Physics Open House 2012

Boston University Physics Department

Condensed Matter Theory

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Physics Open House

Condensed Matter Theory

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3/23/12

Condensed Matter Theory

We are a cohesive group that uses an array of theoretical methods to address relevant experimental problems in condensed matter physics.

Group Members

15 students, 3 post-docs, 7 visiting scientists/yr

David Campbell (nonlinear dynamics) Antonio Castro Neto (graphene, now leave of absence) Claudio Chamon (topological phases, fractionalization) Bill Klein (kinetics and nucleation) Pankaj Mehta (biophysics) Anatoli Polkovnikov (quantum many-body dynamics) Sid Redner (nonequilibrium dynamics, networks) Anders Sandvik (quantum spin systems) Eugene Stanley* (separate group, center for polymer studies)

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Physics Open House

Trustees Presentation

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3/23/12

Condensed matter physics deals with properties of interacting many-particle systems. P. Anderson – more is different (Science, 1972).

“The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” “…the whole becomes not only more than the sum of but very different from the sum of the parts…”

Chaos makes deterministic predictions in large systems fundamentally impossible. Can only predict robust (universal) properties.

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Physics Open House

Condensed Matter Theory, Antonio Castro Neto

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3/23/12

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Physics Open House

Condensed Matter Theory, Claudio Chamon

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3/23/12

Electron fractionalization in graphene-like structures

Transitions in and out of quantum topological phases Quantum Glasses

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Physics Open House

Condensed Matter Theory , Anders Sandvik

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3/23/12

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Physics Open House

Condensed Matter Theory, Sid Redner

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3/23/12

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Physics Open House

Condensed Matter Theory, Anatoli Polkovnikov

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3/23/12

Thermodynamics of driven systems (Luca D’Alessio talk)

Quantum-classical correspondence. Phase space methods. Applications to cold atom systems

3

Observable Ez Q F

vL2 ! 1 116vL

2 7!(3)128 v2L3 1

6144 v2L4

vL2 " 1 0.26#vL 0.0265 vL 0.0276

#vL

TABLE I: Scaling of the excess interaction energy Ez withrespect to the final ground state, the excess total energy Q,and the log-fidelity F with the quench rate and the systemsize for the transverse-field Ising chain.

reduces to known results [8], which were recently verifiednumerically for a particular 1D model [12].Application to the tranverse-field Ising model. To illus-

trate the above general scaling results we consider a linearquench in the one-dimensional (1D) transverse-field Isingmodel with Hamiltonian

H = !!

j

!xj ! J

!

!ij"

!zi !

zj , (10)

where !x and !z are the Pauli matrices and "ij# are near-est neighbours sites. The dimensionless coupling con-stant J drives the system through a critical point atJc = 1, with critical exponents z = " = 1. We considerthe following quench protocol: J(#) = 1 + $ = 1 + v# ,starting in the ground state at #0 = !1/v. Usingthe Jordan-Wigner transformation, the model can bemapped to free fermions. The analysis of the imaginarytime dynamics is straightforward and available in the lit-erature for similar real-time setups [18, 19]. We thereforeonly quote our results in Table I.It is evident that the general scaling prediction (9) in-

deed applies to this example. To illustrate further thefinite-size scaling behavior predicted by Eq. (9), in Fig. 1(left panel) we plot the shift of the interaction energywith respect to the final ground state: Ez = !J%MJ =!J

"

!ij""!iz!

jz# !

"

!ij""0|!iz!

jz |0# versus vL

2. The datafor di!erent system sizes collapse, showing that one canuse the proposed imaginary-time adiabatic approach toextract critical properties of a quantum phase transi-tion. To test our predictions further, we analyze thesquare of the longitudinal magnetization (the order pa-rameter); m2

z = 1/L2"("

j !jz)

2#, which has scaling di-mension "z = 1/4 [20]. This and Eq. (9) imply thatm2

z $ Lv5/8fzJ(Lv2) = L#1/4fzJ(vL2). The large andsmall argument asymptotics of the scaling function fzJare dictated by the equilibrium asymptotics in the dia-batic limit, fzJ (x) % const at x & 1, and by the re-quirement that m2

z % 1/L at vL2 ' 1 when quenchingfrom the disordered phase. If we quench from the or-dered phase, J(#0) > 1, then fzJ(x) % x1/8 at x ' 1 (sothat m2

z % const). The finite-size scaling predictions andasymptotics are in excellent agreement with numericaldata (Fig. 1, left panel) obtained using QMC simulations.QMC method.—A major advantage of the imaginary-

time approach is generalized QMC methods can be ap-plied to evolve a state with the operator (2). Here weuse an approach similar to the stochastic series expan-

100 102 104 106

vL2

10-1

100

101

102

L = 16L = 32L = 64L = 128L = 1024

100 102 104

vL2

0.1

1

L = 8L = 16L = 32L = 64L = 128

Ez mz2L1/4

FIG. 1: (Color online) Excess interaction energy Ez (left) andsquared magnetization (right) of Ising chains graphed accord-ing to our scaling predictions. The lines show the expectedasymptotic forms; the slope is 1 for vL2 ! 1 and 1/2 forvL2 " 1 on the left and $3/8 for vL2 " 1 on the right.

sion method, discussed in the context of the quantumIsing model in Ref. [21]. Evolving from #0 < 0 to # = 0,the exponential in (2) is expanded in a power-series andapplied to an initial state |#(0)#:

|#(#)# =$!

n=0

# 0

!0

d#n

# !n

!0

d#n#1 · · ·

# !2

!0

d#1 (

[!H(#n)] · · · [!H(#1)]|#(0)#. (11)

Writing !H in terms of individual site and bond opera-tors, here denoted Hi, i = 1, . . . , Nop, the operator prod-uct is written as a sum over all strings of these operators.Truncating at some maximum power n = m (adapted tocause no detectable error) and introducing a trivial unitoperator H0 = 1, we can write (11) as:

|#(#)# =!

H

(m! n)!

#m#n0

# 0

!0

d#m · · ·

# !3

!0

d#2

# !2

!0

d#1 (

Him(#m) · · ·Hi2(#2)Hi1(#1)|#(0)#, (12)

where ip ) {0, . . . , Nop},"

H is the sum over all se-quences i1, . . . , im, and n is the number of indices ip *= 0.Expectation values "#(#)|A|#(#)#/"#(#)|#(#)# are

computed by sampling the normalization "#(#)|#(#)#written with (12). For the transverse-field Ising model,which we will apply the method to below, the method isvery similar to the one in [21], the main di!erence be-ing the change in the time boundaries; from periodic atfinite temperature to those dictated by the initial state|#(0)# of the time evolution. Changes in the operatorsequence are made with the times #i fixed. The timesare updated separately by generating ordered time seg-ments to replace existing segments, with the segment sizedetermined to give a Metropolis acceptance rate close to1/2. The scheme is particularly simple when the startingstate is the equal superposition, |#(0)# =

$

i(+i + ,i),which we use below, but other states can be used as well.As always in QMC simulations, we are restricted to sys-tems for which the expansion is positive-definite, which

4

10-2 10-1 100 101 102 103

vL(!+1)/!

0.02

0.030.040.05

0.10

0.20

L = 4L = 6L = 8L = 16L = 32L = 64

10-2 10-1 100 101 102 103

vL(!+1)/!

0.1

1

L = 4L = 6L = 8L = 16L = 32L = 64

Ezv(3!"1)/(!+1) mz

2L1+#

FIG. 2: Same as Fig. 1 for the square-lattice model (quenchingto its critical point Jc = 0.32841). QMC data are scaledaccording to the theoretical results, with the lines illustratingthe predicted asymptotic slopes with exponents z = 1, ! =0.6298, and " = 0.0364 [22] (note that !z = 1 + ").

is the same class for which sign problems can be avoidedin equilibrium simulations.Using this method we first confirmed that the exact re-

sults for the Ising chain are reproduced. We next consid-ered the same model on the 2D square lattice, for whichthe critical coupling is Jc = 0.32841 [23]. In the left panelof Fig. 2 we show the scaling of Ez for L!L lattices withL up to 64, using the known exponent ! = 0.6298 [22](obtained for the classical 3D Ising model, which shouldscale as the 2D quantum model when z = 1). We havedivided Ez by the leading powers of L and v predictedabove and, hence, we should obtain a constant behaviorfor large x. This is not quite seen yet for these systemssizes, but the eventual convergence seems plausible. Forsmaller x the data collapse very well and the asymptoticx " 0 behavior is reproduced. In the right panel we showthat also the squared magnetization scales according toour predictions, over five decades of vL(!+1)/nu.Summary and discussion—We have shown that de-

tailed information on static and dynamic properties ofa system can be obtained by propagating it in imagi-nary time. There are many similarities with real-timedynamics. In particular, we showed that one can useimaginary time to obtain universal exponents character-izing quantum critical points and to measure the fidelitysusceptibility and components of the geometric tensoras response of physical observables to a linear quench.We obtained finite-size scaling expressions characterizingthe response of various observables with the quench rate.In this way we extended the scaling theory of quantumphase transitions to non-equilibrium protocols. A clearadvantage of the imaginary time approach is that one canuse powerful Monte Carlo simulations and circumventcomplications related to real-time simulations. We illus-trated this approach for the transverse-field Ising model.Exact and QMC results show excellent agreement withthe scaling predictions in one and two dimensions. TheQMC method will be useful for studies of a wide rangeof non-trivial models on large lattices.

The ideas presented here apply also to quantum an-nealing, i.e., protocols where, in order to analyze theground state of a complicated classical or quantum prob-lem, one introduces an auxiliary coupling which makesthe Hamiltonian simple and then slowly decreases thiscoupling to zero. This will allow one to address quantumannealing problems using QMC simulations [24].Acknowledgments We acknowledge useful discussions

with V. Gritsev. The work was supported by GrantsNSF DMR-0907039 (AP and CDG), NSF DMR-0803510(AWS), AFOSR FA9550-10-1-0110 (AP), and the SloanFoundation (AP).

SUPPLEMENTARY MATERIAL

We here provide some further details of the calculationsleading up to Eq. (3) and also discuss some additionalproperties of the adiabatic susceptibilities discussed inthe main text and how they are extended to real time.

Adiabatic perturbation theory

Let us discuss the leading non-adiabatic correction tothe imaginary-time Schrodinger equation (1):

""#($) = #H(%($))#. (13)

The natural way to address this question is to use adi-abatic perturbation theory (APT), similar to that de-veloped in Refs. [19, 25] in real time. We write thewave function in the instantaneous eigenbasis {|n(%)$}of H(%):

#($) =!

n

an($)|n(%($))$. (14)

Substituting this expansion into Eq. (1) we find

dand$

+!

m

am($)%n|"" |m$ = #En(%) an($), (15)

where En(%) are the eigenenergies of H(%) correspondingto the states |n$. Making the transformation

an($) = &n($) exp

"# 0

"En($

!)d$ !$

, (16)

we can rewrite Eq. (1) as an integral equation [and notethat &n(0) = an(0)]:

&n($) = &n(0) +!

m

# 0

"d$ ! %n|"" ! |m$&m($ !)

! exp

"

#

# 0

" !

d$ !! (En($!!)# Em($ !!)

$

.(17)

Interaction energy Ising order parameter

Universal quantum dynamics. Quantum geometry (together with A. Sandvik)