Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension

Tensor networks offer a variational formalism to efficiently represent wave-functions of extended quantum many-body systems on a lattice. In a tensor network N, the dimension \chi of the bond indices that connect its tensors controls the number of variational parameters and associated computational costs. In the absence of any symmetry, the minimal bond dimension \chi^{min} required to represent a given many-body wave-function |\Psi> leads to the most compact, computationally efficient tensor network description of |\Psi>. In the presence of a global, on-site symmetry, one can use a tensor network N_{sym} made of symmetric tensors. Symmetric tensors allow to exactly preserve the symmetry and to target specific quantum numbers, while their sparse structure leads to a compact description and lowers computational costs. In this paper we explore the trade-off between using a tensor network N with minimal bond dimension \chi^{min} and a tensor network N_{sym} made of symmetric tensors, where the minimal bond dimension \chi^{min}_{sym} might be larger than \chi^{min}. We present two technical results. First, we show that in a tree tensor network, which is the most general tensor network without loops, the minimal bond dimension can always be achieved with symmetric tensors, so that \chi^{min}_{sym} = \chi^{min}. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that \chi_{sym}^{min} > \chi^{min}. We further argue, however, that in some situations there are important conceptual reasons to prefer a tensor network representation with symmetric tensors (and possibly larger bond dimension) over one with minimal bond dimension.Comment: 12 pages, 13 figure

Finite Density Matrix Renormalisation Group Algorithm for Anyonic Systems

The numerical study of anyonic systems is known to be highly challenging due to their non-bosonic, non-fermionic particle exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behaviour as a result. Meanwhile, the density matrix renormalisation group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging both in terms of programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground state energy of the Golden Chain, which has become the benchmark system for the numerical study of anyons, and is shown to produce results comparable to those of the anyonic TEBD algorithm and superior to the variationally optimised anyonic MERA, at far lesser computational cost.Comment: 24 pages, 37 figure files (25 floating figures). RevTeX 4.1. Minor changes for clarity in Figs. 9 & 11, matching published versio

What are the experiences and outcomes of anti-racist social work education?

This thesis seeks to interrogate the experiences and outcomes of anti-racist social work education and evaluate the pedagogic relevance and practice utility of teaching social work students about ‘race’, racism and anti-racism. A mixed methods research strategy is drawn upon to explore how professional social work training prepares students to work with ‘cultural diversity’ and ‘cultural difference’ and to evaluate the outcomes of teaching and learning which focuses on anti-racism. The methodological position drawn upon in this thesis is a pragmatic one (Williams, 2006), which recognises the role of both nomothetic and idiographic approaches to enabling us to describe and understand how social work students and tutors experience and make sense of anti-racist social work education and the pedagogic challenges and barriers they face to engaging with this discrete area of professional education. Anti-racism is the theoretical and conceptual focus of this thesis and it encompasses a broad coalition of different perspectives and academic interests concerned with actively identifying and resisting racism. It has been characterised as a set of disparate polycentric overlapping practices and discourses (Anthias & Lloyd, 2002), whilst exhibiting a politically committed form of practice (Bhatti-Sinclair, 2011). It has also been described as a radical and oppositional project which emphasises the need to actively identify and resist racism (Bonnett & Carrington, 1996; Tomlinson, 2002). Historically it has been associated with the politics of resistance and social movements in support of decolonialisation, anti fascism and equal rights for immigrant workers (Dominelli, 2008). Frequently, it has been characterised as reflecting a radical dualism between ‘white racism versus Black resistance’ (Gillborn & Ladson-Billings, 2004). Within social work education, anti-racism despite its retrenchment and appropriation into a broad ‘anti oppressive’ practice model (Williams, 1999), continues to be regarded as a progressive educational strategy which has a transformative role. It is viewed as an effective approach to challenging the attitudes and values of individual students (Heron, 2008). It can also lead to ‘perspective transformation’ (Mezirow, 1981), and ‘critical consciousness’ through the process of conscientization (Freire, 1970). Anti-racism is therefore considered to have a valuable pedagogic role in raising awareness of racial inequalities and the processes associated with racial exclusion, whilst also providing a wider critique of the state, its culture, its institutions, ideology, legislation and policy frameworks (Singh, 2006a). The qualitative and quantitative data presented in this thesis suggests that it is possible to discover the situated experiences of teaching and learning on anti-racism and measure how these pedagogic interventions can affect and lead to knowledge, skills and attitudinal change (Carpenter, 2005; 2011). The empirical evidence drawn upon in this thesis identifies important group differences, related to age, ‘race’ and experience of working with a BME service user, which are important for understanding how anti-racist social work education is experienced differently by learners, and how it leaves a complex set footprints which enable us to appreciate how this educational intervention works in different ways for different types of students. Sometimes these differences are subtle, but at other times they are more evident and suggestive of group experiences which go beyond the individual. The empirical evidence also suggests that social work educators experience anti-racist social work education as a challenging and emotionally supercharged area of the curriculum and that their levels of engagement, preparedness and commitment is often dependent upon where they are positioned socially, culturally and politically. This thesis is important because regionally and nationally there have been very few attempts to empirically capture how professional social work training programmes accommodate and evidence ‘race’ equality and cultural diversity issues (Williams et al., 2009; Williams & Parrott, 2013)

Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX

Symmetry protected entanglement renormalization

Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multi-scale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. In this paper we propose the use of symmetric tensors as a mechanism to build a symmetry protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry protected entanglement renormalization produces the proper structure of RG fixed-points, namely a fixed-point for each symmetry protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.Comment: 5 pages, 3 figure

A Review on Lean Manufacturing: A Feasible Solution to Industrial Objectives

In the recent years the manufacturing objectives are very competitive and rapid technological developed process to fulfill the demands of customers and profit objective of an organization. This leads to produce special manufacturing techniques and emerging concept of technology. Earlier high quality and efficiency were the necessary and sufficient conditions for staying in the business. In the past year, the manufacturing industries are to fulfill the customs orders. In the current paper the role of lean manufacturing and its development has been discussed with its objectives and other essential parameters. In the present situation the manufactures must be able to rapidly develop and produce customized products to fulfill the customer requirements. Lean is viewed by many as the latest improvement in the tradition of cost reduction to increase the profit. It has the advantage of a very descriptive active name and has been, in many cases, used like any other cost-reduction approach. This means that the Lean word can be found in many places, projects and proposals. This means that it applies to the same implementation problems as those other approaches which have created a level of cynicism in some quarters about its effectiveness. To achieve all objective the use of lean manufacturing is excellent solution

The economic sustainability of cropping systems in Indian Punjab: A farmers' perspective

Food for all continues to be a key issue, especially in the developing world where every fifth person is chronically undernourished. India, a fast growing developing country has also experienced serious food shortages for example in the mid 1960s. Punjab, a small northern Indian state has developed, particularly since the Green Revolution in the mid 1960s, to be a key agricultural area producing 13% of the food grains of India. Increased productivity brought economic benefits to farmers and led to the establishment of Wheat-Rice Cropping Pattern (WRCP) as the main agricultural system of Punjab which more recently has become reliant on underground water resources, agricultural machinery, chemical fertilisers and pesticides. More recently stagnating yields and increased cost of cultivation of WRCP have squeezed the net farm profitability. However, the WRCP has been, and remains the first choice of farmers, because of its comparative economic advantage, assured marketing and stable productivity level. This paper compares the economic sustainability of WRCP to that of other alternative cropping patterns in Punjab and answers the question “Why farmers continue with the WRCP despite various crop diversification efforts in the past”. Interviews with 120 farmers across Punjab illustrated the economic and risk advantages of WRCP over other potential cropping patterns and concludes that if cropping systems in Punjab are to become more environmentally sustainable then policy makers will need to put mechanisms in place which either encourage a more sustainable WRCP or provide the basis for the growth of alternative, less environmentally damaging cropping systems.agriculture, cropping systems, Punjab, sustainability, Environmental Economics and Policy, Farm Management,

Control bit Based Congestion Control in Mobile Ad-hoc Network using OLSR Protocol

MANET is mobile ad-hoc network having various mobile nodes moves from position to the other position. While moving they can be at different speeds. Such that sometimes they are neighbor of given node and some time they become neighbor of other node. Each time the neighbor lists get changed. So each time each node has to upgrade the neighbor list. Each source node identifies the route by broadcast the route request. In congestion control mechanism in current research control bit is used. Before sending any data packet control signal will be send. If this control bit will be acknowledged then path is assumed to be cleared from congestion and send the packet on the route. Else alternative route will be selected. While this technique performance will be measured on the basis of different parameters like end to end delay, packet Delivery Ratio, Success rate and throughput. Under current research someparameter has improved compared to the previous mechanism. This we have tested over to the OLSR protocol and compared it with AODV based existing technique

Matrix product states for anyonic systems and efficient simulation of dynamics

Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry non-local degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D critical system of (a) Ising anyons and (b) Fibonacci anyons both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.Comment: 18 pages, 15 figue