model in Section 2.2: we will verify that the limiting regimes of the field theory contain excitations whose quantum numbers are in accord with the phases discussed in Section 2.1, and will then use the field theory to describe the critical point.
2.1 Phases and their excitations
Let us first consider the case where is close to 1. Exactly at , is identical to the square lattice Heisenberg antiferromagnet, and this is known to have long-range, magnetic Néel order in its ground state i.e. the spin-rotation symmetry is broken and the spins have a non-zero, staggered, expectation value in the ground state with
where is some fixed unit vector in spin space, and is on the two sublattices. This long-range order is expected to be preserved for a finite range of close to 1. The low-lying excitations above the ground state consist of slow spatial deformations in the orientation : these are the familiar spin waves, and they can carry arbitrarily low energy i.e. the phase is 'gapless'. There are two polarizations of spin waves at each wavevector (measured from the antiferromagnetic ordering wavevector), and they have excitation energy , with the spin-wave velocities in the two spatial directions.
Let us turn now to the vicinity of . Exactly at , is the Hamiltonian of a set of decoupled spin ladders. Such spin ladders are known to have a paramagnetic ground state, with spin rotation symmetry preserved, and an energy gap to all excitations [4]. A caricature of the ground state is sketched in Fig 2: spins on opposite rungs of the ladder pair in valence bond singlets in a manner which preserves all lattice symmetries.
Excitations are now formed by breaking a valence bond, which leads to a three-fold degenerate state with total spin , as shown in Fig 3; this broken bond can hop from site-to-site, leading to a triplet quasiparticle excitation.
Note that this quasiparticle is not a spin-wave (or equivalently, a 'magnon') but is more properly referred to as a spin 1 exciton. For small, but not exactly 0, we expect that the ground state will remain a gapped paramagnet, and the spin exciton will now move in two dimensions. We parameterize its energy at small wavevectors by
where is the spin gap, and , are velocities. Fig 3 also presents a simple argument which shows that the exciton cannot fission into two 'spinons'.
The very distinct symmetry signatures of the ground states and excitations between and make it clear that the two limits cannot be continuously connected. It is known [5, 6] that there is an intermediate second-order phase transition at . Both the spin gap and the Néel order parameter vanish continuously as is approached from either side.
2.2 Quantum field theory
A convenient field-theoretic description of the two phases introduced in Section 2.1, and of the critical point which separates them, is obtained by the coherent state path integral. A detailed derivation of this approach has been provided by the author in Chapter 13 of Ref. [1] which the reader should now consult; we will only highlight a few important points here.
The coherent state path integral is best introduced by considering first a single spin with angular momentum and Hamiltonian
where is a static magnetic field. The eigenvalues of are, of course, , where , and so we also know its partition function at a temperature . The coherent state path integral represents in terms of an integral over all possible closed curves, , on the surface of a unit sphere, each curve representing a possible history of the precessing spin in imaginary time, :
The first term is the so-called 'Berry phase', and is defined to be the oriented area of the spherical triangle defined by , , and an arbitrary reference point (which is usually chosen to be the north pole). We will see that behaves in many respects like the time-component of a gauge field, and this accounts for the suggestive notation. All physical results should be independent of the choice of , and it is easy to see that changes in amount to gauge transformations of . So if we change to , then the resulting is related to by
where measures the oriented area of the spherical triangle defined by , , and .
We now apply the above path integral independently to every site of the coupled ladder antiferromagnet: we obtain a path integral over fields , where is a site index as in (1). As discussed in Chapter 13 of Ref. [1], it is convenient to now take a spatial continuum limit, to allow focus on the important low energy excitations. This proceeds by the parameterization
where on the two subattices, is the lattice spacing, , is a unit length continuum field representing the staggered Néel order parameter, and is the continuum field for the conserved, uniform, magnetization density (the normalization chosen in (7) is that required to make exactly equal to the total spin). Next, a gradient expansion in performed on the effective action for , and after integrating out the fluctuations we obtain the following path integral for the partition function of the coupled ladder antiferromagnet:
where the velocities , , and the coupling constant . We will now view as the tuning parameter, and values of on either side of a critical value place the system in one of the two phases discussed in Section 2.1; notice that is a decreasing function of for , and so the "strong" coupling phase corresponds to the spin gap phase with , while is the Néel phase.
The first term in (8) is a residual Berry phase, and this remains expressed on the underlying lattice: there is no natural way to take its continuum limit. Notice that this term induces complex weights in the partition function and so it has no analog in the theory of classical phase transitions. This Berry phase plays a central role in determining the phases and critical points of many quantum spin systems, and describing its consequences is one of the central purposes of this article. However, our discussions in the following sections will show that at least for the case of the coupled ladder antiferromagnet the Berry phase in (8) can be safely neglected, and so we will drop it in the remainder of this subsection.
Upon omission of the Berry phase in (8), becomes a standard path integral, familiar in well established theories of classical phase transitions in dimensions. For instance, after interpreting as a third dimension (i.e. ), and as the local magnetization in a three-dimensional ferromagnet (e.g. iron), then is simply its classical partition function at a finite 'temperature' , with the coupling constant . From the theory of such ferromagnets it is known that a convenient description of the physical properties of emerges not in a formulation in terms of the fixed length field (the 'hard spin' representation), but in a formulation using a 'soft spin' field () with no length constraint. We connect the two formulations by a coarse-graining transformation, so that represents the average of over some small coarse-graining volume of spacetime. The new form of is now
where keep in mind that the Berry phase in (8) has been dropped, is now the new tuning parameter (replacing in (8)), and the quartic interaction has replaced the fixed length constraint in (8). The two phases of Section 2.1 are now separated by a critical value , with the spin gap phase corresponding to , while the Néel phase appears for .
The well-known properties of the model (9), when translated into the quantum framework of interest here, are easily seen to reproduce the physical properties of the phases discussed in Section 2.1.
For , in (9) has a "ferromagnetic" phase associated with the breaking of spin rotation invariance, and the development of the expectation value ; to lowest order in , the spontaneous magnetization is and is determined by the minimum of the "Mexican hat" potential for the field. After using (7) this "ferromagnetic" order in the classical model is seen to correspond to the antiferromagnetic Néel order (2) in the quantum model. A standard Gaussian fluctuation analysis about this ordered states shows that susceptiblity transverse to the spontaneous moment is given by
where is a frequency associated with imaginary time. After analytically continuing to real frequencies, , this yields the quantum response function to an applied staggered field transverse to the Néel order:
This response function has a pole, representing the two spin-wave excitations, at energy . This is in complete accord with our discussion of the properties of the Néel phase in Section 2.1.
For , the "disordered" phase of in (9) corresponds to the quantum paramagnet with a spin gap. Here, a computation of the susceptibility to lowest order in yields
where we have already analytically continued to real frequencies, , and (for now) the residue . This response function is independent of the direction of the applied staggered field, and so its pole represents the exciton at energy . This is again in accord with the discussion of the spin gap phase in Section 2.1: the exciton dispersion (3) is obtained from the present result in a small momentum expansion. The pole structure in (12) actually holds to all orders in an expansion in [1]: at quantum temperature , such corrections renormalize the values of the spin gap and the quasiparticle residue , and induce additional spectral weight associated with multiple excitonic continua at energies larger than , but they do not broaden the quasiparticle pole. Physically, this happens because the exciton is the lowest energy excitation with spin , and so conservation of total spin prevents implies that this excitation has an infinite lifetime. Only at does a finite lifetime appear, associated with scattering off the thermally excited density of pre-exiting excitons.
The present field theoretical framework can also be extended to obtain a theory for the critical point at (), as has been discussed in some detail in [1]. In its application to the classical ferromagnet, the critical properties of (9) have been understood in some detail in studies of the field theory, and accurate values of the critical exponents have been obtained. In a recent quantum Monte Carlo study of the coupled ladder antiferromagnet , Matsumoto et al. [7] determined the critical exponents at to impressive accuracy, and all values are in accord with the exponents of the field theory. This agreement is independent numerical support for the omission of the Berry phases in proceeding from (8) to (9); separate theoretical arguments for the same approximation will emerge in the following sections. As we approach the critical point with , the pole structure in (12) continues to hold at low frequencies, but the spin gap vanishes as and the residue vanishes as , where , , and are standard critical exponents. Right at there are no quasiparticle excitions, and the dynamic susceptiblity instead displays a branch-cut representing the continuum of critical excitations:
For more information on the physics of this critical continuum, refer to [1].
3 Quantum XY chain
This section will divert from the main subject of this paper by considering some simple examples of quantum phase transitions in one dimension. Our purpose here is to illustrate the consequences of the Berry phases in (8) by considering a simplified model for which they can be evaluated completely; we will also be able to test our method by comparing the results with those obtained by a bosonization analysis of the same one-dimensional model. We will then proceed to apply closely related methods in two dimensions in Section 4.
We will evaluate the generalization of (8) in one dimension for case in which the Hamiltonian has on a global U(1) spin symmetry, and the anisotropies are such that the spins prefer to lie within the - plane in spin space. In this case we can parameterize by a single angle :
with these simplifications, we will find that powerful duality methods enable us to obtain considerable insight into the physics a partition function with Berry phases and complex weights. With the parameterization (14) and the restriction to one spatial dimension, we expect the resulting to provide a description of a large class of spin quantum antiferromagnets in one dimension; to be specfic, we expect it will model the Hamiltonian
where are first/second neighbor antiferromagnetic exchange constants ( is not too large), and the anisotropy induces a preference for the spins to reside in the - plane. We can reasonably expect that the phase diagram of as a function of and is similar to that of as a function of . The former is already known by an earlier bosonization analysis [9] and will provide an instructive comparison with the latter.
Our computation begins by a transformation of the Berry phase in (8) to an alternative form in one dimension. We discretize spacetime into a square lattice of sites, , ( is now a spacetime index, unlike the spatial index in (15), and its interpretation should be clear from the context). For this spacetime lattice we define , with , to be the oriented area of the spherical triangle formed by , , and a fixed reference . Then, as illustrated in Fig 4, we can use the identity
to relate the Berry phase to the flux piercing the plaquettes of the spacetime lattice;
here is a fixed integer field on the sites, , of the dual lattice, and a convenient choice is i.e. is unity on every odd column, and zero on every even column. The symbol represents a discrete lattice derivative, with .
Now we use the in-plane form of in (14) to our advantage. The flux through any plaquette is clearly equal to the surface area of the quadrilateral on the unit sphere formed by the on the four vertices of the plaquette. For planar spins, such a quadrilateral lies entirely on the equator, and its area is generically zero. The only exception occurs when a vortex in pierces the plaquette: in this case, the quadrilateral is the entire equator and the flux is the area of a hemisphere . So we have established that the Berry phases in (8) generate the factor
for easy-plane spin systems in one dimension. As the vortex number is always an integer, we see that the factor in (17) is always unity for integer , and the Berry phases can be ignored for these values of .
The remaining terms in the action of (8) can also be simplified for the present situation. On the discrete spacetime lattice, and with (14), these terms become , which is the usual action of a "classical" XY model in dimensions. As is standard, duality manipulations on this XY model are more conveniently performed in a periodic Gaussian, 'Villain', representation of the weights of the model. In this manner, we finally obtain the generalization of the partition function (8) to U(1) symmetric, quantum spin chains:
where are integers on the links of the direct lattice which, when summed, produce the periodic Gaussian weights for , the represents a sum over plaquettes, and are the vortex numbers of the plaquettes. The expression (18) is amenable to an exact duality transformation which produces a representation in which all weights are positive: we apply the Poisson summation identity to the sum over on each link, and replace it by a sum over integers :
Here, and henceforth, we drop overall normalization constants in front of partition functions. Now the integral over the can be performed independently on each site, and only imposes the constraint . We solve this constraint by writing , with the integers on the sites of the dual lattice. So we obtain the exact mapping of (18) to [8]
where the "heights" on the sites of the dual lattice are restricted to the values
The expression (20) is a canonical height (or 'solid-on-solid') model in two dimensions. The only influence of the quantum Berry phase are the restrictions on the allowed values of the heights of the "interface": for integer the heights take all integer values just as in the standard solid-on-solid model, while for half-integer , the heights are restricted to half-odd-integer (integer) values on the odd (even) columns of the square lattice. As we have made a number of approximations in the deriving from the path integral of spin systems, we shall be interested in a generalized class of height models with additional short-range non-local couplings, apart from the simple terms appearing in (20). With these additional couplings, standard methods from the theory of interface models can be used to show [8] that the generalized height models describe phases not only in the easy-plane, small , limit of (15), but also those in the easy-axis, large limit when the spins prefer to lie in the directions. In all of this analysis it is essential that we not relax the restrictions on the allowed values of the heights, as these are fundamental consequences of the quantized value of the spin . We describe the main results for two sets of values in turn.
3.1 integer
The standard model with integer heights on every site has two phases. For small , the interface is rough; this is dual to the classical or quantum XY phase with quasi-long-range spin order in the - plane, with power-law decay of spin correlations. For the quantum spin chain, we may refer to this phase as a Tomonaga-Luttinger (TL) liquid, as it shares many of the characteristics of other TL liquids.
For large , the interface is smooth; this is dual to the "disordered" phase of the classical XY model. For the quantum spin chain, this large phase has exponential decay of spin correlations in imaginary time, and therefore a spin gap. Its characteristics are rather similar to the spin gap phase discussed in Section 2: the primary excitation is a gapped exciton with an infinite lifetime at low energies. As the spin anisotropy is reduced by increasing in (15), this spin gap phase connects continuously to the well-known Haldane gap state of integer spin chains with SU(2) symmetry.
The transition between the small and large phases is second-order, and is described by the standard Kosterlitz-Thouless theory of the classical XY model in two dimensions.
3.2 half-odd-integer
The staggering of the heights between integer and half-odd-integer values introduces crucial differences in the phase diagram, and several new phases are allowed in the generalized phase diagram of such height models. However, it is believed [8] that the specific height model (20), with only a local gradient squared coupling, has only one rough phase for all values of , and this corresponds, as above, to a TL phase of the quantum spin chain. The other phases appear upon allowing additional short-range couplings in (20), and these can reasonably be expected to arise from terms in the original Hamiltonian omitted in the duality mappings; as we noted above, in this generalization it is essential that we maintain the restrictions on the allowed height values. The new phases are associated with smooth interfaces, and as in Section 3.1 such phases have a spin gap. However, the staggering of the heights induces crucial differences in the structure of such spin gap states, as we describe below.
A crucial characteristic of a smooth interface is that there is a well-defined value of the average height in every pure state. Furthermore, for the staggered height models under consideration here, every value of the average breaks a lattice translational symmetry of the quantum spin chain. For instance if (modulo integers) then the even columns of the square lattice (which have integer allowed heights) have been picked out, and we may expect the average energy, on the links associated with the even columns to be smaller (say) than those on the odd columns. Conversely, an interface with (modulo integers) reverses the role of the odd and even columns. The choice between these two sets of average heights corresponds to a spontaneously broken symmetry in the ground state, which now has 'spin-Peierls' or 'bond-centered charge' (BC) order. The nature of these ordered states, and of their excitations, is illustrated in Fig 5: they have a spin gap, but the lowest-lying excitations are fractionalized spinons which reside on the domain wall between the two BC states.
Other possible values of in the smooth interface phase lead to other discrete broken symmetries in the quantum spin chain. As we describe in Ref. [8], the values (modulo integers) correspond to states with Ising spin order i.e. . All other values of correspond to four-fold degenerate states with co-existing Ising and BC order.
We sketch the phase diagram of the model (20) for half-odd-integer in Fig 6, as obtained by a bosonization analysis [9].
The TL, BC, and Ising phases are all separated by second-order transition lines, and the three lines meet at a single multi-critical point: all the critical properties can be described by an extension of the Kosterlitz-Thouless renormalization group flows. Analysis of the generalized height models discussed here leads to a phase diagram with an identical topology and critical properties [8].
4 Berry phases in two dimensions
We are now prepared to describe the evaluation of the 2+1 dimensional path integral with Berry phases in (8). As in the 1+1 dimensional case in Section 3 the evaluation is performed on a discrete lattice in spacetime: the degrees of freedom are three component unit vectors, , on the sites, of a cubic lattice. On such a lattice we can rewrite (8) as
where the sum over extends over the three directions ,,. In the application to the coupled latter antiferromagnet, the spatial anisotropy and doubled unit cell of led to an anisotropy in the spatial derivative terms in (8). In the discrete form (22) this reduced symmetry requires a corresponding variation in the couplings associated with the couplings between the various sites on the cubic spacetime lattice: we have dropped this variation in (22) and so, strictly speaking, this partition function applies only to quantum antiferromagnets which have the full symmetry of the square lattice and have a single spin on each site. An example of such an antiferromagnet is the so-called - model
with exchange interaction between square lattice nearest neighbors , and between next-nearest-neighbors (nnn). As in Section 3, we exclude large values of and expect that the phases of as a function of will be described by those of in (22) as a function of . Later we will also discuss the consequences of doubling the unit cell and introducing spatial anisotropy in (22), which is required to obtain a description of the coupled ladder model in (1).
For small , we expect to be in the ordered Néel state, with broken spin rotation symmetry. It is not difficult to account for the Berry phases in a spin-wave fluctuation analysis about this ordered state, and they only have minor consequences [8].
The Berry phases are more crucial in the paramagnet state, with full spin rotation symmetry, expected at large . Because this is a strongly coupled system in the 3 dimensions with a non-abelian symmetry, no exact duality analysis is possible (unlike the simpler situation in Section 3). We proceed with the strategy discussed in Ref. [8]: we replace the integral over the by an integral over the the variables that appear in the Berry phases. Formally, this can be done by introducing new 'dummy' variables and rewriting (22) by introducing factors of unity on each link; this leads to
In the first expression, if the integral over the is performed first, we trivially return to (22); however in the second expression we perform the integral over the variables first at the cost of introducing an unknown effective action for the . In principle, evaluation of may be performed order-by-order in a "high temperature" expansion in : we match correlators of the flux with those of the flux evaluated in the integral over the with positive weights determined only by the term in (22). Rather than undertaking this laborious calculation, we can guess essential features of the effective action from some general constraints. First, correlations in the decay exponentially rapidly for large (with a correlation length ), and so should be local. Second, it should be invariant under the lattice form of the gauge transformation (6)
associated with the change in the reference point on the unit sphere from to , with equal to the area of the spherical triangle formed by , and . Finally the area of any triangle on the sphere is uncertain modulo and so the effective action should be invariant under
The simplest local action which is invariant under (25) and (26) is that of compact U(1) quantum electrodynamics and so we have
for large ; comparison with the large expansion shows that the coupling . As in Section 3 our analysis is aided by replacing the cosine interaction in (27) by a Villain sum over periodic Gaussians:
where are integers on the links of the dual lattice, which pierce the plaquettes of the direct lattice.
We will now perform a series of exact manipulations on (28): our final result, in (35), will be a demonstration of its exact equivalence to another height model, but now with the interface 3=2+1 dimensional. As in Section 3, this interface model is obtained after a duality mapping and has only positive weights. This last fact, of course, makes it much more amenable to a standard statistical analysis. This first step in the duality transformation is to rewrite (28) by the Poisson summation formula:
where (like ) is an integer-valued vector field on the links of the dual lattice. Next, we write the Berry phase in a form more amenable to duality transformations. Choose a 'background' flux which satisfies
where is the direct lattice site in the center of the plaquette defined by the curl on the left-hand-side. Any integer-valued solution of (30) is an acceptable choice for , and a convenient choice is shown in Fig 7.
Using (30) to rewrite the Berry phase in (28), applying (29), and shifting by the integer , we obtain a new exact representation of in (28):
The integral over the can be performed independently on each link, and its only consequence is the imposition of the constraint . We solve this constraint by writing as the gradient of a integer-valued height on the sites of the dual lattice, and so obtain
This is the 2+1 dimensional height model in almost its final form. Before we proceed to an analysis of (32), we pause for a paragraph to make some parenthetic remarks on the application of the above approach to one dimension.
Precisely the same approach as above can be applied to SU(2) symmetric spin chains which we can model by the obvious generalization of (22) to 1+1 dimensions. In the large limit, we proceed with the same set of duality mappings, and in place of (32) we obtain the partition function
The zero gradient constraint on (to be contrasted with the zero curl constraint on in 2+1 dimensions) means that it takes a single fixed value through the entire system. For integer , the value minimizes the action (its action per site is , to be contrasted with the two-fold degenerate BC states with with action per site ), and yields a 'featureless' state with no broken symmetry: this is, of course the Haldane gap state expected for SU(2) symmetric, integer , spin chains, as discussed in Section 3.1. For half-odd-integer , there is BC order, as the action in minimized by the two-fold degenerate values . All of these results in 1+1 dimensions are in complete accord with the results obtained in Section 3 for large , and with the known properties of quantum spin chains.
Now we return to the 2+1 dimensional case of interest in this section, and proceed with our interrupted analysis of (32). We can convert the "frustration" in (32) into offsets for the allowed height values (as in Section 3) by decomposing it into curl and divergence free parts and writing it in terms of new fixed fields, and as follows:
The values of these new fields are shown in Fig 8.
Inserting (34) into (32), we can now write the height model in its simplest form [10, 11, 12]
is the new height variable we shall work with. Notice that the have dropped out, while the act only as fractional offsets (for not an even integer) to the integer heights. From (36) we see that for half-odd-integer the height is restricted to be an integer on one of the four sublattices, an integer plus 1/4 on the second, an integer plus 1/2 on the third, and an integer plus 3/4 on the fourth; the fractional parts of these heights are as shown in Fig 8a; the steps between neighboring heights are always an integer plus 1/4, or an integer plus 3/4. For an odd integer, the heights are integers on one square sublattice, and half-odd-integers on the second sublattice. Finally for even integer the offset has no effect and the height is an integer on all sites. We discuss these classes of values in turn in the following subsections.
(It is worth emphasizing here that the height model in (35) only offers a description of the large paramagnetic phase of the model in (22) in 2+1 dimensions. The small Néel ordered state is not one of the phases of , nor is the critical point describing the onset of magnetic order; an extension of to include the Néel state was discussed in [8], but will not be presented here. This should be contrasted with the situation in 1+1 dimensions discussed in Section 3; there height models like in (20) can describe all the phases and critical points of the quantum spin chain, including the TL phase, which is a state with quasi-long-range Néel order).
4.1 even integer
Unlike the two-dimensional case, three-dimensional height models generically have no roughening transition, and the interface is always smooth. With all heights integers, the smooth phase breaks no symmetries. So like the case of integer in quantum spin chains (Section 3.1), square lattice antiferromagnets with even integer can have a paramagnetic ground state with a spin gap and no broken symmetries. This is in accord with the exact ground state for a antiferromagnet on the square lattice found by Affleck et al. [14], the AKLT state. The structure of this state is identical to that of the paramagnetic state of the field theory in 2+1 dimensions (Eqn (9) with ), with a stable spin exciton excitation. So the critical point between the magnetic Néel state and the spin gap state should also be described by the field theory. As we noted above, this critical point is not contained within .
4.2 half-odd-integer
Now the heights of the interface model can take four possible values, which are integers plus the offsets on the four square sublattices shown in Fig 8a. As in Section 4.1, the interface is always smooth, but as in Section 3.2, any smooth interface must break a lattice symmetry with the development of bond-centered charge (BC) order and this allows a number of distinct spin gap ground states of the lattice antiferromagnet. Consistent with these theoretical predictions, numerical studies of the frustrated spin model (23) have produced evidence for BC order in the state contiguous to the Néel state: see [15, 16] for recent results.
It is useful, first, to obtain a simple physical interpretation of the interface model in the language of the antiferromagnet [13]. From Fig 8a it is clear that nearest neighbor heights can differ either by 1/4 or 3/4 (modulo integers). To minimize the action in (35), we should choose the interface with the largest possible number of steps of . However, the interface is frustrated, and it is not possible to make all steps and at least a quarter of the steps must be . Indeed, there is a precise one-to-one mapping between interfaces with the minimal number of steps (we regard interfaces differing by a uniform integer shift in all heights as equivalent) and the dimer coverings of the square lattice: the proof of this claim is illustrated in Fig 9.
We identify each dimer with a singlet valence bond between the spins (the ellipses in Fig 2), and so each interface corresponds to a quantum state with each spin locked in the a singlet valence bond with a particular nearest neighbor. Fluctuations of the interface in imaginary time between such configurations correspond to quantum tunneling events between such dimer states, and an effective Hamiltonian for this is provided by the quantum dimer model [17, 18].
The nature of the possible smooth phases of the interface model are easy to determine from the above picture, and by standard techniques from statistical theory which have been reviewed elsewhere [11, 8, 13, 19]. Interfaces with average height (modulo integers) correspond to the four-fold degenerate BC ordered states in Fig 10a, while those with (modulo integers) correspond to the four-fold degenerate plaquette BC states in Fig 10b.
All other values of are associated with eight-fold degenerate BC states with a superposition of the orders in Fig 10a and b.
It is interesting to note here that the symmetry of the state in Fig 10a is identical to the symmetry of the coupled ladder Hamiltonian illustrated in Fig 1: so the system has spontaneously decided to arrange itself in the form of a coupled ladder. Whether the BC ordering is spontaneous (as it is here) or explicitly present in the Hamiltonian (as in Section 2), the nature of the non-zero spin excitations should be the same. So the confinement argument of Fig 3b applies here too, and there are no free spinons in the BC state. The elementary excitation is a spin exciton. This should be contrasted with the quantum spin chain in Section 3.2, where the BC state had spinon excitations.
We are now finally in a position to explain a central step in the analysis of Section 2: the neglect of Berry phases in the theory of the magnetic critical point for the coupled ladder antiferromagnet. We have seen here that the primary effect of the Berry phases is to induce BC order in the spin gap state. However, such "order" is already present explicitly in the Hamiltonian of the coupled ladder antiferromagnet, which has two sites per unit cell: the Hamiltonian picks out one of the four BC states and "dimer" fluctuations to the other BC states are suppressed. So we need only focus on the fluctuations of the magnetic order, and these are captured by the field theory.
The nature of the crossover between the Néel and BC states for quantum spin models with the full square lattice symmetry (like in (23)) is far more subtle, and was recently discussed at some length in Ref [8]. Here, both the Néel and BC orders can become critical, and so we cannot naively assert the irrelevance of the Berry phases. One appealing scenario [15, 8], supported by recent numerical studies [15], is that the crossover occurs via two transitions: with increasing , there is first a transition at to a state with co-existing Néel and BC order (only the BC fluctuations are critical at are these are described by a clock model), and then a transition at where Néel order disappears. The second transition at is just as in the coupled ladder antiferromagnet, and is therefore described by the field theory: there is well-formed BC order on the either side of , and it is immaterial here whether this order was explicitly present in the Hamiltonian or spontaneously induced. Indeed, this was secretly the reason we chose to consider the coupled ladder antiferromagnet in our early discussion.
4.3 odd integer
This case is similar to that half-odd-integer, and we will not consider it in detail. The Berry phases again induce BC order in the spin gap state, but this order need only lead to a two-fold degeneracy.
5 Magnetic transitions in -wave superconductors
We have so far considered a variety of magnetic transitions in correlated "Mott" insulators, which have an electron density commensurate with the number of available orbitals. Cuprate high temperature superconductivity appears when such a Mott insulator on the square lattice is doped by a finite density of mobile carriers. All of the available experimental evidence supports the proposal that, apart from some inhomogeneous charge ordered insulating states, the ground state of the doped Mott insulator is a superconductor. In particular, there is no clear-cut evidence for a metallic Fermi liquid state up to a moderate density of holes. Consequently, we will focus our attention on the superconducting ground state in the remainder of this paper. This is to be contrasted with the approach of [20] which examines transitions in metallic states at non-zero temperature.
It is now natural to study the interplay of the Néel magnetic order found in the insulator and the superconducting order of the doped system. (The charge order discussed in Section 4 also plays an important role in the superconductor, but we will not focus attention on this aspect here — see [21] for a recent review.) The experimental evidence indicates that this crossover proceeds via an intermediate state with co-existence of magnetic and superconducting order. Furthermore, the magnetic order is no longer in the simple two-sublattice Néel configuration, but oscillates at a wavevector . So instead of expressing the electron spin as , which is implied by (7) and the subsequent coarse-graining of to , we now write
where a 3-component, complex quantum field; is required to be complex as along as the magnetic state does not have a two sublattice structure, which is only the case for ordering wavevectors and . Also, the experimental evidence also indicates that the magnetic order is collinear, and not spiral; this is the case if .
With this reasoning, we are led to consider the quantum transition between a superconducting state with co-existing magnetic spin-density-wave order at wavevector ( in the SC+SDW state) to an ordinary superconductor ( in the SC state). Clearly, the field should be a degree of freedom in any theory for this transition. However, in a superconductor there are additional fermionic excitations which carry spin , the Bogoliubov quasiparticles, and we have to examine their influence on the critical properties. This is especially the case in a -wave superconductor, in which the energy of the quasiparticles vanishes at four 'nodal' points in the Brillouin zone.
To address this issue, let us review the standard BCS mean-field theory for a -wave superconductor on the square lattice. We consider the Hamiltonian
where is the annihilation operator for an electron on site with spin , is its Fourier transform to momentum space, is the dispersion of the electrons (it is conventional to choose , with the first/second neighbor hopping and the chemical potential), and the term is the same as that in (23) with
and the Pauli matrices. We will consider the consequences of the term in (23) for the superconductor in Section 6 below. Applying the BCS mean-field decoupling to we obtain the Bogoliubov Hamiltonian
For a wide range of parameters, the ground state energy optimized by a wavefunction for the Cooper pairs: this corresponds to the choice . The value of is determined by minimizing the energy of the BCS state
where the fermionic quasiparticle dispersion is
The energy of the quasiparticles, , vanishes at the four points at which . We are especially interested in the low energy quasiparticles in the vicinity of these points, and so we perform a gradient expansion of near each of them. We label the points , , , and write
while assuming the are slowly varying functions of . We also introduce the bispinors , and , and then express in terms of while performing a spatial gradient expansion. This yields the following effective action for the fermionic quasiparticles:
where we have rotated our spatial co-ordinate system by , and the are matrices which are block diagonal, the blocks consisting of Pauli matrices. The velocities are given by the conical structure of near the : we have and .
We are now in a position to discuss the quantum field theory of the transition from the SC+SDW to the SC state in term of the and degrees of freedom. First, we can write down an action for along the lines of that for in the insulator: this will have a structure very similar to that of (9) (the complex nature of induces additional quartic terms which we will not discuss here—the reader is referred to another review by the author [21] for details on this point). The action will also contain the terms in in (44), and it remains to discuss terms which may couple the and . The simplest possible terms are cubic interaction terms like etc. However by comparing the momentum dependencies in (37) and (43) with (39) it is easy to see that such terms will survive after averaging over space only if the special commensurability conditions , or , are satisfied. In general this will not be the case, and experimental evidence also does not support this possibility. Therefore, we will assume in the remainder of this section that these commensurability conditions are not satisfied: the theory for the case , was considered in Ref. [22], and has some formal similarities to the theory to be considered in Section 6 in a different physical context.
In the absence of special commensurability conditions, the simplest term which can couple the and is
However a simple scaling argument shows that the coupling is irrelevant at the SC+SDW to SC critical point. To see this, imagine that the critical point is described by the point, where the and fields are decoupled. This point is actually a fixed point of the renormalization group and is invariant under scaling transformations. From (44) we see that the scaling dimensions of is 1, while that of is


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